Robert J. Lang - Twists, Tilings and Tessellations

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-56306-3 (Hardback) International Standard Book Number-13: 978-1-56881-232-8 (Paperback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Lang, Robert J. (Robert James), 1961- author. Title: Twists, tilings, and tessellations / Robert J. Lang. Description: Boca Raton : CRC Press, 2018. Identifiers: LCCN 2017030497 | ISBN 9781568812328 (pbk.) Subjects: LCSH: Combinatorial designs and configurations. | Twist mappings (Mathematics) | Tiling (Mathematics) | Tessellations (Mathematics) | Origami--Mathematics. Classification: LCC QA166.8 .L36 2018 | DDC 516/.132--dc23 LC record available at https://lccn.loc.gov/2017030497 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

To Klaus Peters

Contents Introduction 1 2

Genesis ? . . . . . . . . . . . . . . . . . . . . What to Expect and What You Need ? . . . . .

1 Vertices 1.1 1.2

1.3 1.4

1.5

Modeling Origami ? . . . . 1.1.1 Crease Patterns ? . . 1.1.2 Creases and Folds ? . Vertices ? . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Kawasaki-Justin Theorem ? . . . . 1.2.2 Justin Ordering Conditions ? . . . . 1.2.3 Three Facet Theorem ? . . . . . . . 1.2.4 Big-Little-Big Angle Theorem ? . . 1.2.5 Maekawa-Justin Theorem ? . . . . 1.2.6 Vertex Type ? . . . . . . . . . . . . 1.2.7 Vertex Validity ? . . . . . . . . . . Degree-2 Vertices ? . . . . . . . . . . . . . Degree-4 Vertices ? . . . . . . . . . . . . . 1.4.1 Unique Smallest Sector ? . . . . . . 1.4.2 Two Consecutive Smallest Sectors ? 1.4.3 Four Equal Sectors ? . . . . . . . . 1.4.4 Constructing Degree-4 Vertices ? . 1.4.5 Half-Plane Properties ? . . . . . . . Multivertex Flat-Foldability ?? . . . . . . . 1.5.1 Isometry Conditions and Semifoldability ?? . . . . . . . . . 1.5.2 Injectivity Conditions and Non-Twist Relation ?? . . . . . . . . . . . . . 1.5.3 Local Flat-Foldability Graph ?? . .

xv xv xviii

1 . . . . . . . . . . . . . . . . . . .

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1 5 12 16 17 21 23 25 27 29 30 32 33 34 35 37 37 41 42

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42

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46 46 vii

1.6

1.7

Vector Formulations of Vertices ? ? ? . 1.6.1 Vector Notation: Points ? ? ? . . 1.6.2 Vector Notation: Lines ? ? ? . . 1.6.3 Translation ? ? ? . . . . . . . . 1.6.4 Rotation ? ? ? . . . . . . . . . . 1.6.5 Reflection ? ? ? . . . . . . . . . 1.6.6 Line Intersection ? ? ? . . . . . 1.6.7 2D Developability ? ? ? . . . . 1.6.8 2D Flat-Foldability ? ? ? . . . . 1.6.9 Analytic versus Numerical ? ? ? Terms ? . . . . . . . . . . . . . . . . .

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2 Periodicity 2.1 2.2

2.3

2.4

2.5

79

Repeating Vertices ? . . . . . . . . 1D Periodicity ? . . . . . . . . . . . 2.2.1 Periodicity and Symmetry ? 2.2.2 Tiles ? . . . . . . . . . . . . 2.2.3 Linear Chains ? . . . . . . . 2D Periodicity ? . . . . . . . . . . . 2.3.1 Huffman Grid ? . . . . . . . 2.3.2 Yoshimura Pattern ? . . . . 2.3.3 Miura-ori ? . . . . . . . . . 2.3.4 Miura-ori Variations ? . . . 2.3.5 Barreto’s Mars ? . . . . . . 2.3.6 Generalized Mars ? . . . . . Partial Periodicity ?, ??, ? ? ? . . . 2.4.1 Yoshimura-Miura Hybrids ? 2.4.2 Semigeneralized Miura-ori ? 2.4.3 Predistortion ?? . . . . . . 2.4.4 Tachi-Miura Mechanisms ? . 2.4.5 Triangulated Cylinders ? . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Triangulated Cylinder Geometry ? ? ? 2.4.7 Waterbomb Tessellation ? . . . . . . 2.4.8 Troublewit and Pleats ? . . . . . . . . 2.4.9 Corrugations and More ? . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . .

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3 Simple Twists 3.1 3.2

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........CONTENTS

Twist-Based Tessellations ? . . . . . . . . Folding a Twist ? . . . . . . . . . . . . . 3.2.1 Diagrams versus Crease Patterns ? 3.2.2 A Square Twist Tessellation ? . .

53 54 55 57 58 60 61 63 67 70 73 79 80 80 85 88 90 93 101 106 114 117 121 126 126 128 138 144 152 160 164 174 184 190

193 . . . .

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193 195 202 206

3.3 3.4

3.5

3.6

3.7 3.8

Elements of a Twist ? . . . . . . . . . . . . . . Regular Polygonal Twists ?, ?? . . . . . . . . . 3.4.1 Cyclic Regular Twists ? . . . . . . . . . 3.4.2 Open- and Closed-Back Twists ? . . . . 3.4.3 Rotation Angle of the Central Polygon ? 3.4.4 Iso-Area Twists ?? . . . . . . . . . . . Twist Flat-Foldability ? . . . . . . . . . . . . . 3.5.1 Local Flat-Foldability ? . . . . . . . . 3.5.2 Pleat Crease Parity ? . . . . . . . . . . 3.5.3 Pleat Assignments ? . . . . . . . . . . 3.5.4 mm/vv Condition ? . . . . . . . . . . . 3.5.5 mv/vm Condition ? . . . . . . . . . . . 3.5.6 MM/VV Condition ? . . . . . . . . . . 3.5.7 MV/VM Condition ? . . . . . . . . . . 3.5.8 Cyclic Overlap Conditions ? . . . . . . 3.5.9 Summary of Limits ? . . . . . . . . . . General Polygonal Twists ??, ? ? ? . . . . . . . 3.6.1 Triangle Twists ?? . . . . . . . . . . . 3.6.2 Higher-Order Irregular Twists ?? . . . 3.6.3 Cyclic Overlaps in Irregular Twists ?? . 3.6.4 Closed-Back Irregular Twists ?? . . . . 3.6.5 Open-Back Brocard Polygon Twists ? ? ? Joining Twists ? . . . . . . . . . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . .

4 Twist Tiles 4.1

4.2 4.3

4.4 4.5

Introduction to Twist Tiles ? . . . 4.1.1 What is a Tile? ? . . . . . 4.1.2 Ways of Mating ? . . . . . 4.1.3 Centered Twist Tiles ? . . 4.1.4 Offset Twist Tiles ? . . . . Vertex Figures ? . . . . . . . . . . Vertices and Angles ? ? ? . . . . . 4.3.1 Unit Polygons ? ? ? . . . . 4.3.2 Centered Twist Tiles ? ? ? 4.3.3 Offset Twist Tiles ? ? ? . . Folded Form Tiles ?, ?? . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Centered Twist Folded Form Tiles ?? 4.4.2 Offset Twist Folded Form Tiles ? . . . Triangle Tiles ?? . . . . . . . . . . . . . . . 4.5.1 Centered Twist Triangle Tiles ?? . . 4.5.2 Offset Twist Triangle Tiles ?? . . . .

208 211 212 214 215 216 222 225 227 228 229 230 231 232 234 238 242 243 247 249 255 262 264 268

271 . . . . . . . . . . . . . . . .

271 271 277 280 287 290 297 298 299 304 306 306 311 312 312 316 CONTENTS

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ix

4.6

4.7

Higher-Order Polygon Tiles ?, ??, ? ? ? . . . 4.6.1 Centered Twist Cyclic Polygon Tiles ? 4.6.2 Cyclic Polygon Construction ? ? ? . . 4.6.3 Quadrilateral Offset Twist Polygon Tiles ?? . . . . . . . . . . . . . . . . 4.6.4 Offset Twist Higher-Order Polygon Tiles ?? . . . . . . . . . . . . . . . . 4.6.5 Pathological Twist Tiles ? . . . . . . 4.6.6 Split-Twist Quadrilateral Tiles ? . . . Terms ? . . . . . . . . . . . . . . . . . . . .

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319 319 321

.

326

. . . .

330 332 334 342

5 Tilings 5.1 5.2

5.3 5.4

5.5 5.6

Introduction to Tilings ? . . . . . . . . . . . Archimedean Tilings ?, ? ? ? . . . . . . . . . 5.2.1 Uniform Tilings ? . . . . . . . . . . . 5.2.2 Constructing Archimedean Tilings ? . 5.2.3 Lattice Patches and Vectors ? ? ? . . . Edge-Oriented Tilings ? . . . . . . . . . . . 5.3.1 Centered Twist Tiles ? . . . . . . . . 5.3.2 Offset Twist Tiles ? . . . . . . . . . . k-Uniform Tilings ? . . . . . . . . . . . . . . 5.4.1 2-Uniform Tilings ? . . . . . . . . . 5.4.2 Two-Colorable 2-Uniform Tilings ? . 5.4.3 Higher-Order Uniform Tilings ? . . . 5.4.4 Periodic Tilings with Other Shapes ? 5.4.5 Grid Tessellations ? . . . . . . . . . . Non-Periodic Tilings ?, ? ? ? . . . . . . . . . 5.5.1 Goldberg Tiling ? . . . . . . . . . . . 5.5.2 Self-Similar Tilings ? ? ? . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . .

6 Primal-Dual Tessellations 6.1 6.2 6.3

x

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345 . . . . . . . . . . . . . . . . . .

Shrink and Rotate ? . . . . . . . . . . . . . . . Properties ?? . . . . . . . . . . . . . . . . . . 6.2.1 Twist and Aspect Ratio ?? . . . . . . . 6.2.2 Crease Pattern/Folded Form Duality ?? Nonregular Polygons ?? . . . . . . . . . . . . 6.3.1 A Broken Tessellation ?? . . . . . . . 6.3.2 Dual Graphs and Interior Duals ?? . . 6.3.3 A Valid Rhombus Tessellation ?? . . . 6.3.4 Relation Between Primal and Dual Graphs ?? . . . . . . . . . . . . . . .

345 346 346 348 352 356 356 366 371 371 375 376 381 389 391 393 395 402

405 405 407 407 411 414 414 416 418 422

6.4

6.5

6.6

6.7

Maxwell’s Reciprocal Figures ?, ?? . . . . . 6.4.1 Indeterminateness and Impossibility ? 6.4.2 Positive and Negative Edge Lengths ? 6.4.3 Crease Assignment ?? . . . . . . . . 6.4.4 Triangle Graphs ?? . . . . . . . . . . 6.4.5 Voronoi and Delaunay ?? . . . . . . Flagstone Tessellations ? . . . . . . . . . . . 6.5.1 Spiderwebs Revisited ? . . . . . . . . 6.5.2 The Flagstone Geometry ? . . . . . . 6.5.3 Flagstone Vertex Construction ? . . . 6.5.4 Examples ? . . . . . . . . . . . . . . Woven Tessellations ?, ? ? ? . . . . . . . . . 6.6.1 Woven Concepts ? . . . . . . . . . . 6.6.2 Simple Woven Patterns ? . . . . . . . 6.6.3 Woven Algorithm ? ? ? . . . . . . . . 6.6.4 Flat Unfoldability ? . . . . . . . . . . 6.6.5 Woven Algorithm, Continued ? ? ? . 6.6.6 Woven Examples ? . . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . .

7 Rigid Foldability 7.1 7.2 7.3 7.4 7.5

7.6 7.7 7.8 7.9

The Easy Way or the Hard Way ? . . . . . . . Half-Open Vertices ?? . . . . . . . . . . . . Spherical Geometry ?? . . . . . . . . . . . . A Degree-4 Vertex in Spherical Geometry ?? 7.4.1 Opposite Fold Angles ?? . . . . . . . 7.4.2 Adjacent Fold Angles ?? . . . . . . . Conditions on Rigid Foldability ?? . . . . . 7.5.1 The Weighted Fold Angle Graph ?? . 7.5.2 Distinctness of Fold Angle ?? . . . . 7.5.3 Matching Fold Angle ?? . . . . . . . General Twists ?? . . . . . . . . . . . . . . 7.6.1 Triangle Twists ?? . . . . . . . . . . 7.6.2 Mechanical Advantage ?? . . . . . . Non-Twist Folds ?? . . . . . . . . . . . . . . 7.7.1 General Meshes ?? . . . . . . . . . . 7.7.2 Quadrilateral Meshes ?? . . . . . . . Non-Quadrilateral Meshes ? . . . . . . . . . 7.8.1 Forced Rigid Foldability ? . . . . . . 7.8.2 Non-Flat-Foldable Vertices ? . . . . . Terms ? . . . . . . . . . . . . . . . . . . . .

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423 425 429 433 435 438 442 443 446 447 450 453 454 456 459 461 465 467 471

475 475 477 480 486 486 489 492 495 498 500 506 508 512 515 515 518 528 528 530 533

CONTENTS

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8 Spherical Vertices 8.1 8.2

8.3

8.4

8.5

8.6

Generalizing Vertices ? . . . . The Gaussian Sphere ?? . . . 8.2.1 Plane Orientation ?? . 8.2.2 The Trace ?? . . . . . 8.2.3 Polyhedral Vertices ?? 8.2.4 A Degree-4 Vertex ?? Sector and Fold Angles ?? . . 8.3.1 Osculating Plane ?? . 8.3.2 Binding Condition ?? 8.3.3 Ruling Plane ?? . . .

535 . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?? 8.3.4 Real Space Solid Angle . . . . . . . 8.3.5 Ruling Angle ?? . . . . . . . . . . . . 8.3.6 Osculating Angle ?? . . . . . . . . . . 8.3.7 Adjacent Fold Angles ?? . . . . . . . . 8.3.8 Flat-Foldable and Straight-Major/Minor Vertices ?? . . . . . . . . . . . . . . . 8.3.9 Sector Angle/Fold Angle Relations ?? . More Angles and Planes ?? . . . . . . . . . . 8.4.1 Sector Elevation Angles ?? . . . . . . 8.4.2 Sector Angles ?? . . . . . . . . . . . . 8.4.3 Bend Angle ?? . . . . . . . . . . . . . 8.4.4 Edge Torsion Angle ?? . . . . . . . . . 8.4.5 Midfold Angles and Planes ?? . . . . . 8.4.6 Infinitesimal Trace ?? . . . . . . . . . 8.4.7 What Specifies a Vertex? ?? . . . . . . Networks of Vertices ?? . . . . . . . . . . . . 8.5.1 Huffman Grid ?? . . . . . . . . . . . . 8.5.2 Gauss Map ?? . . . . . . . . . . . . . 8.5.3 Miura-ori and Mars ?? . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . . . . . .

9 3D Analysis 9.1 9.2

9.3 xii

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3D Vectors ? ? ? . . . . . . . . . . . . . . . 3D Vertices ? ? ? . . . . . . . . . . . . . . . 9.2.1 Fold Direction Vectors ? ? ? . . . . . 9.2.2 Vertex from Fold Directions ? ? ? . . 9.2.3 Degree-4 Vertex from Sector Elevation Angles ? ? ? . . . . . . . . . . . . . Discrete Space Curve ? ? ? . . . . . . . . . .

535 536 536 538 541 543 545 545 547 549 550 554 557 558 561 563 568 569 573 576 578 584 586 590 591 591 593 597 603

605 . . . .

605 610 611 612

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615 617

9.4

9.5

9.6 9.7

Plate Model ? ? ? . . . . . . . . . . . . 9.4.1 Folding a Crease Pattern ? ? ? . 9.4.2 Fold Angle Consistency ? ? ? . 9.4.3 Solving for Fold Angles ? ? ? . Truss Model ? ? ? . . . . . . . . . . . . 9.5.1 3D Isometry and Planarity ? ? ? 9.5.2 Explicit Stress/Strain ? ? ? . . . 9.5.3 3D Developability ? ? ? . . . . Time Efficiency ? . . . . . . . . . . . . Terms ? . . . . . . . . . . . . . . . . .

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10 Rotational Solids 10.1 10.2 10.3

10.4 10.5 10.6 10.7

Three-Dimensional Twists ?, ?? . . . 10.1.1 Puffy Twists ? . . . . . . . . . 10.1.2 Folding a Sphere ?? . . . . . Thin-Flange Algorithm ? ? ? . . . . . Thick-Flange Structures ?, ? ? ? . . . 10.3.1 Mosely’s “Bud” ? . . . . . . . 10.3.2 Thick-Flange Algorithm ? ? ? 10.3.3 Specified Gores ? ? ? . . . . . 10.3.4 Generalized Flanges ? ? ? . . Axial Unfoldings ? ? ? . . . . . . . . Variations on the Theme ? ? ? . . . . 10.5.1 Twist Lateral Shifts ? ? ? . . . 10.5.2 Triangulated Gores ? ? ? . . . Artists of Revolution ? . . . . . . . . Terms ? . . . . . . . . . . . . . . . .

626 627 628 633 636 637 641 644 646 647

649 . . . . . . . . . . . . . . .

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649 649 653 659 664 664 668 671 674 679 683 683 695 699 702

Afterword

705

Acknowledgements

707

Bibliography Index

711 723

CONTENTS

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xiii

Introduction ?

1. Genesis Everyone and no one knows what origami, the Japanese art of paper-folding, is. Everyone knows, because they have seen the well-known Japanese crane, or tsuru, an international symbol of peace. Or they have folded schoolyard paper-folding—boats, bangers, and cootie-catchers, possibly not even knowing that these, too, are part of the origami world. But in another sense, no one knows what origami is, because in the latter part of the 20th century, it exploded in variety, complexity, and artistry, with numerous genres and specializations. The word origami is simply the Japanese words for “folding” (oru) and “paper” (kami), combined in a single word used to describe the craft—and sometimes, art—of decorative paper folding. Contrary to popular belief, there are no fixed rules about paper, glue, or use of cuts: traditional Japanese designs, many of which can be reliably dated to be hundreds of years old, used various sizes and shapes of paper, sometimes multiple sheets, and often used cuts. In modern origami, cuts are rare but not unknown, and distinct genres have arisen in which one folds a single sheet or uses multiple sheets (the latter category divided further into composite origami, using multiple sheets to make separate parts of a subject, and modular origami, using multiple sheets to make identical units that are assembled). The most well-known origami genres are representational and figurate; the origami subject looks like something. Indeed, most people who have ever seen or folded origami have only created representational work. But there is a deep history of non-representational decorative paper-folding, both within xv

the world of Japanese origami and coming from many disparate fields of endeavor outside of the Japanese tradition, ranging from napkin-folding of the 15th century in Europe to early-20th-century Bauhaus architecture to late-20th-century computational geometry and mathematics. It is this latter field of non-representational origami that is the focus of this book, centered around the mathematical genre known as origami tessellations. A tessellation is, in general, a division of the plane into a pattern, and the name comes from the Latin tessera, which was the name for a tile making up a mosaic. Like their namesakes, origami tessellations divide the plane into decorative patterns—but using folds to make the subject from a single sheet of paper, rather than dividing the image up into individual units.1 Tessellations have a distinct history within origami, with independent original work by both Shuzo Fujimoto and Yoshihide Momotani in the 1970s and 1980s, but they underwent a renaissance in the late 1990s when the concept was picked up and explored by several artists, notably Paulo Taborda Barreto and Chris K. Palmer in the 1990s, and many more thereafter. Their work, in turn, inspired still further development in the world of origami and the beginnings of research into folded tessellations’ mathematical properties and algorithms for their design, launching a period of growth and exploration that continues to this day. Along with the expansion of origami tessellations has come an expansion of the scope of interest, as origami “tessellators” began exploring other folded forms that bore similarities to tessellations but were not particularly flat, periodic, or reminiscent of any sort of tiling. They weren’t necessarily flat and were often three-dimensional. What these new folded-paper forms had in common were that they were non-representational and often highly geometric. The origami tessellation explorers also discovered something else: mathematical and non-representational folding had a welldefined existence quite outside of the world of origami. Notably, Josef Albers and the Bauhaus school explored folded paper extensively in the 1920s, including flat, polyhedral, and curved forms. 1

In fact, there are modular origami tessellations, patterns that more closely resemble the traditional type of tessellation mosaic in which individual units are folded and then assembled into an overall pattern. I note their existence but won’t describe them further in this work.

xvi

........INTRODUCTION

Many of the shapes they explored have been rediscovered by modern artists quite independently. The universality of mathematics almost guarantees that a simple elegant form will be discovered over and over by multiple artists. Paper artists, too, have discovered and created geometric and mathematical forms through the years, again, quite independently of the origami tradition. Paper is quite a versatile medium: it can be cut, folded, scored, bent, dampened, reshaped, and altered in a variety of ways. It can be made to take a crease, giving it a memory and a desire to take on certain forms; yet, the springiness of paper can make it resist certain forms as well. This versatility, coupled with its generally low price, has made paper the medium of experimentation for artists and designers of all stripes. It has also been the inspiration of mathematical research into the properties of folded shapes, again, outside of the traditional world of Japanese paper-folding. Some 40 years ago, computer scientist and artist Ron Resch began designing and folding paper forms using mathematical and computational algorithms. Resch was followed by another computer scientist, David Huffman, who, beginning in the 1970s, not only built on Resch’s ideas but developed many new concepts of his own and eventually wrote one of the seminal papers on the mathematics of folded paper. Their work launched a new thread of research into geometric paperfolding, in which mathematics and art were combined in equal proportion: algorithms, existence, and complexity paired with statement, expression, and aesthetic. The world of geometric origami preceded that subset that we now call origami tessellations, and it has, in recent years, grown far beyond mere tessellations. The repeating patterns of tessellations, however, have a particular appeal to me, and, for the purposes of this book, provide a unifying theme for a study of primarily geometric, single-sheet folding. And so with this book, I have placed origami tessellations in both the title and at the heart of the book. Tessellations are beautiful, but they also provide a structured way to introduce the mathematical laws governing origami—laws that govern in so many ways what structures, forms, and shapes can be created by folding. But this book covers more than tessellations; I will range over many different geometric forms. It is not possible to be truly comprehensive, because the field of geometric origami is growing actively, in many directions at once. I hope to pull together here a sampling of many of the possibilities and to provide INTRODUCTION

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xvii

you with tools, both artistic and mathematical, that you can use to reproduce the artworks and patterns within this book and to build and fold your own original creations. ?

2. What to Expect and What You Need Mathematical origami is an extremely diverse field with many branches, only some of which are represented in this book. (It is also a fast-growing field; by the time you read this, there will likely be many new shoots.) Thus, you can, and indeed are invited to, jump among the various chapters, trying out things that look interesting and skipping what makes your eyes glaze over or is packed with forbiddingly complex expressions. It’s no fun at all to be sampling the treats at the table and unwittingly find yourself with a mouthful of something only a connoisseur should appreciate. This metaphor applies particularly to things mathematical. The mathematics in this book covers a wide range of topics and requires a wide ranges of skills. To help you decide what to jump into and what to skim past, I have marked the subsections with from one to three stars, indicating the level of mathematics required: ? Basic. The simplest geometry, requiring little more than an appreciation of shapes, the ability to construct and/or measure an angle with a protractor, and an ability to count, add, and subtract angles. We will use some letters to represent quantities, but we will keep the algebra to a minimum. Suitable for early high school students. ?? Intermediate. Uses algebra (equation-solving), trigonometry, and more advanced concepts from high school geometry. Suitable for upper high school students. ? ? ? Advanced. Uses concepts from linear algebra, vectors and operators. Suitable for college students in technical fields (mathematics, science, engineering) and possibly some advanced high school students.2 2 In the ? sections, angles will be given in terms of degrees. In ?? and ? ? ? sections, we will use radians.

xviii


You will also find origami instructions for several figures at various places. Traditional origami instructions are given in terms of a folding sequence: a step-by-step series of drawings showing a linear progress from the unfolded paper to the fully folded result. However, more often than not, mathematical origami has no folding sequence, which creates its own set of challenges when it comes to presenting instructions for such works. And it’s not just a case of the designer being “too lazy to draw diagrams.” As we will see, many mathematical folds are composed of large irreducible blocks of folds—structures that cannot be broken down into isolated steps. Historically, most origami designs were discovered by sequential manipulations performed on a sheet of paper, and so no matter how long and/or convoluted the path to the end result, it was pretty much guaranteed that a folding sequence existed; the challenge of diagramming it lay primarily in remembering (or reconstructing) the most efficient sequence, leaving out all of the exploratory dead ends. But with the modern age of “technical origami,” or origami sekkei, in which the final form is designed before one ever puts hand to paper, there is no reason to believe that a simple path from start to end exists—and, in many cases, it is possible to show mathematically that such does not exist. In this regard, mathematical origami shares a property with what in some ways is its exact opposite: highly sculpted representational origami, most famously typified by the works of the late Eric Joisel. Joisel called his work “jazz origami,” because the vast majority of the folds were improvised on the spot based on aesthetic considerations. Here, too, there is no set sequence, no set of diagrams that can provide instruction; instead, Joisel simply moved around the paper, bending, shaping, curving, adding folds, nudging it ever closer to the ideal he visualized, but in no set order. Surprisingly, many mathematical folds require a very similar approach: since tens or hundreds of creases may need to come together at once, the artist must simply work his or her way around the crease pattern, the design, bending each fold in the proper direction but in no particular order, until they all (or a large subset) can come together. One nice property that many mathematical folds have is a “tipping point”—a point at which the number of creases going in the right direction reaches a critical mass and the fold, instead of resisting, starts to come together, almost with a life of its own. INTRODUCTION

........

xix

So mathematical origami breaks new ground in its design and in how it goes together, which is to say, even if you don’t have much past experience with origami, that is not much of a handicap. The field of mathematical origami is so new that, in some sense, none of us is very far from the beginning. It is that beginning to which we now turn.

xx


1

Vertices ?

1.1. Modeling Origami The term origami refers to something very specific: Japanese paper-folding. But mathematical origami is much broader than the traditional craft: it isn’t necessarily Japanese, it involves materials other than paper, and it involves actions other than just folding— bending and crumpling, for example, although both could be said to be a form of folding. What we will focus on in this book, though, are those aspects of mathematical folding that are characteristic of most origami: the use of a non-stretchy sheet-like material, manipulated in three dimensions, with few or no cuts. Mathematical folding doesn’t require that you use paper—in fact, in real-world applications of mathematical folding, one can use materials as diverse as plastic, Mylar, Kapton, leather, cloth, and even mats of carbon nanotubes. But throughout this work, for simplicity of language, I will generally refer to the material being folded as “paper,” and paper is often the ideal material to work with: inexpensive, widely available in diverse forms, and possessed of mechanical properties that make it particularly suited for folding. Part of the beauty of origami in general and mathematical folding in particular is that it is tactile and visual; you can feel the paper, you can see the result, and integration of hand-eye experience builds an intuition of what is possible more effectively than any set of mathematical formulas or algebraic description. Nevertheless, there are limits to intuition, and mathematics can provide powerful tools to understand the possibilities of paper and to design specific structures and forms. And so, throughout this work, I will attempt to provide a mathematical description of the topic at hand.

1

There are many ways to describe folding mathematically, and the most natural way depends in large part on the level of abstraction that one chooses in the description. Is the folded form flat or three-dimensional (3D)? Are surfaces straight or curved? Are creases straight or curved? Do we care about effects of material thickness, tensile forces, mechanical yield, creep, and plastic deformation? There is no single “correct” mathematical description of folded paper; there are only various approximations that idealize, emphasize, and/or ignore different aspects of the folding process. Two properties stand out above others as necessary to describe what is recognized as origami and that play a role in nearly all mathematical descriptions: • Non-stretchy paper. The folded shape is a 3D deformation of a planar surface that does not appreciably stretch (or compress) in any direction. • Non-self-intersection. The paper cannot intersect itself in the folded form, or in any intermediate stage. Any mathematical description of paper-folding must include these two properties in some way or another. These two properties—non-stretchiness and non-self-intersection—are at the heart of the folding arts. It is a little awkward to describe the properties of paper by what it is not; better to have a positive term. There are terms for both non-stretchiness and non-self-intersection. When we say that the paper is not stretchy, we mean that if we draw a line on the paper, fold the paper, and then measure the length of the line along the paper, that is, following the path of the paper, the length is unchanged. This property is a quality called isometry—taken from the Greek iso, meaning “same,” and -metry, meaning “measurement.” So the essence of origami folding is that it is isometric: distances along the surface of the paper are preserved going from the flat to the folded state (and, ideally, in all intermediate states). The second property, that the paper cannot intersect itself, also has a mathematical name: injectivity. In the language of mathematics, a mapping from one set (the domain) to another (the range) is an injection if no two points in the domain map to the same point in the range. In real physical origami, we cannot have

2

........CHAPTER 1. VERTICES

two points on the paper occupy the exact same point in space when the paper is folded. Even if you fold two layers together, one layer must lie above or below the other. If two layers switch places— here layer 1 lies above layer 2, there layer 2 lies on top—the rearrangement must happen without the paper penetrating itself, neither in unfolded layers nor at a fold. So injectivity is the quality of non-self-intersection. These two qualities are what define the mathematics that are particular to origami. This is not to say, however, that every mathematical model of origami must strictly have these two qualities. In fact, as we will see, it is frequently convenient to model origami paper as a zero-thickness surface, in which case a stack of layers may very well violate injectivity by occupying the exact same position in mathematical space. The important thing in such cases, though, is that in such a model, we know that the mathematical idealization violates one or the other of the fundamental properties of origami. Frequently, we will patch up such an ideal mathematical model to recover the lost properties. A mathematical description of origami must also make some assumption about the folding process, that is, the way that the paper gets from its initial flat state to its final configuration, the folded form. In standard origami books, that process is a relatively linear sequence of small steps: fold the paper in half; unfold; squash-fold; petal-fold; and so forth, where each term (“squashfold,” “petal-fold,” etc.) refers to a specific manipulation involving a small number of folds at a time. While this linear step-by-step process was historically the most common form of origami, it is not the only way a folded figure can take form. In fact, as we will see, many of the creations of mathematical folding come together only with tens, or even hundreds, of creases moving at once. When we take the process of formation into account, mathematical descriptions and modeling can get very complex indeed; there are folds that “don’t exist,” meaning that within some mathematical system, the motion going from a valid form at step A to a valid form at step B takes an intermediate state that somehow violates the assumptions of the mathematical system and thus, according to the mathematical model, could not be folded so. There are also folds where the folded state exists (within the mathematical system), and the unfolded state exists (within that same system), but there is no smooth progression from the unfolded state to the folded state within the same mathematical sysCHAPTER 1. VERTICES

........

3

Model Flat-Foldable Origami Polyhedral Origami Curved Origami Thick Origami

Description All facets are flat and coplanar; creases have fold angle of 0◦ or ±180◦ ; paper has zero thickness. Facets are flat, creases are straight, but fold angles can vary continuously; paper has zero thickness Facets and creases can be curved; paper has zero thickness Paper thickness is explicitly included.

Table 1.1. Hierarchy of mathematical models of origami.

tem. An example might be the mathematical model in which all surfaces apart from the folds are flat and planar. It may be that the only way to actually fold the paper into the finished state is to curve and/or bend some regions of the paper. If our mathematical system does not allow curving or bending, then we would say that, within that mathematical system, the folded state is “impossible” to fold. Whether a folded state or folding process is “impossible” or not depends, of course, on the mathematical model that one uses to describe it. If we can fold the object in the real world, then surely it exists, whatever the mathematical model might say! We must, of course, always realize that a mathematical model of folding is at best an approximation of what really happens in the physical world. The value of such a model, even as an imperfect approximation, comes when it can provide a reasonably accurate prediction of the folded state, and usually, the simpler the model, the better. We can construct something of a hierarchy of origami modeling of increasing complexity as we relax the rules of folding, as shown in Table 1.1. In general, as one moves down this hierarchy, the mathematical complexity increases—sometimes dramatically. We will explore this hierarchy, but we will move through it gradually, building base camps along the way and scheduling copious rest days as needed. And we will begin with the simplest possible model, which, surprisingly, covers a great deal of both historical and modern paper-folding. The first description we will consider is what for many years was the most common description within mathematical origami, and it is very simple indeed. In this description, we make these simplifying assumptions:

4


1. The paper has zero thickness. 2. The folded form is flat. 3. We don’t care about any intermediate configuration, i.e., whether it is flat or 3D or theoretically possible within our model. We call this model of origami flat-foldable origami. Such a model is, of course, an approximation of reality; there is no such thing as zero-thickness paper, and there is no way that an unfolded crease pattern can discontinuously transform itself into a folded state. Indeed, it is possible to contemplate folded configurations for which there is no practically achievable folding sequence. Nevertheless, this simple model can accurately describe a great deal of historic and modern folding, and it contains surprising richness and depth. This model can provide practical recipes and algorithms for the construction of folded shapes that are beautiful, interesting, and practically useful. ? 1.1.1. Crease Patterns A feature of this simplest type of origami, what we call flatfoldable origami, is that in the folded form, all surfaces are flat, except along straight lines, which are the creases, and the creases meet in groups at points, called vertices. The flat regions bounded by the creases are facets. There is a one-to-one mapping between points in the original paper and points in the folded form, and we can identify each point in the original paper as to whether it ends up in a facet, a crease, or a vertex. Logically enough, we call the points facet points, crease points, or vertex points, respectively. We can then, if we like, decorate the paper with identifying information, coloring each point and line according to its status in the folded form. Such a decoration is called the crease pattern associated with the folded form. The crease pattern is, essentially, a minimal description of the origami figure. For flat origami, often the crease pattern alone suffices as a guide for how to fold the shape. The crease pattern has a long history within origami; Figure 1.1 shows an origami crease pattern (and folded form) from 1845 (reprinted in [13, p. 58]). In historical origami works and works of the early 20th century, crease patterns were not uncommon (see, e.g., [130, pp. 24–26]), but with the growth of step-by-step instructions, they began to fall

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5

Figure 1.1. Crease pattern and folding instructions for “Ono no komachi” (a female poet), from the Kayaragusa, a collection of paper-folding instructions from 1845.

6


Figure 1.2.

Top: left and middle, the fold and result for a valley fold; right, the crease pattern for a valley fold. Bottom: left and middle, the fold and result for a mountain fold; right, the crease pattern for a mountain fold.

out of favor. With the resurgence of mathematical folding and systematic design toward the end of the 20th century [68], though, crease patterns have returned as the blueprint of all of the folding that is to follow, and they will be a key concept throughout this book. In representational folding, the crease pattern (often referred to simply as the CP) is rarely a map of all of the folds in the design; usually, it is a selected subset, chosen by the artist to convey the important properties of the structure and/or internal symmetries. In geometric folding, by contrast, the CP is quite often comprehensive, containing every crease in the finished work. Even so, it often does not provide a full description of the origami figure. It may contain all the folds, but it says nothing about the order in which the folds are made. And many crease patterns, including the one in Figure 1.1, don’t even tell which direction the paper folds. In a flat origami figure, every fold can go in one of two directions, as shown in Figure 1.2. In conventional origami terminology, when you fold a flap toward you, the resulting fold is called a valley fold. When the flap is folded away from you, the resulting fold is called a mountain fold. Historically, valley and mountain folds were not distinguished in any way (as in Figure 1.1), but in the mid-20th century, Akira

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7

Yoshizawa in Japan and Robert Harbin and Samuel L. Randlett in the West adopted a standard for diagrammatic origami instruction in which valley folds were indicated by a dashed line and mountain folds were indicated by a chain line (dot-dot-dash). These conventions are now widely established in step-by-step origami instructions and have become the international language of origami instruction. In a crease pattern, every fold line can be specified as to whether it is a valley fold or mountain fold in the folded form. This specification is called a crease assignment (or just assignment) of the crease pattern. It would seem natural to use the standard dashed and chain lines that are used in origami diagrams, but for crease patterns, they don’t work as well as they do in step-by-step instructions. Dashed lines and chain lines stand out when there are only a few of them, but for complex crease patterns, which arise in both figurate and geometric origami, they dissolve into a visual morass of indistinguishable strokes. For crease patterns, which can contain hundreds of folds, we need to adopt drawing conventions that provide a much stronger visual distinction between mountain and valley lines. In contrast to step-by-step origami diagrams, there is no standard convention yet for crease patterns—in part because the many variables of line pattern, thickness, hue, and saturation can be used to convey a wide range of information beyond simple valley or mountain status. No single attribute is ideal: varying the line weights degrades when a pattern is photocopied; color also doesn’t copy well (and, depending on choice of color, can fail for color-blind readers). If one does use dash patterns, they need to be strongly contrasting, even when viewed at a distance. The most robust convention would be to use all available attributes: line weight, color, saturation, and dashing. There is starting to be a consensus that in complex crease patterns, mountain folds should be dark, less saturated, and solid, while valley folds should be lighter, possibly more saturated in color, and, ideally, dashed, for the color-blind or otherwise visually impaired.1 1

Why are mountains the creases that are dark and solid in crease patterns? Generally, a mountain fold crease—the crease obtained by unfolding a mountain fold—exhibits greater contrast with the surrounding paper than a valley fold crease, so we give mountain folds the line style with stronger contrast in crease patterns.

8


Figure 1.3.

Left: a crease pattern using conventional mountain-valley line patterns. Right: the crease pattern using CP coloring.

The conventions I will use throughout this book are shown in Figure 1.2 on the right. I call this scheme CP coloring. The difference in visual perception and comprehension between the old and more recent representation systems can be striking; Figure 1.3 illustrates the same crease pattern with the two different drawing conventions. I will use this convention throughout for crease patterns. For step-by-step diagrams, however, I will continue to use the conventional dashed (valley) and chain (mountain) lines. The crease pattern can serve as a plan for the folded figure (though that is not its only role). Even as a plan, though, it is not a complete plan, in the sense of providing a complete description of the folded form. Not only does it fail to specify the temporal order in which one might form the creases, it doesn’t necessarily fully specify the stacking order of the facets in the folded form. One could, of course, just make up a stacking order for the facets, but if we choose a stacking order, that will imply a particular crease assignment. It might also imply that the paper intersects itself—which, in the real world, is not allowed. If a crease pattern can be folded with physical paper, i.e., with no stretching or selfintersection, then it is a valid crease pattern. Similarly, a stacking

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9

Figure 1.4.

Left: a crease pattern of two valley folds. Middle, right: two different stacking orders of the facets in the folded form.

A

B

C

A B

C

Figure 1.5.

Left: a crease pattern of two valley folds. Right: one of the two stacking orders in the folded form.

order on a crease pattern is valid if and only if it does not imply any self-intersection. Even with a complete crease pattern, determination of a valid stacking order can be computationally extremely challenging, even intractable. While a fuller analysis of this point requires analysis from the world of computational complexity [11], I would like to point out a very simple example that hints at the potential difficulties. Consider, for example, the crease pattern shown in Figure 1.4, consisting simply of two valley folds. There are two possible stacking orders for the facets, even though the creases are exactly the same. In this pattern, the two possibilities are rather obvious, but in complex crease patterns, there can be subtle and long-range interactions between parts of the crease pattern that limit potential stacking orders. Consider, for example, the crease pattern in Figure 1.5, similar to the preceding, but in which the two vertical valley folds divide the strip evenly into thirds. This pattern, too, admits two stacking orders, in which either facet A or facet C can wind up on top. But if facet A is just the tiniest bit wider than facet B, one of the two stacking orders is no longer possible; facet C can’t wind up on top because that would force facet A to penetrate the right crease. Similarly, if facet C were just a bit

10


wider, that would ensure that facet A could not wind up on top because that would force facet C to penetrate the left crease. And if both facets A and C were wider than B, this would be an invalid crease assignment: it would not be possible to make both folds at the same time in the specified direction. This relationship should set off some warning bells. What happens at the crease between facets B and C on the right depends critically on details at the far left of the crease pattern—namely, how far to the left facet A extends. The same logic applies to the crease between facets A and B. In general, every crease pattern has the potential for such non-local interactions between its constituent parts. The foldability of the pattern, and/or the validity of its crease assignment, can depend on relationships between far-flung features of its crease pattern. We will encounter many such examples in our explorations of tessellations and other mathematical folds. One more note on my schematic representations of origami forms: it is customary in origami to fold from paper that is colored on one side and white (or contrastingly colored) on the other. Although this practice is by no means necessary, it is often helpful to distinguish between the two sides of the paper. So, as I have done in Figures 1.4 and 1.5, I will usually show the two sides of the paper in contrasting colors and will refer, where appropriate, to the “white side” and “colored side” of the paper. Crease patterns will usually be drawn on the white side, by convention, and also, for better legibility and contrast. Although it is an incomplete description of a fold, a crease pattern is a very useful tool for concisely describing the structure of a folded shape, and while in principle the stacking order may be difficult to discern from the crease pattern, for the vast majority of folds of practical interest, the preferred stacking order is readily found. Thus we begin a long and fruitful relationship with crease patterns in this book. I provide many crease patterns as illustrations throughout, and I encourage you to reproduce them and try folding them up as you work your way through the book. Not just any pattern of lines can serve as an origami crease pattern. In fact, there are several highly restrictive conditions that apply that determine whether a crease pattern can be folded at all, whether it keeps the facets flat or forces them to bend, and whether it allows the paper to be entirely flattened. Such conditions are important: they tell us what is possible and impossible,

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11

and from among the possible, they provide guidance to accomplishing desired objectives—design rules, in other words. We start our journey with the simplest, most ideal form of origami: flat-foldable origami and their crease patterns. And we will start our study of such crease patterns with the building block of crease patterns: the crease. ? 1.1.2. Creases and Folds Crease patterns are made up of two types of geometric objects: points and lines. The points, or vertices, are places where lines come together. The lines, of course, are the creases themselves. We can identify a higher level of structure in crease patterns: crease lines outline facets. In the same way that crease lines are bounded by vertices (one at each end), facets are bounded by crease lines, by the border of the paper, or a combination thereof. As we have seen, there are constraints on crease patterns: not all crease assignments give valid, i.e., physically foldable, forms. There are also constraints on the angles of the creases relative to one another, which determine whether the origami figure can, in fact, truly fold flat. And there are constraints on the stacking order of the facets. All of the constraints arise in order to satisfy isometry and injectivity (from Section 1.1). In flat origami crease patterns, a crease line can take on one of three states: it can be a valley crease, a mountain crease, or an unfolded crease, i.e., one that is flat. Figure 1.2 showed a valley and a mountain fold. The third possibility is, simply, no fold at all. Figure 1.2 illustrates the difference between a valley and a mountain fold. If you are looking at the white side of the paper, a valley fold brings the moving part of the paper toward you, while a mountain fold moves it away from you. But we can also define valley and mountain folds in terms of the fold angle. By convention, the fold angle is defined as the deviation from flatness of the intersection between the paper and a plane perpendicular to the fold, as illustrated in Figure 1.6 on the left. By this convention, an unfolded crease has a fold angle of 0◦ , which fits with the concept of “not folded.” Similarly, mountain folds have the same magnitude of fold angle as valley folds, but opposite sign. Swapping the parity of folds—changing all mountain folds to valley folds and vice versa—is the same as changing the sign of all folds.

12


g >0

+180¡

0¡ +180¡ g =0

0¡ +360¡

g <0

-180¡

Figure 1.6.

Definition of the fold angle γ. Left: a flat-folded crease has a fold angle of (top to bottom) +180◦ for a valley fold, 0◦ for an unfolded fold, and −180◦ for a mountain fold. Right: a flat-folded crease has a dihedral angle of 0◦ for a valley fold, +180◦ for an unfolded fold, and +360◦ for a mountain fold.

A closely related measure of angle is the dihedral angle, which is typically taken to be the angle measured between two facets, as illustrated in Figure 1.6 on the right. The fold angle and dihedral angle are simply related: fold angle = 180◦ − dihedral angle. In origami analysis, the fold angle is usually the more natural way to characterize angles: • An unfolded crease has a dihedral angle of 180◦ and a fold angle of 0◦ , with the latter value corresponding to the idea of “no fold.” • Mountain and valley folds have dihedral angles of 360◦ and 0◦ , respectively, and fold angles of −180◦ and +180◦ , respectively, with the latter capturing the idea that a mountain fold is the opposite of a valley fold.

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13

Normally, a crease is identified as a mountain or valley fold based on the perspective of the viewer: a fold is a valley fold if it folds toward the viewer, whether or not the viewer is looking at the white or colored side of the paper. I will usually draw crease patterns as viewed from the white side of the paper, to provide greater contrast and visibility for the crease lines. In a flat origami crease pattern, all creases are one of mountain, valley, or unfolded. For convenience, we will often label these M, V, or U, respectively. A flat origami crease pattern whose lines have been labeled with their fold angle by color and/or line pattern is said to be crease-assigned (or just assigned, for short). One might wonder why one would include unfolded (U) creases at all; if the paper is unfolded everywhere within the facets, then what’s the distinction between an unfolded crease and no crease? It turns out that in the world of origami design, it is not uncommon to construct a crease pattern in two phases: first, compute the locations of all possible creases; second, assign those creases to be mountain, valley, or unfolded, depending on factors that relate to layer ordering, flap position, and the like. So we will consider the possibility of unfolded creases. If a crease pattern consists of only mountain and valley creases (no unfolded creases), we will call it a fully folded crease pattern. Many of the laws of crease assignment, such as the Maekawa-Justin Theorem, apply only to fully folded patterns. When this is the case will usually be clear from context; if the situation is ambiguous, I will make it explicit. We will often consider crease patterns in which the crease lines have not been assigned; such a pattern is called an unassigned pattern, naturally enough. In unassigned patterns, all crease lines will be drawn in the same way, as unfolded crease lines, as shown in Figure 1.7 (and sometimes we will show them as heavier lines, if greater contrast is desired). We also point out a property visible in the pattern in Figure 1.7 that is a universal property of flat origami crease patterns: if two facets are incident on a common folded crease in the crease pattern, then in the folded form, one of the facets must be white-up (white side facing the viewer) and the other facet must be color-up (colored side facing the viewer). Two facets incident to a common unfolded crease must, of course, have the same orientation in the folded form.

14


Figure 1.7.

Top: an unassigned crease pattern. Middle: the assigned crease pattern. Bottom: the folded form corresponding to this assignment.

Often when one is developing an algorithm related to the analysis of crease patterns, it is useful to consider the border of the paper to be a (special) type of crease: a border crease. This assumption means that every facet is bounded by some type of crease: mountain, valley, unfolded, or border. (Since border creases are incident to only a single facet, the fold angle for a border crease would be undefined.) We could, if we wished, mark each facet of the crease pattern in such a way as to indicate which way it faces in the folded form, whether it is white side up or colored side up. An example of this coloring for a simple origami model (Sam Randlett’s “New Flapping Bird” [102, p. 126]) is shown in Figure 1.8. Here we are looking at the white side of the crease pattern, but I have given a slightly darker tint to those facets that end up colored side up in the folded form. Whenever we cross a fold in the crease pattern, we must be moving from a white-up facet to a color-up facet or vice versa. Thus, for any flat origami crease pattern, if we color the facets according to whether they are white-up or color-up, this coloring has the property that no two facets of the same color meet along a common fold line. This is called a two-coloring of the crease pattern. Every flat origami crease pattern can be two-colored (remember, you only include crease lines that are actually folded);

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15

Figure 1.8.

Left: crease pattern; darker facets are colored side up in the folded form. Right: folded form.

the marking of which facets are white-up or color-up provides such a coloring. For any given two-colorable pattern, there are only two possible two-colorings: one is just the reverse of the other. Thus, we can go the opposite direction as well; given a flat origami crease pattern, each of the two possible two-colorings of that pattern automatically gives a map of which facets are white-up and which are color-up in the folded form—this without having to actually fold the pattern up, or even know which folds are mountain or valley. ?

1.2. Vertices Within a crease pattern, crease lines come together at points called vertices, and it is there that the conditions of flat-foldability begin to apply. First, we should define a bit of terminology. We have already talked about fold angles, the angles made between the facets on either side of a crease. At vertices within a crease pattern, we are concerned about the angles between the crease lines themselves. We call these the sector angles at the vertex. A hypothetical vertex is shown in Figure 1.9; each of the sector angles is labeled with the Greek letter theta (θ), subscripted by i, where i is the index of the sector angle. By convention, we will both number and index angles going counterclockwise (CCW) around any vertex, as we have done here. That is, we number the fold lines 1, 2, . . .. The ith sector angle is then the angle between the ith crease line and the next crease line going around the vertex.

16


f3

q3

f4

q4

f2 q2 q1

f1

Figure 1.9.

É

Schematic of a vertex with labeled sector angles {θ i } and fold direction angles {φi }.

f5

It is also possible to characterize each crease line by its fold direction angle, i.e., its angle measured with respect to some reference, typically an imaginary horizontal line emanating to the right. We will denote this fold direction angle by φi for the ith crease line (using the Greek letter phi). In Figure 1.9, the fold direction angle of the first crease line, φ1 , is 0◦ , since this crease line runs horizontally. The fold direction angles of all other crease lines are measured with respect to this reference. The sector angles are simply the difference in angle between two consecutive crease lines, and so for most of the sectors, we have that θi = φi+1 − φi . (1.1) This is except, of course, for the last sector angle, when the fold direction angles wrap around from 360◦ to 0◦ . We can handle this case by modifying the definition of sector angle: θi = (φi+1 − φi )mod 360◦,

(1.2)

where “mod 360◦ ” means that we add or subtract multiples of 360◦ to the value until it lies within the range [0, 360◦ ). (Incidentally, that’s not a typo. In mathematics, using a square bracket means the range includes the endpoint, while using a parenthesis means it doesn’t, so [0, 360◦ ) means that 0 is included in the range but 360◦ isn’t.) Now that we know how to talk about angles around a vertex, we are ready to say something about those angles. ? 1.2.1. Kawasaki-Justin Theorem The first property of vertices in a crease pattern ultimately derives from the condition of non-stretchiness (isometry), which manifests

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17

Figure 1.10.

Left: we cut off a circular region around a vertex in the folded form. Middle: the circle-cut corner. Right: the crease pattern of the vertex after unfolding.

itself at a vertex in the property that all sector angles are unchanged in moving from the crease pattern to the folded form. Let us consider a small thought experiment. Suppose that we have a complex folded origami figure (see Figure 1.10); we identify a single vertex, cut off a small circular arc around that vertex, then unfold it. The unfolded pattern becomes a circle when it’s flattened out. What can we say about the angles of the crease pattern, merely from the knowledge that it came from a flat-foldable form? We can do this for any vertex of any folded form, and we call the resulting circular crease pattern the vertex crease pattern. The angular region between consecutive pairs of folds is a sector; the angle of each sector is, of course, the sector angle, already defined. Each sector in the crease pattern appears in the folded form and vice versa, and they are connected to each other in the same way in both the crease pattern and folded form; that is, sector θ 1 is connected to sector θ 2 , which is connected to sector θ 3 , and so forth. However, while in the crease pattern the sectors are all white side up and are counterclockwise ordered, in the folded form, some of the sectors are white side up while others are colored side up, and this property has important ramifications. Consider a circular folded vertex: its crease pattern and its folded form are illustrated in Figure 1.11, in which I have distorted the folded form so that all of the circular edges are visible, and I have assigned a consistent direction to each sector angle. That direction is indicated by a tiny black arrow in the crease pattern

18


3

4 q3

6

2

q2

3

q1

q4

5

4

2

1 q6

q5

Figure 1.11.

q3 q1 q5

6

1 5

Left: a vertex crease pattern with sector arcs assigned a direction. Right: the folded form of the vertex.

and folded form, and it points consistently; that is, the arrow for sector 1 (angle θ 1 ) points from fold 1 toward fold 2, and so on, all the way around the circle. By following the arrows around the circular arcs, we traverse a complete circle in the crease pattern. By following the same path in the folded form, we no longer traverse a circle, but we still follow a closed path. Now, in the crease pattern, all of the directed sector angles run counterclockwise and must add up to a full circle, so we have an obvious relation on the sector angles, which can be generalized for N crease lines and sector angles: θ 1 + θ 2 + . . . + θ N = 360◦ .

(1.3)

If the sector angles around every vertex sum to 360◦ , then the vertex is said to be developable. For a planar crease pattern, this condition on the sector angles must hold for every vertex in the interior of the paper. In the folded form, though, some of the sector angles are turned over, and the directed arcs of those sectors run clockwise, rather than counterclockwise. Since they all connect up in a closed loop, the sum of the clockwise sector angles must be equal to the sum of the counterclockwise angles, so that as you traverse the loop, you end up in the same place that you started. As we go through the sectors in order, we see that they alternate: white-up, color-up, white-up, color-up, and so forth; and so all of the odd-numbered sector angles must have the same side up in the folded form, and all of the even-numbered sector angles must have the other side up in the folded form. In the example shown in Figure 1.11, the odd-numbered sectors are white side up and the even-numbered sectors are colored

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CHAPTER 1. VERTICES

19

side up. So the total angle of the two sets must be equal; thus Õ Õ θi = θ i, (1.4) i odd

i even

and this relation must hold no matter where we start the numbering—or how many vertices are incident to the vertex. Thus, this brings us to a very powerful and general result, which applies to any vertex in the interior of the paper of a flat-foldable origami form, which is commonly stated as follows: Theorem 1 (Kawasaki-Justin Theorem). Let v be a vertex in an origami crease pattern, and let θ 1, θ 2, . . . , θ N be the angles between consecutive creases going around the vertex (N must be even). Then the vertex can fold flat if and only if θ 1 − θ 2 + θ 3 − θ 4 + . . . − θ N = 0.

(1.5)

The Kawasaki-Justin Theorem was described in the 1980s by Japanese mathematician Toshikazu Kawasaki [60], a prolific origami artist and mathematician, and Jacques Justin [54], a French mathematician who developed much of the mathematical theory of origami. (Actually, the theorem was proven even earlier, by S. A. Robertson in 1978 [104], but it is so widely associated with Kawasaki and Justin that I will continue to use the common name for it.) The number of creases incident on the vertex—the quantity N in the theorem—is called the degree of the vertex. Why must the degree be even? Well, as we travel around the vertex in the folded form, each time we cross a fold the paper switches from white side up to colored side up. In order to end on the same side where we started after going around the circle, we have to go through an even number of flips. Hence, the number of folds must be an even number. And why make the stipulation that the vertex must lie in the interior of the paper? If the vertex lies on the border, there is no way to create a closed loop, on which this result depends. The Kawasaki-Justin Theorem can be stated in many equivalent ways. One useful variation is the following: Theorem 2. Let v be a vertex in an origami crease pattern, and let θ 1, θ 2, . . . , θ N be the angles between consecutive creases going

20


around the vertex (N must be even). Then the vertex can fold flat if and only if θ 1 + θ 3 + θ 5 + . . . + θ N−1 = θ 2 + θ 4 + θ 6 + . . . + θ N = 180◦, (1.6) i.e., the sum of alternating angles around a flat-foldable vertex is equal to 180◦ . We call the condition on the angles of the Kawasaki-Justin Theorem the Kawasaki-Justin Condition. The Kawasaki-Justin Theorem and its variations can also be proved in many ways, and while the preceding demonstration was more of a hand-waving exercise, we will encounter more rigorous formulations later on. We do note that the Kawasaki-Justin Condition is a necessary condition for flat-foldability, not sufficient; there are other conditions that must be satisfied as well, and we will encounter them shortly. The Kawasaki-Justin Theorem is one of the major tools in the arsenal of creating flat-foldable origami; many design rules boil down to ensuring that the Kawasaki-Justin Theorem is satisfied at every vertex of the crease pattern. We will have ample occasion to make use of the Kawasaki-Justin Theorem, so we will give it an abbreviation, KJT, which we will use later on. ? 1.2.2. Justin Ordering Conditions The mathematician Jacques Justin gave a concise set of mathematical conditions that must apply to the stacking order of a set of origami facets [56], which we will call the “Justin Non-Crossing Conditions.” These conditions can be expressed formally and algebraically (as Justin did in [56], and as we will see later), but they are perhaps best appreciated pictorially, as shown in Figures 1.12–1.14. The Justin Non-Crossing Conditions describe the three stacking order configurations that are valid and forbidden in a description of valid flat-folded origami. Imagine you are looking at a cross section of the paper. From top to bottom, they are the following: (a) If two creases overlap each other so that their facets overlap, then the facet pairs incident to the two creases cannot be interleaved, as in Figure 1.12.

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21

A B B A

A B A B

A A B B

Figure 1.12.

Left: allowed stacking orders for facets around two overlapping folded creases. Right: a forbidden stacking order. B

B A A

A B A

B

A A B

B

Figure 1.13.

Left: allowed stacking orders for facets around a folded crease that overlaps an unfolded crease or facet. Right: a forbidden stacking order.

(b) If a layer of paper overlaps a crease, it cannot lie between the facets incident to the crease, as in Figure 1.13. (c) If one facet lies above another on one side of an unfolded crease, it cannot lie below the other facet on the other side of the same line, as in Figure 1.14.

A B

Figure 1.14.

A B

A B

Left: allowed stacking orders for facets around two overlapping unfolded creases. Right: a forbidden stacking order.

22


A B

The Justin Non-Crossing Conditions apply to the stacking order of the facets away from the crease, but the figures make it obvious why these configurations should be forbidden as a description of origami: they all involve the paper passing through itself. In all three cases, facet A passes through facet B at the dotted line. If we could see this sort of cross sectional picture of the folded form, it would be obvious whether there is a self-intersection and whether the picture describes a valid form of origami. The problem is that in the most common model of origami, we don’t have a picture in which the layers are spread apart like this. Usually we have just the crease pattern, and we have to find a stacking order that is consistent with the crease pattern—i.e., the folds drawn as mountain folds are ordered the way mountain folds are supposed to be, as shown in Figure 1.2, and the same for valleys—and that avoids any of the facet orderings shown in Figures 1.12–1.14. ? 1.2.3. Three Facet Theorem The Kawasaki-Justin Theorem is a theorem that stems from the non-stretchiness of the paper; more specifically, it follows from isometry, the fact that when we fold the paper, we do not change distances (or angular measures) on the folded form, as long as we’re measuring along the surface of the paper (and don’t jump between layers when they are stacked up). The Justin Non-Crossing Conditions are different: they arise from the non-self-intersection requirement. They let us formulate a simple law that finds use surprisingly often in the design and analysis of folded structures. Let’s return to the simple crease pattern shown in Figure 1.5 and adjust the dimensions slightly, so that when folded, the two side flaps overlap not just each other, but also the opposite creases by just a bit, as shown in Figure 1.15. From the positions of the creases, we know what the silhouette of the folded form must be. The question to consider is, what are the possible crease assignments on the crease pattern, or equivalently, which layer lies on top of which in the folded form? As we have already noted, the stacking order among the overlapping facets provides a deeper description of the folded form than the crease pattern: given the former, we can work out the latter, while there may be more than one possible stacking order for a given crease assignment.

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23

C A

B

C

A B

Figure 1.15.

Left: an unassigned crease pattern. Right: the silhouette of the folded form.

So what are the possible stacking orders here? Could both A and C lie on top of B? If they did, then one of the two must be on top. If A is on top, then C lies between A and B, which means that C would slice through fold AB, violating case (b) of the Justin ordering conditions. Conversely, if C were on top, then A would lie between B and C, and A would slice through fold BC, also violating case (b). In either case, the paper intersects itself. A similar argument would apply if both A and C lay below B. Thus, we can set some constraints on layer order in this situation: Theorem 3 (Three Facet Theorem). Given three adjacent facets A, B, and C, where in the folded form facet A overlaps crease BC and flap C overlaps crease AB, facets A and C must lie on opposite sides of facet B. The proof of this theorem (henceforth, TFT) comes simply from considering all of the possible arrangements of the three facets: only the ones with A and B on opposite sides of B avoid self-intersection. By considering the relationship between facet order, two-coloring, and crease direction, we can establish a similar law that relates to crease assignment: Theorem 4 (Three Facet Crease Assignment). Given three adjacent facets A, B, and C, where in the folded form facet A overlaps crease BC and flap C overlaps crease AB, creases AB and BC must have opposite parity. This two-crease, three-facet arrangement is the simplest configuration where non-self-intersection plays a role in determining whether a crease pattern is flat-foldable or not. There are far more complex arrangements where self-intersection issues matter, and we will encounter many of them.

24


q3

q2

Figure 1.16.

q1

Three consecutive sectors of a vertex, with θ 2 < θ 1 and θ 2 < θ 3 .

? 1.2.4. Big-Little-Big Angle Theorem One of the simpler arrangements of three facets where TFT plays a role is the case where the two creases share a common vertex, as illustrated in Figure 1.16. In this case, the middle of the three angular sectors has a smaller angular measure than the two sectors to either side, so that, in the folded form, the conditions of TFT are satisfied. In general, for any two consecutive creases around a node, there are four possible crease assignments: • two mountain folds, • two valley folds, • mountain fold then valley fold, • valley fold then mountain fold. Following a terminology introduced by Palmer,2 we will call the angular sector where the two creases are of the same type an iso sector, whether they are both mountain or both valley, and we will refer to the two crease as iso creases. If the two creases differ, the sector is called an anto sector and the pair of creases are anto creases, as illustrated in Figure 1.17.

q

q

q

q

Figure 1.17. iso 2

Private communication.

anto

Left: two iso sectors. Right: two anto sectors.

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25

q3

q2

q1

Figure 1.18. Three possible crease assignments for the two creases on either side of a smallest sector and the corresponding folded form. Top: a valid assignment, V M. Middle: a valid assignment, MV. Bottom: an invalid assignment, VV, which leads to a collision of the layers.

q3

q2

q1

! q3

q2

q1

Whatever else is happening with the other folds of the vertex, we can say one of two things definitively about the folds on either side of the middle sector: the creases must have opposite directions. As shown in Figure 1.18, they can be mountain-valley or valley-mountain; but if both creases have the same assignment, then the wider sectors on either side of the short one collide as they try to fold past one another, as shown at the bottom of Figure 1.18. Thus, the only valid crease assignments for the two creases in this sector are the two anto assignments. And so, this gives another fundamental law of flat-foldability that was identified by Kawasaki [61, 60] and, a few years later, by Justin [56] (though I will use a name coined by Hull [51, p. 173]): Theorem 5 (Big-Little-Big Angle (BLBA) Theorem). At any vertex, the creases on either side of any sector whose angle is smaller than those of its neighbors must have anto (opposite) crease assignment. We call the condition of the BLBA Theorem the BLBA Condition. It is important to note that this relation only holds for strict

26


inequality: the sector angle must be absolutely smaller than its neighbors to force the anto condition. Because this situation turns up fairly often, we will give it a corresponding name: a BLBA sector is a sector at a vertex whose angle is strictly smaller than those of the sectors to either side. ? 1.2.5. Maekawa-Justin Theorem The Kawasaki-Justin Theorem deals with the sector angles, but not with the fold types, and follows purely from isometry of the paper. A second property addresses the fold types—mountain/valley status—themselves. As with the previous section, we will give a “plausibility argument” here, rather than a formal proof. Consider a folded vertex, like the example shown in Figure 1.19, oriented with the vertex at the bottom and white side on the outside. There must be a counterclockwise-most crease coming out of the vertex, such as the one labeled A in the figure, and a clockwise-most crease, labeled B in the figure. (If there are two or more creases at the extremal positions, you can pick one of them arbitrarily). Now consider what happens as we move from crease A to crease B along the figure. Since all of the facets we see are white, we must start on a white-up facet; similarly, we must end on a white-up facet. What about what happens in between? Imagine what happens along the circular edge. Every time we encounter a mountain fold, the path makes a 180◦ turn to the left; every time we encounter a valley fold, the path makes a 180◦ turn to the right. We can make two or more mountain or valley folds in succession, but the paper can’t penetrate itself, which means that the paper edge can’t form a complete loop; instead, every turn that is made at some point needs to be unwound by a turn in the opposite direction. And so, traveling across the front of this cone from crease A to crease B, there must be the same number of mountain folds as valley folds. A

B

Figure 1.19. A folded vertex.

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27

Exactly the same argument applies to the back side, of course. So, looking at all of the folds at the vertex, the number of mountain folds and valley folds must be the same—except for the two folds at the edges, which, for a white-outside vertex, are both mountain folds. So there are two more mountain than valley folds. If, however, we had started with a color-outside vertex, we would have ended up with two more valley than mountain folds. But those are the only two possibilities. And so we have a general law about fold directions that applies to any flat vertex, which is called the Maekawa-Justin Theorem, in honor of Jun Maekawa, who first identified the relation, and Justin, who proved it [55]. It states the following: Theorem 6 (Maekawa-Justin Theorem). For any flat-foldable vertex, let M be the number of mountain folds at the vertex and V be the number of valley folds. Then M − V = ±2.

(1.7)

That is, for any vertex, the number of mountain folds and valley folds at that vertex must differ by exactly 2. We’ll use the abbreviation MJT for the Maekawa-Justin Theorem. We’ll call Equation (1.7) the Maekawa-Justin Condition. The argument presented above implicitly assumes non-selfintersection of the paper, because the assignment of mountain fold to both creases A and B is based on the assumption that the facet that reaches the right-most crease B is still white-up—which is a big assumption. For example, we can imagine something mysterious going on in the middle of the arc, as shown in Figure 1.20, in which a color-up layer somehow gets in front of the white-up layer. In this case, crease B becomes a valley fold and the Maekawa-Justin Theorem would not hold. But this type of rearrangement of the layers does not happen; and we can, indeed, rely upon the Maekawa-Justin Theorem to A

B

? Figure 1.20. A mystery folded vertex.

28


hold at every interior vertex—a vertex in the interior of the paper. Like the Kawasaki-Justin Theorem, it does not necessarily hold for vertices on the border of the paper (and usually does not). A related corollary gives a property we have already seen: any flat-foldable vertex must have an even number of creases emanating from it. From Equation (1.7), M = V ± 2,

(1.8)

the total number of creases must be (V ± 2) + V = 2(V ± 1),

(1.9)

which is clearly even. The argument presented above appeals heavily to intuition and so isn’t really a proof; but it turns out that MJT follows readily from a well-known theorem in spherical geometry, Girard’s Theorem, which we will eventually meet. A flat-foldable vertex can be classified by the relationship between the numbers of creases of each type. The crease type that there is more of is the majority type; the other is the minority type. Flat-foldable vertices whose majority type is mountain are said to be mountain-like vertices; otherwise they are valley-like vertices. With duo paper—paper that is colored on one side and white on the other—a mountain-like vertex will be white in the folded form, and a valley-like vertex will be colored. This can be seen, for example, in Figures 1.25 and 1.27–1.29. ? 1.2.6. Vertex Type While the Maekawa-Justin Theorem specifies the number of mountain and valley folds around a vertex, it does not say anything about their relative order around the vertex. We can concisely describe the fold order around a vertex by constructing a word composed of Ms and Vs giving the fold types one encounters as one goes around the vertex (in counterclockwise direction, by convention). We call this the vertex type. Figure 1.21 shows an example vertex of type VVV M MV. Of course, the vertex type is not unique for a given vertex: it depends on where we start counting. The vertex in Figure 1.21 is also VV M MVV, V M MVVV, M MVVVV, and so forth. And it does not fully specify the vertex: we would also need sector angles for a complete specification of the crease pattern, and we

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29

V V

V

Figure 1.21.

M

A vertex of type VVV M MV.

M

V

would need the stacking order to fully specify the folded form. However, it is a useful shorthand for describing vertices, and so we will find occasion to use it as we go forward. ? 1.2.7. Vertex Validity Suppose we have a vertex and crease assignment (and sector angles). Is it valid? It may satisfy KJT and MJT, but it could still force a layer intersection somewhere along the way as we try to bring all of the folds together. We can determine this using an efficient procedure developed by Hull [49, 50] and described by Demaine and O’Rourke [22, p. 207]. Consider first a flat vertex v with sector angles (θ 1, θ 2, . . . , θ N ) and imagine that we begin to fold it up. If it has a smallest sector angle θi , we would start with that sector; it must be anto, and so there are two possible crease assignments, either MV or V M for the two creases on either side. If we form those two creases (but only those), the paper would now form a cone as shown in Figure 1.22, because we have effectively “taken a bite” out of the vertex circle by making these two folds. Ignoring the fact that the cone no longer lies flat, we could, in fact, treat this as a new vertex in which the trio of sector angles θi−1, θi, θi+1 has been replaced by a single sector angle whose value is θi0 = θi−1 − θi + θi+1 . We call this process sector reduction.3 Beginning with the flat, crease-assigned vertex, we ask: “is there a BLBA sector?” Meaning, if there are multiple equal smallest sectors, does any one of them have the anto crease as3

Demaine et al. refer to this process as “crimping” [22, p. 194], although generally in origami, a zigzag fold through one or more layers like this is called a “pleat” [68, pp. 30–31]. To avoid ambiguity, I will give the procedure its own distinct name.

30


qi+1-qi+qi-1

qi

qi qi+1

qiÐ1

Figure 1.22.

Left: a conical vertex in which angle θ i is the smallest sector angle. Middle: the cone resulting from one of the two possible crease assignments. Right: the vertex after reduction.

signment? If so, reduce that sector, i.e., drop the two creases and replace the sector angle trio θi−1, θi, θi+1 with a single sector of angle θi0 = θi−1 − θi + θi+1 . Repeat the process, always looking for the smallest remaining sector angles, until you are left with two equal sectors. If at any point in the procedure there was no smallest angle that was anto, then the vertex was not flat-foldable; otherwise, it is (and, if we kept track of the layer orders at each reduction, the final configuration provides a valid layer-ordered solution). Call this procedure the Single Vertex Flat-Foldable Test (SVFFT). If a crease-assigned vertex satisfies KJT, MJT, and SVFFT, then it is guaranteed to be valid, i.e., there exists at least one folded form that does not self-intersect. We should note, though, that the valid layer-ordered solution found by this procedure is not guaranteed to be unique. A vertex crease assignment does not fully specify the folded form, i.e., does not always determine uniquely the stacking order of the layers, as shown in Figure 1.23, which presents two different folded forms for the same vertex crease assignment. That ambiguity can play a role in determining flat-foldability for multiple sets of vertices— which, of course, most origami crease patterns consist of—as we will presently see. The vertex reduction process and associated crease assignment counting formulas become rather complex for arbitrary vertices of high degree. Most of the crease patterns that arise in mathematical origami, however, tend to have relatively low vertex degree, and thus only a few special cases apply. Now that we know some general properties about vertices, let’s look at some specific types of vertices.

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31

Figure 1.23.

Left: a crease-assigned vertex. Middle: one folded form for this vertex. Right: another folded form for this vertex.

?

1.3. Degree-2 Vertices The smallest flat-foldable vertex is the degree-2 vertex, a vertex that has two creases emanating from it. A generic degree-2 vertex is shown in Figure 1.24, although we can immediately see from the preceding laws that this cannot be flat-foldable as drawn. Since alternating angles must sum to zero from KJT, we must have θ 1 = θ 2 = 180◦ . And since M + V = ±2 from MJT, either both creases are mountain or both are valley. Thus, the only two possibilities for a degree-2 vertex are that (a) the two creases must be collinear and (b) they must both have the same crease assignment, as shown in Figure 1.25. One might well say that there is no vertex there; that this is just a single crease line. But, for completeness, we should recognize that the degree-2 vertex is still a possible vertex, and while the restriction to flat-foldability forces us to just these two configurations, if we allow 3D folding and/or curved or bent facets, even the humble degree-2 vertex (or even a degree-1 vertex!) can

q1

Figure 1.24. A degree-2 vertex with sector angles θ 1 and θ 2 .

32


q2

Figure 1.25. The two possible flat-foldable configurations for a degree-2 vertex.

give rise to shapes of considerable beauty, as can be seen in the work of British artist Paul Jackson in Figure 1.26. ?

1.4. Degree-4 Vertices The next smallest flat-foldable vertex is the degree-4 vertex, a vertex that has four creases coming from it. This type of vertex shows up often in origami crease patterns, particularly in mathematical

Figure 1.26. Single-crease (and single-vertex) three-dimensional folds by Paul Jackson. Left: “He Said, She Said,” two squares of wet-folded 450 gsm watercolor paper. Photo courtesy of the Eretz Israel Museum. Right: “Untitled One Crease Form,” one square of wet folded 350 gsm watercolor paper. Originally published in [52]. Used by kind permission.

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33

folding. In fact, there are very many origami crease patterns that are composed exclusively of degree-4 vertices. The degree-4 vertex also has some special properties, as we will see, and so we will spend a bit of time focusing on this particular creature. Although there are four sector angles and four creases, the four angles are not independently choosable. If we label the four sector angles θ 1 , θ 2 , θ 3 , and θ 4 , then they must, of course, sum to 360◦ : θ 1 + θ 3 + θ 2 + θ 4 = 360◦ .

(1.10)

But also, from the Kawasaki-Justin Theorem, θ 1 − θ 2 + θ 3 − θ 4 = 0.

(1.11)

So, combining these two relationships, we find that θ 1 + θ 3 = θ 2 + θ 4 = 180◦,

(1.12)

that is, opposite angles must sum to 180◦ . So, there are only two degrees of freedom; one can choose any two adjacent sector angles, and then the other two are their respective supplements. Next, according to the Maekawa-Justin Theorem, there are only two possibilities for crease assignment to such a vertex: three mountains and one valley fold, or three valleys and one mountain fold. That means that the assignment of the four creases will be either M M MV, VVV M, or a cyclic permutation of one of these. But not all three-of-one-and-one-of-the-other crease assignments are possible because, as we have seen, the sector angles themselves play a role in what crease assignments are possible. ? 1.4.1. Unique Smallest Sector If there is a unique smallest sector angle, then according to the BLBA Theorem, that sector angle must be anto, i.e., bounded by two creases of opposite assignment. The other two creases, then, must have the same assignment, and so its sector is iso. Thus, there are only four possible crease assignments for such a vertex. The four possibilities for such a degree-4 vertex—which we will call a unique-smallest-sector vertex—are shown in Figure 1.27. We note that if there is a unique largest sector angle, then the sector opposite to it must be the smallest and must also be unique, so we could have easily named such vertices uniquelargest-sector vertices instead. We can say something about the

34


Figure 1.27. The four possible crease assignments for a unique-smallest-sector degree-4 vertex and their folded forms.

crease assignment at that largest sector, as well. Since the sector opposite has an anto assignment (either MV or V M), and the vertex must have three of one type and one of the other (either VVV M or M M MV), the two creases bounding the largest sector must have the same type, either M M or VV, and therefore, it must be iso. Thus, we have a corollary of the BLBA Theorem, albeit one exclusive to degree-4 vertices: Theorem 7 (Unique Largest Angle Theorem). In a degree-4 vertex, if there is a unique largest sector angle, its crease assignment is iso. The important ingredient here is that the smallest (largest) sector angle be unique; the other two sector angles may or may not be equal to each other. (The figure shows an example where they are unequal.) ? 1.4.2. Two Consecutive Smallest Sectors If the smallest sector angle isn’t unique, there are more than four possible crease assignments. Let us first consider what happens if two consecutive sector angles are equal to each other but are CHAPTER 1. VERTICES

........

35

Figure 1.28. The six possible crease assignments for a degree-4 vertex with two consecutive smallest sectors and their folded forms.

both smaller than the other two sector angles. Since opposite sector angles add to 180◦ , then the remaining two sector angles must also be equal to each other, and if the first two sectors were the smallest, then they must both have sector angles less than 90◦ while the remaining pair has angles greater than 90◦ . Filtering all possible crease assignments by the BLBA Theorem to weed out the impossible assignments reveals that there are exactly six possible crease assignments, shown in Figure 1.28.

36


This particular type of degree-4 vertex arises regularly in origami patterns, and so we will give it its own name: a flatfoldable degree-4 vertex with two consecutive smallest sectors is a symmetric bird’s-foot vertex (or just “bird’s-foot” vertex, for short), named for its resemblance to the arrangement of toes on (most) perching birds. What if the two smallest sectors are equal but not consecutive— i.e., they are opposite? Then, since they sum to 180◦ , they must be equal to 90◦ . But if they are the smallest sectors, then the other two sectors must both have angles greater than 90◦ , which—since those other two angles must also sum to 180◦ —is not possible. What if we consider three consecutive equal smallest sector angles? Then the first and third angle of this trio must be opposite angles, and since they sum to 180◦ , they (along with the one between) must all be equal to 90◦ . In this case, the fourth sector angle must also be equal to 90◦ , which brings us to the third possible configuration for a degree-4 vertex. ? 1.4.3. Four Equal Sectors If all four sector angles are 90◦ , then the vertex has fourfold symmetry. We can choose either M M MV or VVV M assignment, and we can pick any one of the four creases to be the “odd” crease in this assignment. Thus, there are eight possible crease assignments for this most symmetric vertex, which we call a right degree-4 vertex. The eight possibilities are shown in Figure 1.29, along with their folded forms. And that completes the enumeration of the possible assignments for a degree-4 vertex. Although this might seem like overkill of pedantry, it is useful to have an explicit list of the possibilities when one is assigning creases to a more complicated crease pattern consisting of degree-4 vertices: it converts the crease assignment problem to a finite combinatorial problem. For each vertex, based on the angles, we can identify the set of four, six, or eight possibilities that apply to each individual vertex. Crease assignment then consists of assigning MV status to each of the creases so that the creases at each vertex form one of the acceptable possibilities. ? 1.4.4. Constructing Degree-4 Vertices The situation regularly arises in origami design that three of the four creases at a flat-foldable vertex are known and the fourth is to be found. An example is shown in Figure 1.30.

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CHAPTER 1. VERTICES

37

Figure 1.29. The eight possible crease assignments for a right degree-4 vertex and their folded forms.

q3 = ?

Figure 1.30. A partial degree-4 vertex. The remaining crease is to be found.

38


q4 = ?

q2 = 60¡

q1 =100¡

3

3

3 4

2

1

2

2

1

1. Extend line 1 across the vertex and duplicate line 2.

1

2. Rotate the wedge between the extension and line 2 so that the edge along line 2 is lined up with line 3.

3. Erase the right side of the wedge; the left side is the desired fourth crease.

Figure 1.31. Geometric construction of the missing crease at a degree-4 vertex when three are known.

We can, of course, solve this problem mathematically by solving for the unknown angles. In a degree-4 flat-foldable vertex, opposite angles sum to 180◦ , so we must have that θ 3 = 180◦ − θ 1 = 80◦, θ 4 = 180◦ − θ 2 = 120◦ .

(1.13) (1.14)

But what if we are drawing the crease pattern directly? Do we have to stop drawing and measure the angles, compute the new angles, then mark them off on the vertex? No, as it turns out. There are several ways to construct the required fold directly from the three existing creases. Perhaps the easiest is brute force: if you cut out the vertex and form the first three creases in the right place, then press it flat, the remaining crease will form automatically in exactly the right place. But we can also construct the remaining crease geometrically without having to measure any angles, as shown in Figure 1.31. Computer drawing programs typically offer tools to rotate selected items about a selected point, and sometimes even offer “snap-to-object” options that will give precise geometric alignments, making the construction of Figure 1.31 both straightforward and precise. An alternative construction devised by Ilan Garibi4 accomplishes the same result, but using reflection of single lines rather than rotation of a wedge. It is illustrated in Figure 1.32. 4

Private communication.

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CHAPTER 1. VERTICES

39

3

3

2

2

Figure 1.32. An alternative geometric construction of the missing crease at a degree-4 vertex when three are known.

1

1

1. Reflect line 2 across line 1 and again across line 3.

2. The desired crease is halfway between the two reflected lines.

In the last step of Garibi’s construction, one must find the angle bisector between two lines. Some programs can create this directly, but if not, there is a simple workaround, shown in Figure 1.33. This still requires the ability to rotate a line by 90◦ about its midpoint, but this is a function that is very commonly built into vector drawing software. Using tricks such as this, it is often possible to construct quite complex and sophisticated flatfoldable crease patterns without ever computing a single angle. However, as we will eventually see, when we enter the realm of 3D folding, computation is nearly unavoidable.

3

3

2

2

1 1. Draw a line between the two radii with endpoints on the same circle.

3

1 2. Rotate the line by 90¡ about its midpoint.

2

1 3. Extend the line to the vertex to get the fourth crease line.

Figure 1.33. Constructing a bisector between two given lines at the center of a circle can be accomplished with a rotation.

40


? 1.4.5. Half-Plane Properties While we’re on the topic of the rotational positioning of creases around a vertex, there are several interesting properties of vertices that all relate to half-planes, some of which we’ve already seen. For example, for degree-2 vertices the following holds: Theorem 8 (Degree-2 Vertex Half-Planes Theorem). In the crease pattern of a flat-foldable degree-2 vertex, the creases divide the paper into two half-planes. This is, of course, just another way of saying that the two creases must be collinear. Things get a bit more interesting with degree-4 vertices. Theorem 9 (Degree-4 Vertex Half-Planes Theorem). In the crease pattern of a flat-foldable degree-4 vertex, every half-plane contains at least one crease of the majority type. We already know from the Maekawa-Justin Theorem that there must be three folds of one type and one of the other; Theorem 9 tells us that the three cannot be excluded from any given halfplane. In fact, more broadly, every half-plane contains either exactly one or exactly two of the majority creases, except for the special case where all four angles equal 90◦ , in which case there is a half-plane that contains all three (with two of them on the border of the half-plane). The folded form also has half-plane relations: Theorem 10 (Vertex Folded Form Half-Planes Theorem). In the folded form of a flat-foldable vertex, every crease lies within a common half-plane. This result turns out to be useful in numerical analysis: when solving for folded forms numerically, the mathematical conditions can sometimes give rise to spurious solutions that can be weeded out by applying this property. Note that this is not restricted to degree-4 vertices; it is a property of every flat-folded interior vertex. All of these are relatively easily proven, and their proofs are left as an exercise for the interested reader. Theorem 9 also generalizes to higher-order vertices and, as well, to not necessarily flat-foldable vertices [1]: CHAPTER 1. VERTICES

........

41

Theorem 11 (Partially Folded Vertex Half-Planes Theorem). In the crease pattern of any vertex for which all creases can be at least partially folded simultaneously, there cannot be both • a half-plane that contains no mountain folds, • a half-plane that contains no valley folds. ??

1.5. Multivertex Flat-Foldability Thus far, I have given several explicit conditions for a single crease-assigned vertex to be flat-foldable: the Kawasaki-Justin Theorem, the Maekawa-Justin Theorem, and Kawasaki’s BigLittle-Big Angle Theorem. For a simple degree-4 vertex, we can enumerate directly the possible assignments, given the sector angles; we can count all possible assignments, and we can use sector reduction to test if a given assignment is valid. The situation becomes a lot more complicated, however, when we start to consider networks of creases that consist of multiple vertices. Since every crease has two vertices and each vertex may place conditions on all of its incident creases, there is a possibility for different vertices to place contradictory conditions upon sets of creases. Even beyond that, it is possible to find crease assignments that are consistent at every vertex, but that result in collisions between layers of paper that, on the crease pattern, are far removed from one another. A crease assignment for which each vertex considered in isolation is valid is called locally flat-foldable. As the name suggest, it ensures that individual vertices can fold flat but provides no guarantee that the entire crease pattern folds flat without selfintersection. In fact, determining global flat-foldability can be a very challenging problem indeed—one that we will, by and large, bypass in our design and analysis.

?? 1.5.1. Isometry Conditions and Semifoldability In general, for a crease pattern to be foldable in the real world, it must satisfy both isometry and injectivity conditions. The extension of isometry conditions from individual vertices to entire networks of creases is generally straightforward; the extension of injectivity conditions (non-self-intersection) can get very complex indeed. It is natural, then, in origami design to take on the easy

42


P

Figure 1.34. A Justin path on a crease pattern.

part first: we solve for a set of creases that gives the right shape (which addresses isometry), then we look for a crease assignment that allows that shape to be folded without self-intersection (which addresses injectivity). In his paper “Towards a Mathematical Theory of Origami” that introduced many of the theorems of origami [56], Jacques Justin introduced the concept of semifoldability: a crease pattern is semifoldable if it satisfies isometry conditions. Thomas Hull has introduced the notion of ghost paper to describe this concept: allowing a model to satisfy flat-foldability but possibly allowing the paper to pass through itself like a ghost passing through a wall. If we allow ghost paper, then we are only addressing isometry, or to use Justin’s terminology, we only address semifoldability. Justin introduced several conditions related to semifoldability that apply to full crease patterns, not just individual vertices. They are all based on a common concept: the notion of a simple closed path on the crease pattern that doesn’t pass through any vertices, as illustrated in Figure 1.34. We call such a path a Justin path. The requirements of isometry come from a simple notion: if you travel along the Justin path in the folded form, in order for the crease pattern to be flat-foldable, you must end up where you started when you’ve come all the way around. Or, put mathematically:

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CHAPTER 1. VERTICES

43

s4

s5

s3

Figure 1.35.

s2

Cut a narrow strip of paper to follow the path, then split the path at a point P.

P

s6

P

s7

s1

s8

Theorem 12 (Justin Isometry Theorem). For any Justin path C, let F1, . . . , FN be the sequence of folds crossed as one traverses the path from some point P, and let σi be the reflection operation in crease Fi . Then the composition of reflections σ1 ◦ σ2 ◦ · · · ◦ σN

(1.15)

is the identity operation. Theorem 12 must hold for any path on the crease pattern. The notion of “composition” of reflections may be unfamiliar, but an example of its application should make it clear, illustrated in Figures 1.35 and 1.36. Suppose we have a Justin path on a crease pattern; we cut a narrow strip of paper to either side of the path, as illustrated in Figure 1.35, and then we split the path at a point P on the path. Composing the reflections {σi } means that we start at the end and work backward, folding the strip of paper at each fold, as shown in Figure 1.36, which is equivalent to reflecting the paper (and point P) in each crease line. Note that we are keeping track of the original point P at the other end of the split. Observe that after the final reflection, the two halves of the point P line up with each other—as do, in fact, the two full edges of the split. This alignment must hold for every possible path on the crease pattern if the result is to fold flat. The isometry condition has several important consequences that are readily verified: Theorem 13. Let C be a Justin path on a crease pattern. Pick a point P on the path; label the creases in the order that they are encountered as you travel around the path, and let θ 1, θ 2, . . . , θ N be the angles between consecutive creases (defined modulo 180◦ )

44


σ5

σ5

σ4 σ2

σ3

σ6

σ1

σ2

σ3

σ6

σ1

σ4

σ6

P

P

σ7

σ8

σ5

σ4 σ2

σ3

σ1

P

σ7

σ8

P

σ5

σ2

σ3

σ6

P

P

σ7

σ5

σ4

σ4

σ4

σ3 σ2

P

σ1

P

σ3 σ2

σ1

P

P

σ1 P

σ3 P

σ2

P

σ1 P

σ2

σ1

P

P

σ1

Figure 1.36. From top left to bottom right: successive composition of the reflections {σi }.

encountered traveling along the path. Then the entire crease pattern can fold flat only if θ 1 +θ 3 +θ 5 +. . .+θ N−1 = θ 2 +θ 4 +θ 6 +. . .+θ N (mod 180◦ ). (1.16) It is also clear that the Kawasaki-Justin Theorem readily follows from Theorem 12; it is the special case when the path encloses a single vertex of the crease pattern. If a crease pattern is semifoldable, the Justin Isometry Theorem (JIT) must be satisfied for every possible path on the pattern, and vice versa: if JIT is satisfied for every possible path, then the pattern is semifoldable, and, apart from the possibility of layer intersection, it can be folded flat. Fortunately, given a crease pattern, we don’t have to check every possible path. Instead, we only need to check a subset. If the paper on which the crease pattern is mapped is a simple polygon—has no holes—then the Kawasaki-Justin Theorem is sufficient; if it is satisfied at every interior vertex, the pattern is CHAPTER 1. VERTICES

........

45

semifoldable. But what if the paper has holes (say, a doughnutshaped piece of paper)? Justin showed the following: Theorem 14. A crease pattern is semifoldable if and only if the isometry condition (Theorem 12) is satisfied at each of its vertices and by all of the closed paths that are the boundaries of any holes within the sheet. We have to be careful with holes in the paper. If the sheet has one or more holes, it is not difficult to construct a crease pattern that is flat-foldable/semifoldable at every interior vertex but that fails flat-foldability as we go around the hole. ?? 1.5.2. Injectivity Conditions and Non-Twist Relation The injectivity conditions that enforce non-self-intersection are nothing more than the Justin Non-Crossing Conditions already seen, but extended to the entire crease pattern. However, satisfying them can be extremely difficult, computationally speaking, because of the ways in which widely separated parts of the crease pattern can interact with one another in the folded form. In fact, there is no simple test that can be applied to an assigned crease pattern to determine whether it is flat-foldable without layer intersections. There are some simple tests that detect the most obvious pathological conditions, and in many origami designs, those simple tests are sufficient. The Three Facet Theorem and its corollary, the Big-Little-Big-Angle Theorem, can detect problems around individual vertices. One can also check larger assemblies of folds along paths, using the following result by Justin: Theorem 15 (Justin Non-Twist Theorem). Let C be a Justin path that visits n creases and no crease more than once. Let θi be the angle between successive creases, M be the number of mountain folds encountered along the path, and V be the number of valley folds. Then θ 1 + θ 3 + . . . + θ n−1 = θ 2 + θ 4 + . . . + θ n =

M −V × 180◦ (mod 360◦ ). 2

(1.17)

?? 1.5.3. Local Flat-Foldability Graph Let us return, for the moment, to the degree-4 vertex. If the sector angles are distinct, then there is a smallest sector angle, which

46


60¡

80¡

100¡

Figure 1.37. A single vertex with creases joined by red lines for an anto pair and blue lines for an iso pair.

120¡

we know must be anto—the two creases on either side must have opposite parity—and a largest sector angle, which we know must be iso—its two creases must have the same parity. For a single vertex, we can illustrate this requirement in graphical form, by placing a dot in the middle of each edge and connecting the two edges across the anto sector by a red line and the two across the iso sector by a blue line, as illustrated in Figure 1.37. Now, what happens if we have two or more degree-4 vertices that share one or more creases? Each vertex contributes an anto (red) and an iso (blue) line to our graphical form, and if two vertices share a crease, the iso/anto lines for each vertex are connected at the shared edge, as shown in Figure 1.38. With two or more vertices, the lines are no longer disconnected segments; they begin to link up. I call the resulting pattern the local flat-foldability graph, or LFF graph for short. The LFF graph now establishes a connection between creases that are not shared between the vertices. Consider the three creases marked A, B, and C in the figure. Since A and B are connected by a blue line, they must have iso (same) parity. B and C are connected by a red line, so they must have anto (opposite) parity. And so, we can tell that A and C must have opposite parity, even though the two creases

C B A

Figure 1.38. Two vertices with their LFF graph superimposed.

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CHAPTER 1. VERTICES

47

are not actually in contact with one another. In order for all of the vertices to be locally flat-foldable, the crease assignments across the entire pattern must obey the parity relations specified by the LFF graph. Using the LFF graph, one can determine the parity relationship between any two creases that are connected by a path on the LFF graph. Since blue segments indicate same-parity, and red segments indicate opposite-parity, it suffices simply to count the number of the red segments between the two creases in question: if the number is even, the two creases have the same parity; if the number of red segments is odd, then the two creases have opposite parity. And if two creases are not connected by a path on the LFF graph, then their assignments can be chosen independently. The LFF graph typically contains multiple disconnected components. So, if you choose any one crease in a connected component of the LFF graph, then the parity of every other crease in that component is determined by the number of anto edges connecting it to the initial edge. For each connected component, there are two choices of assignment (the first crease can be either M or V); thus, the total number of locally-flat-foldable possibilities is 2N , where N is the number of connected components of the LFF graph. Thus, for example, the pattern of Figure 1.38 has three connected components and, thus, 23 = 8 possible crease assignments that are locally flat-foldable. What if a connected component makes a loop? That is entirely possible, and Figure 1.39 shows one example, for a crease pattern with four interior vertices. There are five connected components, one of them a loop, which is highlighted in gray. What does a loop signify? Since for any two creases the relative parity depends on the number of anto lines in the LFF

A

Figure 1.39. A four-vertex crease pattern with an LFF graph that makes a loop.

48


B

Figure 1.40.

Left: a three-vertex crease pattern that has no locally flat-foldable crease assignment. Right: the crease pattern together with its LFF graph.

graph between them, if there is a loop, we can determine relative parity by following either path and counting the number of anto lines in the path. Thus, for example, in Figure 1.39, the upper path is entirely iso, which implies that crease A and crease B must have the same parity. Or, if we follow the lower path, there are two anto lines; the parity reversals cancel each other out, and so, again, we get the same implication: A and B have the same parity. So this LFF graph tells a consistent story. But consistency is not always to be found! The pattern shown in Figure 1.40, discovered by Thomas Hull [48], has an LFF that contains a loop with an odd number (3) of anto lines. This implies that, if you go around the loop, any given crease in the loop has the opposite parity to itself. This is, obviously, impossible to satisfy, and so there is no crease assignment for this crease pattern that is locally flat-foldable at each vertex. This crease pattern illustrates a broader property of LFF graphs, whose basic notion was described by Kawasaki [61, 60]: Theorem 16. If any closed loop in an LFF graph contains an odd number of anto lines, there is no locally flat-foldable crease assignment for the corresponding crease pattern. The LFF graph offers a quick way to analyze local flatfoldability, even for fairly complex crease patterns—provided that every vertex in the network is degree-4 and has a unique smallest sector. Figure 1.41 shows a much more complex crease pattern

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CHAPTER 1. VERTICES

49

A

Figure 1.41.

Top: a complex crease pattern. Bottom: the crease pattern together with its LFF graph.

B

designed by Kawasaki, both bare and together with its LFF graph overlaid. Examination of the LFF graph shows two loops, A and B. Loop A contains 11 anto lines and thus is an odd-parity loop; consequently, there is no crease assignment that gives this pattern local flat-foldability. We can generalize the LFF graph for vertices of higher degree by going back to the MV-checking algorithm described in Section 1.2.7. Begin with a flat vertex. If there is a BLBA sector, then the sector’s crease assignment must be anto (either MV or V M), so we draw an anto line between those two creases. Now we sector-reduce on those two creases (which then go away) and consider the new cone. Again, there may or may not be a BLBA sector somewhere; if there is, we add an anto line between those two creases, sector-reduce again, and continue. If we successfully get down to the point where there are two creases left, those two

50


A H

45¡

A

60¡

B

40¡

F

45¡

G

F

40¡ 40¡

45¡

E

E

A

60¡ 55¡

B

40¡ C

45¡

55¡

D

45¡ 45¡

C

40¡

A

60¡

40¡

45¡

B

60¡ 60¡ B

C

D

D

A H

45¡

60¡

B

40¡

45¡

G

C

45¡ F

40¡ 40¡ E

45¡

D

Figure 1.42.

Top: reduction of a vertex to construct the LFF graph on its creases. In each step, the shaded sector is the BLBA sector that gets an anto edge between its creases, which are then sector-reduced. Bottom: the original vertex with iso and anto lines between its creases.

must be iso (M M or VV), and so we finish up by adding an iso line between the last two creases. This process is illustrated in Figure 1.42. Note that when the vertex degree is 6 or larger, the edges of the LFF graph are not necessarily between adjacent creases, as in Figure 1.43. 40¡ 80¡

80¡

Figure 1.43. 70¡

70¡ 20¡

A degree-6 vertex together with its LFF graph.

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CHAPTER 1. VERTICES

51

120¡ 90¡ 90¡ 60¡ 40¡ 45¡ 45¡ 40¡

45¡ 40¡ 60¡ 45¡

45¡ 40¡

60¡ 45¡ 40¡ 45¡

45¡ 40¡

Figure 1.44.

Left: a crease pattern with two degree-8 vertices and one degree-4 vertex. Right: the CP overlaid with its LFF graph. The odd-anto loop in the central triangle ensures that it is not flat-foldable.

We can combine two copies of the vertex from Figure 1.42 with one vertex from the impossible triangle of Figure 1.40 to get a new “impossible triangle,” shown in Figure 1.44. Without the LFF graph, it might not be so obvious that this is not flatfoldable; there are 17 creases in total, so theoretically 217 possible assignments. But the LFF graph makes clear that none of them can satisfy local flat-foldability at all three vertices, due to the central loop containing one (versus three previously) anto edge. What about borders? For vertices on the border of the paper, there is no requirement that KJT or MJT be satisified: alternating sector angles have no fixed sum, and the numbers of mountains and valleys have no fixed difference. Surprisingly, though, border vertices can play a role in the LFF graph, because a border vertex can still have a BLBA sector (or multiples), and across that sector, the anto condition must apply, with its corresponding anto edge in the LFF graph. Thus, for example, we can move a corner of the central triangle in Figure 1.40 to the border of the paper (and stretch it upward to ensure a strict BLBA Condition), as shown in Figure 1.45. We lose the iso sector at that vertex; but we retain the anto sector, and so, once again, we have a pattern that satisfies KJT and MJT at its two interior vertices, but that is not locally flat-foldable. The complete constructibility of the LFF graph relies heavily on the presence of a BLBA sector at each stage of the sector re-

52


Figure 1.45. A non-locally-flat-foldable crease pattern, together with its LFF graph that includes a contribution from the border vertex.

duction; when a sector is not strictly smaller than its neighbors but may be equal to one or both, then there are more crease assignment options than can be captured by the simplicity of the LFF graph. As we saw with the degree-4 vertex, runs of consecutive sector angles increase the number of possible assignments: in the case of degree-4 vertices, from 2, to 4, to 8 possibilities, depending on the number of equal sectors. ???

1.6. Vector Formulations of Vertices Much of this book will be devoted to how we mathematically construct and analyze origami crease patterns and their folded forms, and how we go about analyzing them depends strongly on the way we mathematically describe the paper. We have already begun such a description of single vertices, using geometric measures such as sector and fold angles and geometric shapes: triangles, circles, and arcs. A purely geometric description of a crease pattern can provided needed insight into its design. A remarkable amount of origami design can be carried out with minimal mathematics and no computation, simply by manipulating geometric shapes and carrying out basic drawing operations familiar to most artists and/or draftspeople. However, there are many origami patterns that are defined by relationships that are sufficiently complex that the only way to find them is to solve mathematical equations. For those forms, the crease pattern is not simply constructed; it is computed. If we wish to compute a crease pattern, then we need a representation that is more amenable to computer analysis, and that CHAPTER 1. VERTICES

........

53

representation is usually vectorial: each vertex of the crease pattern (and of the folded form) becomes an ordered set of coordinates in a 2D (or 3D) space. Vectors, matrices, and their kin (the general field of linear algebra) are usually not encountered until collegelevel mathematics, but they are an essential part of any curriculum in engineering and most sciences. If you have not had any linear algebra, you can still use this book; just skip over any of the ? ? ? sections. If you have had some linear algebra, you have the tools to explore a vast world of origami possibility. In this section and ? ? ? sections of subsequent chapters, I will assume a basic familiarity with linear algebra but will introduce some specific terminology and conventions that are particular to origami. If you are not familiar with vectors and linear algebra, don’t worry; while you probably won’t be able to write your own computer programs to compute origami tessellations, many of the concepts in this book can be implemented with no more than advanced high school mathematics—or not even that! But if you have some linear algebra background, you will have what you need to compute origami twists, tilings, and tessellations, both crease patterns and folded forms. ? ? ? 1.6.1. Vector Notation: Points In a crease pattern, a vertex of the crease pattern is represented by an ordered pair (x, y) of its x- and y-coordinates. I will use the notation p = (p x, p y ) to represent vectors throughout this book, that is, a lower-case boldface letter indicates a two- or three-dimensional vector. Ordinary mathematical italic type represents non-vector (scalar) quantities. For structures whose folded form is flat and therefore twodimensional, the vertices of the folded form can also be represented by a 2-component vector, e.g., p0 = (p0x, p0y ). I will usually use the same letter for a vertex in the folded form as I used for the corresponding vertex in the crease pattern, but I will add a prime to one or the other to distinguish the two, as in Figure 1.46.

54


p5

p6

p7

p4′

p4 p3′

p3

p1

p2

p1′

p6′

p7′

p2′

p5′

Figure 1.46. A crease pattern and its folded form can both be specified by the vertices of the pattern as 2-vectors. Left: crease pattern, with coordinates {pi }. Right: folded form, with coordinates {pi0 }.

If the folded form is 3D, we can still use a vector description, but we will then use 3-vectors, e.g., p0 = (p0x, p0y, p0z ). Whether a 2D or 3D vector is implied by p0 should always be clear from context. For the next several chapters, we will be concerned primarily with flat-folded (or at least, flat-foldable) forms and so will need only 2-vectors, but we will get into 3D folded forms and 3-vectors beginning in Chapter 9. Once we have a vector description of a crease pattern, we can tap into the great arsenal of linear algebra for analysis and manipulations. As it turns out, the mathematical tools needed for manipulating origami crease patterns overlap strongly with the mathematical tools used in computer graphics, and a good working introduction to the field can be found in books on computer graphics, game design, and the like (see, for example, [75] for a good introduction). ? ? ? 1.6.2. Vector Notation: Lines A single ordered set can represent a point in either 2D or 3D. But how might we represent a line? The best way to represent a line turns out to depend on whether we actually mean an infinitely long line or a line segment, a finite section of a line. There is also a third possibility: a ray, which is

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CHAPTER 1. VERTICES

55

semi-infinite. For a line segment, the logical representation seems obvious; a line segment can be defined uniquely by its endpoints, and so a logical way of representing a line segment would be as an ordered pair of its endpoints. If a line segment has endpoints q1 and q2 , then any point along the segment can be expressed as a parameterization: p(t) = (1 − t)q1 + (t)q2,

t ∈ [0, 1],

(1.18)

where t = 0 gives the first endpoint q1 and t = 1 gives the second endpoint. This parameterization has a logical generalization for representing an infinite line; we can represent any point on the infinite line through q1 and q2 with the same expression, but we can now let the parameter t range from −∞ to +∞. So an infinite line could be represented by an ordered pair (q1, q2 ), where q1 and q2 are any two distinct points on the line. There is another way of representing an infinite line, though. Given two points q1 and q2 somewhere on the line, we can rearrange this expression to give p(t) = q1 + tr,

t ∈ [0, 1],

(1.19)

where the first term, q1 , represents a point on the line, and the multiplier of t, r = (q2 − q1 ), is an offset in the direction of the line. So a line could also be represented by an ordered pair (q, r), where q is a point on the line and r is a direction vector that is parallel to the line. There is yet a third way of representing a line, though, which is less obvious than these two but is sometimes more useful: that is the ordered pair (q, u), where q is a point on the line and u is a direction vector that is perpendicular to the given line. This way of representing a line specifies points on the line indirectly. Any point p on the line satisfies the equation u · (p − q) = 0,

(1.20)

which turns out to be useful when you want to establish a condition that a point lies on a given line in the analysis of a crease pattern or folded form; you can apply this equation without having to explicitly parameterize the point in question along the line. Another benefit of this representation, which we call point+ perpendicular, is that it generalizes in a natural way to higher dimensions; in 3D, a plane can be similarly defined, as we will eventually see. All three of these ways are illustrated in Figure 1.47.

56


y

y

y

L

L r

q2 q1

u

q2 q

q1 x

L

x

x

Figure 1.47. Three ways of representating a line in vector notation. Left: from two distinct points on the line. Middle: from a point on the line and a vector parallel to the line. Right: from a point on the line and a perpendicular vector.

We define the norm of a vector r = (r x, ry ) by q krk ≡ r x2 + ry2 .

(1.21)

For a line segment between two points q1 and q2 = q1 + r, krk gives the length of the line segment, i.e., it is the length of vector r. (The norm generalizes in the obvious way to higher dimensions.) We can construct and/or manipulate origami crease patterns whose vertices are represented by vectors using any of three basic operations, which we now describe. ? ? ? 1.6.3. Translation Translation of a vertex—movement in a specified direction—is achieved simply by adding to the vertex a vector whose components are the distances to move in the respective x- and ydirections. A vector can thus be used to represent two distinct types of quantities: it can represent a point at a fixed position, but it can also be used to represent an offset—a directional distance through which a point is shifted. If p = (p x, p y ) represents a point that is to be moved a distance d x in the x-direction and dy in the y-direction, the total offset can be represented by the vector d = (d x, dy ), and the shifted point p0 has coordinate values given by p0 = p + d, as illustrated in Figure 1.48.

(1.22)

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CHAPTER 1. VERTICES

57

y p′ = p + d

d

Figure 1.48.

p

Translation of a point is accomplished by vector addition.

x

? ? ? 1.6.4. Rotation Rotation of a vector is achieved by multiplying it by a rotation matrix R(φ) (and, optionally, translating it before and/or after rotation). Throughout this book, I will denote matrices by boldface capital letters. The rotation matrix for a 2D rotation of a point about the origin through a counterclockwise angle φ is given by

R(φ) =

cos φ − sin φ sin φ cos φ

,

(1.23)

and the rotated point (r in Figure 1.49) is given by r = R(φ) · q.

(1.24)

Quite often, you will need to rotate one point about another point, rather than about the origin. The result of such a rotation can be efficiently computed by translating the first point by an amount that puts the center of rotation at the origin, rotating it, and then translating it back by the same amount. Thus, to rotate a point p about the point q, the resulting point is given by r = q + R(φ) · (p − q).

Figure 1.49.

Left: rotation of a point q about the origin is accomplished by a matrix multiplication. Right: rotation about another point p.

58

y

(1.25)

y

r

r

q

f f


q

p x

x

One application of rotation in 2D is to convert between the different representations of a line—(p, r), where r is a vector along the line, and (p, u), where u is a vector perpendicular to the line. The direction vectors in Equations (1.19) and (1.20) are related by a 90◦ rotation: 0 ∓1 ◦ u = R(±90 ) · r = · r. (1.26) ±1 0 This is a good place to note that many computer programs and packages for graphical manipulation have functions for translation, rotation, etc. built in; it is unlikely that you will have to implement such low-level functions yourself. However, different systems vary in such things as sign conventions (e.g., does a rotation matrix rotate things clockwise or counterclockwise?). Part of the reason for reproducing these functions here is that it will make clear what sign and direction conventions are assumed when the analysis gets more complex later on. We can go the other direction: given two vectors (q, r), we would like to compute the angle φ between them, as illustrated in Figure 1.49, measured counterclockwise. Said angle would lie in the range [0, 2π) (or, perhaps, in the range [−π, π), if we want to allow negative angle values). A well-known vector relation connects two vectors with the cosine of the angle between them. For the configuration on the left in Figure 1.49 (rotation about the origin), this relation can be written as q · r = kqkkrk cos φ. (1.27) This expression can be solved for φ, but there is an ambiguity, since cos φ = cos(2π − φ). If we rotate one of the two vectors by π/2, using the rotation matrix R( π2 ), we get the relation q · (R( π2 ) · r)) = kqkkrk sin φ,

(1.28)

which also has a similar ambiguity, since sin φ = sin(π − φ), but the ambiguity is between a different pair of angles. Using both relationships completely resolves the ambiguity, though. So, we can compute the unique value of φ from the formula φ = tan−1 q · r, q · (R( π2 ) · r) , (1.29) where the function tan−1 (x, y) is the so-called four-quadrant arctangent whose value is either tan−1 (y/x) or π + tan−1 (y/x), depending on the signs of x and y.

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59

In many cases, we don’t need the angle directly; instead, we are working with trigonometric functions of the angle, and so we can use expressions for its sine and cosine: q·r , kqkkrk

(1.30)

q · (R( π2 ) · r) . kqkkrk

(1.31)

cos φ =

sin φ =

If, instead of looking for the rotation angle about the origin, we are looking for the rotation angle about a third point p, as on the right in Figure 1.49, we translate the vectors to the origin and use the same formula as Equation (1.29): φ = tan−1 (q − p) · (r − p), (q − p) · R( π2 ) · (r − p) . (1.32) ? ? ? 1.6.5. Reflection The operation of reflection in a line is one that happens a lot when mathematically modeling origami, for the simple reason that whenever you make a complete fold (fold angle = 180◦ ), everything on one side of the fold is reflected through the fold line. The transformation from crease pattern to folded form (and back) is one long series of reflection operations applied to the vertices of the pattern. So, if you want to reflect a point through a fold line, the first thing you need is a representation of the fold line that’s causing the reflection. As we have seen, there are three different ways that a given line could be represented. It turns out that for reflection, the third way is a clear winner for simplicity of expression. If we represent the line of reflection by the pair (q, u) where q is a point on the line and u is a direction vector perpendicular to the line as illustrated in Figure 1.50, then the reflection of a point p in said line is given simply by p0 = p − 2u

(p − q) · u , u·u

(1.33)

or, if the direction vector u happens to already be normalized,

60


p0 = p − 2u(p − q) · u.

(1.34)

y u

p′

q

p

Figure 1.50.

x

Reflection of a point.

There’s no beating that for simplicity, and so, whatever the original representation of the reflection line, it is almost always simplest to convert it to the point+perpendicular representation to compute the reflected image of a point. As an added bonus, this formula generalizes to higher dimensions. In 3D, instead of reflecting in a line, we reflect through a plane, and as noted above, the point+perpendicular description defines a plane. The reflection of a point in a plane parameterized in this way is given by exactly the same formula, Equation (1.33). Now, if we are representing the vertices of a crease pattern by points {pi }, the lines of reflection are going to be fold lines of the pattern (or reflected images of said fold lines) and so the lines are going to be represented by vertex pairs (pi, p j ) of their endpoints, not (q, u). However, one can use Equation (1.26) to convert the vertex pair to a perpendicular, then Equation (1.33) to perform the reflection. ? ? ? 1.6.6. Line Intersection Very often in the design and construction of a crease pattern, a vertex is defined as the intersection of two preexisting lines. (In the mathematical origami theory of number fields defined by origami constructions, that is the only way that a vertex is defined). It is useful, then, to have a formula for the intersection of two lines. As we saw with reflection, we have a choice of how the line is defined. The most concise expression comes from the point+direction form of the lines. Define the first line by a point p on the line and direction vector r along the line, and define the second line by point q on the line and direction vector s along the line, as illustrated in Figure 1.51.

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61

y

t r

Figure 1.51. Configuration for finding the intersection of two lines.

p

s q x

Then the intersection point t of the two lines is given by qx − p x qy − p y sx sy t=p+ r, (1.35) rx ry sx sy where | · · · | indicates a matrix determinant. This formula is easily modified for other line representations. The result is undefined if the determinant in the denominator goes to zero; this happens if and only if the two lines are parallel (in which case, of course, they do not intersect). As with some of the previous formulas, a function that gives the intersection of two lines is often built into mathematical manipulation packages, but sometimes it is not, and in any event, it can be useful to “roll your own” for specific analyses. Since we are introducing determinants, this will also be a good place to introduce an alternate notation for matrices and determinants. A (linear) vector is an ordered list of values: p ≡ (p x, p y ). In the same way, a square (or generally, rectangular) matrix can be considered an ordered pair of vectors, e.g., px py M ≡ (p, q) = . (1.36) qx qy The representation of Equation (1.36) is said to be row-major. We can also write the determinant of a matrix M as det M. With these two definitions, Equation (1.35) can be written in the somewhat more compact form t=p+ 62


det(q − p, s) r. det(r, s)

(1.37)

This notation saves us from having to write out all of the individual components of a matrix, and it is also closer to how computations are typically expressed: functions acting on vectors as objects, rather than component by component. ? ? ? 1.6.7. 2D Developability Matrices and vectors allow us to concisely and precisely express the relationship between a crease pattern and its folded form. As an example, we’ll consider a very simple condition: developability, which is the condition that a folded shape can be unfolded to a flat sheet of paper. Given a degree-n vertex with sector angles {αi }, i = 1 . . . n, we already know that condition: it is n Õ

αi = 2π.

(1.38)

i=1

This equation is simple enough that it would seem that there is no need to complicate matters by introducing vectors and matrices, but we will do so all the same. We usually think of an angle as a numerical value (of degrees or radians). But we can also think of an angle as an action, specifically, a rotation (about the specified angle). So the statement that the sum of the sector angles is 2π could be expressed in terms of actions thus: The result of rotating about all of the sector angles, in sequence, should be equal to the result of rotating through a total angle of 2π. The composition of rotations can be expressed as a product of matrix rotations, using the 2D rotation matrices from Section 1.6.4. Recall that the matrix that gives a 2D rotation about the origin through angle θ is cos θ − sin θ R(θ) = . sin θ cos θ So the condition that the angles sum to 2π is Ö D≡ R(αi ) = R(2π) = I,

(1.39)

i

where I is the 2 × 2 identity matrix

10 01

.

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63

On the surface, this seems like a needless complication. A 2 × 2 matrix has four components; all four must be satisfied for equality. We have replaced a single simple equation that is linear in the angles with four equations involving complex combinations of trigonometric functions of the angles. For example, the (1, 1) entry of the rotational matrix product D for a degree-4 vertex with sector angles (α1, α2, α3, α4 ) is D1,1 = cos α4 (cos α3 (cos(α1 ) cos α2 − sin α1 sin(α2 )) + sin α3 (sin α1 (− cos α2 ) − sin α2 cos(α1 ))) + sin α4 (cos α3 (sin α1 (− cos α2 ) − sin α2 cos α1 ) − sin α3 (cos α1 cos α2 − sin α1 sin α2 )). (1.40) This quantity must be equal to 1 (since we are equating the matrix D to I). It is not at all obvious that the condition is Í equivalent to i αi = 2π. And, in fact, the condition D1,1 = 1 is not precisely equivalent Í to i αi = 2π. It is only one of four components of the matrix equation D = I, but even that is not equivalent. Recalling the rotational matrix identities R(θ) · R(φ) = R(θ + φ),

(1.41)

R(2πn) = I

(1.42)

for integer n, we can see that Equation (1.39) is actually equivalent to the condition Õ αi = 2πn, (1.43) i

for any integer n. So while the solution set for Equation (1.39) includes properly developable sets of angles in its solutions, it also includes spurious solutions: those for which the sector angles sum to 0, for example, and those for which the sector angles sum to 4π. We can eliminate the former by adding the requirement that αi > 0. Eliminating the latter requires more care. For a degree-4 vertex, if we constrain the sector angles so that αi < π, then it’s not possible for four angles to sum to 4π, and so spurious solutions are indeed eliminated. For higher-degree vertices, however, we cannot so easily eliminate spurious solutions. For a degree-6 vertex, αi = 2π/3 are otherwise perfectly good angular values but they sum to 4π, rather than 2π; such a vertex is not developable.

64


Another question to consider with the matrix form of developability is this: do we need all four entries of the matrix product D, and if not, which one(s)? Rather than diving into algebraic manipulation of its entries, let’s think about what the matrix D is: it is a product of rotations, so D itself is some rotation. Rotation matrices satisfy two important properties: • Their rows and columns are orthonormal. • They are unitary (with unit determinant). If we know that the matrix D is a rotation matrix, then we may only need a single component in order to fully specify the matrix. If, for example, we know that D1,1 = 1, then the unit magnitude of its rows and columns implies immediately that D1,2 = D2,1 = 0. This, in turn, implies that D2,2 = ±1 by the same token. The second requirement of unit determinant then tells us that only one of the two values of D2,2 is acceptable; we must have D2,2 = 1. Thus, even though Equation (1.39) is really four equations on its four entries, we only need one of them. If D1,1 = 1, then the other three entries automatically satisfy their individual equations. The same goes for the other diagonal entry (by the same reasoning): if D2,2 = 1, then the other three entries automatically satisfy their individual equations. What about the off-diagonal entries? Suppose, for example, we have satisfied D1,2 = 0. Is this solution set the same as the solution set for Equation (1.39)? 0 We can see that it is not, by observing that D = −1 0 −1 has its off-diagonal entries equal to zero but is not a solution of the developability system. However, if one off-diagonal entry is zero, the other must be, and so the matrix must be either 10 01 0 or −1 0 −1 . Thus, a single off-diagonal element can get us to the solution, provided that we have another means of eliminating the second spurious solution (for example, by requiring that a diagonal entry be positive). As we will see, despite the potential for such spurious solutions, the off-diagonal terms have considerable utility when searching for solutions via numerical means. Hanging over all of this is the question: why, if we are searching for angles, would we even bother with the complication of products of rotational matrices to establish the developability condition? It seems like a needless complication. And if we are

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65

p3 p2 q3

Figure 1.52. Schematic of a vertex defined by the coordinates of the points surrounding the vertex.

p4

q4

q2 q1

p1

É

p5

expressing a crease pattern in terms of angles, it very often is a needless complication. But we are not always describing crease patterns in terms of their angles. In fact, very often the most convenient description of a crease pattern is not in terms of angles and crease lengths, but in terms of vertex coordinates as 2-vectors. In such a description, the sector angles are not explicit; they are derived from the vertex coordinates. Consider a vertex centered at the origin, as illustrated in Figure 1.52. Instead of defining the creases around the vertex by their fold direction angles, we instead define them by the vertices at their ends, which we will denote by p1, p2, . . . , pn . In order to express the developability, we need to know what the angles {θi } are in terms of the vector coordinates {pi }. From Í there, we can write the developability condition i θi = 2π. Quite obviously, this expression will have inverse trigonometric functions embedded within whose arguments are algebraic combinations of the vertex coordinates. We could also plug those angles into Equation (1.39), our matrix equation for developability—just to complicate things further. But the matrix expression offers the possibility of simplification as well; we can completely bypass the trigonometric expressions by noting that the entries of the individual rotation matrices may be written entirely in terms of the dot products that give the angle cosine and sine in Equations (1.30) and (1.31). Using those relations, the rotation matrix that rotates from pi to pi+1 (or the reverse) may be written as 1 pi+1 · pi ∓pi+1 · (R( π2 ) · pi ) R(±θi ) = (1.44) pi+1 · pi kpi+1 k kpi k ±pi+1 · (R( π2 ) · pi ) 66


Now we can write the developability condition as a matrix equation that is purely algebraic in terms of the vertex coordinates {pi } around the vertex under consideration: no trigonometry required. And, as noted already, we don’t actually need all four entries of the resulting matrix as equalities: only one of the diagonal entries needs to satisfy its equality to be assured that all four entries satisfy their respective equalities. It must be noted, though, that if we have a description of a crease pattern as a 2D plane embedding of its vertices, then developability is automatically satisfied at every vertex! So it might appear that we have gained nothing by this analysis. We have derived a matrix equation for developability that, by its very formulation, is guaranteed to hold. However . . . the points {pi } don’t have to be a crease pattern; they could equally well be the vertices of a description of the folded form. In which case, we could then apply the developability condition, or rather an equivalent expression involving the rotation matrices of Equation (1.44), to the folded form, effectively finding a flat-unfoldability condition. We will do something very much like that in Chapter 6. The concepts introduced here are, in fact, broadly applicable. We can enforce conditions that apply to angles by manipulations of vertex coordinates. And by using rotation matrices composed from those coordinates, we can construct purely algebraic conditions on origami crease patterns—and, as we will see, their folded forms. ? ? ? 1.6.8. 2D Flat-Foldability Developability is automatically satisfied for a crease pattern drawn on a plane where we know the 2D vertex coordinates. But another important condition is not automatically satisfied: the KawasakiJustin Theorem, which was necessary for flat-foldability. KJT applies to the angles around every interior vertex of a crease pattern. In a vectorial model of a crease pattern, this law must still apply. In a vector model, though, the fundamental description of the crease pattern is not in terms of angles; it is in terms of the vertex coordinates. In order to apply KJT to a vector crease pattern, or even a single vertex, we need a way of extracting the relevant angles from the vector coordinates of the vertices. As we have just seen, we now have exactly that.

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67

For the vertex of Figure 1.52, the Kawasaki-Justin Theorem for flat-foldability is n Õ

(−1)i θi = 0,

(1.45)

i=1

or equivalently,

Õ

θi =

i even

Õ

θi = π.

(1.46)

i odd

These can be expressed as matrix equations, using products of the matrices given in Equation (1.44): n Ö

R((−1)i θi ) = I,

(1.47)

i=1

or equivalently, Ö Ö R(θi ) = R(θi ) = R(π) = i even

−1 0 0 −1

.

(1.48)

i odd

We can define these rotation matrix products as the flatfoldability matrices, F≡

n Ö

R((−1)i θi ),

(1.49)

i=1

F(even) ≡

Ö

R(θi ),

(1.50)

R(θi ),

(1.51)

i even (odd)

F

≡

Ö i odd

and then the equivalent of the Kawasaki-Justin Condition is any of 1 0 −1 0 −1 0 (even) (odd) F= or F = or F = . (1.52) 0 1 0 −1 0 −1 In practice, either of the latter two equations is often more useful, since it has half the number of terms in the expanded matrix product. We refer to any of the equalities in Equation (1.52) as the Algebraic Kawasaki-Justin Condition (AKJC). You might notice one difference between Equations (1.52) and (1.48) and the original Kawasaki-Justin Condition: the latter was

68


a single equation, while these algebraic expressions define four individual equations. Which do we choose? As we saw with the developability condition, we only need to choose one of the entries of the resulting product matrix, but the one we choose depends on what we’re looking for and what other constraints there might be on the problem, and, depending on what we choose, we might need to be careful of spurious solutions to any equation system. Remember, the Kawasaki-Justin Condition assumes a particular cyclic ordering of the lines (and hence, vectors) about the vertex in question. One could potentially find a solution that satisfies this algebraic equation but that has a different cyclic order than the assumed one that yielded the equation. Perhaps an example will make this clearer. For a degree-4 vertex, the alternating-angle form of the algebraic KJT can be expressed in an extremely simple form in terms of the individual vertex coordinates. If we take pi ≡ (xi, yi ), then for a degree-4 vertex, the condition arising from either off-diagonal entry of the rotation matrix becomes (x1 x2 + y1 y2 ) (x4 y3 − x3 y4 ) + (x2 y1 − x1 y2 ) (x3 x4 + y3 y4 ) = 0.

(1.53)

This expression is linear in the coordinate values for any one point; if we were given three of the four points, the fourth must lie on a line defined by this equation. So, for example, if we take p1 = (1, 0), p2 = (0.5, 0.6), p3 = (−0.8, 0.3), then the coordinates of p4 must satisfy 0.63x4 + 0.22y4 = 0.

(1.54)

This configuration is illustrated in Figure 1.53. The fourth vertex p4 must lie on the line L defined by the preceding linear equation. y p4(?)

p2=(0.5,0.6)

p3=(Ð0.8,0.3) x p1=(1,0)

L p4

Figure 1.53. Algebraic Kawasaki-Justin Condition. The fourth vertex must lie on the line L.

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CHAPTER 1. VERTICES

69

The point could, of course, be anywhere on this line and satisfy the algebraic condition; both of the points marked p4 in the figure are solutions. However, any choice on the upper section violates the assumption of the cyclic order that we used to construct the equation; only the lower point is a valid candidate for the remaining vertex. Despite the existence of spurious solutions, the simplicity of the algebraic condition both is aesthetically appealing and can be useful when setting up numerical solutions for crease patterns defined by the arrangement of their vertices. With suitable safeguards to avoid spurious solutions, the AKJC can be quite useful. ? ? ? 1.6.9. Analytic versus Numerical When setting up a matrix condition on angles—developability or flat-foldability, for example—the matrix system of equations is commonly overdetermined, as was the case with the two examples shown here. There are four individual equations in a 2 × 2 matrix equation, but only one of them may be needed; once we have solved the equation for one entry, the equations for the others may also be satisfied (depending upon which entry we chose to solve). At the same time, as we have seen, matrix rotation product equations are susceptible to spurious solutions—again, depending upon which entry we have chosen to solve. For the developability condition, we had the matrix equation 1 0 D= . 0 1 For the flat-foldability condition, we had any of 1 0 −1 0 −1 0 (even) (odd) F= or F = or F = . 0 1 0 −1 0 −1 For these systems, the equations arising from the diagonal components are inhomogeneous in the vertex coordinates, while the off-diagonal equations are homogeneous. The off-diagonal equations give rise to spurious solutions that are not part of the solution set of the diagonal equations, since, for example, for 0 is a valid solution for the developability the matrix D = −1 0 −1 off-diagonal components. Thus, if we are seeking an analytic solution to this system, we will generally have fewer spurious solutions if we choose the equations corresponding to the diagonal entries.

70


We may, however, be seeking a numerical solution to a system of equations. A broad general technique for designing origami forms is to take the vertex coordinates as variables, then set conditions that define the desired form and enforce properties like isometry, developability, flat-foldability, and the like, and then solve the resulting system of equations for the desired vertex coordinates (in 2D and/or 3D). When solving for systems numerically that include matrix rotation product conditions, the diagonal terms bring with them a subtle trap. Since any of these matrices is a rotation matrix, its rows and columns are orthonormal vectors, i.e., they have unit magnitude. That means that any given entry of the matrix must have magnitude less than or equal to 1. The equations arising from the diagonal entries of the matrices we’ve seen thus far have magnitude 1 (+1 for the developability matrices, −1 for the even or odd flat-foldability matrices). For a diagonal entry, say, Di,i , our equation will have the general form Di,i (x) = 1, (1.55) for some parameter x that we are seeking. (We will in general be seeking multiple parameters {x j }; that doesn’t change the argument to come.) Because D is a rotation, no component can have magnitude greater than 1, so at a point x that satisfies this equation, Di,i must be a local maximum. The functions that enter into the rotation matrices are generally smooth functions of their arguments, so Di,i (x) will be a smooth function at this local maximum, meaning that it is locally parabolic (or of higher even degree) at any root x. And that means that if x is a root of the diagonal entry equation, it is going to be a double root (or of higher multiplicity), which is significant: it means that the gradient of Di,i will vanish at the root. That behavior may cause great problems for any numerical root-finding routine. Typically, numerical root-finding algorithms use the gradient to establish the direction toward the root. (The conjugate-gradient root-finding technique does exactly that). If the root is a double root, though, the gradient vanishes as one approaches the root, leading to numerical instabilities in root search routines. And the situation can be worse than that: there may not even be a root, numerically speaking. While the mathematically ideal

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71

Di,i (x) may take on the value 1 precisely at the root, numerical round-off errors arising in the evaluation of Di,i (x) may give rise to a function that doesn’t ever reach the value 1; it may fall short by some tiny amount related to the numerical precision of the evaluation. In such a case, a canned root-finding routine might well return the result “no solution found,” when, in fact, a solution may well exist that we missed by a mere 10−17 units in function evaluation. Because of this phenomenon, when numerically solving systems based on matrix rotation products, it is often a more robust strategy to work with the off-diagonal entries of the matrix product whose values are set equal to zero. True, this approach can give rise to spurious solutions, but those can be eliminated by setting inequality constraints on the diagonal entries that move the spurious solutions out of the feasible space for the equation system. Thus, for the 2D developability matrix, rather than taking D1,1 = 1 as our defining equation, a more numerically robust system would be D1,2 = 0

and

D1,1 > 0.

(1.56)

(odd) For the 2D flat-foldability matrix, rather than taking F1,1 = −1 as our defining equation, a more numerical robust system would be (odd) F1,2 =0

and

(odd) F1,1 < 0.

(1.57)

Modern root-finding routines can typically handle mixed equality and inequality constraints. The added complexity of the inequality constraint is generally more than justified by the numerical robustness of using the off-diagonal terms in the root-finding procedure. The 2D developability matrix product is not often used in practical design algorithms; as noted, it is automatically satisfied for any plane embedding of the crease pattern. The flat-foldability matrix product is used, however, and we will see examples where it can be applied to find crease patterns that fold into a desired flatfolded shape. The developability matrix product is not without its uses, though; it is simple enough that we can use it to develop an understanding of the strengths and limitations of the matrix product treatment. And both developability and flat-foldability matrices have their analogs when we turn to fully three-dimensional structures.

72


The vector description and tools of this section can be used to describe and analyze an origami crease pattern, and the same tools can be used to describe and analyze the folded form—when said folded form is flat and two-dimensional. In general, origami figures can be three-dimensional in the folded form. Nevertheless, the vector description can be quite useful to describe them; in fact, a vector formulation is often the only tractable way to describe and analyze a 3D folded form. Many of the 2D vector relations of this section have analogs in 3D, but there is also an entirely new set of useful utilities for 3D folds. We’ll encounter those, and apply them, later in this book. For now, though, we have what we need to begin modeling and solving for crease patterns. We’ve started in this chapter by modeling individual vertices. Let’s now start putting some vertices together. ?

1.7. Terms Algebraic Kawasaki-Justin Condition An algebraic form of the Kawasaki-Justin Condition based on products of rotation matrices. Anto sector A sector bounded by two creases of opposite type. See iso sector. Big-Little-Big Angle Condition If a smaller sector angle is bounded by two larger sector angles, the crease assignments of the two creases bounding the smaller angle are opposite each other. Big-Little-Big Angle Theorem If the assignments of the creases of a smaller sector angle bounded by two larger sector angles does not satisfy the Big-Little-Big Angle Condition, then the creases and sectors are not flat-foldable. Big-Little-Big Angle sector A smaller sector angle that is bounded by two larger sector angles, in which case the BLBA Theorem applies to the crease assignment. Bird’s-foot vertex A degree-4 vertex where the two smallest sector angles are consecutive and equal. Border crease A classification of a segment of the border of the paper that treats it as a crease. CHAPTER 1. VERTICES

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73

Color-up Orientation of duo paper so that the colored side faces the viewer, or, mathematically, the side that the normal vector points away from. CP coloring A color and dashing scheme for crease patterns that maximizes readability, in which mountain fold lines are dark and solid and valley fold lines are lighter, dashed, and (optionally) colored. Crease A linear feature along which a fold takes place. A crease can be unfolded, partially folded, or fully (flat) folded. Crease assignment The association of a fold angle (or equivalently, mountain/valley/unfolded status) with each crease in a crease pattern. Crease pattern A mapping of all of the creases in an origami form, usually with some indication of their fold direction and/or fold angle. Degree The number of creases that come together at a vertex, or the number of vertices (and edges) around a polygon. A degree-N vertex has N creases that come together at the vertex. A degree-N polygon has N edges and N vertices. Developable, developability The property that the sum of the angles around a vertex is 360◦ . If a folded form is developable, it can be unfolded to a flat sheet of paper. Dihedral angle The angle between two facets of the folded form, typically lying between 0◦ and 360◦ , with 180◦ being unfolded. See fold angle. Duo paper Paper assumed to have a white side and a colored side. Facet A polygonal region of a crease pattern or folded form, bounded by some combination of creases and/or the border of the paper. Flat-foldable A pattern of mountain and valley folds that, when formed, can be flattened without any buckling, selfintersection, or creation of new creases.

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Fold angle The angle between two facets, defined as the deviation from flatness. Typically lying between −180◦ for a mountain fold and 180◦ for a valley fold. Fold direction angle The angle of a crease line measured relative to some global reference, typically the positive x-axis. Folded form The folded state. Fully folded A crease pattern is fully folded if every fold is either mountain or valley; there are no unfolded creases. Ghost paper A crease pattern that can fold flat without violating isometry conditions but that may self-intersect. See semifoldable. Globally flat-foldable A crease pattern that can fold flat without self-intersection. Injectivity The property that the paper does not self-intersect. In mathematics, it is the property of a mapping in which no two points in the domain map to the same point in the range. Interior vertex A vertex not on the border of the paper. Invalid crease pattern A crease pattern whose assignment cannot be physically realized in folded paper, at least, not without some form of distortion or self-intersection. Iso sector A sector bounded by two creases of the same type. See anto sector. Isometry The property that the shortest distance between two points as measured along the surface is unchanged in going from the unfolded to folded form; the material neither stretches nor compresses. Justin path A closed path on a crease pattern that does not pass through any vertices. Kawasaki-Justin Condition At any interior vertex of a crease pattern, the sector angles between folded creases satisfy α1 − α2 + . . . = 0. Kawasaki-Justin Theorem Any interior vertex of a flat-foldable crease pattern satisfies the Kawasaki-Justin Condition. CHAPTER 1. VERTICES

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Locally flat-foldable A crease pattern in which each vertex, viewed in isolation, is flat-foldable. Maekawa-Justin Condition At any interior vertex of a crease pattern, the number of creases of each type satisfy M −V = ±2. Maekawa-Justin Theorem Any interior vertex of a flat-foldable crease pattern satisfies the Maekawa-Justin Condition. Majority crease type At a flat-foldable vertex, the majority crease type is the type that there are more of. Minority crease type At a flat-foldable vertex, the minority crease type is the one that there are fewer of. Mountain fold A fold with a fold angle in the range [−180◦, 0◦ ). Mountain-like vertex A flat-foldable vertex that has more mountain folds than valley folds. Sector The paper between two consecutive creases around a vertex. Sector angle The angle between two consecutive creases around a vertex, measured counterclockwise about the vertex if signed. Semifoldable A crease pattern that can fold without violating isometry conditions but that may self-intersect. See ghost paper. Symmetric bird’s-foot vertex A flat-foldable degree-4 vertex whose sector angles come in equal consecutive pairs. Three Facet Theorem In three consecutive facets, if the outer facets both overlap the opposite creases, then the creases must have opposite parity. Two-coloring Coloring a plane graph using only two colors so that every pair of polygons that meet at an edge are oppositely colored. Unassigned crease pattern A crease pattern in which the creases have not been specified as mountain, valley, and/or unfolded. Unfolded crease A fold with a fold angle of 0◦ .

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Unique-largest-sector vertex A degree-4 vertex in which one of the sector angles is larger than all of the others. Unique-smallest-sector vertex A degree-4 vertex in which one of the sector angles is smaller than all of the others. Valid crease pattern A crease pattern that, if folded according to the crease assignment, gives rise to a physically realizable folded form (with no stretching or self-intersection). Valley fold A fold with a fold angle in the range (0◦, 180◦ ]. Valley-like vertex A flat-foldable vertex that has more valley folds than mountain folds. Vertex A point where two or more (usually more) creases come together. Vertex crease pattern A portion of a crease pattern right around a single vertex. White-up Orientation of duo paper so that the white side faces the viewer, or, mathematically, the side that the normal vector points toward.

CHAPTER 1. VERTICES

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2

Periodicity ?

2.1. Repeating Vertices

Origami vertices all by themselves are not terribly interesting. (Although, as we will see in Chapters 7 and 8, there is still quite a lot of a mathematical nature that we can say about isolated vertices.) The infinite variety of origami structures arises when we start bringing vertices together in combinations: a few, tens, hundreds, or even thousands of them in real-world objects. And when we start to contemplate their mathematics, we can even consider structures that contain infinite numbers of vertices and creases—even if we cannot physically fold them. In a network of vertices, two connected vertices cannot be designed independently; the fold between them must have the same assignment at each vertex. If we choose the folds around one vertex, then we have implicitly made a choice of assignment of one or more folds around its adjacent vertices. In a network that contains loops of folds, assignments must be consistent around a loop; if you make a set of assignments at a vertex, then travel around the loop making assignments to folds at each vertex, when you get back to where you started, the assignment on an incoming edge had better match the assignment you started with. If it doesn’t, then somewhere along the way, you’ve made a choice that resulted in an invalid assignment. We saw this phenomenon in the previous chapter, in the analysis of local flat-foldability (see Section 1.5). We will see many more examples of such loop conditions in upcoming chapters. There is one sure way to avoid invalidity resulting from inconsistent conditions around loops, and that is to consider crease patterns that contain no loops of vertices. We will start by looking at a few of these. 79

?

2.2. 1D Periodicity

? 2.2.1. Periodicity and Symmetry Throughout much of this book we will be looking at periodic structures and, in particular, structures folded from periodic crease patterns. A pattern is periodic if it can be shifted through some distance that leaves it unchanged. The distance that it shifts is the period of the pattern. Origami in general, and tessellations in particular, are rife with periodic structures. If a pattern is not periodic, then it is said to be aperiodic. Strictly speaking, no finite pattern can be truly periodic. Even if the pattern of folds seems unchanged by a shift, the edges of the paper will not be left unchanged, as illustrated in Figure 2.1: the places where the pattern starts and stops are not left unaltered by the shift. So when we say that a finite crease pattern is periodic, what we really mean is that it is a finite piece of a theoretically infinite pattern that is periodic. And just to deal with the contrary folk out there who point out that any isolated crease pattern could be made part of an infinite periodic pattern, let us say that we are only considering patterns in which the finite piece contains two or more repetitions of the repeating part. In Figure 2.1, the pattern is periodic in the horizontal direction. If you shift the paper by the distance between the two vertical lines, which is the period, the crease pattern is left unchanged—at least, if we imagine that it continues to the left and right beyond the rectangle of the paper, as indicated by the dotted lines.

...

... period

...

...

Figure 2.1. A periodic crease pattern. The period is the distance by which the pattern can be shifted, leaving it unchanged. For finite paper, we assume that the pattern continues beyond the edges of the paper.

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period

period

Figure 2.2. A periodic origami structure. Left: crease pattern. Right: folded form.

This pattern is periodic; but is it a valid origami crease pattern? That is, can it fold up flat—or even fold partially, in 3D? As it turns out, this pattern does fold flat with no self-intersection; it creates the zigzag shape shown in Figure 2.2. In this case, both the crease pattern and the folded form are periodic. Note, though, that the periods of the two are not the same. The period of the folded form is smaller. This will generally be the case when both the crease pattern and folded form are periodic; if a crease pattern gives rise to a periodic folded form, the period of the folded form is strictly less than the period of the crease pattern. This is a specific example of a broader property of flat origami: in every flat origami fold, the distance between any two points in the crease pattern is the same or larger than the distance between the same two points in the folded form. If a straight line between the two points crosses any folds transverse to the line, the distance is strictly larger. Back to periodicity. Imagine placing two copies of the same pattern on the table, one on top of the other. When we shift the pattern by the period, we bring each point on the top copy into alignment with another point on the bottom copy, which corresponds to a different point on the top copy. For example, the black dot in Figure 2.1 will come into alignment with the gray dot. We can mark the periodicity on a pattern by drawing an arrow from one feature to its image after the shift, as shown in Figure 2.3.

Figure 2.3. The periodicity can be marked by drawing an arrow from one point to its image after translation in both the crease pattern and folded form.

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CHAPTER 2. PERIODICITY

81

The position of the arrow isn’t unique; we can use any feature of the crease pattern as the point for reference. It’s usually convenient to pick a vertex of the pattern, but which vertex is chosen is a matter of personal choice. Whichever vertex you choose as the starting reference, the length and direction of the periodicity arrow are unchanged. The three green arrows in Figure 2.3 are the periodicity arrows for different vertices, but they all have the same length and direction (they are parallel). In the language of Section 1.6, it is a vector—a combination of length and associated direction. If one shift of distance d leaves a pattern unchanged, then two identical shifts will still leave it unchanged. So there is an ambiguity in the definition of the period. If the period is the distance that leaves the pattern unchanged, then d, 2d, and, in general, nd (for integer n) all qualify as periods under the definition thus far. So we’ll modify that definition: the period is the smallest possible distance whose shift leaves the pattern unchanged. (As we will see, even that definition permits some ambiguity when we consider multiple periodicities.) The periodicity of a crease pattern does not necessarily imply periodicity of the folded form. Figure 2.4 shows an example where the crease pattern is periodic but the folded form—at least, this particular folded form—does not need to be; both periodic and aperiodic folded forms are shown. This is because there is information about the folded form that is not fully specified by

Figure 2.4. A periodic origami crease pattern that admits both periodic and aperiodic folded forms. Top: crease pattern. Bottom left: periodic folded form. Bottom right: aperiodic folded form.

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Figure 2.5. The silhouette of the folded form is periodic if we ignore effects of layer ordering.

the crease pattern, namely, the stacking order of the facets. If the stacking order can be chosen in such a way as to break overall periodicity, that will not be reflected in the crease pattern. A pleat is a side-by-side mountain and valley fold pair. The pattern in Figure 2.4 consists of a series of double pleats; within each pair, one pleat must overlap the other, and we can choose the overlap order independently for each pair. So we can choose to make the folded form periodic (as on the left) or, by changing one or more pleat pairs, to break the periodicity. If, however, we overlook the effects of layer order and look at only the silhouette of the fold lines, then periodicity will be restored, as shown in Figure 2.5. The silhouette is fully determined by the crease pattern, and so here periodicity of the crease pattern does imply periodicity of the folded form. Still, there are definitely periodic crease patterns that have aperiodic folded forms, whether we consider layer ordering or not. Figure 2.6 shows one such example, both crease pattern and folded form. You can see the pattern in this figure: the folded form winds up in a circle. As you extend the crease pattern linearly to left and right, the folded form continues to wind around, rotating rather than translating.

Figure 2.6. A periodic crease pattern whose folded form is not periodic. Left: crease pattern, with periodicity vector marked. Right: folded form.

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This deceptively simple pattern displays several very important concepts. First, the obvious: a periodic crease pattern does not necessarily give rise to a periodic folded form. But there’s another: as we fold the crease pattern, we must rotate portions of the paper into the folded form, and the longer the pattern runs, the greater the amount of rotation is required. So this pattern can serve a role as a building block, that of a linear-to-rotary motion converter. Let’s come back to the question of periodicity. True, the folded form doesn’t display periodicity, as we have defined it above. But it certainly displays something repetitive and symmetric. It’s clear that if you take this folded form and shift it some way, the pattern of the folded form remains unchanged. But now, that shift is a rotation, rather than a translation: a different operation. If a system is unchanged by the application of some operation, then it is said to exhibit a symmetry. Symmetries are deeply embedded in mathematics (and, for that matter, in the physical world). A periodic pattern is symmetric under translation; it is said to have translational symmetry. A pattern that is left unchanged by a rotation is said to have rotational symmetry. What Figure 2.6 shows is that a translational symmetry in the crease pattern can give rise to a rotational symmetry in the folded form. A natural question to ask is, can we go the other direction? That is, can we start with a crease pattern with a rotational symmetry and end up with a folded form that has translational symmetry? The answer is yes, and it is not hard to construct such a crease pattern. Looking back at Figure 2.6, we see that the pattern that is getting repeated by the translational symmetry is the pair of angled folds. When we make both folds, that pair imparts a rotation of what’s on one side relative to what’s on the other, as illustrated in Figure 2.7. So if the underlying symmetry of the crease pattern was a translation, then the symmetry of the folded form consists of the translation of the crease pattern, combined with the rotation imparted by the folds of the repeating unit. Figure 2.7. The repeating unit is an angled pair of folds, which imparts a rotation of one side relative to the other.

84

A


A

A

A

Figure 2.8. Construction of a rotationally symmetric crease pattern that gives a periodic folded form. Left: the repeating unit. The repeating part lies between the two vertical creases. Middle: assembled into a periodic folded form. Right: the unfolded crease pattern.

The rotation imparted by the underlying unit adds to the translational symmetry of the overall crease pattern. So if we want to start with rotational symmetry in the crease pattern and end with translational symmetry in the folded form, all we need to do is choose an underlying unit that imparts a rotation that cancels the rotation in the symmetry of the crease pattern. How to ensure perfect cancellation? Well, one way would be to start with the desired folded form and then unfold it to get the crease pattern we’re after. Take the folded form of our repeating unit in Figure 2.7; splice it into a folded pattern with linear translational symmetry; and then unfold the result, as shown in Figure 2.8. Figure 2.8 shows that the relationship between translational and rotational symmetry can go both ways within a crease pattern, but it also illustrates a broader principle: in many situations, we can construct an origami pattern by designing the folded form directly, then unfolding it (conceptually, physically, or mathematically) to discover the desired crease pattern. Translational and rotational symmetry are often linked between the crease pattern and folded form. We will see this phenomenon appear over and over in structures to come—especially when we move into two- and three-dimensional forms. ? 2.2.2. Tiles Figure 2.8 illustrates a general way of building up periodic and/or rotationally symmetric origami structures (whether crease pattern or folded form). We identify a basic repeating unit, then we join

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85

(a)

(b)

(c)

(d)

(e)

Figure 2.9. Tiles creating a periodic folded form. (a) The repeating unit, with tile lines indicated in orange. (b) The folded form tile, with excess paper cut away. (c) Join two tiles along their shared tile line. (d) The joined pair. (e) Removing the tile lines gives the resulting folded form.

copies of that unit to each other. When we do that, we need to clearly identify the boundaries of the repeating unit that get joined up. These are not necessarily the edge of the paper; in Figure 2.8, I said that the repeating unit was the paper between the pair of vertical (unfolded) crease lines. To avoid ambiguity, though, we should not be using fold lines to delineate something that is not a fold. In Figure 2.8, the vertical unfolded crease lines are acting as the boundaries of repeating units that get joined to one another to make up a larger crease pattern or folded form, like the tiles of a mosaic. We will call those patches of crease pattern tiles, and we call the boundaries along which they join tile lines. Going back to Figure 2.8, we replace those unfolded crease lines with orange tile lines, as shown in Figure 2.9. Once we’ve defined the boundaries of the tile by the tile lines, we can eliminate the paper extending outside of the tile lines, leaving a pure, standalone tile, in this case a tile of the folded form. We can build up arbitrarily large sections of an origami structure by simply joining tiles along their corresponding tile lines. The power of the tiling approach is that if we begin with a folded form tile, we can unfold it to get a crease pattern tile that we can use to build up a periodic (or in this case, rotationally symmetric) crease pattern in the same way, as Figure 2.10 shows. There is some freedom in how we choose the tile boundaries; all that’s really necessary is that corresponding tile boundaries have the same shape, so that they fit together. Or, put more

86


(a)

(b)

(c)

Figure 2.10. Converting from a folded form tile to a crease pattern tile. (a) The folded form tile. (b) Unfolding the folded form tile gives a crease pattern tile. (c) Three crease pattern tiles joined to create a single crease pattern.

directly, the translation that defines the periodicity (or rotation that defines the rotational symmetry) must transform one tile line into the other within a tile. So, in Figure 2.11, all three crease pattern tiles are perfectly valid, and all can be combined with copies of themselves to give rise to the same crease pattern (aside from what happens at the ends of the strip). In general, the simpler the tile lines are, the easier the tiles will be to work with. And once one has defined a tile (of either crease pattern or folded form), it is a simple practical matter to create the full pattern by duplicating and pasting copies of the tile side by side, so that the tile lines are aligned. This is especially quick and easy with computer drawing programs, but the technique can even be employed with manual drawing, by making paper copies (a)

(b)

(c)

(d)

Figure 2.11. Three equivalent crease pattern tiles. (a) The crease pattern tile. (b) Same tile, but with curved tile lines. Note that corresponding tile lines must be rotated copies of one another. (c) Same tile, but with the tile lines superimposed on the mountain folds. (d) Same tile, but with one of the mountain folds excluded.

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of individual tiles, then using them as a template to build up the full pattern. A small complication arises if we place the tile lines on the fold lines of the crease pattern, as shown in Figure 2.11(c). It raises the question: does the fold line belong in the tile or not? If we exclude it from the tile, then it will be lost from the overall pattern. If we include it, then we’ll have duplicates of that fold when we join two tiles together. Suppose that the tile lines were slightly displaced to one side or the other. If we do that with the tile of Figure 2.11(c), then one of the mountain folds would be included within the tile and the other excluded; if we then splice two tiles together, all would be well: there would be exactly one mountain fold along each tile line. So, if the tile lines run along folds, we must keep track of corresponding pairs of tile lines and choose one of each pair to include the fold line, as in Figure 2.11(d). There is also the practical issue when drawing that if a tile line and fold line are coincident, then one is going to cover up the other. (I’ve dealt with that issue in Figure 2.11 by making the mountain folds fatter where they are overlapped by a tile line.) To avoid this issue and the need to keep track of which tile line gets the fold that it overlaps, I generally try to define my tiles so that no tile lines run along fold lines or vertices, for which similar choices of inclusion/non-inclusion must be made. This is purely a matter of personal taste, though, and in some case, defining tiles along fold lines will be the natural choice. ? 2.2.3. Linear Chains If a crease pattern consists of a set of crease lines with no vertices in the interior of the pattern, as in Figures 2.4–2.8, then to determine if the crease pattern is valid, we only need to consider the way the layers overlap and the possibility of self-intersection. If, however, the crease pattern contains vertices in its interior, as in Figures 2.1 and 2.2, then we need to ensure that the fold lines between two vertices match in crease assignment and, if the vertices are not fully flat-folded, in measured fold angle. One way to see this matching requirement is to construct the tile for the repeating pattern. Figure 2.12 shows the crease pattern of Figure 2.1 divided up into tiles and a single tile. The tile in the figure contains two interior vertices and has creases that cross the tile border. In order to arrange copies of this

88


Figure 2.12. Identifying the tiles in a periodic crease pattern. Left: the pattern. Right: the tile.

tile into a periodic chain, the crease crossing the left tile line must do so at the same position and angle as the one crossing the right tile line. In this pattern, this is clearly the case: each horizontal crease hits the left or right tile line exactly halfway up, and the angle is 90◦ . With two interior vertices in the tile, this isn’t the simplest possible tile of a linear chain, and it is worth asking: what are the possibilities in the simplest case, with only a single degree-4 vertex in each tile and consecutive vertices connected by folds across a tile border? Without loss of generality, we can choose the vertex to have three valleys and one mountain. Then the two folds that cross the tile line to the left and right must be valleys since we only have one mountain fold to work with. Next, since the direction of the fold on the left must match the direction of the fold on the right after translation, the two folds must be collinear. And for the position where each fold hits the tile line to have the same periodicity, the two folds must be oriented along the direction of periodicity. That, in turn, greatly restricts the other two folds. In order to fold the vertex flat (or even to fold all four creases together by any amount), the two remaining creases must be on opposite sides of the straight-line fold pair, making the same angle with the pair (which we can choose). This gives a family of patterns, characterized by this last angle, one example of which is shown in Figure 2.13, along with its folded form. That particular family is not too interesting. But now let’s consider the possibility that the crease pattern has rotational symmetry, rather than periodicity. In this case, the two valleys can intersect at some angle other than collinearly, and that opens up rather more options. For one thing, there are two distinct families: we can put both of the two remaining creases on one side of the pair, or one on each side. In the latter case, the mountain fold must go in the < 180◦

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Figure 2.13.

Left: a periodic tile containing a single degree-4 vertex. Middle: the resulting crease pattern for three tiles. Right: the folded form.

angle between the pair, with the valley fold going somewhere on the other side. Flat-foldable examples of both cases are shown in Figure 2.14. In both cases, the crease pattern has rotational symmetry, but the folded forms differ; the upper tile gives a folded form that also has rotational symmetry, while the lower tile gives rise to periodicity (translational symmetry). Both of these patterns are flat-foldable, of course, but the second pattern is interesting in the partially folded state, in which it forms a polygonal helix in three dimensions. I encourage you to fold one and try it out. You might also try out different vertex angles, to see how the choice affects the helicity in the partially folded state. That’s only a single vertex per tile. We can extend the idea in several ways; go to vertices of higher degree, for example, or create a tile with two, three, or more vertices within the tile. There are many interesting periodic structures to discover, and later on, we will examine more of them. But not right now. We’ve established some important concepts: periodicity, rotational symmetry, and the use of tiles to create periodic structures. Linear chains of tiles display some interesting structure, but the possibilities are far richer when we extend periodicity into two dimensions. ?

2.3. 2D Periodicity Mathematically, a tiling is a partitioning of the plane into regions—which are the tiles. In the previous section, we partitioned (roughly) linear strips of papers into tiles that are periodic

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Figure 2.14.

Top row: a tile with rotational symmetry in the crease pattern with folds on the same side of the valley pair. Bottom row: a tile with rotational symmetry in the crease pattern with folds on opposite sides of the valley pair. Left: tile. Middle: crease pattern composed of four tiles. Right: folded form.

(or in some cases, rotationally symmetric). A periodic pattern can be characterized by its period and the direction of periodicity: two quantities that can be combined in the vector of periodicity, indicated by an arrow in the plane whose length is the period and whose orientation gives the direction of periodicity. If we have a basic building block of a tile, we can create a linear array by placing multiple copies of that tile at multiples of the periodicity. There are patterns that exhibit periodicity in two different directions, for example, a grid of squares. Such patterns are called doubly periodic. In general, a doubly periodic pattern will be characterized by two different (non-collinear) vectors of periodicity. Often, as in a square grid, the two vectors point in orthogonal directions; they don’t have to, however. We will refer to the vectors as basis vectors for the doubly periodic structure. CHAPTER 2. PERIODICITY

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Figure 2.15. Illustration of a doubly periodic pattern. Tiles are outlined in orange. Basis vectors are shown in green.

Any doubly periodic structure can be partitioned into tiles that are quadrilaterals—in fact, parallelograms—so that the entire pattern can be generated from a single tile by making copies of the tile and translating them by integer multiples of the two basis vectors, as illustrated in Figure 2.15. Doubly periodic patterns are of interest in origami because with a single tile, one can fill an arbitrarily large region of the plane—thereby creating an arbitrarily large folded structure, which can be of both aesthetic and practical interest. The tiles for a doubly periodic pattern have tile lines on all four sides and must obey two sets of matching conditions. The left and right sides must line up, as with our linearly periodic patterns, but now the top and bottom of the tiles must mate in a similar fashion. So, now, can we find a doubly periodic pattern that contains a single degree-4 vertex in each tile? We already saw that for a single direction of periodicity, the two folds that hit the two tile lines must be the same parity and collinear. With a doubly periodic pattern, the two folds in the other direction must satisfy the same condition. That amounts to saying that there are either four folds of one type, or two folds each of both types. In either of those two cases, the Maekawa-Justin Theorem requires at least two more folds, so the vertex must be at least degree-6. So the simplest doubly periodic tile cannot consist of a single degree-4 vertex, but that does leave open the possibility that, say, there is a doubly periodic pattern composed of a single type of degree-6 (or higher) vertex—as we will presently see. But we can still ask the question: is there a doubly periodic pattern that is composed entirely of a single type of vertex? There is, and it is the source of some very interesting folded behavior. 92


? 2.3.1. Huffman Grid The simplest possible doubly periodic crease pattern consists of a single vertex, repeated over and over in rows and columns, so that the mountains and valleys emanating from the vertex join up collinearly with those of its neighbors, matching both in direction and in fold angle (crease assignment, in flat-foldable patterns). The simplest nontrivial vertex that could serve as the tile of such a regular array is the degree-4 vertex, so the question naturally arises: which degree-4 vertices can be tiled with copies of themselves to create a crease pattern that folds along all of its creases so that each vertex folds identically? The somewhat surprising answer is: any of them. The computer scientist David Huffman (with whom we will spend considerable time later on) showed in his landmark paper, “Curvature and Creases: A Primer on Paper” [47], that any degree-4 vertex can be arrayed with copies of itself to produce a doubly periodic crease pattern that folds smoothly with all facets remaining flat during the folding process. Huffman didn’t give this structure a name and wasn’t the first to describe cylindrical grids of degree-4 vertices, but he seems to have been the first person to analyze it in full generality, and so I will call it a Huffman grid. An example of a Huffman grid crease pattern and several stages of the folded form are shown in Figure 2.16. The Huffman grid is a 2D array of degree-4 vertices, all alike, which are then the vertices of a 2D array of quadrilateral facets, also (necessarily) all alike. The construction is straightforward. One first chooses the degree-4 vertex; this will be called the generating vertex of the grid. For the example shown in Figure 2.16, the sector angles around the vertex were chosen to be (90◦, 50◦, 100◦, 120◦ ). We can also choose two lengths, for example, the lengths of the first two creases. For this example, I have chosen lengths of 1 and 2 units, respectively. Now, we must array copies of this vertex in such a way as to achieve a 2D periodic pattern, keeping in mind that every vertex must have the same sector angles and the same fold angles. It’s not enough to have the folds simply match in their mountain or valley assignment; in general, a 3D degree-4 vertex may have all four fold angles differ (in Chapter 8 we will see exactly what the relationships between fold angles are). So that means, for example, as we travel outward from a vertex along any given crease, at the

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Figure 2.16. A Huffman grid, parameterized on the angle of the horizontal valley folds. Top row: left to right, crease pattern and valley folds at 3◦ and 6◦ . Bottom row: left to right, valley folds at 9◦ , 12◦ , and 18◦ .

end of the crease we should encounter a rotated copy of the same vertex. We can see this requirement more clearly by coloring all four creases uniquely, as in Figure 2.17. We choose the lengths of the green and blue mountain folds, as shown in Figure 2.17(a). Then at the far end of each fold there must be a rotated copy of the vertex, as in (b). The two remaining creases intersect; that intersection defines the lengths of the other two creases. Now that we have all four crease lengths, we can replicate the full pattern by placing copies of the vertex (some merely translated, others translated and rotated) to build out the pattern. The crease pattern is clearly doubly periodic; we can check this by identifying a feature of the pattern and its translated copies, as shown in Figure 2.18. I have marked all of the vertices that are translated copies of one another with black dots. Although all of the vertices are alike, half of them are rotated (by the same amount) relative to the black-dot vertices; these constitute a second set and are marked with gray dots. Thus, each tile contains two vertices: one black, one gray.

94


(a)

(b)

50¡ 100¡

(c)

90¡ 120¡

Figure 2.17. Construction of a Huffman grid. (a) The generating vertex. We can choose two crease lengths, the blue and green. (b) At each end of the two creases, we place two rotated copies of the vertex. The intersections of the magenta and amber crease lines define the other two crease lengths. (c) Replicating the grid (at 50% of the size in (a) and (b)).

The tile for this Huffman grid is a rectangle (in this case; more generally, a parallelogram) whose edges have the length and orientation of the periodicity vectors. The position of the tile can be taken anywhere within the crease pattern; it doesn’t have to have its corners coincide with vertices of the crease pattern, but it is elegant and convenient in this case to do so. To ensure that when we replicate the tile, we get no duplicate vertices, each tile contains a single black vertex and a single gray vertex. Then using

Figure 2.18. Identification of the periodicity vectors (green, lower left) and tile (orange) for the Huffman grid. Right: building up the pattern by replicating and translating the tile.

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this tile, an arbitrarily large region of the crease pattern may be built up by replicating and translating copies of the tile. Now that we know how to construct the pattern, let’s look at how it behaves under folding, looking back at Figure 2.16. It is rather surprising: as one starts to fold the pattern, it curls up quite quickly to form a polygonal cylinder. And the curling happens quickly; as can be seen from the figure, by the time the valley folds have reached the quite shallow fold angle of 18◦ , the paper has curled in almost a complete loop and is about to collide with itself. This pattern is never going to fold flat—in fact, it won’t even get close, at least not without self-intersecting. It’s just a little bit surprising that a doubly periodic crease pattern would give rise to a cylindrical folded form, rather than a doubly periodic folded form—but only a little surprising. After all, we saw with linear periodic patterns that periodicity in the crease pattern could give rise to rotational symmetry in the folded form, and vice versa. In the Huffman grid, there are two directions of periodicity. One gives rise to a translational symmetry along the axis of the cylinder; the other gives rise to rotational symmetry, about the cylinder. In fact, neither symmetry is exact; if you look closely at Figure 2.16, you’ll see that the pattern of polygons along the cylinder is not pure translation but instead has a slight helicity. The same goes for the rotational symmetry; vertices are not strictly rotated about the axis of the cylinder, but instead they are slightly offset as one goes around. There are two directions of symmetry in the folded form, but both are a mixture of translation along the axis and rotation about the axis. Both are helical, but one is closer to translation, the other to rotation. It might seem that this behavior could be peculiar to the particular Huffman grid that I chose here, but no: in fact, every Huffman grid exhibits this cylindrical curling behavior. As one flexes the crease pattern from flat to folded, it curls up with the cylindrical radius getting tighter and tighter. And, in general, the orientation of the axis of the cylinder relative to the crease pattern is not obvious. (It will not be until Chapter 8 that we will discover what this relationship is.) Any Huffman grid composed of a single vertex type is guaranteed to work if the generating vertex itself is a degree-4 vertex that satisfies a few simple conditions. It doesn’t have to be flat-foldable,

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so it doesn’t necessarily have to satisfy the Kawasaki-Justin Theorem. It does, however, have to have three mountain folds and one valley fold (or vice versa); all four sector angles cannot be 90◦ ; and the three folds of the same type cannot all lie in the same half-plane. If the vertex satisfies those conditions, the Huffman grid constructed from it will work, and by “work” I mean that the entire structure can flex with all facets remaining flat—a property known as rigid foldability.1 In fact, the structure has a single degree of freedom, meaning that if it flexes at all, all of the folds must flex together with the fold angles taking on a prescribed oneto-one relationship with each other. We will learn more about rigid foldability and degrees of freedom in Chapter 7. If you fold a Huffman grid from paper, this single-degree-offreedom property may not be obvious, because the facets in paper are not constrained to flatness; the quadrilaterals can usually flex easily along their diagonals, and so the entire structure will be somewhat “squishy” (depending on the stiffness of the paper). It is fairly easy to distort the cylinder away from cylindrical symmetry, for example, twisting one end more tightly than the other, to make it somewhat conical. But this is a distortion that relies upon bending of the facets. If you make the structure from truly rigid material (say, sheet metal with actual hinges for the folds), the constraint to cylindrical folded form will be more clearly evident in the behavior of the physical model. And I do encourage you to fold and experiment with physical models: not just the crease pattern of Figure 2.16, which you can easily replicate on a larger scale, but also with changing the angles of the generating vertices and the lengths of the edges of the quadrilaterals. In general, the symmetry of a Huffman grid will be helical, neither pure translation nor pure rotation. By varying the angles of the generating vertex, you can vary the direction of the symmetry. There is one version, however, that gives rise to pure translation and rotational symmetry: the grid that comes from using the mirror-symmetric bird’s-foot vertex as the generating vertex. An example of one of these patterns is shown in Figure 2.19, which is based on a generating vertex with sector angles (120◦, 60◦, 60◦, 120◦ ) and crease lengths (1, 2, 3, 2). I will call this 1

Although the name sounds oxymoronic (how can something be rigid and foldable?), the term means that the facets remain flat and rigid and all flexing happens along the folds.

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Figure 2.19. The chicken wire pattern, a Huffman grid constructed from a mirror-symmetric vertex, again parameterized on the angle of the horizontal valley folds. Top row: left to right, crease pattern and valley folds at 6◦ and 18◦ . Bottom row: left to right, valley folds at 24◦ , 30◦ , and 36◦ .

the chicken wire pattern, and versions of it have been around for a long time. (It is closely related to the Yoshimura pattern, which we will shortly meet.) Martin Gardner described a method of achieving the hexagonal pattern of mountain folds (though not the valleys) by coiling paper into a tube and pinching it, and he noted that one of his correspondents reported seeing the method in the 1930s [35, pp. 83, 92]. The chicken wire pattern has pure translational symmetry along the axis of the cylinder and pure rotational symmetry about the axis. Furthermore, as can be seen in the last subfigure, as the pattern curls up, the jagged edge of one side appears to approach mating with the corresponding edge of the other. This is more than appearance; there is going to be a “magic” fold angle at which the teeth meet, and because of the rotational symmetry, if the vertices at the tip of the teeth line up with their corresponding notches, the edges all along the teeth will line up exactly. The value of the magic fold angle will depend on various parameters of the geometry: the angles and distances of the generating vertex, of course, and also upon the number of repetitions in the pattern.

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If you examined the crease patterns of Figure 2.16 and 2.19 side by side, you might not even think of them as the same structure. They are, however, the same fundamental concept; it is only the symmetry of the generating vertex in Figure 2.19 that makes it special. This symmetric form, in fact, predates Huffman’s work; it was identified in 1970 by Japanese engineer Koryo Miura [86], of whom we will see more shortly. The general case of this pattern uses a non-symmetric generating vertex that, in general, gives helical symmetry to the grids of quadrilaterals. The general case can also exhibit this behavior that as the valley fold angle is increased, the two serrated edges are brought together, but having the teeth of one side fit into the notches of the other is not guaranteed for all vertices. Rather, there are going to be magic combinations of vertex angles, fold lengths, and numbers of repetitions that give precisely the right helicity for one side to line up with the other. How many different things can we vary when designing a Huffman grid? We can choose the angles of the generating vertex and the various fold lengths, but as we have already seen, we cannot choose all of them independently. For the general degree4 vertex, since there is no requirement for flat-foldability, we can choose the first three sector angles (α1, α2, α3 ) nearly arbitrarily, but then the fourth sector angle α4 is determined. It is given by α4 = 2π − (α1 + α2 + α3 ), so the first three sector angles must sum to less than 2π. And, of course, the crease assignment must be valid. So that gives three degrees of freedom for angles, plus the number of possible assignments, which, as we saw in the previous chapter, can range from four to eight, depending on what the sector angles are. For distances, we can pick lengths of two folds of the vertex, but then, as we have seen, the lengths of the other two folds are determined by the intersection of their rotated copies. And, as with the sector angles, we don’t have complete freedom to choose fold lengths. Just as the sector angles must all be greater than or equal to zero, in order to create a Huffman grid composed of degree-4 vertices around quadrilateral facets, all four fold lengths must be greater than zero. Figure 2.20 shows what happens as we decrease the length of one of the two folds of the generating vertex. The intersection of the other two folds determines the other two fold lengths, and there comes a point where one of the fold lengths goes to zero—or worse, goes negative.

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(a)

(b)

(c)

Figure 2.20. Varying the fold lengths. (a) All four fold lengths are positive. (b) The amber mountain fold length has gone to zero. (c) The amber mountain fold has a negative length, which results in a crossing embedding.

Negative fold lengths are not allowed. In fact, the concept of a “negative fold length” may seem nonsensical, if we are thinking of fold lengths as absolute distance between two vertices. But if we think of a fold length as a signed distance, that is, a distance measured in a particular direction, then a negative fold length just means that you’re moving in the opposite direction from what you intended, and a negative distance between two vertices simply means the vertices have the wrong order along the line connecting them. That would mean, in general, that the other creases around the vertices intersect with each other in unplanned ways—they cross—and since we have already stipulated that creases must meet only at (predefined) vertices, a solution that gives rise to a negative fold length between two vertices is inconsistent with our definition of what a crease pattern is. Using the language of graph theory, the crease pattern graph has a crossing embedding. So negative fold lengths will be banished from our crease patterns. But something else interesting can happen: the fold length can go to zero, in which case the fold between the two vertices vanishes and the two vertices merge into a single vertex. This is something that can give rise to a valid (and indeed, foldable) crease pattern. If we merge two degree-4 vertices, then we will be left with a degree-6 vertex. We can apply this process to the Huffman grid, and in doing so, we achieve a new pattern composed of degree-6 vertices and triangular facets that also has a rich history in the world of periodic origami: the Yoshimura pattern.

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(a)

(b)

(c)

(d)

Figure 2.21. Evolution of the Yoshimura pattern from a Huffman grid. (a) A quadrilateral of the Huffman grid. We shorten the blue fold until the yellow fold length goes to zero. (b) The quadrilateral collapses to a triangle. (c) Extending the pattern. (d) A larger patch of the pattern, at 50% scale and rotated to make the valley folds horizontal.

? 2.3.2. Yoshimura Pattern Let’s take a generating vertex similar to that of Figure 2.20 and choose fold lengths that take the orange mountain fold length down to zero. In this case, the Huffman grid will collapse to the pattern shown in Figure 2.21, which I have rotated so that the valley folds now run horizontally. This pattern is called the Yoshimura pattern after the Japanese researcher Yoshimaru Yoshimura, who observed it in the buckling pattern of longitudinally stressed cylinders back in the 1950s [129]. Yoshimura wasn’t exploring this pattern as a development of the quadrilateral grid; rather, it was a naturally occurring pattern. If you take a cylinder and compress it along its axis, it will eventually buckle. Depending on the material properties and stresses, one of the failure modes is the formation of a diamondlike pattern of folds: the Yoshimura pattern. To be absolutely precise, though, the original Yoshimura pattern was not the one shown in Figure 2.21; it was one that was rather more symmetric, shown in Figure 2.22. And the original Yoshimura pattern was observed in a closed cylinder, not a flat sheet. The pattern is undoubtedly much older; it forms naturally in the draping of fabric and can be seen in the left sleeve of the Mona

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Figure 2.22. The original Yoshimura pattern. Left: crease pattern. Right: folded form.

Lisa, among other places, as shown in Figure 2.23. In fact, the right sleeve looks suspiciously like the chicken wire pattern shown in Figure 2.19. Figures 2.21 and 2.22 are the same basic crease pattern, the difference being that one is more symmetric than the other. Both display the same behavior, one reminiscent of the Huffman grid; as the valley fold is flexed, the pattern curls cylindrically or helically, as shown in Figure 2.24, which shows the folded form for both patterns. In general, the Yoshimura pattern is going to show some form of helical symmetry (a mixture of translation along the axis and rotation about the axis) like the Huffman grid. And also like the Huffman grid, a symmetric vertex will give rise to symmetry that is pure rotational in one direction and pure translational in the other. Unlike the Huffman grid, though, which has a single degree of freedom, the Yoshimura pattern has multiple degrees of freedom

Figure 2.23. The Mona Lisa, by Leonardo da Vinci. A close-up of the sleeve reveals the cylindrical buckling pattern that we now call the Yoshimura pattern, mixed, a bit, with the chicken wire pattern.

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Figure 2.24.

Left: the symmetric Yoshimura pattern, slightly opened from Figure 2.22. Right: the asymmetric pattern, showing its helical symmetry.

if its edges are unconstrained, i.e., if it is folded from a sheet, rather than a closed cylinder. When the edges are joined to make a cylinder, it becomes rigid, but when the edges are unconstrained, the pattern is quite bendy, even if all of the triangular faces are individually rigid. This additional flexibility comes about because each of the vertices has multiple degrees of freedom. The number of degrees of freedom in a vertex of degree n is n − 3. A degree-4 vertex, such as that of the Huffman grid, has a single degree of freedom. In the Yoshimura pattern, the vertices are degree-6, which would suggest that each vertex has a total of three degrees of freedom. But wait: suppose we are seeking symmetric solutions—those with translational, rotational, or helical symmetry. Then all the vertices need to work together, connected to one another by folds whose angles must be consistent. If we assume a single vertex that is replicated everywhere, then we must make sure that fold angles and the sector angles between them are consistent from vertex to vertex. Let’s look at the question of just how general the Yoshimura pattern can get. Figure 2.25 shows a patch of the Yoshimura pattern composed of identical vertices but with all four mountain folds colored distinctly (and with different sector angles around the vertex). This pattern can be built up from a building block that is a quadrilateral of four mountain folds whose diagonal is crossed by a single valley fold. The mountain folds can all be different lengths and have different fold angles. But there is only one valley fold in the building block, which shows up twice in each vertex.

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(a)

(b) g3 a3 a2 a1 a4 a a6 5

g4 g5

g6

(c)

g2 g1 a3 a2 a1 a4 a a6 5

Figure 2.25.

(a) A general Yoshimura-pattern vertex, with sector angles α1, . . . , α6 and fold angles γ1, . . . , γ6 . (b) A patch of a general Yoshimura pattern. Sector angles are labeled; periodicity vectors are shown in green. (c) Four quadrilaterals come together at each vertex.

We don’t have three degrees of freedom at each vertex if the structure is periodic, because the two valley folds γ1 and γ4 must have the same fold angle. That takes away a degree. In general, for a given pattern, we can choose two fold angles (say, γ2 and γ6 ); then every other angle is determined. (In Chapter 8 we will see how to calculate those other fold angles.) In Figure 2.25, those two valley folds are collinear. Is this coincidence, or must it always be so? Looking at Figure 2.25(c), we see that if each vertex is built up from copies of the same quadrilateral, then the geometry of the construction dictates that the valley folds are all parallel to one another. And that means that folds γ1 and γ4 must indeed be parallel. This, in turn, places constraints on the sector angles. We cannot choose them arbitrarily; rather, we must have that α1 + α2 + α3 = 180◦

and

α4 + α5 + α6 = 180◦ .

(2.1)

So, in the design of the pattern, we can choose any two of the first three angles and any two of the second three, and at that point, all six sector angles will be determined. What about the fold lengths? Now, in the Huffman grid, we could choose the lengths of two folds, but the other two were determined by fold intersections. Here, we have six folds around each vertex, so potentially six lengths to choose. But we also have many requirements on the intersections between folds. In fact, if we choose the length of the valley fold and choose all of the sector angles, all four mountain fold lengths are deter-

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Figure 2.26. The asymmetric Yoshimura pattern for different values of the two mountain folds γ2 and γ6 . Top left: (γ2, γ6 ) = (25◦, 25◦ ). Top right: (γ2, γ6 ) = (25◦, 35◦ ). Bottom left: (γ2, γ6 ) = (35◦, 25◦ ). Bottom right: (γ2, γ6 ) = (30◦, 30◦ ).

mined. Take a look at one of the quadrilaterals in Figure 2.25(c). Extending upward from the endpoints of the valley fold are folds γ2 (from the left) and γ3 (from the right); their intersection defines the lengths of both mountain folds. The same goes for the two folds extending downward, γ6 and γ5 . So a single length, that of the valley fold, and four of the six sector angles entirely define the crease pattern. Now, in the folded form, even if we demand periodicity, we still have two degrees of freedom; we can choose, for example, γ2 and γ6 independently. Varying these two angles affects the amount of curl in the pattern, but also the helicity. Figure 2.26 shows the pattern of Figure 2.21 with four different values of the two angles. Comparing the figures in the upper right and lower left in particular, you can see that the primary difference is a helical shift. The teeth coming from each edge of the paper are about the same distance apart, but they are shifted laterally with respect to each other. That means that for any Yoshimura pattern, there is going to be some set of fold angles that brings the edges together, thereby creating a tube (and if the edges are joined, a rigid tube). If we join the ends of the sheet, it becomes quite rigid—there are no degrees of freedom at all. In fact, it becomes considerably stronger than a smooth cylinder of the same diameter from the same material. This property means that this pattern has practical applications. It was used for a soft drink can for a Japanese brand, shown in Figure 2.27.

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Figure 2.27. A Japanese soft drink can based on the Yoshimura pattern.

The pattern in the soft drink can has several beneficial effects. First, there is the purely decorative one; it’s a lovely pattern. Not clear from this picture is the fact that with carbonated beverages, the can is under pressure and the pattern is smoothed out; when the can is opened, the pattern pops into high relief. More practically, the strengthening provided by the truss-like structure of folds means that a thinner gauge of metal may be used, allowing for savings of both weight and cost—vital in the consumer goods industry. Yoshimura identified this pattern in buckling cylinders, but its application to soft drink technology came from someone else: Koryo Miura, who, as already noted, studied and analyzed this pattern and its relatives (like the chicken wire pattern). He did a lot more, discovering, describing, and lending his name to an entire family of structures used for deployable origami. It’s about time that we meet him, and them. ? 2.3.3. Miura-ori In Miura’s study of the Yoshimura pattern, he noted that all of the varieties of the pattern exhibited cylindrical curvature as the fold pattern was flexed. If the vertices were mountain-like, as in Figure 2.25, then the curvature would be convex toward the viewer; if the fold parities were flipped, then the curvature would be concave toward the viewer. He further observed that one could splice together a convex and a concave portion of the crease pattern, giving rise to a pattern that curled both ways, as illustrated in Figures 2.28 and 2.29.

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Figure 2.28.

Left: Koryo Miura demonstrating his original model of the Yoshimura pattern including a change of curvature. Right: close-up of the model.

There is an interesting detail in the way the two halves of the pattern are spliced together. It is not as simple as reversing the crease parity on one side of the Yoshimura pattern. Rather, the pattern is split; half of it is crease-reversed; and then the two halves are offset relative to one another by half of the vertical periodicity before being rejoined, as illustrated in Figure 2.30. The reversal/offsetting/splice operation creates a new set of shapes within the pattern, a vertical row of parallelograms, which you can see in the very middle of Figure 2.30(d). It also alters the vertices in the middle, changing their degree from 6 to 4.

Figure 2.29.

Left: crease pattern for the reversing Yoshimura pattern. Right: folded form.

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(a)

(b)

(c)

(d)

Figure 2.30. Sequence for reversing half of a Yoshimura pattern. (a) Divide the pattern in half. (b) Invert the crease parity in one half. (c) Offset one half of the pattern. (d) Rejoin the two halves, extending and removing the pattern to keep the paper rectangular.

This construction allows one to reverse the curvature of the Yoshimura pattern. We focus our attention on the diagonal creases. On the left side, each zigzag diagonal mountain fold imparts some large-scale convex cylindrical curvature to the pattern. On the right side, each zigzag diagonal valley fold imparts large-scale concave curvature. Where the two zigzags occur side by side in the middle, the two curvatures cancel each other out, and so the middle of the paper appears to have no net curvature. Miura realized that if one built a pattern consisting of alternating mountain and valley zigzags, i.e., reproducing just the pattern in the middle of Figure 2.30, it would produce a pattern with no net curvature, resulting in a pattern that opens and closes while maintaining a roughly planar configuration. And indeed, that is precisely the case, as Figure 2.31 shows. This pattern, the Miura-ori, has become famous within the world of applied origami and has become a regular workhorse of deployable structures. Indeed, Miura himself proposed it for use in a solar array, in a mission for JAXA, the Japanese space agency, that flew in 1995 [87]. A key feature of the Miura-ori is that it folds rigidly with a single degree of freedom, meaning that if the facets of the pattern were perfectly stiff, then the entire pattern could fold and unfold

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Figure 2.31.

Left: crease pattern for a Miura-ori pattern. Right: partially folded form (turned over).

in a single motion. Flexing any one panel would be sufficient to actuate every crease in the pattern. In practice, small deviations from ideality—stiffness of the folds, bending of the facets, and so forth—prevents the perfect coupling of all of the folds, but the tendency of them to all move together greatly facilitates the deployment of mechanisms based on the Miura-ori. Why this is the case—that the pattern is rigidly foldable, and that it has a single degree of freedom—will have to wait until Chapters 7 and 8—but we can explore many of the properties of Miura-oris by analyzing their static unfolded and folded forms. The Miura-ori is more than this specific pattern; it is a family of patterns, because we may vary several parameters while preserving the basic properties of in-plane deployment motion and flat-foldability. The crease pattern is composed of parallelograms, arranged in rows and columns, with the creases forming a set of zigzag lines that run one way and collinear lines that run the other. The zigzag folds are all the same parity along a single zigzag line, with the lines alternating mountain and valley from one to the next. The horizontal folds, by contrast, while collinear, alternate mountain/valley along each segment of each horizontal line. By convention, we will call the same-parity-along-the-fold zigzag folds of a Miura-ori the major folds of the pattern, and the alternating-parity straight folds the minor folds. (We will meet the justification for these names in Chapter 7.) It turns out that the fold angles along the major folds are the same all along the fold, and the mountain and valley major fold angles are equal and opposite. This is also the case for the minor fold angles (mountain and valley fold angles are equal and opposite), but in general, the

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major folds

Figure 2.32. Characteristic dimensions and terminology of a symmetric Miura-ori crease pattern.

minor folds

d1 a

d2

d2

a d1

major and minor fold angles are different from each other for any non-flat state of foldedness. With these constraints on folds and the assumption that the pattern is doubly periodic, there are only three additional parameters that, when chosen, fully define the crease pattern: • the length d1 of a minor fold, • the length d2 of a major fold, • the angle α between horizontal and zigzag folds, as shown in Figure 2.32. By convention, we can choose the angle α to be the acute angle, so that the four angles of each parallelogram are α (two each) and 180◦ − α (also two each). Any of these parameters can be varied, and they will affect the dimensions of the crease pattern and folded form (of course); less obviously, they affect the way that the mechanism deploys. A detailed analysis of the latter must await further analytical development, but it is useful to construct different examples from stiff paper (scoring the creases) and play with them to get a feel for the differences. The particular parameters (d1 = 1, d2 = 1, α = 60◦ ) give rise to a very symmetric version of the Miura-ori that we may adopt as a “baseline” form. A 6 × 6 version of this pattern is illustrated in Figure 2.33 as a crease pattern, partially folded form with a minor fold angle of 90◦ , and nearly completely folded form with a minor fold angle of 170◦ . Figure 2.34 shows another 6 × 6 array with α = 80◦ . As the characteristic angle α gets larger, several things happen, all visible

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Figure 2.33.

Left: crease pattern for a Miura-ori pattern with (d1 = 1, d2 = 1, α = 60◦ ). Middle: partially folded form. Right: nearly fully folded form.

in the figure: the individual parallelograms approach rectangles, and the fully folded form becomes shorter along its length. Conversely, at any given point in the fully folded form, there are more layers as more of the facets mutually overlap. So, as a deployable structure, larger α gives a more efficient stowed form. There is, however, a tradeoff, which can be seen in the middle subfigure of Figure 2.34. While the minor creases are partially folded (with a fold angle of 90◦ ), the major creases are nearly fully folded. As the characteristic angle increases, the folding motion takes on two distinct forms. As you start to fold the crease pattern, most of the action happens on the major folds. Then, as they are nearly folded, the minor folds undergo most of their motion. This two-stage motion decreases the coupling between the major and minor folds, and it can give rise to undesired compliance in a folding structure with very large values of α.

Figure 2.34.


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Figure 2.35.


In fact, the limiting value of α is α = 90◦ , at which point the facets become purely rectangular and the major and minor folds become entirely uncoupled. An α = 90◦ Miura-ori can be created by pleating the paper in one direction, like a fan (making the major folds), then pleating the fan in the opposite direction (making the minor folds). This process gives the smallest possible package— every facet overlaps every other facet precisely—but at the cost of completely decoupling the major and minor folds and losing the desirable single-degree-of-freedom motion offered by the Miuraori. We could also go the other direction, making α smaller. An example with α = 30◦ is shown in Figure 2.35. Not surprisingly, the pattern is stretched out, rather than compressed, and the major and minor folds vary at closer to the same rate. The other variable parameters are the two distances d1 and d2 . These can be independently varied, although what matters is their ratio. Figure 2.36 shows the effect of varying their ratio; compare these to Figure 2.33. It is a very common occurrence in the world of origami that one may discover a structure or mechanism, only to find that someone else came up with the exact same thing by an entirely different route. As already noted, the Yoshimura pattern forms naturally via compressive deformation of a cylindrical surface. It should not be too surprising that the Miura-ori concept has also been found before. Figure 2.37 shows a figure from a 1959 patent by Henry Hochfeld for “Process and Machine for Pleating Pliable Materials" [46]. Not only does it show what is clearly a Miura-ori pattern, but it even describes a machine for creating the same. In fact, similar folded patterns may be seen in much older forms, going back to 16th-century decorative napkin-folding [106]. Mattia Giegher’s 1639 work Li Tre Trattati (Three Trea-

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Figure 2.36.

Top row: a Miura-ori pattern with (d1 = 2, d2 = 1, α = 60◦ ). Bottom row: a Miura-ori pattern with (d1 = 2, d2 = 2, α = 60◦ ). Left: crease pattern. Middle: partially folded form. Right: nearly fully folded form.

Figure 2.37. A figure from Henry Hochfeld’s 1959 patent on folding pleated materials, showing a version of the Miura-ori.

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Figure 2.38. How to fold a pleated sheet into a Miura-ori, from Li Tre Trattati, 1639.

tises) on table decoration using napkin-folding shows how one folds a pleated napkin into what is unmistakably a Miuri-ori, as seen in Figure 2.38. (One might cry foul, that the folding of starched napkins isn’t the same as paper-folding. However, Giegher taught these techniques at the University of Padua, and he almost certainly taught his pupils using paper as a practice material. So, not only was this an example of early European “origami,” Giegher’s course might have been the original university origami curriculum!2 ) The Miura-ori pattern shows up again and again over time, whenever people have been exploring the folding of paper. Examples of Miura-ori appear in the work of Josef Albers’s students of the Bauhaus school of the 1920s [128, p. 435] and in quite a few patents from 1958 to the present [121, 46, 36, 16, 64]. It was Miura, however, who recognized the generalization and application of the pattern, and it is still perhaps appropriate that his name is the one permanently attached to the concept. ? 2.3.4. Miura-ori Variations I started the previous subsection by pointing out the relationship between the Miura-ori and the Yoshimura pattern; if we join two Yoshimura patterns with the same angle α but opposite parity, the building block of the Miura-ori provides the “glue” to create the joint. The relationship goes deeper than that, though, because there is a fortuitous seeming coincidence involved in such a splice: when we cut and reassemble the two half-patterns with offsets, we create a new set of vertices along the join line—and there is no a priori reason to expect that vertices assembled in such a way 2

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I am indebted to origami historian Joan Sallas for background on Giegher.


Figure 2.39. Two back-to-back degree-4 Miura-ori vertices, when merged, give rise to a degree-6 Yoshimura pattern vertex.

would be flat-foldable, let alone possessing of compatible fold angles at all stages of partial folding. But the vertices of the Miura-ori and those of the Yoshimura pattern are compatible; if we chop a Yoshimura pattern in half, the sector angles and fold angles around the newly created vertices are such that after reassembly, the former add up to 360◦ and the latter match along the joints. We can see this by approaching such a joint from another direction: by placing two degree-4 vertices of a Miura-ori at opposite ends of a line and collapsing the line segment between them, similarly to how we went from the Huffman grid (of quadrilaterals) to the Yoshimura pattern (of triangles) as was shown in Figure 2.20. This process of collapse is illustrated in Figure 2.39. We take two Miura-ori vertices of the same type (here, mountain-like) so that a common minor fold connects them back to back. If their common fold has the same position within each vertex relative to the other folds at the vertex, then we can be assured that the fold angles of the two vertices match at all positions from unfolded to folded. If we now shorten the length of the fold between the two vertices, we will arrive at a degree-6 vertex, in which the upper and lower triangles (that touch at only a vertex) have the same angular relationship to each other as the two corresponding connected facets of the Miura-ori. The other three folds of each Miura-ori vertex then become the six folds of the Yoshimura vertex, and since the fold angles were mutually compatible before the merge, they must be similarly compatible after the merge. Thus, the fold angles of a Miura-ori with characteristic angle α form a set of compatible fold angles of a Yoshimura vertex based on the same sector angle. (It is important to note, though, that we can’t necessarily go the other way. A degree-6 vertex has two degrees of freedom in

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Figure 2.40. The periodicity vectors (green) and tile (orange) for the Miura-ori.

its partially folded state, and every possible configuration cannot necessarily be split into two degree-4 vertices.) The Miura-ori, like the Yoshimura pattern and the Huffman grid, is 2D periodic, and it is instructive to identify both the periodicity vectors and a single tile of the pattern. Figure 2.40 shows both for the (d1 = 1, d2 = 1, α = 60◦ ) Miura-ori. A single tile of the Miura-ori contains two each of mountainlike and valley-like vertices. That allows for still more variation while preserving 2D periodicity. For example, we can choose all four lengths around each vertex to have different values if we are careful to construct the mountain and valley vertices in pairs with the same relative lengths, as in Figure 2.41. This pattern has a significant qualitative difference from the previous Miura-oris. First, in all of the previous examples, all of the major folds lay in one of two common planes along the top and bottom in all stages of partial folding. In this example, though, the

Figure 2.41.

A Miura-ori pattern with four fold lengths around each vertex: (d1 = 1, d2 = 1.5, d3 = 2, d4 = 2.5, α = 60◦ ). Left: crease pattern. Middle: partially folded form (turned over). Right: flat-folded form.

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Figure 2.42. Changing the length of a single row or column of a Miura-ori. Top: any column can be widened by adding the same amount of length to each horizontal edge in a given column. Bottom: any row can be widened by adding the same amount of length to each diagonal edge in a given row.

facets are “stair-stepped,” with sets of corners, but not full creases, residing in common planes. In the fully folded state, this gives rise to a scale-like overlapping pattern. It is also clear from this pattern that if we are willing to give up strict periodicity, we can vary this pattern further because we can independently add or subtract length to the folds of any single row or column of quadrilaterals, as illustrated in Figure 2.42. In fact, as this construction makes clear, we should be able to arbitrarily specify the width of each column and height of each row independently. And so we can. But there is yet one more degree of freedom open to us in the design of Miura-ori-like fold patterns, which we will explore first. ? 2.3.5. Barreto’s Mars As already noted, in all of these varieties of the Miura-ori, the vertices have the same geometric configuration: the sector angles are the same around each vertex, in the same order; the crease assignments are mountain-like for half of the vertices and valleylike for the other half; and the fold angles are the same (except

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Figure 2.43. James Minoru Sakoda’s “Staircase” pattern (from [105]). Left: crease pattern. Middle: partially folded. Right: flat-folded (different scale).

for sign) for all of the major folds, and the same (except for sign) for all of the minor folds. In all the examples shown thus far, the vertex is a symmetric bird’s-foot vertex, and that makes the vertex, and hence, the entire pattern, flat-foldable. We saw in the previous chapter that a flat-foldable degree-4 vertex does not need to be bird’s-foot with two pairs of identical sector angles; in fact, in the general case, the four sector angles can be different, as long as they satisfy the Kawasaki-Justin Condition that opposite angles sum to 180◦ . That raises the question: is it possible to create a Miura-ori-like pattern, but by using nonsymmetric degree-4 vertices? The answer is yes, and examples of both were presented by James Minoru Sakoda [105] and, especially, Paulo Taborda Barreto [4], at the Second International Meeting of Origami Science and Scientific Origami in 1994. A small patch of Sakoda’s “Staircase” pattern is shown in Figure 2.43. Sakoda’s “Staircase” is a specific example of a more general family. Barreto gave several recipes for constructing such patterns, both periodic and non-periodic, which he dubbed the “Mars” family of designs. Both Sakoda’s and Barreto’s patterns were similar to the Miura-ori in crease assignment—major crease chains that alternated as all-mountain and all-valley, and minor crease chains consisting of alternating mountain and valley—but instead of using a symmetric bird’s-foot vertex, they were constructed from a different type of flat-foldable vertex, one illustrated in Figure 2.44.

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g+ gÐ

Figure 2.44.

a 90¡ gÐ 90¡ 180¡Ð a

The generating vertex of Sakoda’s “Staircase” and Barreto’s “Mars” patterns.

g+

The generating vertex for a Mars pattern consists of two opposite 90◦ angles and two opposite angles that sum to 180◦ . We can define one of them as α; the other then must be 180◦ − α. The vertex is, obviously, flat-foldable. In Sakoda’s “Staircase,” we have α = 45◦ , but in general, α can take on any value in the range (0, 90◦ ). As we did with the symmetric bird’s-foot vertex of the Miuraori, we will call the two opposite creases of the same fold type the major creases and the two opposite creases of opposite fold types the minor creases. As with the symmetric bird’s-foot vertex, the fold angles of the two major creases are equal, and we will denote its magnitude by γ+ and its sign by the fold line style. Also as with the symmetric bird’s-foot vertex, the fold angles of the two minor creases are equal and opposite, and we will denote their magnitude by γ− . To understand the construction of such a pattern, let us try constructing a small patch—a single quadrilateral. We start with a single generating vertex; we then arrange this with copies of itself to create a fully self-consistent crease pattern, and by consistent, I mean one where the crease directions and fold angles leaving one vertex match up with those entering the adjacent vertex. Such an assembly is shown in Figure 2.45. 90¡

a 180¡Ð a 90¡

gÐ

180¡Ð a 90¡ 90¡ a

g+ gÐ

a 90¡ 90¡ 180¡Ð a g+

g+ 90¡ gÐ

180¡Ð a a 90¡ gÐ

g+

Figure 2.45. Four copies of the generating vertex (two of each type) can be assembled into a quadrilateral with matching fold directions and angles.

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If we are aiming for a flat-folded periodic pattern, then the only thing that matters is the fold direction: mountain or valley. But if we are seeking a structure that will be used in the partially folded state, then we need to match in both fold parity and in the actual value of the fold angle, which is somewhere between 0◦ and +180◦ for a valley fold and between 0◦ and −180◦ for a mountain fold. We’ll learn how to compute the fold angles and their relationship to one another in subsequent chapters, but we can get the matching right from a purely qualitative consideration, ensuring that we only line up major creases with other major creases of the same fold angle and parity, and similarly with minor creases. We can build up the crease pattern by arranging copies of the vertex—some mountain-like, some valley-like—so that their fold angles line up with each other by (a) being collinear, (b) having the same fold parity, and (c) matching in fold angle. In order to form a closed polygon, we need two copies of each type of vertex: two each of mountain- and valley-like versions of the original vertex. As we continue to build up the crease pattern, it will consist of rectangles and parallelograms, as you can see in Figure 2.43. Both types of polygon have the property that opposite edge lengths are equal. That means that, as in the Miura-ori, within a single column of quadrilaterals, every crease that cuts across the column has the same length (these were the horizontals in the Miura-ori; in this pattern they are horizontal and tilted, in alternating columns). Similarly, within a single row, every crease that cuts across the row also has the same length. This property means that we don’t have a lot of freedom to choose crease lengths in the pattern; once we’ve chosen a single length within a row or column, all of the corresponding lengths within that row or column are forced to be the same value, as illustrated in Figure 2.46. That, in turn, means that we can construct a complete crease pattern by choosing just the edge lengths along the bottom and left side—the two zigzagging green lines in Figure 2.46. Once we have chosen those two lines, which we will call the generating lines of the crease pattern, the complete pattern is fully determined. Better yet, it is very easily constructed. Looking closer at Figure 2.46, we see that all of the chains of major crease lines are identical in shape; each is just a shifted (and crease-reversed) version of its neighbor. The same goes for the chains of minor crease

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d

d d

d d

Figure 2.46. d

Distance propagation within the “Staircase” pattern. Every vertical crease within the darker row has the same length d.

d

lines. This property gives rise to a simple geometric construction algorithm that can be carried out with nothing more than pencil, paper, and protractor, as illustrated in Figure 2.47. In Barreto’s algorithm described in [4], the generating lines are drawn on a grid, which automatically sets α to be 45◦ , and the second generating line is chosen to be a rotated copy of the first. Using this algorithm, Barreto created many beautiful works. A computed reconstruction of one of them, “MarsJoker” (1994), is shown in Figure 2.48. It is not necessary to make one of the generating lines a rotated copy of the other; distances along both can be chosen arbitrarily. The only requirement for the algorithm in Figure 2.47 is that the angles at the vertices along both generating lines have the same value, which is based on the generating vertex for the pattern. A similar algorithm works for a generalized version of the Miura-ori. In this case, the construction is simpler, because the minor creases are purely horizontal and the major creases zigzag back and forth. A simple example of a varying-distance Miura-ori is shown in Figure 2.49. ? 2.3.6. Generalized Mars The next logical step in exploring variations of the Miura-ori would be to consider flat-foldable vertices that are not bird’s-foot nor have a 90◦ sector angle. The basic concept is the same as for Mars patterns; we create mountain-like and valley-like versions of the generating vertex, then arrange them so that major and minor fold angles match up in

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90¡Ð a a 90¡ 90¡ 90¡Ð a

90¡Ð a

1. Start with the generating vertex at the dot and draw two generating lines, initially at right angles. You can add the bends anywhere you like, but all of the alternating bend angles should be equal.

2. Create copies of the left generating line and place one copy at each vertex of the lower generating line, assigning them to be alternately mountain and valley folds.

! 3. Create copies of the bottom generating line and place one copy at each vertex of the left generating line, alternating mountain and valley within each copy and from one copy to the next.

4. If, as in this example, the acute angle at each vertex has the (non-flat-foldable) iso assignment, you can swap the crease assignment of all of the bottom-generatingline copies to fix it.

Figure 2.47. Construction sequence for a Mars-type origami pattern.

Figure 2.48.

Left: crease pattern for Barreto’s “MarsJoker.” Middle: partially folded form. Right: flat-folded form (turned over).

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Figure 2.49.

Left: crease pattern for a varying-distance Miura-ori. Middle: partially folded form (turned over). Right: flat-folded form.

both angular value and fold direction. Figure 2.50 shows a general flat-foldable vertex and an arrangement of four copies that meets these requirements. There are two notable differences from the corresponding configuration for the Mars construction. If the generating vertex is valley-like, then the two mountain-like vertices must be the mirror images of the valley-like vertices (they are also rotated, of course). You can see the mirror reversal in the figure; in the valley-like vertices, the sector angles α1 –α4 circulate counterclockwise around their vertex, while in the two mountain-like vertices, they circulate counterclockwise. a4

a3 a1 a 2

g+ gÐ

a2 a 1 a3 a 4

g+

gÐ

gÐ

a1 a 2 a4 a3

g+ a2 a 1 a3 a 4

g+

gÐ

a3 a4 a2 a 1

Figure 2.50. Constructing a generalized Mars pattern from a generating vertex. Left: the generating vertex. Right: four versions of the vertex can be arranged to create a closed quadrilateral where the folds match in both direction and fold angle.

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This construction is a generalization of the construction in Figure 2.45; in point of fact, the mountain-like vertices in that construction must also be mirror-images of the valley-like generating vertex. However, since a flat-foldable degree-4 vertex with a 90◦ angle is its own mirror image (plus a rotation), one can simply use rotated copies of the original vertex when constructing a Mars pattern—or for that matter, when constructing a Miuraori, since the symmetric bird’s-foot vertex is also its own mirror image. For a general vertex, though, the mirror images must be explicit, as in Figure 2.50. There is another difference, though. In Figure 2.45, the polygons were parallelograms and so every column-crossing crease had the same length within a single column and every row-crossing crease had the same length within a single row. But now, in Figure 2.50, opposite edges of each quadrilateral have different lengths. This has ramifications for the construction of the crease pattern. Opposite edges of each quadrilateral are no longer necessarily equal, but they are still determined; if we have two edges, then the other two are fully specified, just as we saw in the Huffman grid. So, to construct such a pattern, we can still start with two generating lines, as before, as illustrated in Figure 2.51. But now the edge lengths are found by projecting the lines from the two adjacent vertices and finding their intersection, as illustrated in the figure for the vertex at the gray dot. By constructing each vertex from its neighbors, one can iteratively build up the crease pattern, but instead of copying and moving lines, as in Figure 2.47, one must build up the pattern one vertex at a time. Figure 2.52 shows a small patch of such a pattern. For this pattern, the segments of the generating lines were taken to have unit length, but as the pattern propagates from the lower left toward the upper right, some edges get longer, some shorter, and had we continued this pattern further to the right, two or more chains of minor folds would have eventually crossed, breaking the pattern. So, while it is technically feasible to construct generalized Mars patterns, the deterministic variation in the quadrilateral dimensions limits the flexibility and, in some cases, the sizes of the possible patterns.

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a4 a3 a1 a 2

gÐ a1 a 2 a4 a3

g+

g+

a2 a 1 a3 a

gÐ

4

a3 a4 a2 a 1

a4

a3 a1 a 2

gÐ

a1 a 2 a4 a3

g+

a4

a3 a1 a 2

gÐ

a1 a 2 a4 a3

g+

a2 a 1 a3 a 4

gÐ

a3 a4 a2 a 1

g+ a2 a 1 a3 a 4

g+

gÐ

a3 a4 a2 a 1

Figure 2.51. Construction of a generalized Mars pattern from a non-mirror-symmetric generating vertex (black dot) and two generating lines (green). Each vertex is at the intersection of two lines emanating from previouslydefined vertices.

Figure 2.52. A generalized Mars pattern for a generating vertex with sector angles (130◦, 60◦, 50◦, 120◦ ). Left: crease pattern. Right: partially folded form (turned over).

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(a)

(b)

Figure 2.53. Building block tiles for creating hybrid Yoshimura-Miura patterns. (a) Joined Yoshimura patterns, divided into vertical tiles. (b) Four types of tiles. (c) Four ways of joining tile pairs.

(c)

? ?? ???

2.4. Partial Periodicity

? 2.4.1. Yoshimura-Miura Hybrids The Miura-ori building block arises from necessity if we try to glue together two Yoshimura patterns that have opposite curvature. If we divide the joined pattern of Figure 2.29 into vertical stripes, we see that all of the vertical stripes fall into one of four distinct types, as illustrated in Figure 2.53(a) and (b). There are four types of tile, but each can mate with only two of the others in a way that gives rise to a flat-foldable pattern. Four of the eight possibilities are shown in Figure 2.53(c). One type of mating gives rise to a “Yoshimura-like” section of the pattern; the other type gives rise to a “Miura-like” section of pattern. Yoshimura patterns curl; Miura patterns are straight. By mixing and matching different combinations of tiles, one can create folded patterns that display quite varied large-scale curvature, as shown in Figure 2.54. Yoshimura-Miura hybrid patterns are a family of designs, defined by several different independently variable quantities:

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Figure 2.54. “Gentle Waves” by the author, a hybrid Yoshimura-Miura pattern consisting of varying combinations of Yoshimura and Miura vertical tiles. The tilt angle is α = 30◦ ; the minor folds are folded at 70◦ . Left: crease pattern. Right: folded form (turned over).

• the choice of tiles along the horizontal direction in the crease pattern (subject to matching rules); • the number of repetitions in the vertical direction; • the tilt angle α, which is common to all of the diagonal folds; • the degree of foldedness. The degree of foldedness is something that bears comment. As we already saw, the Yoshimura pattern has two different ways it can be deformed when partially folded: it can be curved tighter or looser, and its ends can be shifted relative to each other (i.e., varying the helicity of the minor folds). For each type of motion, all of the fold angles are affected if we perform a large-scale shift in the pattern. The Miura-ori, by contrast, has only a single degree of freedom; it can open or close with all folds moving together in a prescribed way. If we splice together a Yoshimura pattern and Miura-ori, the result will have the reduced freedom of the Miuraori, so that the entire pattern has the single degree of freedom of the Miura-ori. At least, that is the case in purely mathematical terms. However, if you actually fold any of the patterns in this section, you will find that, empirically, they have a lot more freedom than the above paragraph would suggest; they can be twisted, compressed at one CHAPTER 2. PERIODICITY

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end and expanded at the other, and, in general, deformed in many ways. These additional deformations happen because with most folding materials, individual panels can twist, and folds can shift their position slightly, which permits these additional motions. So, my comments about Miura-oris having only a single degree of freedom must be taken with a grain of salt; the model in which that is true is a mathematical approximation of the real world, and ultimately, it is the real-world folding behavior that matters! Mathematical approximations can be a useful tool for design, but we must always be aware of the limits of such approximations. Within the approximation where we assume that individual panels do not twist, in any Yoshimura-Miura hybrid pattern, all minor folds (the horizontals in the crease pattern) will have the same magnitude of fold angle (whether mountain or valley), and similarly, all major folds (diagonal creases in the crease pattern) will also have the same magnitude of fold angle (albeit one that differs from that of the minor folds in every non-flat state). In all four of the tiles in Figure 2.53(b), the diagonal folds have the same tilt angle α, which we can choose to be anything between 0 and 90◦ . But can we mix tiles that have different tilt angles? It turns out that we can. If you take two tiles with different values of α, they mate in the same way as tiles with the same α. Most importantly, the resulting closed vertices formed at the tile boundaries not only are flat-foldable but they have fold angles that are compatible across the full range of folding from unfolded to flat. (We will learn how to prove this claim in Chapter 7.) ? 2.4.2. Semigeneralized Miura-ori One of the variables in the design of Miura and Yoshimura patterns is the number of vertical repetitions in the pattern. They certainly look more interesting with a lot of repetitions, but if we choose a single repetition (two rows), the result is suggestive. A row of Figure 2.55 is shown in Figure 2.56. This single row captures the essence of the pattern, and conceptually, it is a very simple object: a rectangular strip of paper, reverse-folded at varying angles and varying distances. Such a pattern, on its own, both is flat-foldable and can exist partially folded with no bending of the facets with any value of the fold angle running down the middle. But, as we have seen in the generalized Yoshimura-Miura hybrid, we can array such strips to create a surface that follows the path traced out by the single strip.

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Figure 2.55. “Double Spiral” by the author, a hybrid Yoshimura-Miura pattern, consisting of two varying-angle Yoshimura patterns joined by a Miura-ori splice. Left: crease pattern. Right: folded form (turned over).

This suggests a different way of designing such forms. Rather than starting with a blank crease pattern, picking angles, and then seeing what shape we get, we could instead start with a strip, fold it into a pattern that gives the surface cross section that we want, then unfold it and use it as a template to construct the full crease pattern. A nice thing about this algorithm is that it can be carried out entirely by folding and/or drawing: no computation needed, with one important reservation, which we’ll get to. We’ll start with the path that the minor fold (the center of the strip) follows in the flat-folded form. We will call this path the generating line for the surface; it represents the desired cross section in the direction perpendicular to the direction of periodicity. We’ll create a strip of the crease pattern that represents a single repetition of the periodic surface, and we will call this the generating strip for the pattern. Figure 2.57 shows a sequence for graphically constructing a generating strip from a generating line.

Figure 2.56. A single row of “Double Spiral.” Left: crease pattern. Right: folded form (turned over).

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1. Begin by drawing the generating line that the desired strip (and surface) will follow.

2. Draw pairs of lines parallel to and equally spaced from each of the segments of the generating line. Make sure each line of each pair is long enough to intersect both of the next pair.

l r l

r

l

r

3. Now picture yourself traveling along the generating line (from left to right, in this example). Draw along the line lying on the left side of the generating until you hit the line to the right of the next segment; follow it until you hit the left line of the next segment; and so on, until you get to the end.

4. Do the same thing, starting with the other line, again alternating between right and left.

5. Connect corresponding corners of both pairs of lines and erase all of the guidelines.

6. The line drawing is the silhouette of the folded strip that follows the desired path.

Figure 2.57. Graphical construction of a strip that follows a specified path when it is folded flat.

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We begin by constructing a drawing of a folded form (or rather, the silhouette of the folded form) that follows the path and is a valid folded form for some flat-sheet crease pattern. Next, to get the crease pattern for the generating strip from the drawing of its folded form, we take each of the overlapping polygons in the drawing and arrange them into a single rectangular strip, as shown in Figure 2.58. This gives the locations and orientations of all of the creases, which we can then assign using our existing known rules for Miura-oris and their kin: minor folds (horizontals) alternate in sign; major folds (zigzag verticals) have the same sign. Once a single row is constructed, it can be arrayed with copies of itself to make a full surface. Figure 2.59 shows the resulting complete crease pattern and a folded form for the sample generating line. It is also possible to carry out the design using a folded strip of paper, as shown in Figure 2.60. Fold the doubled strip to follow the path, then unfold it, and use it as a template to transfer the creases to the paper to create an array. I call such a surface a semigeneralized Miura-ori (SGMO). We can think of such a surface as being created by splicing together individual slices of Miura-ori, in either of two orientations, with varying angles and distances between them. A semigeneralized surface can take on any arbitrary cross section in one direction, but it exhibits strict periodicity in the other (which is the reason for the “semi” part of “semigeneralized”). The concept of the semigeneralized Miura-ori, like so many other origami structures, has deeper roots (though not by that name). Conceptually, the SGMO can be created by making a series of pleats in a sheet of paper in one direction, then repeatedly reverse-folding it at several angles in the other direction to create the 3D shape. This technique can be found in a centuries-old magic trick, called “Troublewit” [63], which we will come back to. The design recipe above is a very straightforward way to create a semigeneralized Miura-ori with any desired cross section; in fact, you could create a surface whose cross section is your own signature! I’ve not done that, but in 2012, I was asked to create an origami version of the “Google Doodle” to honor the great 20th-century folding master Akira Yoshizawa. I created each letter as an instance of a semigeneralized Miura-ori. The silhouettes, crease patterns, and resulting “Doodle” (decorated with Yoshizawa’s “Butterfly”) are shown in Figures 2.61 and 2.62.

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1. Rearrange the polygons to form a strip, turning over the colored ones.

2. Create a mirror image of the pattern of polygons above the original strip, and make creases at the boundaries between the polygons.

3. Assign the center creases to alternate between mountain and valley (you can choose which one you start with).

4. Assign the diagonal creases so that each vertex becomes a birdÕs-foot vertex.

5. Array two (or more) copies vertically, alternating the sign of the horizontal creases in each column.

Figure 2.58. Graphical construction of the unfolded strip.

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Figure 2.59. A periodic array of the computed strip. Left: crease pattern. Right: partially folded form. The horizontal folds all have a fold angle of ±150◦ .

1. Lay a folded strip of paper down on the desired path. Fold it over at each bend so that it follows the path.

2. Unfold the strip.

3. Use the unfolded strip as a template to draw the array of creases.

4. For small arrays, you can precrease the entire paper by starting with a pleated strip.

Figure 2.60. Folding sequence to create the desired creases by manual folding.

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Figure 2.61.

Top row: crease patterns for the letters of the “Google Doodle.” Bottom row: the desired letterforms.

Figure 2.62. The Google Doodle for March 14, 2012.

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Figure 2.63. Two bends in the same direction impose a minimum width on the strip because consecutive vertices collide. Top: the path and guidelines for two different widths. Middle: a narrow strip. Bottom: a wide strip. The two bird’s-foot vertices have merged into a Yoshimura vertex.

There is great variety possible in semigeneralized Miura-oris; one can choose the path to be followed by the surface to be almost any piecewise continuous sequence of connected straight lines; it can even double back on itself (which you can see in the “Doodle” patterns). Still, there are a few issues that may arise in their design. The first is that for any given generating line, there may be a minimum strip width whose value depends on the lengths and angles in the generating line. If two consecutive bends in the desired path go the same direction (like the “roof” in Figure 2.57), that gives rise to two back-to-back birds’s-foot vertices, which, as the strip grows wider, must move toward each other, as illustrated in Figure 2.63. As the strip becomes wider, the two consecutive bird’s-foot vertices move toward each other and eventually collide, merging into a Yoshimura vertex. If we made the strip wider, the two vertices would cross, as would their connected creases, and we would have to introduce additional vertices and creases into the pattern to create a valid foldable crease pattern.

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Figure 2.64. Creating a semigeneralized Miura-ori with shallow bends in the surface. The solid green line is the desired generating line. The thin black line is the actual path we use, which forces all bends to be 90◦ or sharper.

The point of merging sets a natural limit on the strip width for such a pattern; the permissible width is the minimum width set by any of the segments of the surface. If we choose the width to be exactly one of the minimum values, then at least some of the vertices of the pattern will be Yoshimura vertices. This can be seen in the crease patterns for the Google Doodle letters in Figure 2.61. You can see from Figure 2.63 that the width limitation will arise sooner for short segments than long ones and, especially, for segments bounded by shallow bend angles. That makes it difficult to approximate smooth surfaces using this technique. Subdividing a surface into many short segments with slight bends—what one would desire for such an approximation—will force an extremely small strip width; one smaller than might be desired from considerations of the aesthetics or functionality of the target surface. However, there is a nice trick we can use to overcome this problem: wherever there is a shallow bend in the surface, we can replace it by a series of sharper bends, as illustrated in Figure 2.64. Instead of having a shallow bend of angle β where β is small, we replace it by three consecutive bends of values (−90◦ + β/2, 180◦, −90◦ + β/2), so that all three angular bends are large. I call such a feature—a bend and doubling-back in order to realize a shallow bend—a shallow-angle divot. The realized implementation of this example is shown in Figure 2.65. In this example, I’ve placed the divots to all point down, which makes the top surface relatively smooth, and I’ve angled them so as to equalize the bend angles to either side. I should point out, though, that we can point the divots at any angle on either the top

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Figure 2.65. A semigeneralized Miura-ori with shallow-angle divots. Left: crease pattern. Right: folded form (minor fold angle of ±150◦ ).

or bottom surface. If we place the divot on the outside of the bend, we can angle it so that one side of each divot is collinear with one of the two sides of the path and, effectively, remove one of the major creases, reducing the number of creases at the bend to two, as in Figure 2.66. The structure in the bottom row of Figure 2.66 is familiar to most origami artists; it is an example of what would be called a crimp, a very common maneuver in representational folding.

Figure 2.66. A comparison of divots angled in different directions for the same desired generating path (a single bend of −120◦ ). Top left: divot on the bottom, evenly divided. Top right: divot on the top, evenly divided. Bottom left: divot collinear with the right side. Bottom right: divot collinear with the left side.

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Figure 2.67. The crease pattern of Figure 2.59 for different values of minor fold angle. Top row: left to right, −180◦ (flat-folded), −150◦ , and −120◦ . Bottom row: left to right, −90◦ , −60◦ , and −30◦ .

In the figure, the folded form is shown partially folded, i.e., not pressed flat, and there is an interesting detail: although these were designed for the same generating line, with the same bend angle when folded flat, the bottom edges are bent at slightly different angles from one another—and none of them are bent at precisely −120◦ , the design angle. In semigeneralized Miura-ori, the angle between consecutive sections of pleats is not fixed but varies with the degree of foldedness. This property has several ramifications, as we will now see. ?? 2.4.3. Predistortion It should not be surprising that the angles between consecutive sections of pleats would vary with the state of foldedness; after all, when the pattern is fully unfolded, all angles are zero. We should expect that the angles would vary from zero at the unfolded state to some fixed value at the flat-folded state. Indeed, we can see this behavior in action in Figure 2.67, where we show our test case of Figure 2.59 for a range of different minor fold angles. If we are going to design a surface that follows a particular path, we will need to take into account the fold angles at which it will be displayed. To do that, we will need to understand the

138


gÐ

p-a a a p-a

z -gÐ

g+

b

-gÐ

g+

g+ q

gÐ

g+

g+

b

g+

gÐ

gÐ

Figure 2.68. Geometry of a partially folded bird’s-foot vertex. Left: crease pattern. The sector angles are, in order, (α, α, π − α, π − α). The major and minor fold angles are, respectively, γ+ and γ− . Middle: the partially folded form; ζ is the angle between the two minor creases, and β is the bend angle of the minor folds. Right: the partially folded form turned over; θ is the angle between the two major creases.

relationship between the desired fold path and the sector and fold angles at any state of partial folding. Every vertex in a generalized Miura-ori is some form of bird’s-foot vertex (albeit for possibly varying characteristic angle α), so we should look at all the various angles of such a vertex—the sector angles, the fold angles, and the angles in 3D between the various folds. A general bird’s-foot vertex with angles labeled is shown in Figures 2.68. We have already seen the sector and fold angles. For our purposes, we need one more angle: the angle between the two minor creases of the folded vertex, which we denote by ζ and which (for reasons we will learn) we call the ruling angle of the vertex. In the case of fold angles, we find it is usually more convenient to work with the fold angle (deviation from straightness) rather than the dihedral angle (angle between planes), and the same situation will arise here; it is a bit more convenient to work with the deviation from straightness of the minor fold line. In this case, we will give this quantity its own variable, β, and call it the bend angle of the pair of minor folds incident upon the vertex. We also introduce, for completeness, the angle between the two major folds. We denote this angle by θ, and we call it the osculating angle of the vertex. The fold angles, sector angles, and angles θ, ζ, and β are all related to one another. We will work out their relationships in general in Chapter 8, but for now, we will simply present the special case of a bird’s-foot vertex as a given.

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First, the major and minor fold angles are related to one another through the sector angle α: tan 12 γ+ tan 12 γ−

= sec α,

(2.2)

which means that, given one of the two angles, we can derive the other: γ+ = 2 tan−1 sec α tan 12 γ− , (2.3) γ− = 2 tan−1 cos α tan 12 γ+ . (2.4) The osculating angle θ satisfies3 sin 12 θ = sin α cos 12 γ− .

(2.5)

The ruling angle ζ satisfies4 cos 12 ζ = sin α sin 12 γ+,

(2.6)

which exhibits a pleasant symmetry with Equation (2.5). Note that each of the trigonometric functions in Equations (2.5) and (2.6) is strictly nonnegative for all values of the various angles that appear within them. Using the fact that ζ = π − β, Equation (2.6) is equivalent to sin 12 β = sin α sin 12 γ+ . (2.7) We would ultimately like to compute the value of α from the minor fold angle γ− , not γ+ . Using Equation (2.2), though, we can find that tan 12 β = tan α sin 12 γ−, (2.8) which gives the desired relationship. Given a desired path for the surface to follow, using Equation (2.8), we can work out the sector angles we need at each vertex of the crease pattern from the bend angles at each corner of the path and the desired minor fold angle γ− , which we can choose to be any nonzero angle—with certain limits, as we will see. To make this concrete, let us define the desired path as a series of segments of length di, i = 1, . . . , N, with angles θi, i = 1, . . . , N − 1, between consecutive segments, as illustrated in Figure 2.69. 3 4

140

This can be derived from Equation (8.37). This can be derived from Equation (8.23).


z2

Figure 2.69.

-b2 z1

d2

d3

Notation for a specified generating line to be the cross section of a semigeneralized Miura-ori.

z3

b1

d1

Denote the sector angle for the ith vertex by αi . Then, from Equation (2.8), each of the desired sector angles is given by " # 1 tan β i 2 αi = tan−1 . (2.9) 1 sin 2 γ− Note that the right side of this equation can be positive or negative, depending on the signs of βi and γ− . Negative solutions should be shifted by the periodicity of tan−1 , i.e., by adding π to negative values to bring them into the range (0, π). What about the distances between consecutive vertices? This will depend upon how we position the folded surface relative to the desired generating line. In Figures 2.57 and 2.60, we positioned our strip so that it extended equally above and below the generating line in the fully flat-folded state. If we choose the same approach for a partially folded semigeneralized Miura-ori, then the distances between consecutive vertices will be a bit longer or a bit shorter than the distances di . If we imagine cutting the folded surface by the target surface, then, independently of the minor fold angle γ− , the target surface cuts through each panel halfway across its width, as illustrated in Figure 2.70.

Figure 2.70. The target surface cuts through the middle of each horizontal panel as portions of the folded surface extend above and below the target. Left: looking along the direction of periodicity. Right: 3D view.

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141

w

w

di p-aiÐ1 p-aiÐ1

aiÐ1 aiÐ1

di¢ di

p-ai p-ai

ai ai

Figure 2.71. A portion of a single strip of the crease pattern; di is the distance between consecutive vertices of the path, and di0 is the distance between consecutive vertices along the chain of minor folds.

Compare this now to a portion of the crease pattern, as in Figure 2.71; if the target surface cuts each panel along its midline, then we can trace back the line of intersection to the crease pattern and, from that, work out the relationship between the lengths of the segments of the path that defines the target surface and the distances between the vertices of the crease pattern. If we choose the width of each vertical panel to be w, then the distance between consecutive vertices is given by di0 =

w w cot αi−1 + di + cot αi . 2 2

(2.10)

And that completes the design algorithm: we now know all of the distances and angles in the crease pattern. If we choose to have m repetitions of the pattern along the direction of periodicity, then the overall width in that direction in the folded form will be Wtot = 2mw cos 12 γ− .

(2.11)

A typical method of folding such a pattern would be to precrease all of the creases, press it fully flat, then open it back up to the desired minor fold angle. The fully flat-folded form does not have the same cross section as our desired path, as shown in Figure 2.72. In general, the bend angles will all be sharper than their corresponding angles in the desired path (although to varying degrees). Thus, I call this design technique predistortion; we are intentionally distorting the flat-folded form so that when it is partially unfolded, it takes on the desired path in 3D. Once we’ve got a toolkit for creating semigeneralized Miuraori with arbitrary cross section, there are many possibilities for forms both in the artistic realm and with functional applications. As an example of the latter, Figure 2.73 shows an architectural

142


Figure 2.72. A semigeneralized Miura-ori designed for a right-angled folded form with a minor fold angle of 90◦ . Top left: folded form. Top right: same thing, viewed along the direction of periodicity. Bottom left: crease pattern. Bottom right: flattened form, now distorted.

barrel vault designed using this technique. The cross section is a semicircle, and I have introduced divots at each joint to allow for the shallow bends in the overall surface. There is much more that can be done with semigeneralized Miura-ori concepts, and we will explore them further, but I would like to pause to make a comment on the name: why only “semi”generalized Miura-ori?

Figure 2.73. A cylindrical barrel vault, implemented from a semigeneralized Miura-ori with a minor fold angle of 90◦ . Left: crease pattern. Right: folded form.


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The semigeneralized Miura-ori has the topological folding pattern of a Miura-ori with the cross section taking on some arbitrary form in one direction while remaining strictly periodic in the other. We can envision the possibility of varying the cross section arbitrarily in both directions; that would be a fully generalized Miura-ori. Because the fold angles and sector angles are all related, the analysis of such a pattern becomes rather complex; one must choose fold and sector angles so that they are consistent at every vertex. Such a construction has been carried out by University of Tokyo professor Tomohiro Tachi, who has worked out the underlying mathematics [111] and written design software [119] for creating such patterns, which, in general, make use of arbitrary degree-4 vertices, not just the highly symmetric (and much simpler) bird’s-foot vertex. We will develop the descriptive mathematics necessary to handle such structures a bit later on, in Chapter 8. ? 2.4.4. Tachi-Miura Mechanisms As we saw in Equation (2.8), at each bend in the surface of a semigeneralized Miura-ori, the bend angle varies with the minor fold angle in a nonlinear way. If we have a chain of several different bend angles, the angular difference between the inclinations of the first and last segment will also vary with the minor fold angle, as illustrated in Figure 2.74.

Figure 2.74. The crease pattern of Figure 2.59 for different values of minor fold angle, viewed along the direction of periodicity. Top row: left to right, −180◦ (flat-folded), −150◦ , and −120◦ . Bottom row: left to right, −90◦ , −60◦ , and −30◦ .

144


Figure 2.75.

A folded strip with two consecutive vertex sector angles of 45◦ for different values of minor fold angle, viewed along the direction of periodicity. Top row: left to right, crease pattern, −36◦ , and −72◦ . Bottom row: left to right, −108◦ , −144◦ , and −180◦ (flat-folded).

While the relationship between bend angle and minor fold angle is not linear, there is a symmetry with respect to the sector angle α; if we replace α with its supplement, 180◦ − α, then the bend angle has the same magnitude, but opposite sign, and this is the case over the full range of minor fold angles. And since the minor fold angle changes sign across a vertex, if we have two consecutive vertices with the same sector angles α, then the two segments on either side of the pair will remain parallel across the range of minor fold angles, as illustrated in Figure 2.75. This behavior can be exploited. If we combine such a strip with its mirror image, then in the resulting mechanism, the bottom edges remain parallel and in the same plane across the full range of minor fold angles, as in Figure 2.76. What’s more, if we replace each sector angle α with 180◦ −α and adjust the distances between the vertices, we can obtain a strip that displays the same behavior, but in which the middle buckles downward, as in Figure 2.77. The vertex-to-vertex distances in Figures 2.76 and 2.77 were chosen so that the generating paths (the red lines in Figure 2.70) were mirror images of one another. This ensures that not only the two folded forms have their leftmost and rightmost panels remaining parallel to each other across the folding range, but the upward- and downward-pointing forms also have the same lengths across their full folding ranges. And this, in turn, means that one could, in principle, glue the two sheets together by their horizontal flanges, and the entire assembly would remain flexible, as shown in Figure 2.78.

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Figure 2.76.

A folded strip with vertex sector angles of (45◦, 45◦, 135◦, 135◦ ) for different values of minor fold angle, viewed along the direction of periodicity. Top row: left to right, crease pattern, −36◦ , and −72◦ . Bottom row: left to right, −108◦ , −144◦ , and −180◦ (flat-folded).

Figure 2.77.

A folded strip with vertex sector angles of (45◦, 45◦, 135◦, 135◦ ) for different values of minor fold angle, viewed along the direction of periodicity. Top row: left to right, crease pattern, −36◦ , and −72◦ . Bottom row: left to right, −108◦ , −144◦ , and −180◦ (flat-folded).

The Tachi-Miura polyhedron is a flexible tube that can extend and contract while keeping its facets planar without stretching. It is rigidly foldable, to use a term we will explore more deeply in later chapters. This mechanism has applications in the technological world; for example, it could be used as an extensible boom or shroud, as part of a deployable structure. It might seem that one could achieve the same result with a simple semigeneralized Miura-ori, i.e., a shape obtained by repeatedly reverse-folding pleats to form a loop and then joining the ends. However, as Figure 2.79 shows, the tube obtained by

146


Figure 2.78. A Tachi-Miura polyhedron based on sector angles of 60◦ at four different minor fold angles. Top left: 170◦ . Top right: 120◦ . Bottom left: 60◦ . Bottom right: 10◦ .

that strategem will not stay closed as the minor fold angles flex; rather, as the minor folds unfold, the entire tube uncurls. Tomohiro Tachi and Koryo Miura have developed several variations of this concept [88, 120] (hence the name “Tachi-Miura” polyhedron). For example, one can construct the full polyhedron from a single sheet, by joining the two halves along one of their shared edges. Conversely, one could cut away the double-layered regions of paper and re-glue the cut edges, to produce a polyhedron with no doubled edges (albeit at the expense of creating some non-developable vertices, interior vertices whose sector an-

Figure 2.79. A tubular semigeneralized Miura-ori at four different minor fold angles. Top left: 170◦ . Top right: 120◦ . Bottom left: 60◦ . Bottom right: 10◦ .


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147

Figure 2.80. A Tachi-Miura polyhedron with no excess paper (but some non-developable vertices). Rendering courtesy of Tomohiro Tachi.

gles sum to less than or greater than 180◦ ). An example, generated by Tachi, is shown in Figure 2.80. An additional family of structures based on the same concept was also demonstrated by Tachi and Miura: by layering and attaching folded sheets that individually have the same structure that gives rise to the Tachi-Miura polyhedron, one can achieve a cellular mechanism that is rigidly flexible and that has the interesting property that its overall dimensions change in different proportion to one another as the mechanism is flexed. An example of such a material is shown in Figure 2.81. These objects stray a bit from the single-sheet philosophy of origami, as the various stacked layers should be glued together for best effect. They have the interesting behavior that when you expand them in one direction, they can expand in one of the other directions (or in both). With ordinary materials, if you stretch the material in one direction, it will typically get smaller in the other direction. The ratio between expansion in the one direction and shrinkage in the other is called the Poisson’s ratio for the material; if a material expands in both directions, it is said to have a negative Poisson’s ratio (at least, for that pair of directions).

148


Figure 2.81. A Tachi-Miura cellular form, composed of a stack of eight layers (four each of two opposite-polarity sheets). Left: crease pattern of a single sheet. Middle: folded form at a minor fold angle of 90◦ , oblique view. Right: folded form at a minor fold angle of 90◦ , viewed along one of the directions of periodicity.

Many origami mechanisms can be viewed as a type of bulk material where the fine structure of the folding pattern gives rise to large-scale mechanical properties that are analogous to those of more homogeneous materials. Mechanisms that behave like bulk materials but whose mechanical properties differ from those of the underlying material are called mechanical metamaterials; “meta” (Greek for “beyond”) because they exhibit properties that go beyond the underlying materials from which they are made. Many origami mechanisms can be considered to be mechanical metamaterials, and several of them display a negative Poisson’s ratio—stretch them in one direction, they expand in another. In fact, the conventional Miura-ori is a mechanical metamaterial; if you stretch it along its length, it also expands across its width. But it also gets slightly shorter in height, so it has a negative Poisson’s ratio in one direction, but a positive (ordinary) Poisson’s ratio in the other. A single Miura-ori can easily be scaled in length and width by adding rows and columns, but its height remains limited by the size of a single quadrilateral facet. By stacking Miura-oris or other folding patterns, though, one can build up three-dimensional metamaterials of arbitrary length, width, and height. The Tachi-Miura cellular structures are a class of mechanical metamaterials that exhibit negative Poisson’s ratio in at least one direction, as can be seen in Figure 2.82, which shows the object from Figure 2.81 at four different minor fold angles. These objects

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149

Figure 2.82. The Tachi-Miura cellular form from Figure 2.81, composed of a stack of eight layers (four each of two opposite-polarity sheets) at four different minor fold angles. Top left: 170◦ . Top right: 120◦ . Bottom left: 60◦ . Bottom right: 10◦ .

have the unexpected property that they change state from almost entirely flat in one direction to almost entirely flat in the other. In general, Tachi-Miura cellular forms will expand (or contract) in two directions while contracting (or expanding) in the other one. By careful choice of bend angle and operating range of minor fold angle, though, it is possible to obtain simultaneous expansion along all three axes. The object shown in Figure 2.83 has been designed to exhibit negative Poisson’s ratio in all direc-

150


Figure 2.83. An origami cellular structure that exhibits near-isotropic negative Poisson’s ratio at four different stages of flexing.

tions, meaning that all three directions expand or all three contract. From smallest size to largest, it expands by a factor of about 1.9 in all three axes. These mechanisms (and all such mechanisms based on the Miura-ori) exhibit a single degree of freedom in their motion: as one fold angle is flexed, all of the others flex in lockstep. At least, that’s the theory—but that theory only describes materials where the facets are perfectly stiff and non-stretchable and the hinges are perfectly flexible. These conditions rarely hold in practice. There is almost always a little bit of “give.” Facets can bend, hinges can CHAPTER 2. PERIODICITY

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151

exhibit residual stiffness that can impose flexing on facets, and folds can soften and deform in ways that mimic stretching and/or compression of facets. Consequently, if you build objects like the ones described in this section, you may find that they don’t behave precisely the way the mathematics would predict. Surfaces, objects, and mechanisms can twist and distort in unexpected ways. That doesn’t mean that the mathematics is wrong; but it may mean that we have attempted to apply it beyond its regime of validity because we have not taken into account all of the non-idealities of paper, or whatever our folding construction medium may be. In many cases, the non-idealities of paper can be undesirable; our mechanism doesn’t behave the way we want or expect it to. In others, though, we can make use of the non-idealities of paper to achieve interesting and useful forms and behavior. We will see a few examples of this phenomenon in the next and coming sections. ? 2.4.5. Triangulated Cylinders We saw earlier that the Huffman grid naturally curls up to form a cylinder (see Figure 2.16) as well as the Yoshimura pattern (Figure 2.24), which can curl in various cylindrical and/or helical ways. In fact, as Tachi has shown [118], every 2D periodic folding pattern displays either in-plane motion (like the classical Miuraori) or some combination of two different helical motions (like Yoshimura patterns), which includes pure cylindrical motion. The freedom to flex only happens when the edges are free, however. If we take such a pattern and join its ends, the resulting form becomes quite rigid. This follows intuitively from the observation that as we open and close the vertices of the pattern, the edges move toward and away from each other; the two motions are coupled. By joining the edges, we eliminate both opening/closing of the edges and their ability to slide past one another; if there are only two possible motions, we’ve eliminated both of them. While connecting the ends makes a Yoshimura pattern into a rigid tube, we do have some freedom in how we connect the ends, as illustrated in Figure 2.84. If we follow a chain of valley folds (as viewed from the colored side of the paper)—one is shown in red in Figure 2.84—it traces out a polygonal circle or polygonal helix on the surface of the tube. If the chain joins to itself, then it closes into a polygon, as in the left subfigure. But, as shown in the

152


Figure 2.84. Three different ways of joining the edges of a Yoshimura pattern with different helical offsets of a chain of crease pattern valley folds (marked in red). Left: no offset. Middle: offset by one. Right: offset by two.

middle and right subfigures, it can be offset by varying amounts along the joint as it wraps around the cylindrical axis. As we saw earlier in Section 2.3.2, there are at most three distinct values of fold angle in a periodic folded Yoshimura pattern, so the creases can be grouped into sets that share the same fold angle. Each set of creases with a common fold angle forms linear chains that wrap around the pattern. In general, each of the chains of valley folds and mountain folds with the same fold angle will form some type of helix; the helicity—offset from one turn to the next—of each chain will vary with the sector angles of the vertices and the mountain fold angles at each vertex (or equivalently, how you join the ends). If the ends of a chain of valley folds are offset sufficiently far from one another, then instead of a chain of valley folds closing on itself, one of the chains of mountain folds can close on itself, and this occurrence gives rise to a variety of closed tube that has a new and interesting property. Although the closed tube is rigid (as is any closed tube of a Yoshimura pattern), there can be two distinct folded states of the same crease pattern, as shown in the example of Figure 2.85. The Yoshimura pattern, broadly speaking, is composed of a grid of degree-6 vertices, formed by the intersections of three parallel sets of line segments. As we saw, because of the two degrees of freedom in the crease pattern mechanism, we have different ways of joining the edges to form a tube. If we think

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153

Figure 2.85. A closed tube formed from a Yoshimura pattern having two folded states with left and right edges joined. Left: crease pattern. Middle: one folded state. Right: the other folded state.

of the pattern as a periodic collection of quadrilaterals (outlined by mountain folds in Figure 2.85) diagonally crossed by folds of the opposite type (valley folds in Figure 2.85), then joining the ends of a row of quadrilaterals creates a tube with a polygonal cross section with faces subdivided into triangles, and we call it a triangulated cylinder. Like so many other periodic folding patterns, this pattern has been discovered and re-discovered repeatedly by various researchers. While it seems likely to be quite old, the earliest mathematical analysis of this pattern was carried out by Simon Guest (now a professor at Cambridge University) during his Ph.D. research [41, 42, 43, 44]. Guest credits the concept to a cardboard model he saw of a bacterial flagellum constructed by one of his professors, C. R. Calladine. Guest coined the name “triangulated cylinder” for this structure, and I have adopted his usage. It was also popularized by an influential article by Biruta Kresling [65], who noted that it arose naturally as a buckling mode of cylinders under compression. Kresling called the triangulated cylinder pattern “the Kresling pattern,” and it has become relatively well known by that name. Just as the Yoshimura pattern was a buckling mode of a cylinder under pure axial compression, this mode arises naturally by compressing the end of a cylinder while also applying a twisting force. In fact, it takes far less force to create the triangulated

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Figure 2.86. Twisting and compressing an aluminum beverage can naturally creates the triangulated cylinder pattern.

cylinder pattern than the Yoshimura pattern; growing up in the 1970s, when beverage cans switched from steel to aluminum, I found amusement in the ease in which a twist-press could create this pattern (see Figure 2.86). Triangulated cylinders were also extensively explored by Taketoshi Nojima in his master’s thesis [95] and a paper [94] that carried out a wide-ranging exploration of Huffman grids, Miuraoris, Yoshimura patterns, and more. More recently, the concept and further structural variations have been explored by Tomoko Fuse [33]. As with the patterns we’ve seen thus far, there is considerable variation possible in the pattern. In the crease pattern, we can choose • the number of columns, • the height of each row (which determines the height of the folded form), • the lateral shift of each row (which determines the rate of twist in the folded form). Several more examples are shown in Figure 2.87. In general for these tubes, there will be two stable states that have two different heights. If we seek to design such a structure, we would likely wish to choose the heights and then work backward to find the crease pattern that gives those particular heights. There are two limiting cases to contemplate. The tallest that the tube could possibly be is the height of the crease pattern itself, and that could only occur if there is no twisting at all; in this case, the facet outlined by mountain folds would simply be rectangles

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Figure 2.87. Three examples of triangulated cylinders, each with two stable states, one half the height of the other. Left: 5-fold rotational symmetry. Middle: 6-fold symmetry. Right: 8-fold symmetry.

crossed by an (unfolded) valley fold along the diagonal, as in Figure 2.88. The other limiting case is when one of the states is completely collapsed flat, as in the right subfigure of Figure 2.88. This is suggestive: the two states are not just distinct; they are very, very different, and that suggests application as a deployable structure, as well as for artistic effect. Although there are only two stable states—with rigid panels and undeformed creases—if these

Figure 2.88. A triangulated cylinder pattern of maximum height. Left: crease pattern. Middle: one folded state (maximum height, no twist). Right: the other stable state (fully flattened).

156


Figure 2.89. A triangulated cylinder pattern, sequentially collapsed.

structures are fabricated with somewhat pliable materials, such as paper or plastic, then they can be deformed from one state to the other. The deformation happens by introducing strained deformations into the system; the strains create stresses; the stresses push the structure toward one or the other stable state; and the net result is that the tube can “click” from one state to the next. Or, more precisely, the rows will click from one state to the next (usually, in an unpredictable order), as illustrated in Figure 2.89. Many origami artists have explored this twist-tube concept in their art. As the figures suggest, each row of the crease pattern is somewhat independent of its neighbors; you can give each row a different amount of twist, or even reverse the twist from one row to the next. Artist Tomoko Fuse, who has also extensively explored twists and has an entire book on the subject [33], devised an elegant and clever way to exploit this phenomenon; by alternating twist directions, one achieves a tube that, by twisting one direction and then the other, can expose and conceal alternate layers of the tube. By coloring the clockwise and counterclockwise sections separately, a striking color-change effect can be created, as shown in Figure 2.90. I encourage you to transfer the crease pattern to a large sheet, cut it out, glue the top and bottom edges together (using the tabs), and then try it out. A similar concept was also discovered independently by Vietnamese-American artist and engineer Uyen Nguyen, whose company has used such twists in the world of high fashion, notably in a series of small handbags, as shown in Figure 2.91. This pattern has also found application in deployable structures. Professor Stavros Georgakopoulos at Florida International

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Figure 2.90.

Top: crease pattern for the bidirectional tube. Bottom: two states.

Figure 2.91. A twist-tube-based handbag, designed by Uyen Nguyen. Photo by Bao-Khang Ngoc Nguyen.

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Figure 2.92. An origami quadrifilar helical antenna on Kapton substrate, based on the triangulated cylinder pattern. Photo courtesy of Stavros Georgakopoulos and Xueli Liu, Department of Electrical and Computer Engineering, Florida International University, Miami, FL.

University and his students have developed deployable microwave antennas using this family of patterns; an example is shown in Figure 2.92. You can make these tubes by simply picking values for the width-to-height ratio of a single parallelogram panel and the base angle of the parallelogram, then gluing the ends together. You will find that the tube naturally clicks into its stable positions and different tubes resist switching between the two states to varying degrees; some (like the square tube in Figure 2.89) are very resistant to collapse; others switch readily from one state to the other. Empirically, you will find that the closer the heights of the two stable states are to each other, the more readily the tube twists from one state to the other. In application, though, we would like to specify dimensional parameters such as the folded height: in particular, if we want the tube to collapse flat, we would like to specify one of the two design heights (and perhaps the other, if we want to control the stiffness against collapse). In order to do that, we need to carry out a parameterized analysis of a single level of the twist. This requires a bit more mathematics than we have needed up to this point, and it may be safely skipped if you wish. For a comprehensive mathematical analysis, see Guest [42, 43, 44].

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? ? ? 2.4.6. Triangulated Cylinder Geometry We will begin by considering a single level of the triangulated cylinder with the geometry shown in Figure 2.93. We assume m-fold rotational symmetry (i.e., m = 4 for a square, m = 5 for a pentagon, and so forth). There should be two stable folded states, which we will distinguish by indices i = 1, 2. The heights of the two states are hi , i = 1, 2, and we will denote the height of the crease pattern as h0 . We consider a single panel with corners p, q, ri , and si . The panel consists of two facets, joined by a crease from p to si . We assume that the bottom remains fixed, but the top two vertices ri and si are different between the two stable states. For simplicity, assume the edge from p to q is unit length. We also introduce the angle φ = π/m for convenience. Both the bottom and top polygons are regular m-gons, but the top is going to be twisted relative to the bottom by some angle. We define δφi as the angular twist of the top relative to the bottom. The problem is, then: given the rotational order m and the two desired folded form heights h1 and h2 , what are the dimensions of the parallelogram for that tube? Specifically, what are the base angle α, side length r, and altitude h0 of the parallelogram, as illustrated on the right in Figure 2.93? By setting up a three-dimensional coordinate system and solving for dimensions that give the same crease pattern for the two folded form heights h1 and h2 , we can find expressions for the parameters that define both crease pattern and folded form. We

dfi si ri

hi

r

Figure 2.93. Geometry of the triangulated cylinder for m = 5. Left: folded form. Right: crease pattern.

160

a q

p


f

h0 a 1

find that

α = cos

v t r=

h22 r

h0 =

© ª ® x2 (x2 − cot(φ)) ® r ®, ® x22 + 1 h22 x22 + 1 + x22 csc2 (φ) « ¬

−1

+

x22 csc2 (φ) x22

+1

(2.12)

,

2 h22 x22 + 1 + x23 cot(φ)(x2 cot(φ) + 2) + x22 x22 + 1

,

where the quantities x1 and x2 are given by q sin φ cot2 φ csc2 φ − (h12 − h22 )2 − cos φ x1 = 2 sin φ , (1 + (h12 − h22 )) + (1 − (h12 − h22 )) cos 2φ q sin φ cot2 φ csc2 φ − (h12 − h22 )2 − cos φ x2 = 2 sin φ . (1 − (h12 − h22 )) + (1 + (h12 − h22 )) cos 2φ

(2.13)

The twists of the top polygon relative to the bottom in the two states are given by δφ1 = 2 tan−1 x1, (2.14) δφ2 = 2 tan−1 x2 . Although these expressions are complex, there is some useful information in them. First, if the two height values h1 and h2 are chosen to be equal, then the two polygonal twist angles δφ1 and δφ2 are equal and the pattern becomes monostable—there is only a single folded state. Otherwise, it is bistable. The two different heights are the heights of the stable states; in between, the folding pattern must deform in some way, via stretching or buckling of the material. (How, precisely, it stretches or buckles depends very much upon the material from which it is made and the properties of the folds that act as hinges.) If we make the two heights differ, then the tube can switch between the two heights by twisting from one δφi value to the other, depending on the pliability of the material from which it is made. CHAPTER 2. PERIODICITY

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2f

ri

q

dfi

Figure 2.94. Top view of a twist tube just shy of the critical point where the layers start to collide in the middle.

si

p

If you make a set of tubes with varying height differences, you will find that the more different the two heights are, the harder it is to “click” the tube from one height to the other because the greater will be the required material deformation in the intermediate state. The height difference cannot be too great, though; because of the square root appearing in the definition of x1 and x2 , there is the potential for an imaginary solution, i.e., no real solution, if the term (cot2 φ csc2 φ − (h12 − h22 )2 ) goes sufficiently negative, which it does if the difference between h1 and h2 becomes too great. We can find the boundary of the solution set by setting the argument of the square root to 0, which happens at h12 − h22 = ± cot φ csc φ.

(2.15)

That sets an upper limit on the difference in the two design heights; if |h12 − h22 | > cot φ csc φ, then there is no solution for either height. There is another limitation on the range of possible parameters, set by the amount of twist. If we look straight down the tube from the top, as illustrated in Figure 2.94, we see that as the relative rotation from one layer to the next increases, the valley folds approach the center of the regular polygon. Eventually they touch, and so for rotation angles δφi larger than that critical value, the layers will intersect each other somewhere in the middle of the twist. Clearly from the figure, the valley folds of the parallelograms will collide when the total rotation angle from p to si is equal to π and the valley fold passes through the center. Thus, we must have (for a right-handed twist with δφi > 0), δφi < π − 2φ.

(2.16)

Since both values of δφi depend on h1 and h2 , Equation (2.16) is implicitly a limitation on the values of the two design heights.

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Figure 2.95. A twist tube that touches at the center. Top left: crease pattern. Top middle: first stable state. Top right: view down the (hollow) center of the tube. Bottom middle: second stable state. Bottom right: the edges touch in the middle of the tube.

It defines a critical value of the rotations δφi , δφi,crit = π − 2φ,

(2.17)

which, in turn, creates a critical value on the parameters xi , xi,crit = tan π2 − φ , (2.18) which, in turn, places another constraint on the relationship between the two heights. We find that |h12 − h22 | ≤ cot2 φ.

(2.19)

Comparing Equations (2.15) and (2.19), we see that the latter is always stricter, since the right side contains an extra factor cos φ. So this condition sets the actual limit on height difference between the two states. If the two heights satisfy Equation (2.19) at equality, we call this a critical design. When we choose heights at the critical value, both the crease pattern and folded form become interesting and distinctive. The example shown in Figure 2.88 was critical, as it turns out. Another example with sixfold rotational symmetry is shown in Figure 2.95. In this case, the critical configuration is not flat, but three-dimensional and quite solid. One could imagine using such a structure for its mechanical stability. Figures 2.88 and 2.95 share two interesting properties: (a) the parallelograms of the crease patterns are actually rectangles (i.e.,

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angle α is π/2), and (b) the second stable state is a polygonal tube with the valley folds at flat angles. This is more than coincidence: it is, in fact, the case for any combination of rotational symmetry and height parameters at criticality. Many artists and designers have made use of the triangulated concept already, but considering the number of things you can vary—rotational order, number of segments, parameters of each segment, twist directions, twist heights—there are undoubtedly many possibilities still to be discovered and explored. Artists like Nojima and Fuse have explored conical forms, which give self-similar spirals reminiscent of seashells. The Yoshimura pattern has two continuous degrees of freedom in its motion, but, as we have mentioned, joining the edges dramatically drops its theoretical flexibility to the two bistable forms. If we don’t join the edges, then we’re back to two degrees of freedom—at least, considering only the mechanics of the vertices. However, such mechanisms are further constrained by self-intersection avoidance: the paper can’t pass through itself. Self-intersection avoidance can be used to constrain a Yoshimuralike mechanism to a lower degree-of-freedom behavior. One of the most interesting and surprising cylindrical mechanisms is a model called “Spring Into Action,” designed by the late British artist Jeff Beynon. It has become iconic in the world of origami, and it is tremendously fun to fold and play with. With Jeff’s kind permission, I give folding instructions for it on pages 165–168. ? 2.4.7. Waterbomb Tessellation The patterns possible with semigeneralized Miura-oris are almost endless, but they do all share two properties: (1) they are strictly periodic in one direction at all folded states, and (2) they have a single degree of freedom in their folding motion (at least, if we don’t allow bending of the quadrilateral facets). We saw, though, that the Yoshimura pattern and its variants exhibit two degrees of freedom: they can move in two distinct ways (and mixtures thereof). This extra freedom arises from the degree6 vertices in the pattern. If we take a semigeneralized Miura-ori and, by eliminating edges in the pattern, allow some of the degree4 vertices to coalesce into degree-6 vertices, we might expect to pick up some of the additional flexibility of the Yoshimura pattern. And indeed, this is the case.

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Spring Into Action by Jeff Beynon 25/8 in

52 mm

57/8 in

158 mm

11 in

297 mm

1. Begin with a 15:8 rectangle. You can make this from a sheet of American letter paper (81/2 by 11 in) by cutting a strip 25/8 inch wide off of the long side.

2. Alternatively, you can make this from a sheet of A4 European letter paper (297 by 210 mm) by cutting a strip 52 mm wide off of the long side.

3. Fold the paper in half from side to side and unfold.

4. Fold the paper along a crease that runs between the upper left corner and the lower right corner, making a pinch about 1/3 of the way along the fold.

5. Fold the upper left corner down along a crease that runs from the lower left corner to the middle of the upper edge and unfold, pinching at the crossing.

6. The intersection of the two creases is 1/3 of the way down from the top and 1/3 of the way from the left edge to the right. Fold the bottom edge up to touch the crease intersection and unfold.

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7. Fold the top edge down to touch the crease you just made and unfold.

8. Fold the right edge over to the crease intersection and unfold.

9. Fold the left edge over to the crease you just made and unfold.

10. Fold the side edges in to touch the existing creases and unfold.

11. Divide each of the horizontal panels in half.

12. Divide each of the horizontal panels in half again.

13. Turn the paper over.

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14. Crease the upper left rectangle along the diagonal with the crease running from lower left to upper right.

15. Repeat on the remaining eleven panels in the column.

16. Repeat on the remaining 60 panels, reversing the direction of the diagonals in alternating columns.

17. Curl the paper into a tube and turn it over.

18. Twist the end counterclockwise using the existing creases.

19. Now twist the end clockwise, again using the existing creases. You'll find it very difficult to get all the creases going at once, but when they're all started, it will pop into place.

20. Without undoing the twists, bring the white layer out from inside the end.

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The middle disk slips inside itself here.

21. The twists should now form two separate disks on the end of the tube.

23. Twist the right end clockwise on the existing creases, this time making sure that the lower edge goes inside the upper edge.

22. Twist the right end one-half turn counterclockwise using the existing creases; make sure you tuck the upper edge inside the pocket as you do.

24. Repeat steps 22Ð23 on the remaining two segments.

26. Finished Spring into Action. Squeeze the center of the model to make the sides spring out. The model works best if you use somewhat heavy paper and store it compressed (for example, under a heavy book) before springing it.

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25. Squeeze the middle disk to activate Spring Into Action.

Figure 2.96. Evolution of the Waterbomb tessellation. Top row: a semigeneralized Miura-ori, which flexes with a single degree of freedom. Bottom row: reducing the distance between selected pairs of vertices. Left: crease pattern. Middle: partially folded form. Right: nearly flat-folded form.

One possible way of performing this coalescence is illustrated in Figure 2.96, in which the top row shows a particular semigeneralized Miura-ori and how it changes as we eliminate the shortest segments within the crease pattern. As long as those segments have nonzero length, the pattern has a single degree of freedom and is linearly periodic in the direction transverse to the minor folds—that is, it expands in a straight line perpendicular to the minor folds (horizontal creases) as it is flexed. Note from the middle image that in the intermediate state, it is curved along the minor fold direction, straightening out only as it approaches flat-folded. However, when we completely eliminate those edges, coalescing pairs of degree-4 vertices into degree-6 vertices, something almost magical happens: the pattern acquires a new type of motion, illustrated in Figure 2.97. Instead of being curved along the minor-fold direction and linear along the perpendicular direction, this motion is straight along the former and curved along the latter. This pattern can move in both ways—and, as well, in mixtures of the two.

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Figure 2.97. Second periodic symmetry of the Waterbomb tessellation. Top left: crease pattern. Remaining figures: evolution of the pattern from unfolded to nearly flat-folded.

This new pattern is, in fact, not that new and is known by several names, depending on the context in which is was discovered (or re-discovered, as the case may be). It was described in an origami context by the great Japanese master of geometric folding Shuzo Fujimoto in his 1976 masterwork Rittai Origami [30], but the pattern has, perhaps, achieved its greatest renown in a design by Yuri Shumakov, which we will shortly meet. One of this pattern’s names, and the one I prefer for its descriptive value, is the Waterbomb tessellation, because it is composed of arrays of square units, each of which is the traditional Waterbomb base. Alternate columns of Waterbomb bases are offset up or down by one-half unit. The Waterbomb tessellation exhibits two periodic modes of motion that can be explored as it is flexed. The first, shown in the

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Figure 2.98. Reversing motion of the Waterbomb tessellation, looking down the cylindrical axis of symmetry, as the pattern folds from near-unfolded (top left) to near-flat-folded (bottom right). The red line traces the rightmost corner of the initial pattern.

bottom row of Figure 2.96, is a linear motion—the pattern expands linearly in the direction transverse to the minor folds of the crease pattern. The second, shown in Figure 2.97, is cylindrical; the pattern is rotationally symmetric about a cylindrical axis that runs in the direction of the minor folds of the crease pattern. What is even more interesting is that this cylindrical motion is not uniform: it actually reverses direction over the course of the folding motion. This can be seen in the sequence of images in Figure 2.98, showing the motion from unfolded to flat-folded, looking “down the barrel” of the cylindrical axis. As you can see, with four rows to the pattern, it nearly closes on itself, and in fact, for five or more rows, the pattern does collide with itself during the course of the motion. This does not mean that such patterns are impossible to fold; only that they must be distorted into a non-periodic and/or bent form at some point during their construction and flexing. Because all the vertices are degree-6, the overall pattern actually has many, many degrees of freedom (we will learn more

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Figure 2.99.

Crease pattern for Yuri Shumakov’s “Magic Ball.” This is the Waterbomb tessellation (rotated 90◦ from Figure 2.97), with a touch of Miura-ori at the top and bottom.

about this in Chapter 7). A physically folded model will have a preferred state, however. The actual configuration that any real folded object takes comes from an equilibrium found by balancing the springiness of the folds and constraints upon the motion determined from the folding pattern. In the Waterbomb tessellation, that motion is what is called synclastic: if you bend the pattern into a curve without tightly constraining it, it curves transversely in the same direction as the original bend, forming a shape like the surface of a sphere. The Miura-ori, by contrast, is anticlastic; if you bend it in one direction (forcing the quadrilateral facets to bend), it also curves transversely, but in the opposite direction, like a saddle. The Waterbomb tessellation, like the Miura-ori, is another example of a mechanical metamaterial, in which the pattern of folds gives the overall surface, on average, mechanical properties that are very different from those of the unfolded material. The synclastic behavior of the Waterbomb tessellation gives rise to a lovely origami design, the “Magic Ball” of Yuri Shumakov [108], shown in Figures 2.99 and 2.100. In this design, a Waterbomb tessellation is formed into a tube and two opposite edges joined. The resulting springy surface is both beautiful to look at and oddly beguiling to play with. Yuri and his wife Katrin Shumakov have developed a wide range of variations of this concept, and online video instructions for some of them are readily

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Figure 2.100. Three configurations of Yuri Shumakov’s “Magic Ball.”

available. I give a crease pattern here, leaving the folding and assembly as an exercise for the reader. The “Magic Ball” is squishy: compressing it at its end makes it bulge out in the middle. This is an illustration of the synclastic behavior of the Waterbomb tessellation. The Shumakovs have developed an enormous variety of shapes based on this pattern [108], both single-sheet and modular forms and variations shaped like balloons, trees, and more. As just one example of the pattern’s versatility, Figure 2.101 shows a set of lampshades designed by them that make use of it.

Figure 2.101. Lampshades by Yuri and Katrin Shumakov, based on the Waterbomb tessellation. Image courtesy of Yuri and Katrin Shumakov.

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The concept of joining the ends of the Waterbomb tessellation into a tube was conceived even earlier, by Fujimoto [30], who showed that many periodic patterns can be similarly treated, by stretching into a cylindrical form. We will come back to this concept. Now, if we constrain an unjoined Waterbomb tessellation pattern to be periodic along the cylindrical axis (no localized bulging allowed) and only permit expansion perfectly transverse to the axis of rotation, as illustrated in Figure 2.97, then the resulting motion has only a single degree of freedom, and the angular fraction of a cylinder that the pattern subtends varies continuously with the motion. Hence, if you were to join the ends into a tube—and you would need at least five rows to do that—that joining would freeze the motion and the tube would be rigid, at least, theoretically. In practice, however, small deviations of the creases from their theoretical positions—which can occur naturally with softly rolled creases—allow a great deal more flexibility than a simple theoretical model would suggest. The “Magic Ball” can expand and contract cylindrically both with and without bulging by making use of such small distortions of the pattern. And this ability of the tube to expand and contract in diameter thereby makes it useful in the real world. One of the more interesting applications of the Waterbomb tessellation was developed by Oxford University professor Zhong You and his postdoc Kaori Kuribayashi-Shigetomi [131]. They developed an aortic stent based on this tessellation, shown in Figure 2.102. The stent is fabricated from shape-memory alloy that is compressed to a smaller size on the fold pattern, which allows it to be guided into place in the circulatory system. Once in the desired location, it is warmed via a catheter, and it expands out, holding the blood vessel open. ? 2.4.8. Troublewit and Pleats Different periodic patterns exhibit different flexural motions. The Huffman grid has a single degree of freedom, and its flexing motion is always cylindrical. The Waterbomb tessellation has two degrees of freedom and both of its flexing motions are purely cylindrical. The Miura-ori and its periodic generalizations, though, always exhibit straight-line motion transversely to the minor folds of the pattern—at least, if we force all of the facets to remain straight.

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Figure 2.102. You and Kuribayashi-Shigetomi’s aortic stent, based on the Waterbomb tessellation along helical lines. Image courtesy of Zhong You.

If we allow the quadrilateral facets of a semigeneralized Miuraori to flex along their diagonals—which is quite common if we’re working with paper, even fairly stiff paper—then such patterns can often be bent cylindrically in the direction transverse to the minor folds, adding a new level of diversity to the structures and mechanisms that can be formed with such patterns. This versatility has not gone unnoticed. During the Victorian period in England, a popular magic routine involved the manipulation of a pleated sheet of paper into a wide range of (usually) cylindrical forms. The routine is called “Troublewit.” It begins with a large sheet of paper, pleated first one way, then the other, as shown in Figure 2.103. This folded form is an example of a semigeneralized Miuraori. By shifting the angles of the various pleated segments and then stretching the pattern into a cylindrical form, it can be manipulated into a wide variety of surprisingly different shapes, a few of which are shown in Figure 2.104. With a bit of practice, the transformations can be made smoothly and quickly, and when worked into a story, provide an entertaining interlude as part of a magic routine. “Troublewit” is a classic routine [63] and is well known among magicians. Versions of the routine has been traced back as far as 1676 [85]. As we have seen, people were pleating

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Figure 2.103. The Troublewit. Left: crease pattern. Right: initial folded form.

Figure 2.104. Some of the Troublewit shapes. Top row: left to right, “Dumbbell,” “Vase,” and “Christmas Popper.” Bottom row: left to right, “Parasol,” “Hat,” and “Rosette.”

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Figure 2.105. Several of Shuzo Fujimoto’s rotationally symmetric pillars, from Rittai Origami.

napkins and paper back in the 1600s, so it is not surprising that manipulations of multiply pleated forms also have a long heritage. Also unsurprising is that the concept of stretching pleats into rotationally symmetric forms has been repeatedly rediscovered by many people in the course of folding paper. The students of Josef Albers (who we have already met) developed stretched pleated forms in their 1920s Bauhaus development, and stretched pleats have a long history in Japanese origami: not just in the simple paper fan, but also in representational folding. The technique was explored in purely geometric (non-representational) forms within the Japanese folding tradition by the great geometric folding artist Shuzo Fujimoto in his book Rittai Origami [30], in which he showed many examples of geometric shapes created via this technique. A few reconstructed examples are shown in Figure 2.105. Fujimoto describes more than 50 different designs in his book that show a wide variety of shapes and textures.

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Figure 2.106. Design of a rotationally-stretched goblet. Top left: a strip of paper folded into the cross section of the desired shape. Top right: the unfolded strip. Bottom left: the crease pattern of the strip, mirrored and then repeated. Bottom right: the resulting shape after bending it cylindrically and joining the ends.

Remarkably, one can construct patterns for this family of shapes using a very simple procedure that involves almost no mathematics at all. The symmetries in the crease patterns themselves suggest a method of construction. In Figure 2.105, each crease pattern consists of a vertical strip paired with its mirror image; these pairs are then replicated horizontally to make up the full crease pattern. If we were to cut and flat-fold a single one of these strips, we would find that the shape of the strip is approximately the cross section of one side of the cylindrical form, as shown in Figure 2.106. To design such a form, you take a strip of paper and flat-fold it into the cross section of the desired shape, then unfold it. The creases left in the paper provide the positions and angles of the

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creases needed for the folded form. You can use the unfolded strip as a template to replicate the pattern on a larger rectangle to create the desired folding pattern. This method also works to design a semigeneralized Miuraori without doing any angle/distance calculations. In fact, each of these rotationally stretched pleated forms is just a semigeneralized Miura-ori; any SGMO may be (in principle) stretched into a cylindrical form, at the cost of having some bent quadrilateral facets and some distortion of the form. Note that the shape taken by the flat-folded strip is an approximation of the cross section of the rotational form, but it is not exactly the same. This mirrors the situation with SGMOs that we saw earlier. When we stretch out a flat-folded form, the minor fold bend angles open up, as in Figure 2.72, which changes the shape. With SGMOs, it was possible to precisely calculate the amount of distortion needed in the flat-folded form to give a desired 3D cross section. With rotationally stretched pleats, the situation is more complicated, because the amount that any given fold is stretched varies with its radial distance from the axis of rotation and the number of repetitions in the pattern—and the bending of quadrilateral faces adds further complication. Nevertheless, as Figure 2.106 shows, the flat-folded shape is usually a pretty good approximation of the finished cross section, good enough to be used as the basis for design. You might have noted that the crease assignment in the folded strip is not the same as in the periodic pattern in Figure 2.106. There is obviously an ambiguity in assignment when forming the cross section, because for a given set of crease positions, every non-self-intersecting assignment will give the same cross section. But not every assignment will allow a non-self-intersecting 3D form, and, in fact, it is possible to create flat-folded strips that cannot be transformed into a non-self-intersecting SGMO or rotationally stretched pleated form. There is, though, a simple method to determine the proper assignment (if it exists) for the folds that cross the strips (i.e., the non-vertical creases in the crease patterns of Figure 2.106). If we label the two long edges of the strip A and B, as in Figure 2.107, the desired crease assignment for the cross-creases is the one for which on one side of the strip, edge A is never covered by the interior of a facet, and if you turn the strip over, edge B is never covered by the interior of a facet.

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A

B

A

B

B A

A

B

B

B A

A

Figure 2.107. Determining the crease pattern for a semigeneralized Miura-ori or rotationally-stretched pleat. Left: alternating mountain and valley folds won’t work because there are regions (indicated by the amber circles) where edges A and B are covered by the interior of facets. Right: changing a few of the folds gives a valid crease assignment for the strip, which can then be used to build a repeated pattern. Note that the A path (on the left) and B path (on the right) are both uncovered along their full length.

Why the distinction about “covered by the interior”? That’s because it is allowed for an A edge to be covered by the B edge or vice versa, if the two edges are collinear. As for why the edge is allowed to cover but the interior isn’t: this follows directly from the Justin Non-Crossing Conditions. For the vertical creases, what about their assignment once we’ve arrayed the strips into a rectangular crease pattern? There is a similarly simple rule for determining their assignment, but I will leave the discovery of that rule as an exercise for the reader. I’ll give a hint, though: if you use two-colored paper, as in Figure 2.108 to fold the strip, the crease assignment is related to the exposed colors of the folded strip. The path of the folded strip does not need to strictly follow the outline of the desired shape; by incorporating short “detours” perpendicular to the path, one can create additional folded edges that add texture and beauty to the folded shape, as in the two examples in Figure 2.109 by Israeli artist Ilan Garibi.

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A

B

B

A

B

A

B

A

B

A

B

A

B A

A

Figure 2.108. Arraying the folded strip. Left: folding the strip from two-colored paper gives a simple rule for the assignment of the vertical folds. Right: the arrayed crease pattern, composed of alternations of the strip and its mirror image.

Figure 2.109. Rotationally stretched forms by Ilan Garibi. Left: “Faberge Egg” (2011), flat egg configuration. Right: “Faberge Egg,” vase configuration.


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Figure 2.110. “Tavolini” (2013), by Ilan Garibi.

Garibi has applied folding techniques to many materials: not just paper, as shown here, but also wood, metal, and other materials. Figure 2.110 shows a rotationally-stretched form folded from laser-scored wood veneer laminate. Note that the vertices of this pattern includes both degree-8 and degree-5 vertices; the latter are clearly not flat-foldable, but many non-flat-foldable patterns can be used to create 3D surfaces. An even simpler construction method for rotationally stretched pleated forms was developed by British artist Paul Jackson and used in numerous works whose style is now inextricably associated with his name. Instead of flat-folding a strip and using it as a template, Jackson cross-pleats a rectangle with folds at 90◦ , then stretches the pleats individually to form the curved cross section. He has used this technique (along with dry pastels to accentuate the folds) to create a wide variety of beautiful forms, two of which are shown in Figure 2.111.

Figure 2.111.

Left: a cross-pleated pattern stretched into a curve. Photo originally published in [53]. Used by kind permission. Middle: “Brown Bowl,” by Paul Jackson (from the Organic Abstract series. Folded paper and dry pastel). Right: “Pod,” by Paul Jackson (from the Organic Abstract series. Folded paper and dry pastel). All photos courtesy of Paul Jackson.

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Figure 2.112. You can fold a triangle into a strip that defines the cross section created by a circular crease pattern.

Now, the technique of using a strip as a template works for other shapes of strips, not just rectangular ones. If you use a triangle instead of a rectangle, then when you join copies of the strip into an array, the resulting crease pattern will curve around onto itself as you add units. If you start with a triangle whose tip angle is an integral fraction of a half-circle, then you can build up a complete circular crease pattern that, when folded, will automatically stretch into a rotationally symmetric form, as illustrated in Figures 2.112 and 2.113. When joining rectangles, it is relatively easy to create the 3D form by flat-folding the entire pattern, then stretching it into shape. With a circle, though, it is not possible to flat-fold the pattern, at least not with the proper crease assignment. However, you can fold the pattern in half, flat-fold the double-layered half-circle, then unfold and reverse the fold direction of half of the creases to get the proper form.

Figure 2.113. Arraying the triangular strip as alternations of the strip and its mirror image gives a complete circular crease pattern and folded form.

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Figure 2.114. A circular rotationally pleated origami Bundt™ cake mold. Left: crease pattern. Middle: paper mold (with parchment liner). Right: baked cake.

David Morgan in the Industrial Design Department of Brigham Young University and his students have developed a particularly tasty application of this folding procedure: an origami version of a Bundt™ cake mold, shown in Figure 2.114. ? 2.4.9. Corrugations and More Tubes, triangulated cylinders, Troublewits, rotationally-stretched pleats, semigeneralized Miura-oris, and more: these are all examples of a genre of origami known as corrugations. While there is some discussion about just what, precisely, constitutes a corrugation, most of the folds called corrugations are geometric forms in which the majority of the creases are only partially folded, as opposed to flat-folded. The scope of corrugations is vast; the ones shown in this section are only a small sampling of the possibilities. Corrugations are often highly symmetric, exhibiting combinations of rotational symmetry and/or one- and two-dimensional periodicity, but they need not be symmetric. In fact, some of the most visually striking corrugations arise when an obvious periodicity is broken on a different length scale from the periodicity. This effect can be seen in the works of Japanese paper artist Yuko Nishimura, whose works, commonly from 100 × 100 cm squares, take the regularity of pleats but interrupt them by superimposing larger-scale, swooping boundaries between regions of different periodicity and/or orientation, as in Figure 2.115. The technique exemplified by “Troublewit” and rotationally stretched pleats, of flat-folding a shape repeatedly then stretching it into three-dimensionality, can be applied to much more complex

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Figure 2.115. Artwork by Yuko Nishimura. Top left: “Organic,” 2006. Top right: “Organic,” 2006. Bottom left: “Stir,” 2006. Bottom right: “Thick,” 2006.

paths and pleats than shown above. One of the modern masters of this technique is Ray Schamp, who has created many lovely corrugations based on both linear and rotational stretching. A few are shown in Figure 2.116. Rotational stretching poses two additional challenges: first, the pattern often cannot be fully flat-folded (though it can sometimes be flat-folded in sections to set the creases a few at a time). One way around this challenge, which has the side benefit of creating a more interesting 3D state, is to add slits to the paper. Schamp’s design “S-curve” in Figure 2.116 takes this approach.

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Figure 2.116. Artwork by Ray Schamp. Top left: “3rd Degree Corrguation,” 2012. Top middle: “Marble Wave,” 2007. Top right: “Equidistant Weave,” 2010. Bottom left: “Around and Between,” 2007. Bottom middle: “S-Curve,” 2007. Bottom right: “Figure 8,” 2007.

A second challenge is that, even if a rotationally stretched form is flat-foldable in sections, the full pattern may not be mathematically self-consistent in any state other than fully flat and unfolded. However, once the paper has been given “memory” of the folds by forming a crease, the balance between the strains of folded creases and small distortions throughout the fold can allow the form to still take on a three-dimensional form reminiscent of a bas-relief sculpture, as in Figure 2.117. The partially folded creases in corrugations make the resulting surfaces visually interesting, usually much more so than when they are collapsed into the flat-folded state. At least, that’s the theory. In the real world, though, real paper has thickness and springiness, which can bring life and form to ostensibly flat-folded patterns. The three-dimensionality of “flat-folded” patterns was displayed by Paul Jackson in a work he titled “Bulge,” formed by alternating flat-folded pleats; the residual springiness of the paper popped it into an elegant curved organic form. This concept was taken up by Croatian-American artist Goran Konjevod, who developed the genre into a wide variety of three-dimensional shapes, several of which are shown in Figure 2.118.

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Figure 2.117. “Nine Centers” (2007) by Ray Schamp.

Figure 2.118. Cross-pleated forms by Goran Konjevod. Top left: “Bowl:32, fancy” (2007). Top right: “Wave:32” (2008). Bottom left: “Bowl:32 locked” (2006). Bottom right: “64-grid pureland improvisation” (2006).


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In these works, the theoretical model says it should be “flat and uninteresting,” but the non-idealities of paper—finite thickness and springiness—actually give rise to beautiful and unexpected structures. This is a reminder that we must always be aware of the limitations of our theoretical models! The Miura-ori pattern is flat-foldable, but a simple modification of it gives a non-flat-foldable variation that has several desirable mechanical properties, notably, a “hard stop” that prevents it from collapsing to flatness. Such patterns have been investigated by Yves Klett and his colleagues at the Institut für Flugzeugbau (Institute of Aircraft Design) at the University of Stuttgart as structural elements within sandwich panels, except instead of using standard honeycomb cores, he and his colleagues are using Miura-oris and their kin. In order to fold these materials in high volume from high-performance materials—paper, resin-impregnated textiles, carbon fiber, and more—they have developed automated machines for folding these patterns: a modern update on the machine described in Henry Hochfeld’s patent (Figure 2.37). These patterns can be incredibly strong: Figure 2.119 shows one such modified Miura-ori pattern supporting the weight of a car. The notion of stretching pleated patterns into curved forms is not restricted to abstract geometric shapes; several artists have incorporated such patterns into representational designs. Two particularly beautiful such examples are Jun Maekawa’s “Peacock,” introduced in Kunihiko Kasahara’s landmark 1983 book on Maekawa, Viva Origami [58], and a more recent example, a lovely “Butterfly” by the young Russian artist Andrey Ermakov, both shown in Figure 2.120. Because they are three-dimensional, corrugations are challenging to design and analyze mathematically, requiring mathematical techniques that we will explore in later chapters. Flatfoldable origami designs are often considerably easier to develop mathematically (though they, too, can call for sophisticated mathematics). It might seem that restricting consideration to flat-foldable origami patterns would limit designs to relatively simple folded patterns, but this is not the case; flat-folded geometric patterns offer remarkable complexity and beauty. In the next few chapters, we will explore another vast genre, this time of flat-folded forms: that of twist-fold-based tessellations.

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Figure 2.119.

Top left: a Miura-ori folded from carbon fiber textile. Top midde: a Miura-ori folded from vellum. Top right: a modified Miura-ori folded from aluminum. Bottom left: a machine-folded modified Miura-ori folded from resin-impregnated aramid fiber. Bottom middle: use of the modified Miura-ori as the core of a structural panel. Bottom right: the modified Miura-ori can support the weight of a car.

Figure 2.120.

Left: “Peacock” (2010) by Jun Maekawa, incorporating Miura-ori for the tail. Based on a 2000 revision of the original ca. 1980–1983 design. Right: “Butterfly” (ca. 2009) by Andrey Ermakov, incorporating Miura-ori to pattern the wings.

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?

2.5. Terms Anticlastic A pattern that when bent in a curve along one direction curves in the opposite direction, forming a saddle shape. Aperiodic A pattern that is not periodic. Basis vectors Two vectors that describe both translation distances and directions for a doubly periodic pattern. Bend angle The 3D angular change between the two minors folds in a folded Miura-ori. Bistable A folded pattern that has two unstrained folded states but that cannot switch between them without undergoing some form of strain and/or distortion. Chicken wire pattern A version of the Huffman grid constructed from a mirror-symmetric bird’s-foot vertex. Corrugation An origami pattern, usually geometric, in which the majority of the creases are partially (not flat-) folded. Crimp A pair of opposite-parity creases roughly perpendicular to a fold; a combination of two pleats. Crossing embedding A choice of vertex positions for a crease pattern (or any plane graph) that allows edges to cross each other at points other than the defined vertices. Doubly periodic A pattern that is translationally periodic in two different directions. Generating line A line used to define a periodic pattern, such as the Mars-type crease pattern or semi-generalized Miura-ori. Generating vertex A vertex that can be replicated into a periodic pattern, such as the Huffman grid or Yoshimura pattern. Huffman grid A 2D periodic grid composed of a single type of degree-4 vertex. Kresling pattern A periodic pattern of identical triangles and degree-6 vertices around a closed cylinder. See also triangulated cylinder.

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Major fold (Miura-ori) In a Miura-ori, the two folds at each vertex that are opposite one another and have the same crease assignment. Mechanical metamaterials Fine-grained mechanisms that give a bulk mechanical behavior that is different from that of the constituent materials, such as a negative Poisson’s ratio. Minor fold (Miura-ori) In a Miura-ori, the two folds at each vertex that are opposite one another and have the opposite crease assignment. Miura-ori A fold pattern described by Koryo Miura consisting of a doubly periodic array of parallograms and their mirror images. Monostable A folded pattern that has only a single unstrained folded state. Osculating angle The angle between the two major folds at a vertex of a folded Miura-ori. Period The distance that a periodic pattern can be translated that leaves it unchanged. Periodic A pattern that can be translated some distance that leaves it unchanged. Pleat A mountain and valley fold next to each other, roughly (or exactly) parallel. Poisson’s ratio The amount by which a material or mechanism shrinks in one direction when it is stretched in a perpendicular direction. Predistortion Designing the flat-folded path of a semigeneralized Miura-ori with sharper angles than the desired trajectory so that when it is partially folded, it takes on a desired trajectory. Rigid foldability A property of a crease pattern that can fold with all flexing happening along creases; the facets remain flat and vertices and creases do not move within the paper. Rigidly foldable An origami crease pattern is rigidly foldable if it can be continuously transformed between two different CHAPTER 2. PERIODICITY

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states (e.g., unfolded to flat-folded) without bending or buckling of the facets or movement of the vertices and creases within the paper. Rotational symmetry A property of an object that is unchanged after rotating it through some nonzero angle. Ruling angle The angle between the two minor folds at a vertex of a folded Miura-ori. Semigeneralized Miura-ori A crease pattern similar to the Miuraori that is periodic in one direction but not necessarily periodic in the other. Shallow-angle divot A pattern within a semigeneralized Miura-ori that allows small-angle bends. Symmetry A property of an object that it is unchanged after applying some non-trivial transformation. Synclastic A pattern that when bent in a curve along one direction curves in the same direction along the opposite direction, forming a spherical shape. Tile A patch of crease pattern that can be joined with other tiles to create a complete and valid crease pattern. Tile line A border of a tile along which it can be joined with other tiles. Translational symmetry A property of an object that is unchanged after translating it some nonzero distance. Triangulated cylinder A periodic pattern of identical triangles and degree-6 vertices around a closed cylinder. See also Kresling pattern. Vector A combination of a length and direction that can describe a direction of periodicity of a pattern. Waterbomb tessellation A crease pattern consisting of an array of Waterbomb base patterns with alternate rows offset from each other. Yoshimura pattern A periodic pattern of identical triangles and degree-6 vertices. 192


3

Simple Twists ?

3.1. Twist-Based Tessellations

In the previous chapter, we put vertices together into periodic and near-periodic arrays, which allowed the creation of quite beautiful patterns. In these patterns, the repeating units were relatively simple, one or two vertices per repeating unit. It’s time now to start putting vertices together into more complex assemblages that give rise to interesting folded forms. As we’ve seen, once we start putting together multiple vertices, new rules arise from the relationships between adjacent vertices, and, more interestingly, rules arise from the potential of long-range interactions between the facets of paper between the creases. The structures we will look at are origami tessellations: dissections of the plane into interesting patterns where the borders are formed by folded edges and/or variations in the numbers of layers. While they have roots extending back decades, the field of origami tessellations has exploded since the turn of the 21st century, and we will devote the next several chapters of this book to their study: both the underlying mathematics and tools for their construction. Figure 3.1 illustrates several origami tessellations by one of the pioneers of the genre: Yoshihide Momotani (b. 1928). Momotani with his wife Sumiko wrote an enormous number of books of origami representational designs for which they are best known, but throughout the 20th century, Yoshihide developed a wide variety of geometric patterns, a few of which are shown in the figure. Another mid-20th-century Japanese origami artist, Shuzo Fujimoto (who we have already met) also developed a wide variety of periodic patterns of the form we now call tessellations, many of 193

Figure 3.1. A collection of Yoshihide Momotani’s origami tessellations ca. 2009.

which he published in his Neijiri Origami (Twist Origami) series [31, 32, 29]. His work inspired American artist Chris K. Palmer in the mid to late 1990s to explore the genre, and Palmer’s work led to an explosion of interest worldwide. The terms “corrugation” and “tessellation” are both used to describe geometric (usually non-representational) origami that takes the form of a surface that is subdivided by folds. Corrugations usually have some surface relief—the folds are mostly partially folded, rather than flat-folded—while tessellations are, more often than not, flat-folded. But this difference is not absolute; there are tessellations that include 3D elements, and some people (myself included) consider corrugations to be a subset of the broader category of tessellation. Many flat-folded tessellations display several common features. They are often composed of a grid of creases in a regular array, and the folded form is similarly regular and periodic—that is, it repeats in two different directions. And they often contain repeated groupings of folds, which can be thought of as units, or building blocks. One such building block can be seen in various guises in many origami tessellations; several examples are shown in Figure 3.2. This pattern is one of the fundamental building blocks of origami tessellations. It is called a simple flat twist. Simple flat twists are ubiquitous within the world of origami tessellations, and they make an excellent introduction to the mathematics of flat-foldable multi-vertex origami. They are simple enough to be easily folded, yet complex enough to display un-

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........CHAPTER 3. SIMPLE TWISTS

Figure 3.2. Details from several origami tessellations. Top left: “Paper Spirals,” by Chris K. Palmer. Top right: “Unknown #8,” by Alex Bateman. Bottom left: “37 Hyperbolic Limit,” by the author. Bottom right: “Square Dance,” by Ralf Konrad (from a design by Alex Bateman).

usual and surprising behaviors and to call upon all of the laws we identified in the previous chapter that govern flat-foldability. In this chapter, we will focus on this simple structure and examine its properties. ?

3.2. Folding a Twist The best way to start building an appreciation for flat twists is to fold one and play with it, so let’s do that right now. Take a square of paper and follow the folding sequence for a square twist shown in Figure 3.3 on pages 196–197. This pattern is a simple flat twist—perhaps not the simplest, but one of the easiest to fold. The overall pattern of lines looks twisted, as if you could somehow reach into the paper, grab a central square, and give it a turn. In fact, it is twisted. Take the square twist you just folded and draw the letters A–D in the four corners and E in the center. Then carefully unfold the paper by grabbing two opposite edges and pulling them apart until you can flatten the crease pattern as

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CHAPTER 3. SIMPLE TWISTS

195

1. Begin with a square, white side up. Fold in half and unfold both vertically and horizontally. Then turn the paper over.

2. Fold the lower left corner up so that the crease connects the upper left corner to the middle of the bottom edge; press firmly along its middle third and unfold.

4. Mountain-fold the top edge behind.

5. Fold the right side behind.

7. Fold the top right corner down to lie along the fold you just made.

8. Fold the right side to the left so that the bottom edges are aligned and unfold.

Figure 3.3. Folding sequence for a square twist.

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3. Fold the lower left corner up to touch the line you just made so that the new fold hits the bottom right corner; pinch along the left edge and unfold.

6. Fold the top down through all layers so that the left edges are aligned and unfold.

9. Unfold to step 5.

10. Spread the near and far layers a bit and dent the shaded region downward, forming a valley fold across the center of the resulting diamond-shaped region.

12. Fold the upper right layer down.

11. Swing the vertical flap in front to the left and the one behind to the right. Flatten firmly.

13. Mountain-fold the far layer upward as far as possible.

14. The finished square twist.

Figure 3.3 (Continued). Folding sequence for a square twist.

shown in Figure 3.4. If you have kept the four corners in the same orientation as before, you will see that the orientation of the letter E in the center has changed: it is rotated by 90◦ . The hallmark of an origami twist is that as the structure is folded, some part of the paper maintains its original orientation while a localized region undergoes a rotation. There are many types of twist in origami: some are flat, some are 3D. They can be simple, consisting of only a few folds, or extremely complex,

A A

B

Figure 3.4.

E

E D

B

C D

C

Unfolding this square twist reveals that the central square is rotated by 90◦ in the folding process.

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and they can even occur in arrays. What is common to them all is this localized rotation, which, when we see it happen, is always surprising and often quite beautiful. Squares are not the only shape that can be folded into a twist; hexagons, too, have twists, as the next folding sequence shows. Figure 3.5 shows how to cut the largest regular hexagon from a square. Figure 3.6 gives the folding sequence for the twist. Once you have a hexagon, follow the folding sequence in Figure 3.6 to construct a hexagonal twist. This, too, has the center twisted relative to the outside edges in the folded form. If you label the inside and outside of the twist as you did with the square twist, you will see that in moving from the folded form to the crease pattern, the central hexagon rotates—this time, through an angle of 120◦ . Figure 3.7 shows this rotation; here I have used an arrow in the center, rather than letter, to make the rotation angle clearer. Twists come in multiple forms, too; by changing the directions of some of the folds you can rearrange the layers of a twist and dramatically change its appearance. Try this with the square twist you folded back in Figure 3.3, following the procedure shown in Figure 3.8. This form of the square twist has an additional property: it opens and closes very easily. In fact, if you grasp the sides and then pull them apart, you will see that the pattern opens and closes smoothly and can do so without bending any of the facets—a property with the name rigid foldability. A mechanism with this property is said to be rigidly foldable. (We briefly touched on this property in Chapter 2; we will explore it in more detail in Chapter 7.) Rigid foldability is a relatively uncommon property in origami structures, and indeed a special condition applies to this twist. A different rearrangement of the layers can give yet a different appearance. If you arrange the layers as shown in Figure 3.9, you get a pattern that is not rigidly foldable but has another unusual property. The first two arrangements look very different on the front and back, but this pattern looks the same on the back side, as Figure 3.9 shows, albeit reflected about a 45◦ axis and with the colors reversed. This is a symmetry, and one that extends to the crease pattern as well, which exhibits the same property, known as iso-area symmetry. This, too, is a special property that we will encounter again as we proceed.

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1. WeÕll start by folding a hexagon from a square. Begin with a square, white side up. Fold in half along the diagonals and unfold.

4. Fold the top corner down to this point, making a new mark along the right edge.

7. Fold the lower right corner up to lie along this fold.

2. Fold the top down to the bottom, making a short crease along the right side.

5. Fold the paper in half along the diagonal.

8. Cut from corner to corner.

9. Unfold completely.

3. Fold the bottom right corner up to this mark, making the new fold hit the bottom left corner. Pinch the crease along the right edge and unfold.

6. Fold the upper left corner down to hit the edge at the mark you made in step 4.

10. A regular hexagon.

Figure 3.5. How to cut the largest regular hexagon from a square.


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1. Begin with a regular hexagon, oriented so that the sides are vertical. Fold in half vertically and unfold.

2. Repeat in the other two directions. Then turn the paper over.

3. Fold the indicated corner down to the middle of the adjacent edge; make the crease sharp only to the crease shown, then unfold.

5. Repeat steps 3Ð4 on the next side, and continue, all the way around, for a total of 6 folds.

6. Turn the creases around the central hexagon into mountain folds by pinching them one at a time and unfolding.

1/6

4. Rotate the paper 1/6 turn.

B A

B

7. Mountain-fold the creases radiating from the center of the hexagon and dent the center along the indicated valley fold; then shift corners A and B toward, and then past, each other.

8. Fold one group of layers down in front and the far group of layers up behind.

Figure 3.6. Folding sequence for a hexagonal twist.

200

A


9. The finished hexagonal twist.

A

B

A

F

B

E

C

C

F

D E

D

Figure 3.7. Unfolding this hexagonal twist reveals that the central hexagon is rotated by 120◦ in the folding process.

All simple flat twists have a few features in common: there is a central polygon, and there are pairs of parallel folds radiating outward from each side of the central polygon (see the crease patterns in Figures 3.4 and 3.7). The parallel folds that radiate outward from a twist (and the paper between and to either side of them) are called pleats. One can create arrays of twists in which the pleats of one twist connect up with the pleats of its neighbors, creating a much larger pattern. The rearranged twist of Figure 3.9 lends itself to an array with a fun folding sequence and a surprising result, the “Wall” of Yoshihide Momotani. This figure was one of the first recognizable origami tessellations—though not the first 2D periodic origami—and it remains as fun to fold today as when it was presented by Momotani over 30 years ago [89]. If you have not folded the simple twists yet in this section, I would urge you,

1. Begin with the square twist. Bring the upper left layer in front of the central square by changing fold directions, but donÕt make any new creases.

2. Do the same thing with the lower left layer, bringing it in front of the central diamond.

3. The rearranged twist.

Figure 3.8. Rearranging the layers of a twist by changing its folds can dramatically change its appearance.

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1. Begin with the square twist. Bring the upper left layer in front of the central square by changing fold directions, but donÕt make any new creases.

2. Do the same thing with the lower right layer, bringing it in front of the central diamond.

3. The rearranged twist. Turn the paper over.

4. Same thing, but color-changed and reflected.

Figure 3.9. Rearranging a different pair of layers results in a shape that has a similar appearance on the front and back.

before going any farther in this book, to take a sheet of paper and fold this most enjoyable origami design, following the instructions on pages 203–205. Momotani’s “Wall” is a fascinating design that demonstrates many ideas that have become widespread in origami. It is composed of an array of simple flat twists in which the pleats of one twist connect to the pleats of its neighbors. It is an example of a flagstone tessellation, one in which the visible facets butt up against one another rather than overlapping. It exhibits the isoarea property, in that its crease pattern remains unchanged upon mountain-valley inversion plus another symmetry operation (reflection, in this case); and, if you fold one from stiff paper, you will find that it is rigidly foldable–that is, if you tug on two opposite sides, the pattern will open and close while all of the facets remain flat and undistorted. Quite impressive for such a simple fold! It is also easily expanded, and I encourage you to try to fold one from a larger sheet, creasing in a 32 × 32 grid rather than the 16 × 16 shown in the preceding instructions. ? 3.2.1. Diagrams versus Crease Patterns Momotani’s “Wall” is unusual in another regard: it has a relatively simple step-by-step folding sequence. This is, in fact, quite unusual in the world of tessellations, and in the world of mathematical

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MomotaniÕs Wall by Yoshihide Momotani

1. Begin with a square, white side up. Fold the bottom edge up to the top and unfold, dividing the paper in half.

2. Divide the top and bottom panels in half, which divides the paper into quarters.

3. Divide the paper into eighths.

1/4

4. Turn the paper over from side to side.

5. Divide the eight panels in half as shown.

7. Repeat steps 1Ð4, this time beginning from the colored side (so youÕll end with the white side visible).

8. Pleat the paper using the existing creases.

6. Rotate 1/4 turn.


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A

10. Pleat the same way you did in step 8, but horizontally.

A

12. Lift the layer above edge A and swing edge A upward, pulling out some loose paper.

A

13. Press the layers flat.


204


14. Repeat steps 12Ð13 with three more edges along the left.

17. Fold the indicated edge to the left, carefully disentangling the layers at its top and bottom.


15. Repeat steps 12Ð13 in eight more places.

18. Press the layers flat.

19. Repeat steps 17Ð18 on the lower edge.

20. Repeat steps 17Ð18 on six more edges.

21. Finished MomotaniÕs Wall, front and behind.

origami in general. People who have come to mathematical folding from the world of origami have come to expect step-by-step diagrams: after all, for over 50 years, since the introduction by Yoshizawa of the modern diagramming notation, every origami book has provided such step-by-step instructions, with the only change over time being the number of steps (which has grown steadily). But a step-by-step folding sequence presupposes that the final design can be somehow broken down into individual clusters of folds, with each cluster consisting of no more than a handful of creases. This is rarely the case with tessellations and mathematical folds, in which, more often than not, the only way to create the shape is to crease each and every fold individually and then bring them all together. In mechanical terms, the folds typically form CHAPTER 3. SIMPLE TWISTS

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a strongly coupled system: no one fold, or set of folds, can be isolated from the others. And so, in most cases, the only way to create a tessellation pattern is to precrease and then bring all of the folds together at once—a step that is now commonly called the collapse. If you have folded the test examples so far in this chapter, you have encountered a collapse. Step 7 of the hexagonal twist is, in fact, a collapse, in that a large number of folds must be made simultaneously. In that example, we didn’t have to make all of the folds of the collapse in a single step—a few of them were left until step 8. But it will be relatively common in origami tessellations that, in fact, all folds must come together at once. Another way in which tessellations and mathematical folds differ from their traditional origami counterparts is that the folds often cannot be constructed via folding alone—at least, not without covering the paper with extraneous creases. (Although we will see that one style of tessellations, grid tessellations, offers constructionby-folding). The two twists we folded and Momotani’s “Wall” are constructible, but in many cases, the crease patterns are mathematically computed, and the only practical method of getting the creases in the right place is by measuring and marking. Or photocopying, as the case may be. Once one has drawn, computed, or printed the pattern, it can be mechanically transferred to the folding paper, by photocopying or other instrument.1 Throughout this book, I will show crease patterns for structures, twists, tessellations, and other patterns. I would encourage you to, at the very least, photocopy them (and possibly blow them up to larger size), then try folding along the lines as indicated to develop a feel for how tessellations flex and go together. ? 3.2.2. A Square Twist Tessellation An example of one of these crease patterns is shown in Figure 3.10 with its folded form in Figure 3.11. This, like Momotani’s “Wall,” is a pattern formed from square twists—many identical twists, as it turns out. I encourage you to photocopy, cut out, and fold this pattern. This is a tessellation composed of the rigidly foldable square twist shown earlier, and if you fold this pattern from stiff material, you will find that the entire pattern opens and closes in a single mo1

206

Both sign cutters and laser cutters have been used for this purpose.


Figure 3.10. Crease pattern for a square twist tessellation.

tion. However, the different facets of the pattern move in different ways as the pattern is flexed. The large squares move toward and away from each other (translation). The parallelograms flip over and back (reflection). The small squares are perhaps the most interesting; as you collapse and expand the crease pattern, each of the smallest squares undergoes a 90◦ rotation, as we have already seen in the square twist.

Figure 3.11.

Left: folded form when folded from opaque paper. Right: folded form when folded from translucent paper.


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At this point, we have now sampled all of the basic properties needed to construct a wide variety of tessellations. There are simple patterns of folds, called twists, that come in at least square and hexagonal versions (and more). These twists can be assembled by connecting the pleats of one to the pleats of its neighbors. If it is possible to make these connections in a two-dimensional array, we can create a larger pattern of twists. Within this simple definition, there are many opportunities for variation: we can use different polygons, crease assignments, pleat widths, angles, and patterns of connection. Over the next chapter or few, we will explore all of these variations. They all come down to different arrangements of twists, though, so before we go into how we put twists together, let us explore the properties of individual twists in all their manifold splendor. ?

3.3. Elements of a Twist Let us start by developing a classification scheme for simple flat twists (SFTs)—that is, defining what, exactly, makes a pattern of creases a simple flat twist, as opposed to something else. We will take the following as our definition: A simple flat twist is a flat-foldable crease pattern consisting of a closed polygon surrounded by parallel pairs of creases that, when folded, rotate the polygon through some angle in the folded form. Each parallel pair of creases emanates from the endpoints of its corresponding side of the central polygon. The simple flat twist is the first, and easiest, example of a twist fold. We will see that there are many, many more complex twist folds—and ones that don’t necessarily fold flat. But because it’s the simplest, the SFT is a good starting point for a study of twists. And, also as we will see, twists are basic building blocks of many tessellation patterns. We will also identify the parts of a simple flat twist for purposes of future discussion, shown in Figure 3.12. They are the following: • Central polygon. This is the heart of the twist. • Pleats. Each pleat consists of two parallel creases that emanate from the endpoints of a single side of the central polygon.

208


central polygon pleats wedges

¥

Figure 3.12. Anatomy of a simple flat twist. The parts are the central polygon, the pleats, and the wedges.

• Wedges. Each wedge consists of two angled creases that emanate from a single endpoint of the central polygon. The creases that make up the pleats are the pleat creases. Of course, they make up the wedges as well. In a single isolated twist, the pleats continue until they run off the edges of the paper, but in complex tessellations made up of twists, most of the pleats connect to other twists. When we are considering a single pleat, the shape of the paper is relatively unimportant; in fact, it is helpful in some ways to think of an isolated twist as existing in the middle of an infinite (or at least arbitrarily large) sheet of paper. As we will see, there are some limitations on this assumption: there are configurations impossible for infinite-paper-twists that are, in fact, foldable for finite-paper twists. We’ll get to those later. We can construct a twist by a very simple procedure, illustrated in Figure 3.13: 1. Draw a convex polygon. 2. Pick an angle α. We will call this angle the twist angle for this twist. 3. Pick any side. Draw two parallel lines from the endpoints of that side making angles α and (180◦ − α) with this line segment.

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a

a 180-a

a

a

a a

1. Begin with a polygon.

2. Add two parallel lines to one side.

3. Add parallel lines to the remaining sides.

4. Assign mountain and valley folds.

Figure 3.13. Construction of a simple flat twist.

4. Repeat on the other sides. 5. Assign creases (M or V) to the central polygon and the pleats. And then you fold. The appearance of a twist depends on several things: how you assign the creases, for example, and whether or not the paper is opaque or translucent. With translucent paper, as can be seen in Figure 3.14, the appearance can be quite complex with the varying thicknesses of the layers. There is also interesting structure around the edge of the folded form; this, of course, depends on the particular shape chosen for the twist.

Figure 3.14.

Left: crease pattern of a twist. Middle: folded form from opaque paper (not to the same scale). Right: folded form from translucent paper, revealing the layers underneath.

210


Figure 3.15.

Left: crease pattern of a twist embedded in an arbitrarily large sheet. Right: folded form from translucent paper.

The border of the paper plays a role in the appearance of a twist, of course. As we will shortly see, it even plays a role in determining whether the twist is flat-foldable for a given crease assignment. In order to separate what happens at the border from what happens at the central polygon—which is where I would like to focus initially—I will, for much of this chapter, draw twists as if they were embedded within an arbitrarily large sheet of paper, and then show only a circular region around the central twisted polygon, as in Figure 3.15. Now, in the construction algorithm illustrated in Figure 3.13, I have specified that one must use the same pleat angle for every side of the polygon. As we will see, this requirement follows directly from the requirement of flat-foldabilty. But I have not stipulated what type of polygon is in the center. It will turn out that we can use any convex polygon. And I have not said anything about the crease assignment—which is a more serious matter: not all assignments lead to a flat-folded form. The analysis of which pleat angles and which assignments are allowed is considerably simplified if we consider only regular polygons for the central polygon, and so we will restrict our attentions to regular SFTs to start. ? ??

3.4. Regular Polygonal Twists The first step in understanding what twists are possible is to develop a classification scheme for twists. Simple flat twists can be classified by properties of the central polygon, the angle of the surrounding pleats, and the assignment (mountain or valley) of the creases around the central polygon.

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? 3.4.1. Cyclic Regular Twists The most highly symmetric twists have a regular polygon in the middle surrounded by angled pleats, one emanating from each side of the polygon, and we call these regular SFTs. Regular polygons have the property that all sides and all corners are the same: if you rotate a regular n-gon by 1/nth of a turn, it remains the same. The crease lines of a regular twist have the same property; if you rotate a regular n-gonal SFT by 1/nth of a turn, the positions of all of the crease lines, pleats included, remain the same. However, we have the option of assigning creases to be either M or V, and the crease assignment doesn’t necessarily have the same rotational symmetry. The most highly symmetric twists are rotationally symmetric with respect to both the crease line positions and their crease assignments. We call these patterns cyclic twists.2 Because of the rotational symmetry, a cyclic twist must have the central polygon entirely surrounded by creases of one type, either M or V. Thus, there are exactly two types of cyclic SFTs for each regular polygon. Figure 3.16 shows twists for an equilateral triangle, square, pentagon, and hexagon, all with the same angle between the pleats and the side from which the pleats emanate and with the central polygon outlined by mountain folds. Once the assignments for the creases of the central polygon are chosen, the crease assignment for the two creases of each pleat can be fully defined. The base of each pleat contains an acute and an obtuse angle. We’ll assign the crease adjacent to the obtuse angle the same fold direction as the adjacent side of the polygon and give the other pleat crease the opposite assignment. This has the effect that the acute angle will always have an anto crease assignment, ensuring that if this acute angle is the smallest angle at the vertex, it satisfies the BLBA Theorem for flat-foldability. (And if it isn’t the smallest angle? Well, we’ll see.) As Figure 3.16 shows, twists can vary according to the number of sides of the central polygon. They can also vary by the angle formed between each side and the adjacent pleat, as shown in Figure 3.17. For an arbitrary simple flat twist, the twist angle can vary from zero degrees up to a maximum value that is determined by the 2

Why the name “cyclic”? It’s because there is a cycle in the order relationship among their pleats, e.g., a lies on top of b, b lies on top of c, and so forth, around to something lying back on top of a.

212


Figure 3.16. Regular polygonal simple flat twists. Top row: crease patterns. Bottom row: folded forms from translucent paper. Left to right: triangle, square, pentagon, and hexagon.

Figure 3.17. Three versions of a regular hexagonal twist crease pattern. Top row: crease patterns. Bottom row: their respective folded forms. Left to right: twist angles of 30◦ , 45◦ , and 60◦ .


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213

geometric properties of the central polygon. For a regular SFT, the maximum angle is a function of the number of sides of the polygon and the specifics of the crease assignment. But for cyclic twists, the maximum pleat angle depends only on the number of sides of the polygon, and the limit comes from considerations of the folded form. ? 3.4.2. Open- and Closed-Back Twists The maximum twist angle is reached when the layers of paper begin to interfere with one another. Figure 3.17 shows the progress of a hexagonal twist as the twist angle is increased. As the twist angle α increases, the edges of the pleats move toward each other and toward the center of the polygon. When they meet in the center of the polygon, the twist angle can no longer be increased without the layers of the pleats intersecting one another, which is forbidden. The limiting pattern, in which the pleats emanating from each side meet at a single point in the folded state, is called a closed-back twist. When there is a gap between the edges, as in the first two examples in Figure 3.17, it is an open-back twist. The closed-back twist is the limiting case for a flat-foldable cyclic twist as the twist angle is increased. We’ll define this twist angle at which the closed-back condition is obtained as αcrit (and will solve for it shortly). At higher twist angles, α > αcrit , the layers of the pleats will interfere with one another in the center and the crease pattern will not lie flat. However, if we look at the angles at the base of each pleat, one is acute and the other obtuse; as the twist angle α rises above 90◦ , the roles of acute and obtuse angle swap, and the roles of α and (180◦ −α) also swap. This means that if α rises above (180◦ −αcrit ), the pleat crease assignments swap and the pleats no longer interfere with one another. And so this twist is valid again. Thus, there is a “forbidden gap” in the values of allowed twist angle. Cyclic twists exist for α ∈ (0◦, αcrit ] or α ∈ [180◦ − αcrit, 180◦ ). We can characterize these two sets simply by labeling a twist clockwise (CW) or counterclockwise (CCW), according to the direction of twist of the central polygon as the shape is folded. In a clockwise twist, as the paper is folded, the central polygon moves in a clockwise direction. Examples of CW and CCW twists are shown in Figure 3.18. 214


Figure 3.18.

Left: a counterclockwise heptagonal twist. Right: a clockwise heptagonal twist. Top row: crease patterns. Bottom row: folded forms.

? 3.4.3. Rotation Angle of the Central Polygon When a twist moves from the folded to unfolded state, you can see that the central polygon rotates with respect to the rest of the paper. Actually, we must be a bit careful with what we mean by “the rest of the paper.” All parts of a twist move in some way: the pleats flip over; the wedges between the pleats translate toward each other. The central polygon translates and rotates with respect to the wedges. As the width of the pleats approaches zero, the wedges become “the rest of the paper.” So a more precise statement of the question is, through what angle does the central polygon rotate with respect to the wedges? We can answer this by considering just a single wedge. Figure 3.19 shows how the central polygon moves with respect to one of the surrounding wedges. The figure shows that if the twist angle is α, the polygon rotates through an angle of 2α. Since the pattern is rotationally symmetric, the central polygon rotates through the same angle with respect to each of the other wedges. Of course, one could equally keep the central polygon fixed in position; in that case, as the twist is folded, each of the wedges would appear to rotate through an angle of 2α in the opposite direction.

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215

Figure 3.19.

Left: the central polygon before folding. Right: clockwise rotation of the polygon after folding.

a

aa

?? 3.4.4. Iso-Area Twists The Japanese mathematician Toshikazu Kawasaki (of KawasakiJustin Theorem fame) has coined the term iso-area [59] to apply to an origami model in which “both sides of the paper are exposed to the same amount on both sides of the paper” [57, p. 96]. A simple example of an iso-area fold (in fact, it is an iso-area twist) is shown in Figure 3.20. Both sides display the same amount of white and colored paper in the pattern. The crease patterns for many iso-area folds display an interesting property that is visible in the pattern in Figure 3.20. If you turn every mountain fold into a valley fold and, at the same time, turn every valley fold into a mountain, the crease pattern is the same as what you started with, rotated 1/4 turn. This property, or something like it, is common to many isoarea folds: the “something like it” is the following: if you convert

Figure 3.20.

Left: crease pattern for an iso-area fold. Right: the folded shape, obverse and reverse.

216


all mountain folds to valley folds and vice versa, the resulting crease pattern is the same as what you started with in some other orientation: rotated, reflected, or translated in some way. This is itself a type of symmetry that links the crease assignment with other geometric properties of the crease pattern, and it is a property of the crease pattern itself. This property may or may not relate to the amount of color exposed on each side of the fold. Jun Maekawa recognized the attractiveness of this property linking symmetry and crease inversion and suggested that the term “iso-area” be used to describe any crease pattern that exhibited the symmetry that it was unchanged under n-fold rotation plus crease inversion [77]. The attractiveness of this more general symmetry is testified by the fact that the term “iso-area” is commonly applied to crease patterns with this property whether or not they show equal amounts of both sides of the paper (the original definition of the term). However, rotational symmetry is also not the only possible symmetry that could apply to a crease pattern: one could imagine a crease pattern that remains unchanged after some combination of translation and crease inversion (such a crease pattern must necessarily be infinite in extent) or, perhaps, after some combination of magnification and rotation (again, such a pattern must be infinite for this to strictly hold). Maekawa redefined the term “iso-area" to apply more broadly than Kawasaki’s original notion. I will beg both men’s indulgence to redefine it yet one step more broadly: A crease pattern is iso-area if the assigned crease pattern remains the same after interchanging mountain and valley folds and performing some combination of translation, rotation, reflection, inversion, and/or magnification. That is, in the most general sense, an iso-area crease pattern is one where some symmetry operation maps an unassigned crease pattern to itself in a way that is not the identity (that is, every crease maps one-to-one to a different crease), and where said operation has the effect of interchanging mountain and valley folds. The allowed operations include not only the rotations identified by Maekawa but all isometric transformations (translation, rotation, reflection, glide) and the additional conformal transformation that preserves straight lines (magnification). For most iso-area crease CHAPTER 3. SIMPLE TWISTS

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217

patterns, the particular isometric transformation is a rotation or reflection. The first requirement to have an iso-area crease pattern is that there are the same number of mountain and valley folds. Cyclic simple flat twists are clearly not iso-area. Although the pleats contain the same number of mountain and valley folds (one each for each pleat), the central polygon is outlined only by mountain folds. You can, however, reverse the crease assignments of some of the central polygon folds to make certain twists iso-area. This is only possible for polygons with an even number of sides—squares, hexagons, octagons, and so forth—because you need to turn exactly half of the central polygon creases into valley folds. In addition to reversing some of the creases of the central polygon, you also have to reverse some of the pleats as well. The vertices of the central polygon are degree-4. Recall that a degree-4 vertex must have either one or three mountain folds and the largest angle at the vertex must be bounded by creases of the same type. This means that for flat-foldability, for each mountain fold turned into a valley fold, you need to reverse the polarity of the adjacent pleat. This is exactly what we did in our two folded examples in Figures 3.8 and 3.9. This reversal is illustrated for a single pleat of the square twist shown in Figure 3.21. Note that we must reverse all three creases: the edge of the central polygon and the two creases on either side of the pleat. To create an iso-area twist, half of the pleats of a cyclic twist must be reversed in such a way that the crease pattern after reversal is a rotated version of the same thing before. In the square twist, there are two fundamentally different ways of doing this: you can reverse two adjacent pleats or two opposite pleats. Figures 3.22 Figure 3.21.

Left: a cyclic square twist. The shaded pleat is the one that will be reversed. Right: the same twist with a single pleat inverted.

218


Figure 3.22.

Left: a cyclic square twist. Middle: a square iso-area twist. Right: a second square iso-area twist.

Figure 3.23. The folded forms of the three square twists shown in Figure 3.22. Hidden edges are indicated by dotted lines.

and 3.23 show the basic square twist and the two iso-area versions, with shaded regions indicating the two pleats that are reversed. Figure 3.23 shows the folded versions of the crease patterns from Figure 3.22. The patterns of the lines are the same; all that differ are the stacking orders of the layers. The rightmost iso-area pattern has the pleasant property that it can continuously deform between the folded and unfolded state while keeping each of the flat surfaces—the facets—flat, i.e., it is rigidly foldable. Both the regular square twist and the square twist with adjacent pleats reversed do not have this property—you can easily verify that at least one of the facets must flex to move from the folded to the unfolded state. The two iso-area forms shown in Figure 3.22 were obtained by reversing the (left, bottom) edges and the (left, right) edges,

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219

Figure 3.24. The two distinct hexagonal iso-area twists. The shaded pleats are reversed from the cyclic form.

respectively, of the central polygon. We could also have reversed the (top, right) edges, or the (top, bottom) edges. However, these would not be fundamentally different from the two iso-area twists shown in Figure 3.22 since they are merely rotated versions. We will say two twists are considered to be distinct if one cannot be rotated to be made identical to the other. Similarly, hexagonal, octagonal, and in general, 2n-gonal twists all have iso-area forms. The hexagonal twist also has two distinct iso-area forms for a given twist angle, which are shown in Figure 3.24. Any polygonal twists can be labeled by the type of crease encountered as one goes sequentially around the central polygon. Using M for mountain fold and V for valley, the cyclic twists would be M M M M and VVVV for the square twists, for example. The two square iso-area twists have the patterns M MVV and MV MV, respectively. The hexagonal iso-area twists have types M M MVVV and MV MV MV. We can abbreviate these as M 2V 2 and (MV)2 , respectively, for the square iso-area twists, and M 3V 3 and (MV)3 , respectively, for the hexagonal iso-area twists. The symbols for a 2n-gonal iso-area twist are the possible patterns of n Ms and n Vs that are invariant under the combination of the transposition M V and some cyclic permutation (a rotation of the shape). Every 2n-gonal twist has iso-area analogs of the two types shown in Figures 3.22 and 3.24. That is, there is an M nV n and (MV)n form of every 2n-gonal iso-area twist. Polygons of order higher than 6 add additional iso-area twist forms. The octagon, for example, admits iso-area twists of the forms M 4V 4 , (MV)4 and a third form, M MVV M MVV, which we abbreviate as (M 2V 2 )2 . Figure 3.25 shows the iso-area forms for the even polygons with 4 through 12 sides. Above 12, the possibilities grow steeply: 14-, 16-, and 18-sided twists have 10, 20, and 30 iso-area forms.

220


n=4

M 2V 2

( MV )

2

M 3V 3

( MV )

3

M 4V 4

M 2 VMV 2 MV

(M V )

( MV )4

M 5V 5

M 3VMV 3 MV

M 2 VM 2 V 2 MV 2

( MV )

M 6V 6

M 4 VMV 4 MV

M 3VM 2V 3 MV 2

M 3V 2 MV 3 M 2 V

(M V )

M 2 (VM ) V 2 ( MV )

(M V )

( MV )6

n=6

n=8

2

2 2

n = 10

5

n = 12

3

3 2

2

2

2

2 3

Figure 3.25. Regular iso-area twists for 4 through 12 sides.


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221

n 3 4 5 6 7 8 9 10 11 12

Table 3.1. Number of distinct crease assignments for regular n-gonal simple flat twists.

N(n) 4 6 8 14 20 36 60 108 188 352

The enumeration of iso-area crease assignments for regular polygon twists leads to the broader question: how many possible crease assignments are there for a given n-gon twist? With certain restrictions on twist angles, this problem is equivalent to asking how many different ways there are to string n black and white beads on a circular necklace. (Black beads would correspond to, say, valley folds on the inner polygon, and white beads would then correspond to mountain folds.) The necklace problem is one that has been addressed by mathematicians in the past and the solution is well known [109]. The number N(n) of possible crease assignments for an n-gon twist is given by the formula N(n) =

1Õ d 2 φ(n/d), n

(3.1)

d|n

where the summation is over the divisors d of n, and the function φ(k) is the number of integers less than k and relatively prime to k (φ(k) is called the Euler totient function). The values of N(n) up to n = 12 are listed in Table 3.1. But there was an important qualification: “with certain restrictions on twist angles.” Not all crease assignments are possible for every twist angle, and we are about to see why. ?

3.5. Twist Flat-Foldability It seems like a simple, straightforward question: “What are the crease assignments of a twist that lead to a flat-foldable folded state?" We know the basic rules that apply from the previous chapter. The angles at each vertex must satisfy the Kawasaki-Justin

222


Figure 3.26. Three non-flat-foldable hexagonal twists. Each becomes flat-foldable if you trim the paper to the smaller hexagon indicated by the dotted lines.

Condition (that alternating angles sum to 180◦ ). The degree-4 vertices must have three creases of one type and one of the other. A smallest angle must be anto; a largest angle must be iso. And the Justin Non-Crossing Conditions, which prevent self-intersection of the paper, must be met. That’s all there is to it, and the crease patterns themselves are relatively simple. Despite that, however, the question of twist flatfoldability can be surprisingly subtle. As an illustration of this, consider the three hexagonal twists shown in Figure 3.26. None of the three are flat-foldable with the crease assignments shown. And yet, they are only slightly un-flat-foldable; if you reduce the size of the paper to the dotted-line hexagon in each figure, then each one becomes flat-foldable. These three patterns are not flat-foldable; but they fail for distinctly different reasons. The size of the paper surrounding the twist is clearly important, but it’s not the only thing that matters; the specific crease assignment is certainly part of the picture as well. For example, the middle pattern in Figure 3.26 is not flat-foldable at the original size; but inverting the parity of the central polygon makes it flat-foldable, as shown in Figure 3.27. There are twists for which small differences in the twist angle make all the difference. In Figure 3.28, the twist angles differ by only a few degrees, but that is enough to make the difference between flat-foldability and not. And then there are twists that have no possible crease assignment; there is no assignment for the pattern of Figure 3.29, first described by Hull [48], that leads to a flat-foldable form.

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Figure 3.27. Two hexagonal twists with the same twist angle but different crease assignments. Left: not flat-foldable. Right: flat-foldable.

Figure 3.28. Two hexagonal twists with slightly different twist angles and the same crease assignment. Left: not flat-foldable. Right: flat-foldable.

And so, the question of what crease assignments are possible is, indeed, a complex one. We can divide the world of twists into two rough categories, though: there are those where the paper size—or more generally, the surroundings of the twist—matter, and those where it doesn’t. What makes the patterns of Figure 3.26 non-flat-foldable in the larger size is that as the paper around the central polygon gets larger, layers of the pleat begin to interfere somewhere—with each other or with creases around the polygon. If we make the paper smaller, though, we can avoid such interferences. Thus, we can identify a minimum set of conditions on the crease assignments of regular twists by considering the situation where the paper around the twist polygon extends only a small distance farther outside the

Figure 3.29. A triangular twist for which there is no crease assignment that permits flat-foldability.

224


polygon. In such a case, layers only have the potential to interfere with other layers around the same vertex. In other words, we need only consider local flat-foldability. ? 3.5.1. Local Flat-Foldability Even in such a minimal twist, there are constraints on the possible crease assignment, depending on the twist angle, and for regular polygon twists, we can identify these conditions because every vertex is the same. For a vertex with twist angle α on an n-gon twist, the four angles around the vertex, proceeding counterclockwise, are 360◦ 360◦ α, , 180◦ − α, 180◦ − . n n This labeling is illustrated in Figure 3.30, drawn as if for a vertex of a hexagonal twist, but the labeling applies to any twist. If the angle with value 360◦ /n is the sole smallest angle at the vertex, then it is an anto sector, and the opposite sector, with value 180◦ − 360◦ /n, is iso. If that is the case, then the two polygon sides adjacent to the vertex must have the same assignment, either both M or both V. But since this condition applies to every vertex, they must all be M or V, leading to the following general result: Theorem 17. For any regular n-gon twist with twist angle α, if min(α, 180◦ − α) > 360◦ /n, then the only possible crease assignments are the cyclic twists M n and V n .

180¡ -

a 360¡ n

360¡ n

180¡-a

Figure 3.30. The angles around a vertex of a regular n-gon twist.

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225

Figure 3.31.

A cyclic twist for n = 12 and twist angle α = 40◦ . Left: crease pattern. Right: folded form.

As n grows, the value of 360◦ /n shrinks, so for a fixed twist angle α, as the number of sides on the polygon increases, eventually the only possible assignments will be the cyclic ones. For n = 12, for example, if the twist angle is larger than 30◦ , the only possible twists are cyclic, such as the n = 12, α = 40◦ twist shown in Figure 3.31. What if the 360◦ /n angle is the sole largest angle at the vertex? In that case, the interior angle of the polygon must be the sole smallest, and so the two adjacent sides of the polygon must have opposite assignment; the creases must alternate going around the polygon, M, V, M, V, . . .. But that presents a problem for odd n; somewhere, there will have to be two Ms or two Vs in a row. That leads to another theorem: Theorem 18. For any regular n-gon twist with twist angle α, if 180◦ − 360◦ /n < min(α, 180◦ − α) and n is odd, there is no possible crease assignment that is flatfoldable. The question is, though, for what n could this situation arise? The first case, n = 3, is clearly affected, and we see for twist angles α > 60◦ , this situation applies and there is no possible crease assignment. This explains Figure 3.29 above, which has a twist angle of α = 90◦ . This pattern would remain impossible for twist angles down to (but not including) 60◦ . For the next value, n = 5—and, for that matter, for all higher n—we can see that the inequality condition will never be met. And so, the n = 3 twist is, in this regard, a special case.

226


The situation gets more complicated, though, as we start to consider larger paper. In principle, we can place a twist in the middle of an arbitrarily large region of paper, simply by extending the pleats out to infinity. Once the pleats extend far enough, new possibilities for layer interactions occur that begin to impose limits on the possible crease assignments, as we will now see. ? 3.5.2. Pleat Crease Parity Let us now consider what happens in twists where the paper extends sufficiently far from the central polygon that every possible layer-ordering interference that might happen must be considered. (We leave open, for the moment, the question of just how far out one must go for this condition to apply.) We will call such a twist a maximal twist. Consider a rectangular region of paper that runs perpendicular to a pleat as illustrated in Figure 3.32. If we go sufficiently far away from the central polygon, we can find such a strip where the two end regions A and C contain no other creases and are longer than the middle region B, which is the width of the pleat. In order for the two ends A and C not to intersect each other (or their pleats) in the folded form, the two creases must have opposite assignments: either M and V or V and M. So, whatever the crease assignment of the central polygon is, the following must be true: Theorem 19. For any flat-foldable maximal twist, the two creases of every pleat around the twist have opposite assignment from one another.

A B

C

Figure 3.32. In a maximal twist, it is possible to find a rectangular region of paper perpendicular to the strip whose ends would overlap if the pleat creases did not have opposite assignment.

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This cuts down the number of possible assignments of the complete twist considerably. ? 3.5.3. Pleat Assignments With each pleat we can associate three creases: the two creases of the pleat itself, and the crease from the central polygon that connects the two pleat creases. Taking into account the preceding result, there are four possible assignments of these three creases for each pleat. The polygon edge can be M or V; the two creases of the pleat can be MV or V M. Let us label a pleat primarily by the assignment of the polygon edge. If the acute angle at the base of the pleat is anto, we will call the pleat an anto pleat and will label it with a capital M or V. If the acute angle at the base of the pleat is iso, we will call it an iso pleat and will label it with lower case m or v. We illustrate the four possibilities in Figure 3.33. Now, every possible assignment of the creases around a potentially flat-foldable twist could be expressed as a string of symbols, one for each of the pleats of the twist, selected from the alphabet {M, V, m, v}. Each vertex of the central polygon is the junction of two pleats. Thus, we can examine the flat-foldability conditions at all possible vertices by considering all possible pairs of these four symbols and the implied crease assignment. There are 4 × 4 = 16 possibilities. We can dispense with eight of them, however, quite readily. Consider the four combinations {Mm, Mv, mM, mV }, i.e., those

a

Figure 3.33. The four possible crease assignments for a single pleat. Top row: acute angle is anto. Bottom row: acute angle is iso.

228

M

a

V

a

m


a

v

a

Mm

a

Mv

a

a

mM

mV

Figure 3.34. Crease assignments for two consecutive pleats where one is anto and the other is iso.

combinations where an anto pleat follows an iso pleat or vice versa, as illustrated in Figure 3.34. In these assignments, there are either two mountain/two valley or four mountain/zero valley. These vertices therefore fail the Maekawa-Justin Condition and so cannot be flat-foldable for any possible assignment. The same is true if we invert all of the crease assignments, which takes care of the four heterogeneous possibilities {V v, V m, vV, vM }. Thus we have the following: Theorem 20. If a twist is flat-foldable, then all of its pleats are anto or all of them are iso. That leaves eight possible pairs: {M M, VV, MV, V M, mm, vv, vm, mv}. We now consider each of these individually. We note that since any crease pattern is flat-foldable if and only if the crease pattern obtained by reversing every crease is also flatfoldable, we only need to consider half of the combinations: {M M, MV, mm, mv}, since the same conditions will apply to their respective parity inversions. ? 3.5.4. mm/vv Condition Figure 3.35 shows an mm and an mv crease assignment for the polygon creases at a vertex of a regular polygonal twist. We will assume going forward that the pleat angle α is nonobtuse (i.e., α ≤ 90◦ ). Each vertex has two iso and two anto sectors; according to the BLBA Theorem, one of the two anto sectors must be the smallest angle at the vertex. For the mm vertex, the angle labeled 180◦ − α can’t be smallest (it’s larger than α); thus, the angle marked 360◦ /n must be as small or smaller than

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229

180¡ -

360¡ n

180¡-a

a

Figure 3.35.

180¡ -

180¡-a

a 360¡ n

360¡ n

Left: an mm vertex of a polygon twist. Right: an mv vertex of a polygon twist.

360¡ n

mm

mv

any other angle at the vertex. Consequently, we have the following requirements: 360◦ 360◦ ≤ 180◦ − n n

and

360◦ ≤ α. n

(3.2)

The first condition doesn’t even depend on the twist angle, and so it sets an absolute condition on the number of sides, which is n ≥ 4.

(3.3)

The second condition sets a bound on the twist angle α, namely that it is greater than 360◦ /n. Since we have already made the assumption that α ≤ 90◦ , we can see that for there to be any possibility of an mm (or vv) vertex, the range of possible values of α is going to be rather narrow indeed. ? 3.5.5. mv/vm Condition Now considering the other half of Figure 3.35 for an mv or vm vertex. According to the BLBA Theorem, either 180◦ −α or 180◦ − 360◦ /n must be the smallest angle. As before, the assumption that α ≤ 90◦ means that the former can’t be the smallest, and so we must have 180◦ −

360◦ 360◦ ≤ n n

and 180◦ −

360◦ ≤ α. n

(3.4)

Once again, we have two inequalities. The first sets a limit on the absolute number of sides, which is

230


n ≤ 4,

(3.5)

which is exactly the opposite result from the mm/vv condition, and, in fact, is much more restrictive: there are only two possible values of n: 3 and 4. So, consider the case of an n = 3 iso twist. Whether or not it contains any mv or vm vertices, because the number of sides is odd, there must be at least one mm or vv vertex. For that vertex, the mm/vv condition applies, which requires that n ≥ 4, which is a contradiction. Thus, there is no n = 3 regular twist composed of iso pleats. For the remaining case, n = 4, we can have all four types of iso vertex, since n = 4 is permitted by both conditions. For all four types of vertex, the twist angle condition becomes α ≥ 90◦ .

(3.6)

But then, we have already made the assumption that α ≤ 90. So the only possibility for n = 4 is a single twist angle: α = 90◦ . At this twist angle, an M pleat is equivalent to an m pleat, since both angles at the base of the pleat are 90◦ . An mvmv α = 90◦ twist is exactly the same as an MV MV α = 90◦ twist, just with the base angles redefined. So anything that there is to say about an n = 4, α = 90◦ iso pleat assignment can be addressed by considering its anto equivalent. So, the mm/vv and mv/vm conditions turn out to be quite restrictive. They stipulate the following: • For n = 3, there are no iso pleat assignments. • For n = 4, the only iso pleat assignments have a twist angle of 90◦ , which will have the same flat-foldability as their anto pleat assignment equivalents. • For n ≥ 4, the only possible iso pleat assignments are the fully cyclic ones, mn and v n , with α ≥ 360◦ /n. We now turn our attention to the all-anto pleat assignments. ? 3.5.6. MM/VV Condition Now, let us consider the case of an all-anto pleat assignment around a twist. Figure 3.36 shows an M M and MV crease assignment for the polygon creases at a vertex of a twist polygon. We continue with the assumption that α ≤ 90◦ .

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231

180¡ -

Figure 3.36.

Left: an M M vertex of a polygon twist. Right: an MV vertex of a polygon twist.

360¡ n

180¡ -

180¡-a

a

360¡ n

180¡-a

a 360¡ n

360¡ n

MM

MV

Let’s look at the case of an M M vertex. Now the two anto sector angles in the figure are the angles marked α and 360◦ /n. So one of two things must be true: either (1) α is smaller than 360◦ /n, or (2) 360◦ /n is smaller than α. In this case, we don’t know which angle is the smallest: it depends on the value of n. But we can consider both cases. For case (1), if α is the smaller of the two anto angles, then it must be smaller than the two iso angles as well, and so, α ≤ 180◦ −

360◦ . n

(3.7)

For case (2), 360◦ /n is the smaller of the two anto angles, so it must be smaller than the two iso angles as well, and so, 360◦ ≤ 180◦ − α, n

(3.8)

which, when rearranged, gives exactly the same equation. Thus, no matter which of the two anto angles is smaller, for an M M vertex, we must have that α ≤ 180◦ − 360◦ /n. And since the same analysis applies if we interchange M and V, this is true for either M M or VV vertices. ? 3.5.7. MV /VM Condition Finally, let’s look at the case of an MV vertex. Again, the smallest angle in the figure must be one of the two anto angles, which now are α and the interior angle of the polygon, 180◦ − 360◦ /n. (Of course, the way the figure is drawn, this interior angle doesn’t look like the smallest angle, but for small n, the interior angles can be small—for n = 3, the interior angle is only 60◦ .)

232


Again we have two cases to consider: either (1) α is smaller than 180◦ − 360◦ /n, or (2) 180◦ − 360◦ /n is smaller than α. For case (1), if α is smaller, then it must be smaller than the iso angles, and specifically, α<

360◦ . n

(3.9)

For case (2), if 180◦ − 360◦ /n is the smallest angle, it must be smaller than every other angle, and specifically, 180◦ −

360◦ ≤ 180◦ − α, n

(3.10)

which can be rearranged to give exactly the same condition. So in either case, for an MV vertex, we must have that α < 360◦ /n . And since the same analysis applies if we interchange M and V, this equation must be satisfied for both MV and V M vertices. So, in summary, the anto pleat twist angle conditions are much less restrictive than the iso pleat twist angle conditions: • If any vertex is M M or VV, then the twist angle must satisfy the M M/VV condition: α ≤ 180◦ −

360◦ . n

(3.11)

• If any vertex is MV or V M, then the twist angle must satisfy the MV/V M condition: α<

360◦ . n

(3.12)

For a twist in cyclic form, M n or V n , all vertices are M M or VV, and the M M/VV condition applies. If the twist consists solely of alternating M and V, i.e., it is of the form (MV)n , then only the MV/V M condition applies. And if it is a general mixed form, both conditions must apply to the twist angle α. The four conditions we have looked at so far are necessary, but not sufficient, to establish flat-foldability for an isolated twist. We have examined the conditions that apply to individual vertices and to the layers around individual pleats; now we will turn our attention to other longer-range interactions. CHAPTER 3. SIMPLE TWISTS

........

233

Figure 3.37. Crease pattern and folded form for two hexagonal 70◦ twists with two different fully cyclic crease assignments. Hidden layers are indicated with dotted lines. Top: v 6 (iso) assignment. Bottom: V 6 (anto) assignment.

? 3.5.8. Cyclic Overlap Conditions If flat-foldability at each and every vertex—so-called local flatfoldability—were all that mattered, then as long as the four conditions of the previous subsections were satisfied, any crease assignment around a twist would work. But local flat-foldability is not all that matters; we must be concerned with the interferences between layers that are not necessarily all incident to the same vertex or pleat. Consider the two cyclic twists shown in Figure 3.37. The crease pattern for both is a hexagonal (n = 6) twist with a twist angle of α = 70◦ . One is a v 6 assignment; the other is V 6 . According to the local flat-foldability conditions above, both should be foldable; and indeed, if you cut them out and fold them, you will find that they both fold flat, as shown in the figure. The crease patterns are exactly the same with respect to the crease positions and angles; they only differ in their assignment. That difference translates into a difference in the stacking order of the facets. (We could, of course, consider the arrow of causality in the opposite direction: the two folded forms differ in their facet

234


Figure 3.38. The iso cyclic twist. A hidden line shows one lengthened pleat. The yellow highlight indicates where it crosses the edge.

stacking orders, and that, in turn, translates into a difference in the crease assignment.) Clearly, both of these crease patterns fold flat. But just as clearly, if the hexagon from which we were folding were just a little bit larger, they wouldn’t; layers of paper would begin to collide with one another. They do so in different ways, however. Let us consider what would happen if we lengthened just one of the pleats in the iso (v 6 ) assignment, as shown in Figure 3.38. If the bottom pleat were longer, it would extend toward the upper left, as shown in the figure, crossing the edges of the central polygon and then extending onward. In this iso cyclic arrangement, the pleat is trapped between the layer to its left and the central polygon. This layer, in turn, is similarly trapped, and so on, all the way around the polygon. This would force a self-intersection between the pleat and the valley folds at the yellow highlighted line in Figure 3.38. This sets an absolute upper limit on the size of a cylic iso twist, and it tells us as well that there is no maximal (arbitrarily large) iso cyclic twist that avoids self-intersection. With regular polygon twists, we must also consider whether the layers interfere in the very middle of the twist. This situation arises, as we have seen, in cyclic twists. The maximum twist angle in a cyclic twist is the angle that gives a closed back, which we now derive. Consider a single pleat, as shown in Figure 3.39. (A hexagon is shown, but this analysis applies to any regular polygon.) The critical twist angle (and closed back) occurs at the twist angle where all of the pleats meet at the center of the polygon. As the figure shows, the critical twist angle is thus the same as the angle between the side and a line running to the center of the polygon.

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235

a a

Figure 3.39. Schematic of a pleat in a closed-back twist. The shaded region shows the position of the pleat in the folded form.

For an n-sided regular polygon, that angle is αcrit = 90◦ −

180◦ , n

(3.13)

and we call this angle at which the pleats touch in the center the critical angle for the twist. In a regular polygon twist, for pleat angles larger than this critical angle, all of the pleats overlap the point in the middle of the polygon. This sets up a question of stacking order: which layer is on top? In a cyclic twist, i.e., one whose pleats are all of the same type, each pleat will lie on top of the next pleat in the clockwise (or counterclockwise) direction, which, in turn, lies on top of the next pleat, which lies on the next . . . which eventually comes around and lies on top of the pleat we started with. At the point of overlap, that results in the situation where one of the pleats must ultimately lie on top of itself. This is, of course, not possible, and so such a configuration cannot possibly lie flat. Thus, for a cyclic crease assignment in a regular polygon twist, the maximum twist angle must satisfy the cyclic overlap condition, α ≤ 90◦ −

180◦ . n

(3.14)

The cyclic overlap condition arises from considerations of layer order, which depend on the crease assignments of the central polygon, but not the specific assignment of the parallel pleat creases. Thus, everything that can be said about M or V pleats applies equally to m or v pleats. And now, we recall from Section 3.5.5 that the only possible iso pleat assignments for a regular polygon twist were the fully

236


C

B

B

A

A

Figure 3.40.

Left: three V pleats not all in the same half polygon form a non-transitive set: A is on top of B is on top of C is on top of A. Right: in an even-order polygon, two opposite V pleats form a nontransitive pair: A is on top of B is on top of A.

cyclic ones, mn and v n for n ≥ 5, which must therefore satisfy the mm/vv condition, α ≥ 360◦ /n. Combining this with the cyclic inequality, we have that 180◦ 360◦ ≥α≥ . (3.15) n n The cyclic overlap condition applies even with a less-thanfully cyclic crease assignment. Consider, for example, a V pleat for an n-gonal twist at a twist angle just slightly above the critical angle, as shown in Figure 3.40. This pleat will cover up half of the vertices of the central polygon, and so it will lie on top of any V pleats emanating from any one of those vertices. Each of those V pleats, in turn, covers up a different set of half of the vertices and any of their associated V pleats. And the same goes on around the polygon. If there is any trio of V pleats whose acute vertices do not all lie within the same half of the polygon, then we will have the situation where each pleat of the trio covers up one pleat and is covered by the other. But at the center point of the polygon, this cannot be: one of the pleats has to wind up on top. So, in any twist, if there are three V pleats not all in the same half of the polygon, the cyclic condition sets a bound on the twist angle. The same is true if there are three M pleats not all in the same half, or three m, or three v. In addition, for even-order polygons, it takes only two pleats to create this situation. If two V pleats are directly opposite one 90◦ −

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237

another, then for α > αcrit , each of the pleats wants to cover up the other, which is again, impossible. Thus, the cyclic condition applies when either of the following is true: 1. There are three M, V, m, or v pleats whose acute-angle vertices are not all within the same half-polygon; 2. The polygon is even and there are two M, V, m, or v pleats opposite each other. Of course, if the twist is fully cyclic, then both of the conditions are true. For an arbitrary twist, if either of these two conditions apply somewhere around it, we will call the twist partially cyclic (so that a fully cyclic polygon is at least partially cyclic). Even if the polygon is not fully cyclic, it is hard to avoid partial cyclicity, to wit (for anto pleats): • For n = 3, the only partially cyclic assignments are the fully cyclic assignments M M M and VVV. • For n = 4, M MVV and its cyclic permutations are the only assignments that are not partially cyclic. • For n = 5, M M MVV, M MVVV, and their cyclic permutations are the only assignments that are not partially cyclic. • For n = 6 and higher, every crease assignment is at least partially cyclic. Exactly the same possibilities apply for the iso pleat configurations. We can now put all three conditions together to determine the bounds on the twist angle of a regular polygonal twist. ? 3.5.9. Summary of Limits In each case, we have set an upper bound on twist angle α, but this is all assuming that α is the smaller of the angles at the base of the pleat, i.e., α ≤ 90◦ . For larger twist angles, the twist runs the opposite direction, and instead of there being a lower bound on α of the form α ≤ αcrit , there will be a corresponding upper bound, i.e., α ≥ 180◦ − αcrit , with pleat angles between αcrit and

238


180°

OK for all

135°

no MV,VM

Twist Angle(°)

no cyclic

no MM,VV

90°

no cyclic no MV,VM

45° OK for all

Figure 3.41. 0°

2

4

6

8

10

12

Number of Sides

180◦ − αcrit being forbidden and the value of αcrit depending on the specific crease assignment of the central polygon. These three constraints are summarized in the plot in Figure 3.41. For most regular twists, those with n ≥ 6, every crease assignment is partially cyclic, and so the maximum twist angle is given ◦ by the critical angle, α ≤ 90◦ − 180 n (red dots in the figure). But for n = 3, 4, and 5, it is possible to achieve a larger twist angle for certain non-cyclic twists. We call these twists supercritical, since their twist angles lie beyond the critical angle. Figure 3.42 displays the crease assignments and the folded forms for three examples of such supercritical twists. A twist angle of 90◦ is forbidden by the M M/VV condition for a triangular twist and is forbidden by the MV/V M condition for a pentagonal twist, but is allowed (and is indeed possible) for a square twist; a 90◦ square twist is simply two crossing pleats. Which of the three conditions applies depends, of course, on the specific crease assignment. However, it is clear from Figure 3.41 that for every value of n, there is a twist angle below

Ranges of twist angles that give valid regular twists.

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239

Figure 3.42. Supercritical twists. Top row: crease patterns. Bottom row: folded forms. Left: a supercritical triangular flat twist (α = 40◦ ). Middle: a supercritical square twist (α = 70◦ ). Right: a supercritical pentagonal flat twist (α = 62◦ ).

which every possible crease assignment works. We call this the safe twist angle, αsafe for the polygon. For unit regular polygons, we have ◦ 90 − 180◦ /n if n ≤ 6, αsafe = (3.16) 360◦ /n if n ≥ 6. So, for CCW twists, α ≤ αsafe (or for CW twists, α ≥ 180◦ − αsafe ) is the range of twist angles that works for every possible crease assignment. Table 3.2 shows the safe twist angles for n = 3 up to n = 13. Note that the largest safe angle occurs for hexagons (n = 6) and that a twist angle of 30◦ is safe for all regular polygons up through n = 12. For larger polygons, the safe angle grows steadily smaller.

Table 3.2. Safe values of twist angle versus number of sides of the central polygon.

240


n 3 4 5 6 7 8 9 10 11 12 13

αsafe 30◦ 45◦ 54◦ 60◦ 51.4◦ 45◦ 30◦ 36◦ 32.7◦ 30◦ 27.7◦

Figure 3.43. A square twist beyond the critical cyclic angle that is flat-foldable. Left: crease pattern. Right: folded form.

The M M/VV and MV/V M conditions on twist crease assignments come from considerations of the vertices around the twist: they are a local condition. The cyclic condition, by contrast, arises from considerations of layer overlap involving points that might be widely separated: they are a global condition. And, strictly speaking, the cyclic limit only applies to isolated twists surrounded by arbitrarily large amounts of paper. A cyclic twist whose twist angle is greater than the cyclic limit might well be foldable if the paper ends before it crosses the point where collision occurs in the center, as illustrated in Figure 3.43. In fact, it is not even necessary that the paper actually ends; if the twist is part of a larger crease pattern, there could be other creases that change the position of those layers of the paper that would cause an unresolvable layer overlap. Such an example is shown in Figure 3.44. Here, as in Figure 3.43, the central square twist is clearly a cyclic square twist beyond the cyclic limit, but the additional creases outside of the twist terminate the pleats before

Figure 3.44. a

a

a

a

A square twist with twist angle α = 60◦ that is beyond the critical cyclic angle, as part of a larger crease pattern that is flat-foldable. Left: crease pattern. Right: folded form.

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241

they reach an unresolvable state. Because of the possibility of this sort of situation, it is not possible to create universal rules for twist crease assignability without considering the entire crease pattern in which it is embedded. But, as we have seen above, one can stake out a region of twist angle, bounded by the safe angle αsafe , within which every assignment of the central polygon is possible. ?? ???

3.6. General Polygonal Twists Thus far, we have considered only twists based on regular polygons—polygons with equal sides and angles. Irregular polygons—with unequal sides and/or angles—can also serve as the central polygon of a simple flat twist. In fact, one can construct a simple flat twist from any convex polygon by the process described in Section 3.3, of adding a parallel-sided pleat characterized by a common twist angle to each side of the polygon, as shown in Figure 3.45. The twist shown in Figure 3.45 has the same twist angle for every pleat. This is the case for every twist formed by parallel pleats. I have mentioned this before as being a necessity: here, in the most general case, we can see why, illustrated in Figure 3.46. If the pleat creases are parallel, then the angles at their base must sum to 180◦ , so if one is α, the other is 180◦ − α. But then, since both vertices must fold flat and the vertices are degree-4, opposite angles at each vertex must also sum to 180◦ . And so the pleat angle at the base of the next pleat must be α; and so it goes, all the way around the polygon. Flat-foldability and parallelness of pleats enforce the commonality of the twist angle for all pleats.

Figure 3.45. An irregular hexagon twist. Left: crease pattern. Right: folded form.

242


a a

a

a 180¡Ð a

a

Figure 3.46.

a

The constancy of the twist angle α propagates around the twist due to the flat-foldability of each vertex.

Another way of seeing this is our earlier observation that when the twist forms, the central polygon rotates through twice the twist angle of the adjacent pleat with respect to each of the surrounding wedges. This can only happen if the twist angle is common to all pleats—otherwise, different pleats would call for different rotation angles for the central polygon. Irregular polygonal twists, like their regular counterparts, exist in cyclic form, where the central polygon is outlined by all mountain or all valley folds, as well as versions in which some of the surrounding creases and their attached pleats are inverted. Like regular polygon twists, there will be limits on twist angle that are analogous to the M M/VV condition, the MV/V M condition, and the cyclic condition; the specific limits will, of course, depend on the specific angles and, even in some cases, the lengths of the sides of the central polygon. ?? 3.6.1. Triangle Twists Of course, we have already encountered one form of triangle twist: the regular triangle twist, whose central polygon is an equilateral triangle, which has a critical angle αcrit = 30◦ , which is also the safe angle αsafe . One can easily create triangle twists from isosceles or scalene triangles, as illustrated in Figure 3.47. As with the regular twists, we can ask the questions about crease assignments: which are possible, and what are the angular limits on the twist angles? It is clear from the figure that these examples are fairly close to a critical twist angle, due to the nearcollision of the pleat edges. For a given triangle, what might that critical twist angle be?

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243

Figure 3.47.

Left: an isosceles triangle twist. Right: a scalene triangle twist. Top row: crease patterns. Bottom row: folded form.

Let us first look at the constraints imposed by local flatfoldability and, specifically, the BLBA Theorem. Consider a general triangle twist, as illustrated in Figure 3.48. We will define the sides of our triangle to be a, b, and c and the three angles opposite their respective sides as A, B, and C. What are the relations among these angles that determine whether there is a crease assignment that gives a flat-foldable result? Assume, for generality, that the angles are ordered such that A ≤ B ≤ C. Recall that for the equilateral triangle twist, if the interior angle of the triangle (A, B, or C here) was smaller than the twist angle (α), then the BLBA Theorem required that all three

C a A

Figure 3.48. Geometry of a general triangle twist with twist angle α.

244


b

a a

c

a

B

Figure 3.49.

The four possible crease assignments for an irregular triangle twist with A = 40◦, B = 60◦, C = 80◦ , and α = 50◦ . Top row: crease patterns. Bottom row: folded forms.

triangle corners were anto, i.e., the crease parity was opposite between any two adjacent sides of the triangle, and that, in turn, implied that there was no crease assignment that satisfied local flat-foldability. The same situation holds here: Theorem 21. For a general triangle twist with central angles A, B, C and twist angle α, if α > max(A, B, C), there is no valid crease assignment that results in flat-foldability. Since the three angles differ, it is possible for the twist angle to be larger than only some of the interior angles. If α > A, B but α < C, then only two of the triangle corners are required to be anto, and that potentially allows a solution. Suppose, though, that α > A but α < B (and α < C). That makes A the smallest angle at its vertex, so the two adjacent folds, b and c, must be opposite parity, either MV or V M. The third fold, a, can then take on either parity (you should fold and verify this for yourself). So that leaves only four possible crease assignments for this situation, illustrated in Figure 3.49. Supose α is greater than both A and B. Then both A and B are the smallest angles at their respective vertices; the adjacent folds of both must be opposite, and that leaves only two possible crease assignments for the twist, those shown in Figure 3.50.

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245

Figure 3.50. The two possible crease assignments for an irregular triangle twist with A = 40◦ , B = 60◦, C = 80◦ , and α = 70◦ . Top row: crease patterns. Bottom row: folded forms.

If α is smaller than all of A, B, C, then we are free to give the central triangle any crease assignment while satisfying the BLBA Theorem. Any of the four assignments shown in Figure 3.49 will be valid. In addition, we should also be able to assign the cyclic forms, M M M or VVV. But, as we have seen, with cyclic twists, we have an additional condition: the existence of a critical twist angle. That angle is defined as the angle for which all three pleats intersect at a point. For a general triangle twist with angles A, B, C, it is given by the elegantly symmetric sin A sin B sin C −1 αcrit = tan . (3.17) 1 + cos A cos B cos C For α = αcrit , the point of intersection lies somewhere inside the triangle, which means that αcrit is, of course, smaller than any of A, B, C. Thus, αcrit may be taken as αsafe , the twist angle below which every possible crease assignment of the central polygon is allowed. In summary: • For 0 < α ≤ αcrit , all crease assignments are possible. • For αcrit < α ≤ max(A, B, C), only not-all-equal assignments are possible, with two or four assignments, depending on the sizes of the angles relative to α. • For α > max(A, B, C), no assignments are flatfoldable.

246


Figure 3.51.

The two cyclic crease assignments for an irregular triangle twist with A = 40◦ , B = 60◦, C = 80◦ , and α = αcrit = 27.2◦ . Top row: crease patterns. Bottom row: folded forms.

?? 3.6.2. Higher-Order Irregular Twists In the general case of an n-sided polygon, if the twist angle α is constant all the way around the polygon, then the crease pattern is isometrically flat-foldable, meaning that it satisfies conditions on lengths and angles that allow it to lie flat without consideration of layer intersection. With real paper, though, layer intersection must be considered. At minimum, each vertex must be locally flatfoldable; it must satisfy the BLBA Theorem and the MaekawaJustin Theorem. As we saw in the case of the triangle twist, if at a vertex the angle of the central polygon is smaller than the twist angle, that sets a condition on the crease assignment of the two incident creases from the central polygon: they must be different from one another. In the triangle twist, if the twist angle is larger than all three angles of the central triangle, an impossibility condition is set up: each crease must be different from its neighbors on either side, and that is not possible with a central polygon with an odd number of sides. The next potential opportunity for such an impossibility would occur for n = 5 sides—in principle, at least. But, on average, the interior angles of a polygon get larger as the number of sides increases. For a general polygon with interior angles {θi }, i = 1, . . . , n, the angles satisfy n Õ

(π − θi ) = 2π,

(3.18)

i=1


........

247

which implies that for the average interior angle n

θ avg ≡

1Õ 2π θi = π − . n i=1 n

(3.19)

By the time we get to a pentagon, n = 5, the average interior angle is 108◦ (that of a regular pentagon), and if we take the twist angle as the smaller of {α, π − α}, then it is clear that we could never get the twist angle to be larger than all five interior angles and yet smaller than π/2. So the crease assignment impossibility enforced by the BLBA Theorem is unique to the triangle twist. That is not to say that impossibilities cannot arise for higherorder polygons, though—far from it. Each of the conditions we derived in Section 3.5—the mm/vv, mv/vm, M M/VV, and MV/V M conditions, or rather, their generalizations—must apply at each vertex, and since each vertex may have a unique interior angle, one must evaluate all of the angles individually and taken together, to determine the possible crease assignments that satisfy local flat-foldability at all vertices. The value π − θi is the exterior angle of the polygon at each vertex; the “deviation from straightness,” or “bend angle,” of the polygon’s corner. In many situations, a more natural measure of an angle between two line segments is its deviation from straightness, rather than the angle between the segments themselves (just as the natural measure of fold angle is deviation from straightness). For example, Equation (3.18) arises naturally from a simple idea: as you travel around the polygon, you make a complete circle, and so the total of all the turns you make should sum to 2π. The conditions for valid crease assignment for a general ngon are precisely the same as those of Section 3.5, but we must replace the value 360◦ /n (which was the exterior angle of the regular n-gon) with each of the exterior angles π − θi (or, if you prefer, 180◦ − θi ), in turn, for each vertex. Four of the eight possible assignments with angles labeled are shown in Figure 3.52. The other four possible assignments are the parity inversions of these. Now, following the analysis of Section 3.5, we can derive conditions linking each of the interior angles of the polygon with the twist angle α (which, as we did there, we assume is non-

248


a

qi

p-a

a

p-qi

qi

p-a

a

p-qi

mm

qi

p-a

a

p-qi

mv

qi

p-a

p-qi

MM

MV

Figure 3.52.

Four possible crease assignments at a vertex of a general polygon with interior angle θ i .

obtuse). They are the following: mm/vv : mv/vm : M M/VV : MV/V M :

α α α α

≥ ≥ ≤ ≤

π − θi and θi ≥ π2 , θi and θi ≤ π2 , θ i, π − θi .

(3.20) (3.21) (3.22) (3.23)

For the iso pleats, {mm, vv, mv, vm}, and regular n-gons, we saw that there were no mv or vm assignments for n ≥ 4, but for a general n-gon, we can always arrange things so that at least one angle satisfies any of these conditions. We observe that for the anto pleats, {M M, VV, MV, V M }, there is going to be a value of α below which every possible assignment satisfies the BLBA Theorem: α ≤ min {θi, π − θi } . (3.24) i

But, as we have seen before, there is also a cyclic condition to consider, and for general polygons, the cyclic condition can get rather complex indeed. ?? 3.6.3. Cyclic Overlaps in Irregular Twists The cyclic overlap condition in a twist and the BLBA Theorem are both fundamentally problems about layer ordering: choosing a relative ordering of the layers at regions of mutual overlap that avoid intersections between layers and creases or between pairs of creases (as expressed by the Justin Non-Crossing Conditions from Chapter 1; see Section 1.2.2). The BLBA Theorem applies locally at each vertex; one must consider only the facets incident to the vertex. But the types of interferences that occur in cyclic twists can involve facets that are

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249

not incident to a common vertex. In fact, the facets can be quite far removed from one another in the crease pattern but still give rise to an interference in the folded form. This phenomenon should give us pause because it hints at a very deep complication lurking within tessellations (and indeed, within crease patterns in general). While isometric conditions such as the Kawasaki-Justin Condition can be considered locally, vertex-by-vertex, layer-ordering conditions can potentially involve large numbers of widely-dispersed facets, making the resolution of those problems potentially much more difficult. Like their regular brethren, irregular polygonal twists also have a critical twist angle, which is the angle at which it becomes impossible to determine a layer ordering that avoids self-intersection. With regular n-gons, the critical angle was defined when all of the pleats came together at a point. In general, they will not all come together at once to define a single critical angle. Instead, we will find multiple critical angles, each characteristic of a different group of pleats. Consider a general polygon with a V n assignment, i.e., all anto V pleats. Figure 3.53 shows the crease pattern and folded form, with the vertices labeled a–f and the pleats whose acute angles are attached to those vertices labeled a– f , counterclockwise around the polygon. Focus your attention first on pleat a, attached to corner a, and now look at what happens at point b. With this crease assignment, d

c

e f

e

b

d

c

a e

b f

a

Figure 3.53.

Left: an irregular twist. Right: the folded form of the twist. In both figures, the shaded region shows the parallel-edge pleats.

250

b

d

c b

e

f

d

a f

c


a

both the pleat and wedge at vertex b are trapped between the layers of pleat a and the central polygon. So pleat b must lie below pleat a. In the same fashion, pleat c must lie below pleat b, and so forth, all the way around the polygon, so that when we come all the way around, pleat a lies below pleat f . Now, if the pleats had an absolute ordering, such an arrangement would be impossible. If we denoted “a lies on top of b” by a > b, then we would have a > b > c > d > e > f > a,

(3.25)

which would be impossible if these were numbers. But the relative ordering of two facets only matters in the regions where they mutually overlap. This is a perfectly valid crease assignment and pleat ordering, because the pairwise overlaps happen in different places. A problem would arise, though, if there was such a cycle of overlaps all at one spot. That is what happens in cyclic regular n-gons for twist angles larger than the critical angle, and it could happen here, too. (In fact it will happen with this polygon, for sufficiently large twist angle.) The important thing to note is that what matters is not the number of pleats in the cycle: it is that some cycle exists where the pleats mutually overlap at a single point. So, in principle, to identify a cyclic interference, we would need to consider all possible cycles: not just cycles of n pleats, as we saw in regular n-gons, but also cycles of fewer than n pleats. If, for a twist angle α, such a cycle exists, then the cyclic crease assignment for that α is impossible. An exhaustive search might proceed as follows: • First, identify every possible potential cycle of pleats containing between 2 and n pleats. (There are 2n −n−1 of them.) • For each potential cycle, is there a twist angle α < π for which they all overlap at some point? Computationally, that could get intractable fairly quickly, given the exponential dependence of the number of cycles on the size of the polygon. Intuition suggests that if there is going to be a problematic cycle, shorter ones are going to show up first. So let’s look for short cycles to begin with, for example, of length 2. CHAPTER 3. SIMPLE TWISTS

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251

b d

Figure 3.54. Folded form of an irregular twist that possesses a 2-cycle of overlapping pleats.

c

a c d

a

b

Could there be an impossible cycle of just two pleats? Yes, certainly. Consider the configuration of Figure 3.54, which illustrates the folded form of a quadrilateral twist that exhibits a 2-cycle of overlapping pleats. In this twist, pleat a clearly overlaps vertices b and c, and so we must have both a > b and a > c (at least for the illustrated twist angle). But by the same argument, pleat c overlaps vertices d and a, and so c > d and c > a. The two pleats a and c overlap each other in the shaded region. These are the conditions for an impossible overlap, so the illustrated twist angle, at least, is impossible for a cyclic crease assignment. So 2-cycles are, in principle, possible. The problem of 2cycles is a bite-sized piece of the more general problem, because there are only n(n − 1)/2 possible 2-cycles to consider. Actually fewer: no 2-cycle can exist between two consecutive vertices. (Exercise for the reader: why?) So we can analyze them all as follows: • For every ordered pair (i, j) of vertices, compute αi, j as the twist angle at which pleat i passes through vertex j. • Then αcrit,i, j ≡ max{αi, j , α j,i } is the critical twist angle at which there is a 2-cycle between vertices i and j. (2) • And αcrit ≡ mini, j {αcrit,i, j } is the critical twist angle above which there is some 2-cycle impossibility.

These are the conditions for an acute twist angle α < π/2. There are, of course, equivalent relations for twists going the other way that set a lower bound on obtuse twist angles. In general, the acute and obtuse critical angles do not sum to π. Now we can consider 3-cycles: three pleats that mutually overlap. Here we must consider ordered triples (i, j, k) of vertices

252


aint

aint

Figure 3.55. Two polygons with three mutually intersecting pleats. Left: all three pleat angles can increase without causing a self-intersection. Right: the pleat angle cannot increase without causing a self-intersection.

and pleats around the polygon. For any such trio, there is a unique point and twist angle where the three pleats intersect at a point. We define that point as αint,i, j,k for the trio. That angle may or may not be a barrier to further increase of the twist angle. Consider the two polygons shown in Figure 3.55. In the figure on the left, the three pleat angles could be increased without altering the overlap order among the pleats or forcing a self-intersection. In the figure on the right, however, the three pleats are “jammed” against one another; there is no way to increase the twist angle of these three pleats without creating an unsolvable overlap problem. So the three-pleat intersection point (and the twist angle that gives rise to it) does constitute an upper limit on the twist angle, but only if the pleats are arranged as on the right in Figure 3.55, not on the left. The difference between the two is that on the left, the directions of motion as the pleat angles increase—the solid arrows in the figure—all lie within a single half-plane. As the twist angle is increased, the pairwise intersection points can all move into the opposite half-plane without creating a region where there is an overlap cycle. If the three pleat directions do not lie in a common half-plane, then there is a critical angle, which we can define as αcrit,i, j,k =

αint,i, j,k π/2

if the directions of i, j, k do not lie in the same half-plane, otherwise.

........


(3.26) 253

Figure 3.56. An irregular pentagon twist with a critical 3-cycle. Left: counterclockwise twist with (3) αcrit = 42.88◦ . Right: clockwise twist (3) with αcrit = 140.60◦ .

The definition of π/2 is chosen merely for convenience; it allows the 3-cycle critical twist angle to then be written as (3) αcrit ≡ min{αcrit,i, j,k }, i, j,k

where the minimization takes place over all possible triples of pleats. (3) Now αcrit would be the twist angle above which there is some 3-cycle impossibility. Figure 3.56 shows such a twist at its critical twist angles for both counterclockwise and clockwise twists. What about higher-order cycles? Should we consider those? Could the twist angle be limited by an order-4 cycle of pleats? It could . . . but if there is an order-4 cycle of mutually overlapping pleats, the only such cycles that matter are those whose direction vectors do not all lie in the same half-plane (for the same reason that only non-same-half-plane 3-cycles mattered). And any 4-cycle whose direction vectors don’t all lie in the same half-plane and whose pleats overlap at the same point is going to either contain a 3-cycle whose direction vectors don’t lie in the same half-plane, or (if two of the direction vectors are parallel) it contains a 2-cycle. Since we’ve already found all possible 2and 3-cycles and the limits that they place on twist angle, we can safely say that any limits imposed by higher-order cycles are already accommodated by the limits imposed by 2- and 3-cycles. Thus, for any cyclic twist, for acute twist angles α (counterclockwise twists), the crease assignment is valid for (2) (3) α ≤ min{αcrit , αcrit },

(3.27)

with an equivalent condition setting an upper bound for obtuse twist angles (clockwise twists). And finally, we can put all these limits together to define a safe twist angle for which every possible crease assignment around the

254


central polygon with anto pleats is valid. Taking into account the limits for M M and VV vertices and the cyclic form assignments for 2- and 3-cycles,

αsafe

 θi     π − θi  = min αcrit,i, j     αcrit,i, j,k 

for all i, for all i, for all distinct pairs i, j, for all distinct triples i, j, k,

(3.28)

with equivalent expressions for the upper bound. ?? 3.6.4. Closed-Back Irregular Twists In general, for cyclic irregular n-gon twists, as we increase the twist angle, we will usually reach a limit where either a 2- or 3-cycle is created; three of the n pleats come together at a point, but the other n − 3 pleats will, in general, miss the point of intersection. However, there is a certain elegance and beauty when they all come together at a point as we saw for regular polygon twists in Section 3.4.2. When they do, it is a closed-back twist. Such a twist is shown in Figure 3.57. Technically, the term “closed-back” would only seem to apply for the M n assignment, (we might call a V n assignment a closedfront twist), but for the moment, let us simply call them all closedback, and then ask the question: when does this situation arise? When will all the pleats of an irregular simple twist in cyclic form meet at a point? Figure 3.57 shows an example of an irregular counterclockwise twist that has a closed back. When the twist is formed, all of the pleats meet at a point. This situation is not easy to achieve. We will now find the condition that gives this situation for an arbitrary twist.

Figure 3.57.

Left: crease pattern for a counterclockwise closed-back twist. Right: folded closed-back twist. Note that all five pleats meet at a single point.

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255

p4

p-a

Figure 3.58.

a a q4

q3

q1

q2 q2-a a p-a a p2

a a q -a 1 p1 p-a

Geometry of a closed-back irregular twist with corner angles {θ i } and twist angle α.

p-a p3 a a

Figure 3.58 shows the geometry of a closed-back irregular twist. (We show a quadrilateral, but the analysis will apply to any number of sides.) We define the interior angles {θi } of the polygon with vertices {pi } and draw lines from each vertex of the polygon to the point in the center where all of the pleats must touch. These lines are the reflection of the pleat lines in each side of the central polygon, so the twist angle α is mirrored on each side of each polygon edge. Accordingly, the two angles inside of vertex pi have angles, respectively, of α and θi − α. Now, we drop perpendiculars from the central point to each of the sides of the polygon, as shown in Figure 3.59. Label the distance from the central point to the ith vertex ri . All of the angles of the right triangles filling the twisting polygon are defined and we can calculate the length of the perpendicular segments two different ways. Working from the right, the

p4 q4-a


r 2sin a

p-a

r 1sin(q 1Ða)

256

p1

q1-a

q3-a

y3

y4 r1

a a

s3

r4

s4

Figure 3.59. Further details of the geometry of a counterclockwise closed-back twist.

a a

s1

r3 y2 r2

p-a p3 a a s2

a a

q2-a p2

length of the segment that hits the edge running from pi to pi+1 is ri sin(θi − α),

(3.29)

while working from the left, it is ri+1 sin(α).

(3.30)

These must be equal; therefore, ri+1 sin α = ri sin(θi − α).

(3.31)

We write the equations for each vertex sequentially, one atop the other: r2 sin α = r1 sin(θ 1 − α), r3 sin α = r2 sin(θ 2 − α), (3.32) ... r1 sin α = rn sin(θ n − α). Multiplying together both sides of the equations, all of the terms involving any ri drop out, leaving sinn α =

n Ö

sin(θi − α).

(3.33)

i=1

Equation (3.33) depends solely on the polygon interior angles {θi } and the twist angle α. Therefore, for a given set of polygon angles, there are only discrete values of twist angle (one clockwise, one counterclockwise) that permit a closed-back twist. That twist angle is given by the solution(s) to Equation (3.33). Interestingly, the value of the twist angle doesn’t depend on the order of the corner angles of the polygon. If we look at the mirror image of a polygon with vertices pi , the order of its corner angles is reversed, but it will have the same closed-back twist angle for a counterclockwise twist. Once we know the twist angle, the polygon is, apart from a real scaling factor, completely determined. Continuing with the case of a CCW twist, α ∈ (0, π/2), we choose the value of one of the distances {ri }, for example, r1 . Then all of the others may be computed from the recursive relationship ri+1 = ri

sin(θi − α) . sin α

(3.34) CHAPTER 3. SIMPLE TWISTS

........

257

The side lengths si are then readily shown to be si = ri cos(θi − α) + ri+1 cos α,

(3.35)

which, conveniently, reduces to si = ri

sin θi . sin α

(3.36)

Combining Equations (3.34) and (3.36) gives a recursive relation between consecutive sides: si+1 =

sin θi+1 sin(θi − α) si . sin θi sin α

(3.37)

If we define ψi to be the angle between rays ri and ri+1 at the central point, those angles turn out to be simply ψi = π − θi,

(3.38)

i.e., the supplements of the corner angles. For clockwise twists, α ∈ (π/2, π), a similar analysis gives the side lengths and angles. Figure 3.60 shows the geometry. The relationship between the lengths of consecutive radial lines is slightly different: ri+1 = ri

sin(π − α) . sin(θi+1 − (π − α))

(3.39)

This makes the equation for the twist angle n Ö

sin (π − α) = n

sin(θi − (π − α)).

(3.40)

i=1

Figure 3.60. Further details of the geometry of a clockwise closed-back twist.

258

q3-p+a p-a r4 p-a s4

q1-p+a a


y3 y4

r1 p-a p1 p-a

r3

r1 sin (pÐa) r2 sin(q Ð(pÐ a)) 2

p4

p-a p-a

s3

a

s1

p3 a

q2-p+a s2

y2 r2

p-a q2-p+a

p-a

a

p2

This is exactly the same as Equation (3.33) with the substitution α → (π − α). So for a given set of ordered corner angles, if α is the closed-back twist angle for a counterclockwise twist, (π − α) is the closed-back twist angle for a clockwise twist. The side lengths are now given by si = ri cos(π − α) + ri+1 cos(θi+1 − (π − α)),

(3.41)

which similarly reduces to sin θi+1 , sin(θi+1 − (π − α))

(3.42)

sin θi+2 sin(π − α) si . sin(θi+2 − (π − α)) sin θi+1

(3.43)

si = ri and si+1 =

What is interesting about this is that even though the angles {α, π − α} are both solutions to the closed-back-twist equations, the relationships between the side lengths for the counterclockwise (α < π/2) and clockwise (α > π/2) twist are different: the polygons for a closed-back twist may well have a different shape, even for the same sequence of corner angles. Examples are shown in Figures 3.61 and 3.62, for both counterclockwise and clockwise twists. Even though the angles are the same, the polygons do not have the same side lengths. So we cannot create a closed-back twist from an arbitrary central polygon. Once we have chosen the polygon angles, there is only one twist angle that works (CW or CCW), and the side lengths are fully specified, up to a constant scaling factor. Any polygon that supports a closed-back twist has the geometry of Figure 3.59 or Figure 3.60: if you construct lines from all vertices, each line rotated by a specified angle relative to an adjacent side, all lines intersect at a single point. Polygons with this property are known in the world of geometry; they are called Brocard polygons [9]. The point where the lines intersect is the Brocard point, and the angle αcrit is the Brocard angle. For every ordered sequence of corner angles (θi ), there are two unique Brocard polygons: one for a CCW twist with critical angle αcrit,CCW < π/2, and one for a CW twist with critical angle αcrit,CW > π/2, such that αcrit,CW = π − αcrit,CW . In the same way, there are two Brocard points, depending on which way we

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259

Figure 3.61. Counterclockwise closed-back twists: crease patterns, and folded forms, both front and back. Top row: corner angles of 75◦, 85◦, 95◦, 105◦ . Middle row: corner angles of 88◦, 98◦, 108◦, 118◦, 128◦ . Bottom row: corner angles of 95◦, 105◦, 115◦, 125◦, 135◦ .

measure the Brocard angle, as in Figure 3.59 or Figure 3.60. Conventionally, Figure 3.60 describes the positive Brocard point with Brocard angle α; Figure 3.59 describes the negative Brocard point, and the Brocard angle is given by π − α. Brocard polygons show up in closed-back origami twists, as shown here, but as we will see in the next chapter, they have wider applicability. In general, Equation (3.33) for αcrit is transcendental and must be solved numerically. However, there are a few special cases with analytic solutions. We already saw the case for the triangle, in which case αcrit was given by Equation (3.17). For a sequence of four 90◦ angles, the solution is a square (as we have already seen), with a critical twist angle of 45◦ . For a symmetric trapezoid or parallelogram, the four angles can be described by a single parameter, e.g., the acute corner angle θ. A symmetric trapezoid would have corner angles (θ, θ, π − θ, π − θ), while a parallelogram would have corner angles (θ, π − θ, θ, π − θ). Both situations give rise to the same

260


Figure 3.62. Clockwise closed-back twists: crease patterns, and folded forms, both front and back. Same corner angles as Figure 3.61.

equation for the critical angle, which has the analytic solution sin θ αcrit = sin−1 √ . 2

(3.44)

For the trapezoid, if we choose the base (the side between the two θ angles) to be horizontal and have a length of 1, then the two tilted sides have length p 2 − sin2 θ − cos θ,

(3.45)

and the remaining side easily follows. For the parallelogram, if we choose the base side to be unit length (along with the top, of course), then the other two sides have length p 2 − sin2 θ + cos θ. (3.46) Examples of both are shown in Figure 3.63.

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261

Figure 3.63. Counterclockwise closed-back twists: crease patterns, and folded forms, both front and back. Top row: corner angles of 60◦, 60◦, 120◦, 120◦ . Bottom row: corner angles of 60◦, 120◦, 60◦, 120◦ . In both cases, the critical twist angle is 37.76◦ .

? ? ? 3.6.5. Open-Back Brocard Polygon Twists With a Brocard polygon, if the twist angle α is less than the critical angle αcrit , then the pleats do not, of course, meet at a single point. Rather, they leave a hole, a single layer of paper, that forms a polygon outlined by the edges of the pleat. An example is shown in Figure 3.64 for a V 5 cyclic twist. The hole—the central light polygon in Figure 3.64—appears to be a shrunken and rotated copy of the original polygon. In fact, it is—and this is always the case if you start with a Brocard polygon [9]. We can show this by considering the geometry as in Figure 3.65. Since the lines that define the edges of the hole are all rotated from the lines of the original polygon by the twist angle α, the corner angles of the hole must be the same as the corner angles of the original polygon. But are the sides the same length? Figure 3.64. A V 5 twist from a Brocard polygon with corner angles (90◦, 120◦, 100◦, 120◦, 110◦ ) and critical twist angle αcrit = 52.71◦ . The twist angle is α = 30◦ . Left: crease pattern. Right: folded form.

262


a

a

ri

a a a

p1

a

qi

p2

pi

c a qi+1

ri+1 acrit

a

si

pi+1

Figure 3.65. Geometry of the hole in an open-back twist from a Brocard polygon. Left: x-ray view of the twist and hole (red). Right: a single triangle of the twist and hole.

Consider just a single triangle of the hole, as illustrated on the right in Figure 3.65. The lines of the smaller red triangle are rotated-by-α versions of the lines of the outer triangle. One can show that the side lengths of the two triangles are all proportional, with the same proportionality constant ρ≡

|qi+1 − qi | sin(αcrit − α) = . |pi+1 − pi | sin αcrit

(3.47)

Notably, the shrinkage factor ρ does not depend on any of the distances {ri, ri+1, si }, which means that the same shrinkage factor must apply to every triangle in the twist. Thus, all of the sides of the hole have the same shrinkage factor relative to the original polygon, and so the hole is geometrically similar to the original polygon. The similarity property is unique to Brocard polygons. If you set up the construction of Figure 3.65 for various angles α for some arbitrary polygon, then if any one of the following is true, they all are true [9]: • The polygon has a Brocard point (all lines come to a point for some αcrit ). • The original polygon is similar to the hole for some α. • The original polygon is similar to the hole for every α. • Any two holes for different α values are similar to each other.

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263

Figure 3.66. A supercritical VV M M M twist for the Brocard polygon of Figure 3.65 for twist angle α = 65◦ . Left: crease pattern. Right: folded form.

This similarity applies even if the twist angle α is larger than the critical angle αcrit . Of course, the cyclic crease assignment is no longer flat-foldable, but there can be crease assignments with mixed M and V anto pleats that are flat-foldable, as illustrated in Figure 3.66. In such a supercritical twist, the hole is buried inside of the polygon and is multiple layers thick, but if we identify the hole as the innermost polygon outlined by the edges of the pleats, then it, too, is geometrically similar to the outer polygon, with a negative proportionality constant, still given by Equation (3.47). ?

3.7. Joining Twists This wraps up our discussion of simple flat twists. By this point, you may feel that we have beaten the concept of a twist into the ground, but we have laid a nice foundation of understanding. We can construct twists for any convex polygon and can determine what crease assignments are possible for various twist angles. With all that, a single twist is not terribly interesting in isolation. The true beauty of origami tessellations arises when we begin to combine twists into groups, allowing the pleats of one twist to join the pleats of others. The artists who developed the field of twist tessellations— Shuzo Fujimoto, Yoshihide Momotani, and Chris K. Palmer—and the many who came later folded more than isolated twists; they put them together into assemblies of sometimes breathtaking beauty. A few examples are shown in Figures 3.67–3.70 (and we will see more later). Although Fujimoto and Momotani came first, Chris K. Palmer was, more than anyone, the person who brought origami tessella-

264


Figure 3.67. Backlit tessellations by Chris K. Palmer. Top row: left to right, “Ring of Eight Twist Octagons (open back 2),” “Twist Octagons Shift (close back),” and “Twist Octagon (open back 1).” Bottom row: left to right, “Twist Octagons (open back 2),” “Twist Octagons Shift (open back 1),” and “Zillij Octagrams.”

tions to worldwide prominence. He grew up folding origami from books. After receiving a degree in Fine Art, he began to compose his own patterns while living in Granada, Spain, where he found an inspirational home in the Alhambra, sketching the medieval mosaic tilings. Soon after, he found and studied Shuzo Fujimoto’s hira-ori techniques. That, along with collaboration with Jeremy Shafer, led to developing methods to translate these and other tilings into pleats and, eventually, to make the leap from paper folding to folding cloth. Palmer’s work came to wider notice with his presentation at the Second International Meeting of Origami Science and Scientific Origami in 1994 [97] and a subsequent interview and pictorial feature in the short-lived Japanese origami magazine, Oru [93]. CHAPTER 3. SIMPLE TWISTS

........

265

Figure 3.68. “Zillij Eightfold,” design by Chris K. Palmer, folded by Andy Wilson.

Figure 3.69. Tessellations by Ralf Konrad. Left: “Square Dance” (2006), from a design by Alex Bateman. Right: “Sweet Heart.” Both folded from a 30 × 40 cm rectangle.

266


Figure 3.70. Nano tessellations by Ralf Konrad. Top left: “Nano Cube Star Tessellation” (2006). Folded from 34 gsm glassine. Pleat folds are only 1.4 mm apart. Top right: “Orden Tessellation” (2007), Folded from 34 gsm glassine. This contains the “nano cube” pattern in the interior. Pleat folds are 2 mm apart. Bottom left: “Flagstone” (2007), from a design by Joel Cooper. Bottom right: “Hexagon Flagstone” (2006) from a design by Joel Cooper. Pleat folds are 2 mm apart.

One of the more recent tessellations artists is the German folder Ralf Konrad, whose highly precise folding of twist tessellations may be seen in Figure 3.69. As if such complex folding wasn’t enough of a challenge, Konrad specializes is what he calls nano tessellations, folding on a microscopic scale from paper in which the pleats are only 1–2 mm apart. A few of his nano folds are shown in Figure 3.70. As these examples show, twists can join, communicate, and, in general, play together to create marvelous patterns of beauty. But how do we put them together? We now have the bricks: let’s start building things from them.

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267

?

3.8. Terms Anto pleat A pleat around a twist where the acute angle at its base is anto, i.e., the two creases around the acute angle have opposite crease assignment. Brocard angle The angle at which the construction lines of a Brocard polygon intersect at a point. Brocard point The point at which the construction lines of a Brocard polygon intersect. Brocard polygon A polygon for which if a line is constructed from one vertex of each edge at the same angle relative to the edge, there is some angle at which all of the lines intersect at a point. If we construct a twist with the polygon as its central polygon, the twist has a closed back. Central polygon The polygon in the center of a simple flat twist. Clockwise (CW) twist A twist in which the central polygon rotates clockwise relative to the wedges going from the unfolded to folded state, with a twist angle in the range α ∈ (90◦, 180◦ ). Closed-back twist A twist fold in which all of the pleats meet at a single point in the center of the twist. Collapse The act of bringing together a large number of creases that must be folded all at once. Counterclockwise (CCW) twist A twist in which the central polygon rotates counterclockwise relative to the wedges going from the unfolded to folded state, with a twist angle in the range α ∈ (0◦, 90◦ ). Critical twist angle The twist angle beyond which an unavoidable layer collision occurs between two or more pleats in a simple flat twist. Cyclic overlap condition A condition on twist angle that ensures that two or more twist pleats do not mutually overlap at any point. Cyclic twist A simple flat twist whose central polygon is bounded by all mountain or all valley folds.

268


Flagstone tessellation A tessellation in which no visible polygon is overlapped by another. Grid tessellation A tessellation in which most or all of the creases lie upon a regular grid. Iso-area A property of assigned crease patterns such that that under interchange of mountain and valley folds followed by some other symmetry operation (often rotation), the pattern remains unchanged. Iso pleat A pleat around a twist where the acute angle at its base is iso, i.e., the two creases around the acute angle have the same crease assignment. Isometrically flat-foldable A pattern that can be flat-folded without strain or distortion, but ignoring the possibility of selfintersection. See semifoldable, ghost paper of Chapter 1. Maximal twist A twist where the paper outside of the twist extends sufficiently far that all possible non-local layer intersections must be considered in the determination of possible crease assignments. Minimal twist A twist with a sufficiently small region of paper around the central polygon so that the only possible layer intersections occur locally around each vertex. Open-back twist A twist fold that is only a single layer of paper thick in the middle of the folded form. Pleat creases/pleat folds The two folds that make up a pleat. Regular simple flat twist A simple flat twist in which the central polygon is a regular polygon. Safe twist angle The safe twist angle is the angle below which (for CCW twists) or above which (for CW twists) all possible anto crease assignments are flat-foldable. Simple flat twist A twist consisting of a central polygon surrounded by parallel-sided pleats, which folds flat. Supercritical twist A twist in which the twist angle exceeds the critical twist angle. The only regular twists that admit supercritical forms are the n = 3, 4, and 5 cases.

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269

Tessellation (origami) An origami representation of a dissection of the plane into geometric patterns where the borders are formed by folded edges and/or variations in the numbers of layers. Twist An origami pattern in which as the structure is folded, some part of the paper maintains its original orientation while a localized region undergoes rotation. Twist angle In a simple flat twist, the common angle between each of the pleats and the central polygon. Wedge A facet around the central polygon of a simple flat twist that is not a pleat.

270


4

Twist Tiles ?

4.1. Introduction to Twist Tiles

? 4.1.1. What is a Tile? In the last chapter, we looked at origami twists, which come in a wide variety of forms. Any convex polygon can be turned into a twist, simply by erecting parallel pleats, each emanating at the same angle from its side of the polygon. If the twist angle is constant, then the pattern folds flat isometrically (without regard to layer intersection); but when we take into account the possibility of self-intersection (and the need to avoid it), the possible crease assignments of the pleats and the polygon can get very complex indeed, with the possible assignments depending upon the twist angles, the central polygon angles, and the number and lengths of polygon sides. Despite the potential complexity, though, there is always a safe twist angle, αsafe , below which every possible anto crease assignment can fold flat without self-intersection. Still, there is only so much that one can do with a single twist. The power and beauty of origami tessellations arises when we put together twists into groups: two, three, ten, even hundreds. With multiple twists, the pleats going out of one twist become pleats entering others (or pleats that run off the paper). To create designs containing multiple twists, we need to learn how twists can be combined and how their pleats interact. There are many ways to look at the interactions of twists, and we will examine several in the course of this book. But to start with, we’ll turn twists into building blocks that snap together, like Legos™ of origami. We’ll start with a few simple blocks, identify their properties and how they join, then start putting them together into larger assemblies. 271

Figure 4.1. An equilateral triangle, a square, and a hexagon with M n cyclic twist crease patterns on them. Top row: crease patterns. Bottom row: folded forms (at larger scale).

Let us start with three twist crease patterns: a triangle, square, and hexagon twist, each placed within a larger version of that same polygon. Figure 4.1 shows three polygons, each containing a twist with the same number of sides as the polygon. The twists in these figures are cyclic, i.e., M 3 , M 4 , and M 6 , respectively. If we invert the crease assignment, then we get V 3 , V 4 , and V 6 versions, respectively, shown in Figure 4.2. There are, of course, different ways we could place a twist within a related regular polygon. You could imagine rotating the crease pattern to some other angle relative to the polygon. These particular twists are special, though: they mate with one another. We will call them tiles, and we will define presently just what, precisely, makes them special.

Figure 4.2. An equilateral triangle, a square, and a hexagon with V n cyclic twist crease patterns on them. Top row: crease patterns. Bottom row: folded forms (at larger scale).

272

........CHAPTER 4. TWIST TILES

Figure 4.3. Each of the crease pattern tiles mates with a tile of the opposite crease assignment so that the crease lines join up.

What do I mean by “mating”? If you take one of each flavor of tile, say, the two triangle tiles, then you can put the two crease patterns together so that their crease lines connect, as shown in Figure 4.3. The alignment of the crease lines is not simply an aesthetic property of the crease pattern. The edges of the folded forms mate in the same way. If you align the endpoints of a single side of the folded form, then all of the vertices along its jagged edge line up as well, as shown in Figure 4.4. And in fact, these two properties go together. If we join two pieces of a flat-foldable crease pattern into one, so that all of the creases that hit the join line cross in a straight line and match in fold type, then the resulting crease pattern will fold flat, and the resulting folded form will be what you get by joining the folded forms of the individual pieces along their corresponding edges. Figure 4.5 shows the results of this joining.

Figure 4.4. Each of the folded form tiles mates with its opposite number so that the fold lines join up.

........

CHAPTER 4. TWIST TILES

273

Figure 4.5. The folded forms of the three joined pairs of tiles.

This correspondence might seem obvious, especially if we think about going the other direction: cutting apart an existing crease pattern. Obviously if we cut the folded form into two separate pieces, then both the crease patterns and folded forms of the individual parts must mate along their respective join lines. So this result is not particularly earthshaking. (And there are situations when the forward-going relationship isn’t necessarily true, thanks to the problem of self-intersection. It is possible to create two separate crease patterns that fold flat individually but when you try to put them together, a layer collision arises. The BLBA Theorem provides an obvious way to construct such a scenario. For now, let’s just ignore those situations.) We’ll capture this idea in the Principle of Mating: If two flat-foldable crease patterns are joined along a single common line so that all creases that cross the line remain straight and of the same fold type, then the joined crease pattern can fold flat (apart from issues of self-intersection). There are a couple of important qualifications in that statement that I don’t want to let go by unremarked: • We ignore issues of self-intersection because it is rather easy to construct tiles that, when joined, force their layers to collide. But if they could pass through each other (like ghost paper), they would fold flat. • The join line must be a single line; it can’t be interrupted. So if we joined two patterns in such a way as to leave a hole in the result, it is possible that the result might not lie flat, even aside from self-intersection.

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Figure 4.6. Joined arrays of six triangle tiles and four square tiles. Top row: crease patterns. Bottom row: folded forms.

I’ll call this a “principle” for now, rather than a full-fledged theorem, because we’ll make it a bit more precise and a bit more rigorous later on. Its utility arises because this gives us a strategy for building up large, complex crease patterns from small, selfcontained building blocks. And indeed, we can build up larger assemblies of these tiles. Six triangles can be arrayed about a point to make a hexagon; four squares can be arrayed about a point to make a larger square. If we alternate M 3 and V 3 triangular twist tiles around a point, we can build up an arrangement of triangle twists that all fold flat together, and in the same way, by alternating M 4 and V 4 square twist tiles, we can build up a larger square composed of square twists. Examples of both are shown in Figure 4.6. In the crease pattern, I’ve overlaid the boundaries of the original tiles in orange. Hexagons are more problematic. In theory, we should be able to arrange three hexagons around a point. But the tiles shown above join in pairs of opposite parity, so that mountains and valleys meet up with corresponding creases on the adjacent tile, so the triangles and squares can alternate going around the common point. With six triangles, three of each type alternate. With four squares, two of each type alternate. But with three hexagons coming together around a point, we can’t alternate M 6 and V 6 twists.

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Figure 4.7.

Left: a single hexagon tile that mates in groups of three. Right: three such tiles. Top row: crease patterns. Bottom row: folded forms.

But we are not restricted to the cyclic crease assignments, of course. As we saw in the previous chapter, there are many possible crease assignments for a hexagonal twist. In fact, with the hexagon, if we alternate the creases around the central polygon, MV MV MV, then we can create a single hexagonal twist tile for which three copies can join together at a single point, making a group of three twists, as shown in Figure 4.7. The triangles, squares, and hexagons in the figures above were rendered at different scales. But if we scale them all so that the polygon edge lengths are the same, then an even more useful property becomes apparent: not only do they mate pairwise, but any of the three tiles can mate with any of the others—as long as mountains and valleys are chosen to line up. And that lets us put together assemblies of all three types of tile, as shown in Figure 4.8, which is a dodecagon built up from six M 3 triangle tiles, six V 4 square tiles, and one M 6 hexagon tile. This design isn’t the limit; it’s clear that by adding more tiles, one could build up even more complex and intricate folded patterns. These figures should give some indication of the potential of a tiling-based approach to constructing origami tessellations. By creating a library of building blocks that can join together, we can build up quite large and complex patterns that fold flat and that can be very beautiful indeed.

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Figure 4.8.

Left: a crease pattern composed of triangle, square, and hexagon twist tiles. Right: the folded form.

The tiles presented here—triangles, squares, hexagons—hint at the possibilities. But they raise many questions, too: what about other polygons? From which polygons can such tiles be created? (Why didn’t I include the pentagon?) What are the matching rules for putting tiles together? More broadly, what are the quantities that vary, and what are the things that must remain fixed? In the rest of this chapter, we’ll look at these questions in depth. ? 4.1.2. Ways of Mating What is a tile? I’ve been using the word already; let’s attach a definition: A tile is a crease pattern that can be joined with other similar or related tiles to make a single crease pattern, with no holes, that folds together. Note that qualification: “that folds together,” rather than, say, “that folds flat.” In this chapter, everything we do will fold flat (or at least, that’s what we will strive for). It is possible, though, to construct “tiles” that take on 3D forms. Nevertheless, for now, we’re looking for tiles that fold flat and, specifically, those that can be assembled into a single crease pattern that folds flat with no self-intersection. We assemble tiles by joining them pairwise or in groups along join lines—as if you taped together pieces of crease pattern and then folded the result. CHAPTER 4. TWIST TILES

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277

(Taping together individual tiles is, in fact, a very good way of creating and testing tiling-based tessellations.) To join two tiles, they must mate: the joined edges must meet up all the way along the join line with no gaps, and crease lines that cross the join line must match in fold angle and direction— mountains mate to mountains, valleys to valleys, and the lines must be collinear. And of course, the crease patterns within the tiles must fold flat. The tiles in the previous section were carefully constructed so that when we put two together, the fold lines met up with each other in both location and direction, and did so no matter how the tiles were put together (at least, as long as the corners were aligned). This was no accident, of course. I used regular polygons for the tile shapes; in a regular polygon, all of the sides are the same length, and all of the corner angles are the same angle. The crease pattern in each tile was a regular polygon twist, positioned so that the center of the twist was the center of the polygon. That is, perhaps, an obvious choice, one that arises naturally from the aesthetics and symmetry. We could have placed the twist anywhere within the polygon. Figure 4.9 shows some possibilities for a pentagonal tile. Is there a reason beyond aesthetics for choosing the center? Yes, very much: by placing the twist in the center of the polygon, all of the edges look the same, at least, considering the position and angle of the creases that hit the edge. Having them all look the same is an advantage: when we’re trying to put together groups of tiles, there’s only crease assignment to keep track of for this tile. If we know one edge, we know them all. There is another variable to consider, too: the rotational orientation of the twist within the polygon. Figure 4.10 shows three different orientations. Aesthetically, there’s less of a discrimination between these three choices (or others), but there is a very strong functional reason to prefer one: if we try to mate each of these with a rotated

Figure 4.9. Three ways of positioning a twist inside a regular pentagon.

278


Figure 4.10. Three ways of orienting a twist inside a regular pentagon.

copy of itself, then only with the center tile do the creases line up in both position and collinearity, as illustrated in Figure 4.11. The creases mate properly in position and collinearity, but not in type. This was why we needed two types of each tile to create the pairs in the previous section. But, as we have seen, by altering the crease assignment around the central polygon and accordingly in the surrounding pleats, we can change the assignment on any edge. We could invert the crease assignment of one of the two pentagons in Figure 4.11 to achieve perfect mating between the tiles—or even invert the assignment on a single edge. And of course, it may be that by choosing an alternating crease assignment, like the MV MV MV hexagon of the previous section, we can create a tile that can mate with itself, just as long as we choose the proper pair of edges to link up. Since we know that we are going to require changing one of the tiles by inverting its parity, we can consider another change that would allow a different orientation of the twist to create a useful tile. The creases in the third example in Figure 4.11 all hit the edges of their respective tile polygons at 90◦ , so they match in direction, just not in position. If we make the lower tile the mirror image of the upper tile, though, then the creases match up in all ways, as shown on the right in Figure 4.12.

Figure 4.11. Pairing each of the three tiles with a rotated copy.

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Figure 4.12. Pairing each of the three tiles with a reflected copy.

We have two distinct ways of creating tiles that mate with minimally modified versions of themselves (and, perhaps, with similarly defined tiles created from other polygons). The center subfigure of Figure 4.11 shows one: the tile mates with a rotated copy of itself as long as the fold parity is reversed on the mating edge of the mating tile. The right subfigure of Figure 4.12 shows the second: the tile mates with a mirror image of itself. Both of these forms of mating give rise to families of tiles. ? 4.1.3. Centered Twist Tiles Let’s look at that first type of mating: tiles that mate with rotated but parity-reversed versions of themselves. Figure 4.13 shows three matings—or rather, two successful matings and one mismatch.

Figure 4.13. Three pairs of pentagon tiles. Left: mating tiles. Middle: mating tiles with a larger twist. Right: mismatched tiles.

280

a b

w w

b a


a b

w w

b a

In each tile, the edge is divided into three parts by the two folds of the pleat. In the upper tile, let’s call the lengths of the three segments a, w, and b, from left to right. Since the lower tile is a rotated copy of the upper tile, the lengths of the corresponding segments in that tile are, again from left to right, b, w, and a. So in order for the upper and lower tile lines to meet at the same place, we must have the same distance in both tiles, a = b. That means that the segment of the tile edge between the pleat folds must be precisely centered within the tile edge. For this reason, we will call this class of tile a centered twist tile. Centered twist tiles come in different varieties because we can vary the size of the central twist polygon. The center subfigure of Figure 4.13 shows a mating pair with somewhat larger twists. Observe, though, that many of the angles are different, and the middle segment width, w, is different from the pair on the right. These two varieties of tile do not mate with each other, as the right subfigure illustrates. The creases miss each other in both position and angle. If we wish to build up a family of tiles that can all mate pairwise, there must be some common factors among them. Clearly, the distances a, b, and w must be the same from one tile to the next. Suppose for simplicity that the tiles were unit polygons, i.e., their edge lengths are all equal to 1. Then it’s easy to see that we must have a = b = (1 − w)/2. So if w is preserved from one tile to the next, then the folds of mating tiles will meet up in position. If we scale everything to a larger or smaller size, the same rules apply; we just interpret a, b, and w as the fractional lengths of the tile edges. There is a second degree of freedom, though: the angle at which the fold hits the edge. Looking at the right subfigure of Figure 4.13, even if the top folds touched the bottom folds, they’d still meet the edge at different angles and the folds wouldn’t be collinear across the edge. So there are two independent quantities that characterize a particular collection of centered twist tiles: the width of the pleat segment along the edge, w, and the angle at which the creases hit the edge, which we’ll call the tilt angle, or τ. These quantities are illustrated in Figure 4.14. The pleat width/tilt angle pair (w, τ) define a group of mutually mating tiles, as long as they all have the same edge length. Take any two, place them edge to edge, and the creases will line up in

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281

a t 1Ðw 2

w

t 1Ðw 2

Figure 4.14. Schematic of a centered twist tile. The width of each edge where its pleat hits it is w. The tilt angle is τ. These two quantities are sufficient to completely specify the location of all the creases, though not their parities.

position and collinearity. Adjust the crease assignment to match, and you have a fully mated arrangement of tiles that will fold flat—subject to some considerations on layer intersection, which we will get to. And each possible set of values (w, τ) gives a different group of tiles, that all mate within the group, but that, in general will not mate with those from a different (w, τ) set. We will call each distinct (w, τ) set of centered twist tiles a species. In the animal kingdom, the members of a species mate with other members of their own species but not with other species; we’ll use this same notion to describe the mating abilities of tessellation tiles. Extending the biological analogy, we will call the collection of all possible centered twist tiles—all values of (w, τ)—a genus, since they are all closely related but do not mate across species lines. Centered twist tiles, which mate with rotated copies of themselves,1 are one genus. In Figure 4.12, we saw a different way of mating (with reflected copies); those tiles will form a different genus, which we will explore further shortly. Note that the tilt angle τ is not the same as the twist angle α, which is the angle at which each pleat hits the central polygon (also marked in the figure). As we will see, there is a simple and elegant relation relating these three quantities. 1

282

There are limits to how far we should push this analogy.


Figure 4.15.

M n twist tiles for w = 0.15, τ = 75.5◦ . Top row: n = 3, 4, 5 sides. Bottom row: n = 6, 7, 8 sides.

Figure 4.16.


Once we have chosen a set (w, τ), we can construct twist tiles from this species for regular polygons with any number of sides. Figures 4.15–4.17 show examples for n = 3 to 8 sides for M n assignments of three different species of centered twist tile. The (w, τ) way of labeling each species also provides a simple, straightforward way of constructing any tile with nothing more than a pencil, ruler, and protractor (or, nowadays, a computer drawing program), using the procedure illustrated in Figure 4.18. There are some considerations that link crease assignment and the issue of self-intersection. When we are mating two tiles, we must match creases in angle and assignment along their shared edge. That condition sets the crease assignment of one (or the other) pair of pleat folds, and, as we know from twists, that assignment sets the crease assignment of the corresponding central polygon edge.

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283

Figure 4.17.


t 1Ðw 2

1. Construct a regular polygon.

w

2. Divide each edge into three parts, making the central portion a fraction w of the edge length.

4. Connect the intersections of pairs of lines to create the central polygon.

Figure 4.18. Construction process for a centered twist tile.

284


t

1Ðw 2

3. Construct lines emanating from those points, each making angle t with the edge from which it comes.

5. Assign the creases to the central polygon and the surrounding pleats.

But we also know that not all twist crease assignments are flat-foldable without layer intersection, and the valid assignments depend on the twist angle at the central polygon. Examining the three species of twist tiles in Figures 4.15, 4.16, and 4.17, we see that their central polygon twist angles are not all the same. For the first species in Figure 4.15, the twist angle is α = 30◦ for all three tiles; for the second, in Figure 4.16, it is 40◦ ; for the third, in Figure 4.17, α = 55◦ for all three tiles. (And that is an interesting behavior in itself: For every tile within a (w, τ) species, the twist angle α is the same; it has no dependence on n, the number of sides of the polygon. There was no a priori reason to expect that.) As we saw in the previous chapter, for regular n-gons, there is a safe twist angle αsafe , at or below which all possible anto crease assignments are flat-foldable. Checking Equation (3.16) and Figure 3.41, we see that α = 30◦ and α = 40◦ are safe for all of the polygons shown in the first two species of tiles. Any crease assignment will be flat-foldable. But for the third species, w = 0.50, τ = 70.7◦ , the twist angle is α = 55◦ , and for that twist angle, only certain crease assignments in the central polygon are possible: • For n = 3–5, cyclic twists are forbidden. • For n = 6, all assignments are allowed. • For n = 7–8, only cyclic twists are allowed. Depending on how we combine these twists with others, we may or may not be able to find crease assignments that satisfy all of these conditions. But there’s actually more to consider. Remember that the conditions on M M/VV and MV/V M vertices in a twist were local conditions, which depended only on what was happening at the vertices, but the existence of cyclic twists was governed by the possible intersections of layers coming from different vertices. The conditions from the previous chapter applied to isolated twists, but when we put three or more tiles together at a vertex, something new happens, illustrated in Figure 4.19. There are two interior vertices of the orange tiling, each where three tiles come together. The pleats that surround each vertex form a polygon—a right triangle for each vertex. And that triangle also is the center of a twist! It is surrounded by parallel pleats

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285

Figure 4.19. Four mated tiles for w = 0.50, τ = 70.7◦ : two cyclic octagon tiles and two MV MV square tiles.

and has a twist angle of 180◦ − 55◦ = 125◦ . (Or we can think of it as a CW twist with twist angle 55◦ .) This twist, too, is potentially subject to cyclic layer collisions, and the existence of some will depend on the twist angle and crease assignment. This right triangle is not a regular polygon, and so the conditions on twist angles are not as simple as those for a regular polygon. In this particular case, it turns out that the layers do not interfere.2 But we will not always be so lucky. There’s a second issue: even if we created a tile, can we combine it with anything else, even copies of itself? Certainly we can create chains of tiles, mating each tile with one other tile on either side. But if we want to make large closed regions of a tessellation crease pattern, then we’ll need to combine tiles that create interior vertices, as we did in Figures 4.6, 4.7, 4.8, and 4.19. Figure 4.19 hints at the problem: once we have put together two octagons, the only tile that closes the corner is the square tile. But if we had put two heptagons together, no regular polygon tile, or combination of such tiles, would have allowed us to create a closed region around either end of the join line. In the next chapter, we will look systematically at which regular polygon tiles can be assembled at a vertex, and indeed, we will build up some quite large and intricate origami twist tessellations. Fully addressing the question of which crease assignments are permitted requires a bit more mathematics than we have used up to this point, so I will defer that until a bit later in this chapter. But we already have a tool kit to construct quite a few origami 2

I encourage you to construct and fold this pattern. It comes together in a very interesting way.

286


tessellations. The tiles of Figures 4.15 and 4.16 have safe twist angles of α = 30◦ for their central twists; they can be assembled in many ways, with many different crease assignments, to realize many and varied origami twist tessellations. And they are not the only twists, of course. We saw earlier that there was a second genus of twist tiles: those that mate with their mirror image. We now turn our attention to those. ? 4.1.4. Offset Twist Tiles As we saw in Figure 4.12, if we rotate the central twist in a tile so that the pleat folds are perpendicular to the tile edges, we obtain another type of tile, which I will call an offset twist tile. As the name suggests, the pleat is offset from the center of each edge. We characterize such a tile by two numbers: again, we’ll use w as the fractional width of that portion of the edge subtended by the pleat. The second number will be the offset, d, which is the distance from the center of the edge to the center of the pleat, as illustrated in Figure 4.20. Offset twist tiles are a different genus from centered twist tiles. The distances on either side of the pleat, a and b in the figure, are readily shown to be a=

1−w + d, 2

b=

1−w − d. 2

(4.1)

a d a

w b

Figure 4.20. Schematic of an offset twist tile. The width of each edge where its pleat hits it is w. The offset distance is d. These two quantities are sufficient to completely specify the location of all the creases, though not their parities.

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Offset twist tiles can be constructed geometrically in much the same way as centered twist tiles, as was described in Figure 4.18. First, construct the polygon; then divide the edges in fractional distances a, b, w; add the pleats (this time they are perpendicular, so no angle measurements are needed); and connect the intersection points of consecutive pairs to create the central polygon. The offset distance d can be positive or negative. A positive value gives a CCW twist (with twist angle α < 90◦ ), while a negative value gives a CW twist (with α > 90◦ ). As with centered twists, each set of values (w, d) defines a species of tile within which any tile can mate with any other. But while centered twist tiles mated with tiles of opposite crease parity, offset twist tiles mate with the same crease assignment but in mirror image—or equivalently, a tile whose d-value is of the opposite sign. This is an important difference when it comes to putting tiles together to make them mate. With centered twist tiles, if two tiles of the same species come together with the wrong parity, they can be made to mate by switching the crease parity of the pleat along that one single edge. But if two offset twist tiles of the same species come together and are of the wrong parity, then you need to invert the offset distance of the entire tile. That changes the matching rules on all of the other edges. it may not be possible to find a mutually consistent set of tiles and mirror-image tiles for a large array. Or for that matter, at a single vertex. Consider what happens when we bring some number of tiles together at a vertex. We choose parameters (w, d) for the first tile. The next one must be the mirror image, so it gets (w, −d). Then the next is back to (w, d), and so forth, all the way around. The offset distances must alternate going around the vertex, and that means that there must be an even number of tiles at every interior vertex in the tiling. That is a very limiting requirement. It means that there is no offset tile analog of Figure 4.19, because there, three tiles met at each vertex. In the same way, there is no offset tile analog of Figure 4.7, where three hexagons came together. However, we can certainly assemble six triangles or four squares. Figures 4.21 show examples of both, with cyclic mountain twists. Crease patterns made with cyclic twists can look very different on the front and back side, and the figure shows both sides of the folded form.

288


Figure 4.21.

Top row: six offset triangle twist tiles. Bottom row: four offset square twist tiles. Left: crease pattern. Middle: folded form. Right: back side of the folded form.

The arrangement of square tiles is of particular interest: it has the very strong appearance of woven strips, particularly when created in larger arrays and with fully opaque paper, as shown in Figure 4.22. We will explore this concept further in later chapters. As with centered twist tiles, each distinct set of parameters (w, ±d) defines a unique species of tile. Tiles can mate within their own species, but not across species. Figure 4.23 shows the members of the species (w, d) = (0.2, 0.1) for polygons with number of sides n = 3 through 8. Here we see another difference from centered twist tiles: the twist angle α is visibly different from one tile to the next. In fact, it is smallest for n = 3 (α = 30◦ ) and gets steadily larger as the number of sides increases. In fact, for this particular set of parameters, the twist angle turns out to be the critical twist angle for a cyclic twist. In general, then, when choosing a species, we’ll need to check the twist angles in all distinct polygons against the possible crease assignments. As with centered twist tiles, the

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289

Figure 4.22. A square weave pattern, composed of alternating offset twist tiles. Left: crease pattern and tiling. Right: folded form from opaque paper.

Figure 4.23.

M n twist tiles for w = 0.2, d = 0.1. Top row: n = 3, 4, 5 sides. Bottom row: n = 6, 7, 8 sides.

crease assignments can be anything that is flat-foldable without self-intersection. But because the tiles in an offset twist tiling must alternate between CCW and CW twists, if the twist angles avoid self-intersection, an arrangement that is cyclic for all tiles is always possible. Still, there is the rather strict requirement that the number of polygons at a tile vertex be even. There also is the strict requirement that in order to create a large region of tilings, we must have tiles that can be placed with no gaps all around a vertex. Not all regular polygon tiles have this ability; we are about to find out which ones do, and why. ?

4.2. Vertex Figures There is an infinite number of regular polygons, but when we impose the requirement that they must be used to tile the plane, the number of suitable polygons drops precipitously, to a mere handful.

290


If we want to create a large origami tessellation with a lot of structure, we need to use a lot of tiles. If we want it to have a reasonable aspect ratio—say, square, or mildly rectangular—then the tiling is going to have interior vertices, places where tiles come together and completely close up. A large origami tessellation is going to have many such vertices. It is reasonable to ask, then, which of the tessellation tiles can be arranged around a vertex so that they leave no gaps? And for this consideration, we don’t even need to worry about the crease pattern within the tile, whether it is centered twist, offset twist, or something else. All we really need to consider is the bounding polygon, and specifically, its interior angles. Recall that the interior angle of a regular polygon of n sides is given by 360◦ 2 ◦ ◦ 180 − = 180 1 − . (4.2) n n The sum of the angles around a vertex should be 360◦ . So if we have k polygons with n1, n2, . . . , nk sides, all meeting at a common vertex, the sum of the angles around the vertex is 2 2 2 ◦ ◦ ◦ 180 1 − + 180 1 − + . . . + 180 1 − = 360◦ . n1 n2 nk

(4.3)

This simplifies to

1 1 1 k −2=2 + +...+ . n1 n2 nk

(4.4)

This is what is called a Diophantine equation: an equation whose solutions are required to be positive integers. For every integer value of k, if there are integers {n1, n2, . . . , nk } (not necessarily different) that satisfy Equation (4.4), then that means that the k polygons with sides {n1, n2, . . . , nk } fit neatly around a vertex with no gaps, and so, in principle, tessellation tiles with those numbers of sides could be used to create an origami tessellation. It turns out that there are only 17 sets of positive integers that satisfy Equation (4.4). That means that there are only 17 possible sets of regular polygons that can fit around a single vertex without leaving any gaps. For example, the k = 6 set {3, 3, 3, 3, 3, 3} works, since 1 1 1 1 1 1 6−2=2 + + + + + = 4. (4.5) 3 3 3 3 3 3 So do the sets {3, 3, 3, 4, 4}, {5, 5, 10}, {3, 7, 42}, and 13 other sets.

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291

Figure 4.24. A {3.3.3.4.4} vertex (indicated by the black dot) has three triangles and two squares arranged around it.

We can use the listing of the numbers in the set as a description of the polygons around a vertex. We’ll call such a set the vertex figure, and we will separate the numbers by periods. For example, {3.3.3.4.4} is the figure of a vertex that has three equilateral triangles followed by two squares arranged cyclically around the vertex. The order of the numbers specifies the order of the polygons encountered as we travel around the vertex (by convention, we’ll travel counterclockwise). Figure 4.24, for example, shows a {3.3.3.4.4} vertex. Given a vertex, you can construct its vertex figure by starting with one polygon and writing down each polygon number as you go around the vertex. Since the point where you start counting polygons is somewhat arbitrary, we’ll consider two figures equivalent if a cyclic permutation of the list—taking some numbers off the right side and reattaching them at the left—makes them the same. So, for example, {3.3.3.4.4} is considered the same as {3.3.4.4.3}, {3.4.4.3.3}, {4.4.3.3.3}, and {4.3.3.3.4}. And indeed, as shown in Figure 4.25, all of these vertex figures are the same pattern, each merely a rotated version of the others. We will also consider two vertex figures equivalent if the order of one is the reversal of the other. So, for example, {4.6.12} is

{3.3.3.4.4}

{3.3.4.4.3}

{3.4.4.3.3}

Figure 4.25. Five identical vertex figures.

292


{4.4.3.3.3}

{4.3.3.3.4}

{3.3.3.4.4}

{3.3.4.3.4}

Figure 4.26.

Left: a {3.3.3.4.4} vertex. Right: a {3.3.4.3.4} vertex. These vertex figures are considered distinct because there is no way to turn one into the other by simple rotation or order reversal.

the same vertex figure as {4.12.6}. The two orders correspond to whether we go clockwise or counterclockwise around the vertex. By convention, we will use the order that is lexicographically smaller to describe the vertex figure: {4.6.12}, in this example. Some of the sets of polygons admit more than one arrangement that are distinct from one another, even considering cyclic permutations and order reversal. For example, {3.3.3.4.4} is a different arrangement from {3.3.4.3.4}, because there’s no cyclic permutation or order reversal that transforms one into the other. Geometrically, they’re different too; it’s not possible to rotate one vertex pattern to make it equal to the other, as shown in Figure 4.26. When you include all possible distinct arrangements of polygons, there are 21 different vertex figures for vertices surrounded by regular polygons. These, along with their vertex patterns, are shown in Figure 4.27. Any vertex of any tiling of the plane composed of regular polygons must match one of these 21 figures. And that means that if we seek to create an origami tessellation entirely from regular polygons, we can only use polygons that appear somewhere in this figure, i.e., those regular polygons with side number 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 20, 24, and 42. The small numbers—3, 4, 6—are obvious, of course, but some of the others are not. Who would have guessed that one could build a regular polygon tessellation with a complete vertex that includes a 42-gon, but not from anything between n = 25 and 41?

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293

{3.3.3.3.3.3}

{3.3.3.3.6}

{3.3.3.4.4}

{3.3.4.12}

{3.4.3.12}

{3.3.6.6}

{3.6.3.6}

{3.4.4.6}

{3.4.6.4}

{3.7.42}

{3.9.18}

{3.8.24}

{3.10.15}

{3.12.12}

{4.4.4.4}

{4.5.20}

{4.6.12}

{4.8.8}

{5.5.10}

Figure 4.27. The 21 vertex figures possible with regular polygonal tiles arranged edge to edge.

294


{3.3.4.3.4}

{6.6.6}

Figure 4.28. Building a tiling that includes a {3.7.42} vertex. This is a zoomed-in view; the polygonal arc is a portion of the 42-gon.

Of course, this is no guarantee that we can build an arbitrarily large tessellation from regular polygons that include a 42-gon, or for that matter, any other specific vertex figure, because every interior vertex in the tiling must be one of these 21 possibilities. Suppose we wanted to build a tiling that includes a {3.7.42} vertex, as illustrated in Figure 4.28. We start with such a vertex (left black dot in Figure 4.28). Since that’s the only vertex figure that contains either a 7-gon or 42-gon, the other black dot must also be a vertex of that type, albeit in reverse order (that is, {3.42.7}), which forces another 7-gon to fill it out. But now, at the gray dot, there are two 7-gons, and there is no vertex figure that fills that out. Indeed, we can see that the angular gap is too small to accommodate any regular polygon. So we’re stuck; there is no way to extend this tiling any farther outward from the 42-gon, at least, if we’re only using regular polygons. Similar considerations apply to several other vertex figures and side numbers, of course. Many of the examples I’ve shown have been with pentagons; it would be lovely to build up a tiling with pentagonal twists. But there is only a single vertex figure that includes pentagons: {5.5.10}. If we want to build up a tiling from this vertex, we could start with a decagon and surround it with pentagons, as in Figure 4.29.

Figure 4.29. Building a tiling that includes a {5.5.10} vertex. We start with a decagon and then array pentagons around it. But when we try to add another ring, there’s nothing that fits at the marked vertices.

........


295

Figure 4.30. A tiling of centered twist tiles made up of {5.5.10} vertices. Left: crease pattern tiling. Right: folded form.

But then if we want to add more polygons around the outside, the only thing that fits between two pentagons is another decagon, and after that, there’s nothing we can pack into the next vertices going around the pattern. If we want a crease pattern without gaps or slits, we have to stop there. While this pattern does make a very nice tessellation (see, for example, Figure 4.30), it is not readily extended to larger or more complex patterns. The situation is even more limiting when we consider offset twist tiles. Remember that for those, every interior vertex must have an even number of tiles around it—put mathematically, each tile vertex must be of even degree. Going through this list, we find far fewer possibilities. There are only eight vertex figures of even degree: any tiling composed solely of regular offset twist tiles must have every vertex be one of these, shown in Figure 4.31. That is enough to start building tessellations based on a tiling of polygons. We can choose a vertex figure to start with, then add tiles to fill the plane, making sure that we only choose tilings whose vertex figures are contained in Figures 4.27 or 4.31, depending on whether we’re going to use centered twist tiles or offset twist tiles. We then choose species parameters: (w, τ) for centered twists or (w, d) for offset twists, and then choose crease assignments for the

296


{3.3.3.3.3.3}

{3.3.4.12}

{3.4.3.12}

{3.3.6.6}

{3.6.3.6}

{3.4.4.6}

{3.4.6.4}

{4.4.4.4}

Figure 4.31. The eight vertex figures possible with an even number of regular polygonal tiles arranged edge to edge.

central polygons and their attached twists. The choice of crease assignment is made based on a combination of criteria: • Matching rules—So that the crease types are preserved across the tile boundaries, • Aesthetics—The desired appearance of the tiling, • Avoidance of layer intersection—So that the individual tiles, as well as the entire pattern, can fold flat. Of these three criteria, the last is the most complicated and, at this point, may still seem a bit mysterious. We know that possible crease assignments depend on the twist angle, and the twist angle depends on the tile parameters (and, in the case of offset twist tiles, on the number of sides). Just what that dependence is requires a bit more mathematics, which we now dive into. ???

4.3. Vertices and Angles Let us now pull out the vector mathematics introduced in Chapter 1 to construct and analyze centered and offset twist tiles.

........


297

y

Figure 4.32.

v3

c

Schematic of a unit polygon (a pentagon is shown) in standard orientation.

p p n n

v1

0

1

v2

x

? ? ? 4.3.1. Unit Polygons Although we described simple twists by centering their central polygons on the origin, the construction and analysis of twist tiles and their crease pattern/folded form lines is simplified if we adopt a different standard orientation for individual tiles. It turns out to be computationally advantageous to place one of the vertices of the polygon at the origin and one of the edges on the x-axis, as shown in Figure 4.32 for a pentagon. This will be the standard position for a twist tile. We will denote the vertices of a unit n-gon by {vi }, i = 1, . . . , n. We adopt the convention that the polygon is located in the upper half-plane with the first vertex, v1 , at the origin (0, 0) and the second, v2 , on the x-axis at the point (1, 0). Then the center c of the unit n-gon is located at 1 c= 1, cot πn , (4.6) 2 while the n vertices {vi } are located at, respectively, vi = c − R 2π(i−1) · c, (4.7) n where

cos φ − sin φ R(φ) ≡ (4.8) sin φ cos φ is the rotation matrix through angle φ. A regular unit polygon can be drawn by computing the vertices {vi } and then connecting them with lines in serial order. Equation (4.8) has the property that the vertex indices “wrap around”—meaning that we have vn+i = vi for all i. This is computationally convenient, and we will assume going forward, unless otherwise specified, that all indexed points wrap around in the same way, both forward and backward.

298


v4 r4 s4 t q4

v5

t

a

q3

a q5

r5 t

v1

1Ðw 2

r3 v3

a t

q2

r1

t w

t

a

a q1

s5 t

s3

t

t

t s1

1Ðw 2

s2

r2

v2

Figure 4.33. Construction of a centered twist tile, illustrated for a pentagon. Extend lines inward from points along the edge at the pleat angle τ. The intersections of adjacent pleats form the vertices of the inner polygon.

? ? ? 4.3.2. Centered Twist Tiles Let us now construct a centered twist tile, following the procedure illustrated in Figure 4.18. We number and label the relevant vertices as shown in Figure 4.33. To construct the unit twist analytically, we locate the points r j and s j along the polygon edges: r j = (1 − w)v j + (w)v j+1, s j = (w)v j + (1 − w)v j .

(4.9) (4.10)

It is convenient to define the unit vector u(φ) ≡ (cos φ, sin φ).

(4.11)

Then each of r j and s j launches a line at angle (2π( j − 1)/n + τ), whose direction is given by the unit vector u(2π( j − 1)/n + τ). Each of the s j−1 lines intersects the r j line, giving vertex q j , which we can find by using the line intersection formula from Equation (1.35), i.e., 2π( j−1) 2π( j) q j = lineint s j−1, u + τ , r j , u n + τ . (4.12) n The analytic expressions resulting from Equation (4.12) are rather complex, and we omit them here. But once we have such

........


299

expressions, we can use them to calculate the relationship between the pleat angle τ, the pleat width w, and the twist angle α (shown in Figure 4.33), since the twist angle can be given from the rotation angle formula, Equation (1.32), as α = tan−1 (q j − q j+1 ) · (s j − q j+1 ), (q j − q j+1 ) · R( π2 ) · (s j − q j+1 ) . (4.13) With some manipulations, we can show that a very simple relationship relates these three quantities, namely, tan α = w tan τ.

(4.14)

Somewhat surprisingly, the twist angle α depends only upon the pleat width w and pleat angle τ, and not at all upon the number of sides of the unit polygon. Recall from Chapter 3 that there are limits on the twist angles, depending upon the crease assignment, that permit flat-foldability. These same relations now impose limits on the pleat width w and pleat angle τ. Since tan α is a monotonic function, the inequalities translate directly into inequalities on the tilt angle α. Recall that the three conditions on twist angle α for CCW twists (α < π/2) were 1. M M/VV condition, α < π − 2. MV/V M condition, α < 3. Cyclic condition, α <

π 2

2π n ;

2π n ;

− πn .

Which of these conditions applies depends upon the crease assignment of the central polygon, but recall that there was a safe twist angle αsafe that allowed every possible crease assignment of the central polygon (with anto assignment of adjacent pleats). The safe angle was αsafe (n) =

      

π 2

−

π n

if

n ≤ 6, (4.15)

2π n

if

n ≥ 6.

Since the twist angle is common to every tile, the twist limitations must be considered for every tile in the pattern. Consequently, for a tiling consisting of several different values of n, the safe twist angle for the tiling would be the smallest of the {αsafe (n)}

300


for all of the {n} represented in the tiling. For arbitrarily large n, αsafe (n) becomes arbitrarily small, so there is no single value of αsafe that works for all tilings. We note, however, that π6 (30◦ ) is safe for n = 3 to 12, and so for many tilings of unit polygons, this is a good working maximum value. We can turn these into conditions on (w, τ) through Equation (4.14). Since the crease assignment limiting conditions generally depend directly upon the twist angle α, there is some value in using it, rather than the tilt angle τ, as the primary identifier for the species. Using Equation (4.14), we can do this. That is, we can characterize a centered twist tile species by the pair (w, α), where α is the desired twist angle (common to every tile and twist polygon in the tiling). Then, when it comes to constructing the tile, we solve for the tilt angle: τ = cot−1 (w cot α) .

(4.16)

There are a few other interesting parameters worth exploring. The central polygon is a smaller copy of the tile polygon; how much smaller is it? We can easily calculate this. Since the original tile polygon had unit edge length, the shrinkage factor is just the side length of the inner polygon, given by |qi+1 − qi | for any edge. As with the twist angle/tilt angle relationship, it turns out to be surprisingly simple (surprising, considering the algebra that it takes to get it). The shrinkage factor ρ is given by r ρ=

(1 + w 2 ) + (1 − w 2 ) cos 2τ . 2

(4.17)

The central polygon is also rotated relative to the original tile. The rotation angle β is given by β = tan−1

(1 − w) sin 2τ . (1 + w) + (1 − w) cos 2τ

(4.18)

As with the α–τ relationship, these values are independent of the number of sides of the polygon. They also indicate an alternative way of constructing the crease pattern tile: shrink it by a factor ρ, then rotate it through an angle β. The notion of shrinking and rotating polygons to create twist tessellations is a powerful one, and we will come back to this in later chapters. CHAPTER 4. TWIST TILES

........

301

v4 r4 s4

q2

t

v5

Figure 4.34.

s3

t

q3

r3 v3

r5 q4 s5

Construction at the self-contained limit, when the central polygon hits the boundaries of the tile.

q1 t v1

q5 1Ðw 2

s1

r1 w

t 1Ðw 2

s2 r2 2p/n

v2

Equation (4.17) can also be written as p ρ = cos2 τ + w 2 sin2 τ,

(4.19)

which makes it quite clear that for nonzero w, ρ is strictly smaller than 1; the central polygon is, indeed, smaller than the original tile. That does not mean that the central polygon is always confined to the tile polygon; because of the rotation, it may well extend outside of the polygon. The edge case arises when points {qi }, the corners of the central polygon, coincide with points {ri } on the edges of the tiling, as shown in Figure 4.34. A tile for which the central polygon lies entirely within the tile is self-contained. From the geometry of the figure, it is evident that an n-gon centered twist tile will be self-contained only for tilt angles π τ≤ . (4.20) n For tilt angles below the self-contained limit, the central polygon lies outside of the tile, and the vertices {ri } lie inside the central polygon, as illustrated in Figure 4.35. This might seem nonsensical, even useless; but with the proper interpretation, such a tile could still be quite useful in constructing an origami tessellation. The vertices {ri } and {si } are not vertices of the crease pattern; they are vertices of the tiling, which is, in some sense, a fictitious construction: a scaffolding upon which we erect the crease pattern, said scaffolding to be taken away once the construction is complete. For such a tiling, the crease lines {(qi, ri )} lie inside of the central polygon, true; but we can think

302


v3 q2 r3

s2

s3

r2 r1

v1

s1

q3

q1

Figure 4.35. v2

A centered triangle twist tile that is not self-contained.

of those lines as negative length creases, representing distance to be taken away from other positive-length creases with which they are combined. An example might make this clear. Figure 4.35 shows a triangle with w = 1/3, τ = 45◦ (which gives a twist angle of α ≈ 18.4◦ ). I’ve extended the ri lines both inside and outside of the tile (thin lines). We still have creases connecting points ri and si to the triangle vertices qi , but since the ri lines point the wrong direction, I’ve drawn them in red. We can still assemble such tiles into valid crease patterns. Figure 4.36 shows two such triangles. We deal with the negative-

qi

qj

rj

Figure 4.36.

Left: joining two non-self-contained tiles. Creases along tile borders have been extended to a new border. Right: drop the negative crease lengths and construction lines, leaving an assignable crease pattern.

........


303

Figure 4.37.

Left: the assigned crease pattern. Right: the folded form.

edge-length creases in two ways, depending on whether they appear between tiles or along the outside border of the pattern. • For negative-edge-length creases between two tiles, the tilted tile creases are part of a crease that runs between two central polygon vertices—qi and q j in the inset to the figure. The crease should be a straight line between qi and q j , and for properly self-contained tiles, it is. So we simply delete the extra vertex r j and replace each pair of lines (qi, r j ) and (r j , q j ) with a single direct edge (qi, q j ). • If there are negative-edge-length creases on the border of the pattern, we add an extension all the way around that encloses all of the central polygon corners. Then we extend all of the existing creases to the new border, absorbing negative-edge-length creases in the same way. Figure 4.37 shows the resulting crease pattern, fully assigned, along with the folded form. This technique works for any non-self-contained tile. Tiles become non-self-contained when the pleat width w is large and the tilt angle τ is small, and this, in turn, gives rise to small twist angles α and large central polygons. Since small twist angles are more versatile with respect to crease assignment, non-self-contained tiles can be quite useful for constructing easily foldable crease patterns that roughly resemble the original tile one begins with. ? ? ? 4.3.3. Offset Twist Tiles We can construct the vertices of an offset twist tile in the same way that we did for the centered twist tile. The construction is illustrated in Figure 4.38.

304


v4

s4

s3

r4

v5 a

q4

a q 3

r3

v3

a q2

s2 r2

q5 q a 1 a

r5

s5 v1

w r1 s1

v2

Figure 4.38. Construction of an offset twist tile, illustrated for a pentagon. Extend lines inward from points along the edge perpendicular to the edges. The intersections of adjacent pleats form the vertices of the inner polygon.

In terms of the tile parameters (w, d), the vertices r j and s j along the polygon edges are 1−w r j = ( 1+w 2 − d)v j + ( 2 + d)v j+1,

(4.21)

sj =

(4.22)

( 1−w 2

− d)v j +

( 1+w 2

+ d)v j .

Then each of r j and s j launches a line perpendicular to the edge, so that the direction of both lines is given by the unit vector u(2π( j − 1)/n + π/2). Each of the s j−1 lines intersects the r j line, giving vertex q j , which we can find by using the line intersection formula: q j = lineint s j−1, u 2π(nj−1) + π2 , r j , u 2π(n j) + π2 . (4.23) As we did before, we can use the values of {q j } to solve for the twist angle α, which we find to be hw πi α = tan−1 cot . (4.24) 2d n This brings us a significant difference from centered twist tiles: the twist angle depends explicitly on the number of sides. (Of course, we saw this in Figure 4.23.) The greater the number of sides, the larger the twist angle. In fact, Figure 4.23 showed a special case: if w = 2d, then we have h πi π π α = tan−1 cot = − = αcrit, (4.25) n 2 n the critical twist angle.

........


305

We can also calculate other geometric parameters. They, too, depend explicitly on the number of sides. The shrinkage ρ, the size of the inner polygon relative to that of the tile, is q ρ = w 2 + 4d 2 sec2 πn − 1 , (4.26) and the rotation angle is β = tan

−1

2d π π tan = − α. w n 2

(4.27)

The special case of w = 2d makes the twist angle the critical twist angle, giving a closed-back twist, and this will be the case for every twist in the tessellation. This can give a nice effect to the completed tessellation, as will be seen in some of the examples. We can also consider the question of self-contained-ness: when can the central polygon extend outside of the tile? As was the case in Figure 4.34, the central polygon would touch the edge when the crease extending from point si hits the point ri+1 on the next edge around. But since the crease emanating from si is always perpendicular to the edge, it can hit the next edge only if the interior angle of the polygon is acute. This only happens for n = 3. Working out this geometry, the edge case happens when 3w + 6d = 1. An offset twist tile will be self-contained if n > 3 or w≤

1 (1 − 6d) or 3

d≤

1 (1 − 3w). 6

(4.28)

As with centered twist tiles, we can still construct valid flatfoldable tessellations using non-self-contained tiles, but we’ll need to employ the same adjustments to deal with any negative crease lengths that arise as a result. ? ??

4.4. Folded Form Tiles

?? 4.4.1. Centered Twist Folded Form Tiles Up to now, I’ve presented twist tiles primarily as a means of constructing crease patterns. We use the tiling as a scaffold, fill it with crease pattern tiles, weld the tiles together into a single sheet, then fold the result. And then we can see what we get. We can also create tiles that describe the folded form, and if we put those tiles together in the same way that we assembled

306


Figure 4.39.

Folded forms for tiles with w = 0.15, τ = 75.4◦ , with superimposed tile lines.

the crease pattern tiles, then we can obtain a representation of the folded form—without having to actually fold the crease pattern. Of course, folding a tessellation is half of the fun of its creation. But if we can construct a representation of the folded form without having to fold it, we can then use that representation to tweak and select tiles and arrangements for the crease pattern tiling—and have a pretty good idea before we actually fold it what it’s going to look like. With centered twist tiles, while the crease pattern tiles are regular polygons, as we see in the examples shown back in Figure 4.4, the folded forms of these tiles do not result in regular polygons. In general, the edges form a jagged zigzag, due to the angled pleats. When we join such tiles, they don’t meet up edge to edge, either; there is a necessary overlap, again due to the presence of the pleat. However, we can still create tiles of the folded forms. Although the edges of the folded forms do not meet up edge to edge, the vertices certainly do. So we can superimpose a tile polygon over the folded form, as shown for the three folded form tiles from Figure 4.4, now with superimposed tile lines in Figure 4.39. Now, these tiles have bits that stick out beyond the tiles and gaps within the tiles. But if we put two of them together, the sticking-out bits of one will fill in the missing bits of its neighbor— no matter which tiles we put together, as long as we use tiles of the same species and match valleys and mountains in their pleats. So we could clean up the folded form tiles, trimming off the extra bits, filling in the missing bits, and we would then have tiles that fit together edge to edge and that give a very close approximation of the folded form. The result for the three tiles above would be the three tiles shown in Figure 4.40.

........


307

Figure 4.40.

Folded forms tiles with w = 0.15, τ = 75.4◦ .

If we assemble these tiles in the same arrangement as the crease pattern tiles, the result will be a very good approximation of the folded form. In fact, it will be exact everywhere, except along the edges of the paper, where the true folded form edge would be a bit jagged. But if all we’re after is getting an idea of what the folded form looks like, a bit of imprecision along the border might well be acceptable. But how do we construct these folded form tiles? We could, of course, construct each crease pattern tile, fold it up, then rotate it (to make the bottom edge horizontal), enlarge it (to make the folded form tiles also unit polygons), and extend/truncate the pleats lines (so that they end on the tile lines). But there’s something interesting about these folded form tiles: they look an awful lot like the crease pattern tiles. They’re built the same way: there are two vertices along each edge, there are fold lines emanating from them along a tilt angle, and the intersections of consecutive fold lines around the polygon define a rotated polygon, which is a shrunken and rotated version of the tile polygon. In fact, the pure geometric structure of the folded form tiles exactly mirrors the structure of the crease pattern tiles, with one noticeable difference: the order of the crease lines whose intersections define the central polygon. Let’s take a look at folded form tiles next to (and scaled to the same size as) their corresponding crease pattern tiles, as in Figure 4.41. Comparing the two, it is clear that there are similarities, but the pleat width is larger in the folded form (owing to the overall reduction of scale that happens when you fold a twist), and the tilt angle is smaller (because straightening out the jagged edge introduces a small rotation). The big difference is the order of the intersecting creases. In the crease pattern, the crease on the

308


Figure 4.41.

Top row: crease pattern tiles with w = 0.15, τ = 75.4◦ . Bottom row: corresponding folded form tiles.

bottom edge that interacts with a crease on the right edge is on the right side of the bottom edge; in the folded form tile, it is on the left. This is the situation that would arise if we made a sign error in constructing the tile and made the value of w negative. In fact, if we chose a negative w of the right magnitude and chose a slightly different value of tilt angle τ, then built a crease pattern tile, it would be exactly similar to the pattern of lines of the folded form tile. So we don’t need an entirely new set of machinery to create folded form tiles. We can find a set of equivalent (w, τ) parameters (in which the new value of w is negative), construct the lines of a crease pattern tile with those parameters, and we would have the vertices and lines appropriate to the corresponding folded form tile. Even better, it turns out that there is a very simple relationship between the crease pattern and folded form parameters. If we take (w, τ) to define the crease pattern and define (w0, τ0) as the equivalent parameters for the folded form tiles, it can be shown that w , 1 − 2w τ0 = tan−1 [(1 − 2w) tan τ] .

w0 = −

(4.29) (4.30)


........

309

Figure 4.42.

Top left: an offset twist tiling constructed from folded form tiles with w = 0.15, τ = 66.6◦ (α = 30◦ ). Top right: the corresponding crease pattern tiling. Bottom left: the resulting crease pattern. Bottom right: the true folded form.

This relationship makes it quite straightforward to construct the folded form of an origami tessellation. In fact, tessellations can be designed by working with the folded form tiles; put together folded form tiles of a single species in an interesting way, possibly trying out different crease assignments and/or tile parameters. Once you’re happy with the result, swap out the folded form tiles, swap in the crease pattern tiles, and that gives the crease pattern you need to fold it. And when you do fold, you will get a pattern that is the same (except for the border) as the folded form tiling you started with. Figure 4.42 shows this process for a tiling of square and octagon twists.

310


Figure 4.43.

Top row: crease pattern tiles with w = 0.15, d = 0.11. Bottom row: corresponding folded form tiles.

? 4.4.2. Offset Twist Folded Form Tiles As you might expect, we can construct folded form tiles for offset twist tiles as well, and it works even better. Because the pleats are perpendicular to the polygon edges, the edges of the folded form of an offset twist crease pattern tile are automatically aligned to the edges of the tile polygon; no filling-in or trimming-off are needed. Figure 4.43 shows offset twist crease pattern tiles and their corresponding folded form tiles. The bottom row of Figure 4.43 shows the folded form tiles, but these are exactly what you’d get if you folded the crease pattern tiles (although the result would be somewhat smaller in scale, of course). As with centered twist tiles, the geometric pattern of vertices and fold lines may be computed by assuming equivalent tile parameters. They are w w0 = − , (4.31) 1 − 2w d d0 = . (4.32) 1 − 2w And so, once again, one can design a tiling by assembling folded form tiles to get a desired appearance, then swapping in the corresponding crease pattern tiles to build the crease pattern.

........


311

Figure 4.44.

Top left: an offset twist tiling constructed from folded form tiles with w = 0.2, d = 0.1. Top right: the corresponding crease pattern tiling. Bottom left: the resulting crease pattern. Bottom right: the true folded form.

Figure 4.44 shows the process for an offset tessellation tiling. Of course, we are restricted to tilings in which interior vertices have only even degree, but even with that restriction, there are many interesting and beautiful tilings that are possible to construct. ??

4.5. Triangle Tiles

?? 4.5.1. Centered Twist Triangle Tiles It is a little bit frustrating that several of the regular polygons only go together with other regular polygons in a single vertex figure,

312


Figure 4.45.

A centered twist triangle tile with corner angles (50◦, 60◦, 70◦ ) and species parameters w = 0.1, α = 25◦ . Left: crease pattern. Right: folded form.

effectively preventing the construction of larger twist tessellations that include them. As we saw in Figure 4.28, we very quickly reach a position where there is an angular gap that can’t be filled in by a regular polygon. Wouldn’t it be nice if we could fill in such gaps with something else, though—like, perhaps, a triangle that isn’t regular? We could certainly try the same approach for constructing the crease pattern tile that we used for regular polygons: divide the edges with a relative pleat width w, fire off crease lines at tilt angles τ, let them intersect, connect the points, and then see what we get. An example is shown in Figure 4.45 for an arbitrary example: one with corner angles of 50◦ , 60◦ , and 70◦ , and correspondingly different side lengths. Indeed, the crease pattern looks like it has the same twist angle at every vertex, and therefore it must fold flat. And it does; the right subfigure in Figure 4.45 shows the folded form. But did we get lucky, or does it always work? Well, we can set up a computation of the points—triangle vertices {vi }, tile edge vertices {ri, si }, and central polygon vertices {qi }, just as we did for the unit polygons. Only in this case, we won’t use unit polygons; we’ll use an arbitrary triangle, specified by its three corner angles (θ 1, θ 2, θ 3 ). We will adopt the same standard position as we did for unit polygons: vertex v1 is at the origin, v2 is at the point (1, 0), and the triangle lies in the upper half-plane, as illustrated in Figure 4.46. The three sides are different lengths (as, of course, they will be if the angles are different). Instead of making w the absolute width of the pleat along each edge, we instead divide all three

........


313

v3 q3

y r3 s3

Figure 4.46. Geometric construction of a centered twist triangle tile with corner angles (θ 1, θ 2, θ 3 ).

t t

t

q3

v1

q1

q1 1Ðw 2

r1

t

a

a

s2 r2

q2 a

t w

q2

t

s1

1Ðw 2

v2

x

sides in the same proportion and make w the pleat width relative to the length of each side. So the three pleats are, in general, of different widths in absolute terms. But they all work together to give rise to the same twist angle, which turns out to be the same as for unit polygon twists, given by Equation (4.14). Perhaps this is not surprising. If the twist angle did not depend on the number of sides of the polygon, then it seems plausible that it wouldn’t depend on the interior angles of a triangle. Indeed, the other properties of a centered twist tile do carry over. The central polygon is a shrunken and rotated version of the tile polygon, and the shrinkage factor and rotation angle turn out to be exactly the same. And this will allow any two edges of different tiles that have the same length to mate properly, since their pleat widths and angles will still line up. That, in turn, raises a question: what is the center of shrinkage and rotation? In a regular unit n-gon, the center is obvious from symmetry. In an arbitrary triangle, though, it is, perhaps, not so obvious. Triangles have many different types of centers—points that are important for various geometric properties. It is not clear, a priori, which is the center of interest here. With a bit more mathematics, though, we can establish that for any triangle with any set of base angles, the center of shrinkage and rotation of the central polygon is the circumcenter—the center of a circumscribing circle. Equivalently, that is the point that equalizes the three distances from it to the corners of the triangle, as illustrated in Figure 4.47. The three tile vertices v1, v2, v3 lie on the circumcircle; the three central polygon vertices q1, q2, q3 also lie on a common,

314


v3 q3

q3 q2

v1

q1

q1

q2

v2

smaller circle with the same center. In fact, it is easy to see that the ratio between the two circle radii must be ρ, the shrinkage factor already defined. But this behavior highlights something else interesting about this triangle twist. Each edge of the central polygon is shrunken and rotated from the original tile in the same way, and the creases that come from one side of the triangle don’t depend in any way on the other two sides. In Figure 4.47, the red creases come only from the bottom of the triangle. The blue creases come from the right side, and the green from the left. We could have constructed these three sets of creases entirely independently. Yet, when we put them together, the creases and vertices all line up in such a way as to create a flat-foldable crease pattern tile. Now we have a way of “plugging the gaps” in a centered twist tiling that uses some of the less versatile regular polygons. We can assemble regular polygons into a tiling and then fill in the non-equilateral triangles with triangular twist tiles. As a simple example, Figure 4.48 shows a tiling that is the 7-fold rotationally symmetric equivalent of Figure 4.8, which was a hexagon surrounded by squares and equilateral triangles. If we replace the hexagon with a heptagon, we can keep the squares, but now the triangles become isosceles triangles with an apex angle of 360/7 ≈ 51.4◦ . Recall from the previous chapter that for cyclic crease assignments (M n or V n ), there is a critical twist angle αcrit above which some of the pleats must intersect. For the equilateral triangle,

Figure 4.47. Circumcircle about a centered twist triangle tile with corner angles (θ 1, θ 2, θ 3 ).

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315

Figure 4.48.

A tessellation from heptagon, square, and isosceles triangle tiles with w = 0.15, α = 29◦ . Left: tiling and crease pattern. Right: folded form.

the critical twist angle was 30◦ . For an arbitrary set of polygon interior angles, the critical angles may be found by solving a somewhat complex trigonometric equation. For general triangle twists, there was a simple analytic formula for the critical angle, which was given by Equation (3.17). For corner angles θ 1, θ 2, θ 3 , the lower critical angle is sin θ 1 sin θ 2 sin θ 3 −1 αcrit = tan . (4.33) 1 + cos θ 1 cos θ 2 cos θ 3 For the isosceles triangles in the heptagon tessellation of Figure 4.48, the critical twist angle works out to αcrit ≈ 29.6◦ , which I have taken as the twist angle to use for the pattern. The triangle twists turn out to be closed-back; the square and heptagon twists are open-back, with the heptagon having a larger hole than the squares. ?? 4.5.2. Offset Twist Triangle Tiles Our success with creating a centered twist tile for an arbitrary triangle bodes well for trying the offset twist tile in a general triangle. And indeed, it works. Figure 4.49 shows the crease pattern and folded form for an offset twist constructed within a scalene triangle. As we did with the centered twist tile, the species parameters—w and d—are taken to be fractions of the respective

316


Figure 4.49.

An offset twist triangle tile with corner angles (50◦, 60◦, 70◦ ) and species parameters w = 0.09, d = 0.06. Left: crease pattern. Right: folded form.

side lengths. The crease pattern produces a valid flat-foldable twist with the same twist angle on all three sides (necessary for flat-foldability), and, as with the regular polygon offset twists, the edges of the folded form are all collinear, thereby forming a smaller folded form tile. With centered twist triangle tiles, the twist angle for a triangle was independent of the corner angles of the triangle, just as it was independent of the number of sides for regular polygons. With offset twist tiles, the twist angle depended explicitly on the number of sides of a regular polygon, and so it should come as no surprise that in a triangle, it depends on the angles of the triangle. We can set up a construction analogous to Figure 4.46 for the offset twist tile as in Figure 4.50. We then solve for vertex coordinates and angles. We will find that each pleat does indeed make the same angle with the central polygon, creating a valid flat-foldable twist. The twist angle is v3 q3

y

s2

r3 s3

r2

q3

a a

a

v1

Figure 4.50.

q2

q1

q1 1Ðw +d 2

r1

w

s1

q2 1Ðw Ð d 2

v2

x

Geometric construction of an offset twist triangle tile with corner angles (θ 1, θ 2, θ 3 ).

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317

given by α = tan

−1

w sin θ 1 sin θ 2 sin θ 3 , 2d 1 + cos θ 1 cos θ 2 cos θ 3

(4.34)

where, as before, the corner angles are θ 1, θ 2, θ 3 . That expression should look familiar: part of it is shared with the critical angle for a triangle twist in Equation (4.33). So we could write this more simply as h i −1 w α = tan tan αcrit . (4.35) 2d If we would like our triangle twists to be closed-back, then we want α = αcrit , and consequently we should take d = w/2, as was the case with regular polygon offset twists. Since the twist is valid and flat-foldable with a constant pleat angle, the central polygon must be geometrically similar to the outer polygon. It is easily shown that the shrinkage factor ρ is ρ = w csc α,

(4.36)

π − α. 2

(4.37)

and the rotation angle β is β=

With triangular centered twist tiles, the center of shrinkage and rotation is always the circumcenter of the triangle, independent of the parameters (w, τ) (or (w, α)). The center of shrinkage and rotation is a fixed point of the scaling/rotation/offset transformation that takes the outer tile to the central polygon. We can solve for that fixed point for any offset twist tile. It turns out, though, that—unlike the centered twist tile—the center of shrinkage and rotation for an offset twist tile varies with the parameters (w, d); there is not a unique point that works for every set of parameters. Still, we can readily construct offset triangle tiles by simply propagating the pleats in from the triangle sides and finding their intersection. We are guaranteed to get a flat-foldable tile, and such tiles can then be combined with the regular n-gon offset tiles to create flat-foldable twist tessellations. Figure 4.51 shows a heptagon tiling, using the same underlying tiling as in Figure 4.48, but now with offset twist tiles.

318


Figure 4.51.

An offset twist tessellation from heptagon, square, and isosceles triangle tiles with w = 0.15, d = 0.075, and all cyclic mountain twists. Left: tiling and crease pattern. Right: folded form.

? ?? ???

4.6. Higher-Order Polygon Tiles

? 4.6.1. Centered Twist Cyclic Polygon Tiles Let’s look again at Figure 4.47, in which the three differently colored sets of crease lines could be constructed independently, and yet their vertices all aligned with one another when we put them together. Each triangle edge is a chord of the circle. If we subdivide the edge, add tilted pleats, then top the two pleats with a shrunken/rotated edge for the inner polygon, we have what we might call a sub-tile associated with the chord. If three chords connect to make a triangle with vertices on the circle, then the three subtiles associated with each chord will align to create pieces of an overall flat-foldable tessellation tile. What is interesting about this is that nothing in the alignment of chords and subtiles depended on the chords being part of a triangle; if they are chords of the same circle and have the same species parameters, they’ll automatically align together with the same twist angle. This means that we could do the same thing with a larger number of chords, i.e., tessellation tiles with a larger number of sides. What the triangular centered twist tile has shown is more than just a way to plug triangular gaps in tilings. We can, in fact,

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319

use this concept to create centered twist crease pattern tiles that work for polygons with any number of sides of any length and any corner angles—as long as all of the sides are chords of the same circle, which requires that the vertices of said polygon lie on a common circle. A set of points lying on a common circle are concyclic. A polygon whose vertices are concyclic is a cyclic polygon. The centered twist tiling construction works for more than just regular polygons and triangles. It in fact works for every cyclic polygon. That extends the family of centered twist tessellation tiles considerably! (Note that we’re now using the term cyclic in two different ways: crease assignment and circumscribability. In the discussion that follows, we’re using it in the latter sense.) As we’ve seen, every triangle supports centered twist tiles. Since one can draw a circle through any three points, every triangle’s vertices are concyclic. The circumcenter of the vertices is the intersection of the perpendicular bisectors of the three sides, so to find the circumcenter of a triangle, one can construct the perpendicular bisectors of any two sides and find their intersection. That intersection will be the circumcenter, and the perpendicular bisector of the third side will pass through it. For polygons with a larger number of sides, though, there is no guaranteed existence of a circle that passes through all vertices. There are, however, broad classes of polygons that work. For n = 4, we’ve already seen that squares are cyclic, and it’s not hard to see that rectangles are as well. Trapezoids that possess a line of mirror symmetry are cyclic as well, but other symmetric quadrilaterals—rhombuses, kite shapes, and parallelograms—are, in general, not cyclic, although of course specific examples of all three may be constructed. Examples of cyclic quadrilateral crease tiles are shown in Figure 4.52. These crease pattern tiles do indeed fold flat, and they may be combined with other centered twist tiles, appropriately rotated and scaled, to create even more origami tessellations. Every rectangle and mirror-symmetric trapezoid is cyclic, and so the construction technique that we applied to regular polygons and arbitrary triangles will work on them, too. Divide each side with a centered pleat of relative width w, construct fold lines at tilt angle τ, and connect intersections to form the central polygon. The resulting tile will be flat-foldable and can be crease-assigned and/or joined with other tiles to realize still more centered twist

320


Figure 4.52. Four examples of cyclic quadrilaterals and centered twist MV MV tiles for w = 0.15, α = 30◦ , along with circumscribing circles. The circumcenter is shown as a dot in the middle. Top left: square. Top right: rectangular. Bottom left: symmetric trapezoid (base angles of 80◦ ). Bottom right: generic quadrilateral (angles of 80◦, 90◦, 100◦, 90◦ ).

tile tessellations. As an example, in Figure 4.53 I have surrounded a regular nonagon with a ring of trapezoids. As Figure 4.52 illustrates, some other quadrilaterals are cyclic. Which ones? And what about higher-order polygons? Which, if any, are cyclic? To answer that, we need to delve further into the geometry of their construction. ? ? ? 4.6.2. Cyclic Polygon Construction For cyclic polygons with an odd number of sides, if we specify the interior angles of its corners and the positions of two vertices (as we’ve done with our unit polygons, with v1 = (0, 0) and v2 = (1, 0)), then the rest of the polygon—the lengths of the sides and the positions of the vertices—is completely determined, as are the position and radius of the circumcircle.

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321

Figure 4.53.

A centered twist tessellation from a nonagon and nine trapezoid tiles with w = 0.13, α = 30◦ . Left: tiling and crease pattern. Right: folded form.

For even degree, though, if we specify the corner angles, there is a one-parameter family of cyclic polygons that have those angles. Why there is a difference requires a bit deeper exploration of such polygons, which reveals an elegant property of some of their angles. Consider a cyclic n-gon as illustrated in Figure 4.54. We define the interior angles of the corners {θi } and draw radii from the center to each corner. We define the central angles {ψi } as the angles between successive radii. Clearly, if we specify v1 = (0, 0) and v2 = (1, 0) and know the central angles {ψi }, then we can find the circumcenter c and

v3

y

vn

yn

Figure 4.54. Construction of a cyclic polygon with interior angles {θ i } and central angles {ψi }.

322

q1 q1 v1


c y1

y2 q2 q2 v2

x

radius r: c = ( 12 , 12 cot ψ21 ),

(4.38)

r=

(4.39)

1 2

csc ψ21 .

Then the rest of the vertices would be defined: vi+1 = c + R(ψi ) · (vi − c).

(4.40)

So, finding the polygon is equivalent to finding all of the central angles. Examining the individual triangles in Figure 4.54, we see that the corner angles and central angles are simply related: θi = π − 12 (ψi + ψi−1 ).

(4.41)

We can put this in matrix form. Define © θ ≡ «

θ1 ª θ2 ® , .. ® . ¬

so that

© ψ ≡ «

ψ1 ª ψ2 ® , .. ® . ¬

© π ≡ «

π ª π ®, .. ® . ¬

θ = π − Mn · ψ,

(4.42)

(4.43)

where Mn is the circulant matrix © Mn = «

0 . . . 0 12 ª 1 ® 2 ... 0 0 ® .. .. . . .. .. ® . . . . ® . . ® 0 0 . . . 12 0 ® 0 0 . . . 12 12 ¬ 1 2 1 2

(4.44)

The central angles are given by ψ = M−1 n · (π − θ),

(4.45)

which is simple and straightforward, except for one thing: sometimes the matrix Mn is singular. In fact, as it turns out, it is singular for n even and nonsingular for n odd. For n odd, the inverse is easy to find. The (i, j)th element of the inverse is M−1

n i, j

i+ j−1    (−1)  (−1)i+ j =   1 

if i < j, if i > j, if i = j.

(4.46)

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323

For n even, M is singular. There is a single vector in its null space, which we define as θ 0 . It is given (to within a multiplicative constant) by 1 © ª −1 ® ® θ 0 ≡ 1 ®® , (4.47) −1 ® . ® . « . ¬ i.e., a vector of elements with alternating sign. The general solution for θ is then given by ψ = M+n · (π − θ) + ζ θ 0,

(4.48)

where ζ is any constant and M+n is the pseudoinverse (also called the Moore-Penrose inverse) computed from the singular value decomposition of Mn . This is the one-parameter family of solutions; we can choose any value for ζ and for each we obtain a different set of central angles {ψi } and thus, a different cyclic polygon. The pseudoinverse M+n turns out to be relatively simple. Its (i, j)th element is 2 M+n i, j = (−1)i+ j−1 ((i − j + 12 ) mod n) − 2n , (4.49) n and the first two useful pseudoinverses are

M+4

© 1 = 4 «

3 3 −1 −1 ª −1 3 3 −1 ® ®, −1 −1 3 3 ® 3 −1 −1 3 ¬

© 1 + M6 = 6 «

5 5 −3 1 1 −3 ª −3 5 5 −3 1 1 ® ® 1 −3 5 5 −3 1 ® ®. 1 1 −3 5 5 −3 ® ® −3 1 1 −3 5 5 ® 5 −3 1 1 −3 5 ¬

(4.50)

Whenever two sets of variables are related by a matrix equation such as Equation (4.43) and the matrix is singular, there is a conservation law associated with each of the vectors in the null space of the matrix. Since θ 0 is in the null space of Mn , it must be true that θ 0 · (Mn · ψ) = 0 (4.51) for any ψ, and since θ 0 · π = 0 as well, we have that 324


θ 0 · θ = 0,

(4.52)

or θ 1 − θ 2 + θ 3 − θ 4 + . . . = 0.

(4.53)

Equation (4.53) must be satisfied for the corner angles of any cyclic polygon with an even number of sides. It has the same form as the Kawasaki-Justin Condition on the sector angles of a flat-foldable vertex. Each corner vertex is incident to a quadrilateral formed by the perpendicular bisectors of the polygon sides and half of each incident side. The angle opposite to the corner angle is its supplement, θ¯i = π − θi, (4.54) and so it follows that these, too, satisfy a Kawasaki-like condition: θ¯1 − θ¯2 + θ¯3 − θ¯4 + . . . = 0.

(4.55)

But these angles now are the sector angle of a vertex: namely, the vertex formed by the perpendicular bisectors. This is an interesting property: if a cyclic polygon has an even number of sides, the perpendicular bisectors of its sides meet at a point (the circumcenter) and form a flat-foldable origami vertex. For quadrilaterals, this result is equivalent to the well-known property that in a cyclic quadrilateral, opposite corner angles sum to π. For odd-order polygons, the matrix is nonsingular, so there is no corresponding conservation law, nor connection to flatfoldability. Coming back to the construction problem, for an odd number of central angles, the set of corner angles is unique, but for an even number of central angles, the corner angles are unique to within some additive multiple of the null space vector θ 0 . That means that we are free to place some other condition on the polygon. By making it a unit polygon, we’ve already specified that the first side has unit length (setting v1 = (0, 0) and v2 = (1, 0)). We could, for example, choose the side length of one of the other sides. With the central angles parameterized on the scalar ζ, each of the other vertex coordinates will similarly be some expression in terms of ζ, and thus, the side lengths are also dependent upon ζ. Setting one of the other sides to some specified value will fix the value of ζ, and that will fully determine the cyclic polygon. CHAPTER 4. TWIST TILES

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325

Figure 4.55. An offset twist crease pattern for vertices (0, 0), (1, 0), (.9, .5), (.2, 8) and offset twist parameters w = 0.2, d = 0.15. It is not flat-foldable.

?? 4.6.3. Quadrilateral Offset Twist Polygon Tiles Now we look at the question of quadrilateral offset twist tiles. A first inclination might be to simply try out the tile construction on an arbitrary quadrilateral: divide each edge according to pleat width w and offset d, project pleat lines until they intersect, connect up the intersections, and see what we get. Alas, for a general quadrilateral, the resulting crease pattern is not flat-foldable. Figure 4.55 shows an example, and with this example, the four twist angles—the angles each pleat makes with the central polygon—all turn out to be different, ranging from 20.9◦ to 50.5◦ . That is, perhaps, not too surprising. We saw that for centered twist tiles, higher-order polygons need to be cyclic (their vertices concyclic, all lying on a common circle). It seems plausible that some similar restriction would apply to higher-order offset twist tiles. Looking back at the known offset twist tiles, there is one striking common feature: for all of them, there is a critical twist angle αcrit that gives rise to a closed-back tile. Granted, that critical twist angle is different for the various regular n-gons and triangles. But it does suggest that there is some fundamental connection between offset twist tiles and closed-back twist polygons. And indeed, there is. Certainly for triangles, we saw that an arbitrary triangle can be turned into an offset twist tile for any twist parameters, with cyclic crease assignments possible for twist angles up to those that give a closed-back triangle twist, and some mixed assignments for higher twist angles. So perhaps we should look at higher-order closed-back polygon twists. And indeed, if we choose a symmetric trapezoid that is a closed-back trapezoid twist and carry out the offset twist construction, it does create a valid offset twist tile that folds flat (isometrically, not necessarily considering self-intersection) for

326


Figure 4.56. An offset twist crease pattern tile with base angle θ = 70◦ and offset twist parameters w = 0.14, d = 0.07.

any combination of twist parameters (w, d). An example is shown in Figure 4.56. This construction works for any closed-back trapezoid twist, but that stipulation sets the aspect ratio of the trapezoid. Recall from the previous chapter that the diagonal side length relative to the (unit length) base is p s = 2 − sin2 θ − cos θ. (4.56) The central polygon is a geometrically similar copy of the tile polygon with shrinkage factor p w 2 + 2d 2 (3 + cos 2θ) ρ= (4.57) sin θ and rotation angle " β = tan−1

w sin θ 2d

r

# 2 . 3 + cos 2θ

(4.58)

The twist angle is α=

β π+β

if if

d > 0, d < 0.

(4.59)

The twist has a critical angle: sin θ αcrit = sin−1 √ . 2

(4.60)

For twist angles up to the critical angle, cyclic crease assignments are possible; for larger twist angles, only certain supercritical crease assignments are possible. It is easily shown that, as with the other offset twist tiles, α = αcrit happens for w = 2d.

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327

Figure 4.57. An offset twist crease pattern tile with corner angles (95.58◦, 72.75◦, 81.92◦, 109.73◦ ) and offset twist parameters w = 0.14, d = 0.08.

As for other arbitrary quadrilaterals, while it is possible to find specific examples of irregular quadrilaterals that give rise to flatfoldable offset twist tiles, for most of them, we lose the desirable property that every pair of species parameters (w, d) gives rise to a flat-foldable tile. Figure 4.57 shows one example. This tile has a flat-foldable offset twist tile for w = 0.14, d = 0.09, but for each possible value of w, there is only a single value of d that gives rise to a flat-foldable tile. Furthermore, there is no closed-back solution; for this twist angle of α = 36.13◦ , three of the pleats intersect at a point, but the fourth one misses the intersection. Still, we have a versatile set of offset twist tiles: for any given (w, d) combination, there are offset twist tiles for all regular polygons, all possible triangles, and there is a trapezoid for every possible base angle. If we choose species parameters w = 2d, then every twist in the pattern can be closed-back, at least, if the mountain/valley matching rules and self-intersection considerations permit. There is, however, one further limitation; as we make the trapezoid base angles more and more acute, the position of the twist moves down toward the base, as illustrated in Figure 4.58. If a vertex of the central polygon dips below the edge of the tile, it could potentially interfere with whatever it is joined to. Figure 4.58. A trapezoidal offset twist crease pattern tile with base angles of 60◦ and offset twist parameters w = 0.06, d = 0.04.

328


Figure 4.59. An offset twist tessellation of all closed-back M 4 twists composed of squares, equilateral triangles, and trapezoids with base angles of 75◦ and offset twist parameters w = 0.2, d = 0.1.

Whether this situation arises depends on a somewhat complex relationship between the species parameters (w, d) and the trapezoid base angle θ, but, generally speaking, as the base angle dips toward the neighborhood of 60◦ , the range of acceptable (w, d) becomes small (and limited to small values), limiting the size of the possible twist. As the trapezoid base angle opens up toward 90◦ , the range of possible twist parameters opens up, permitting some very lovely patterns. Figure 4.59 shows a tessellation of a square and equilateral triangles with 75◦ trapezoids acting as the “glue” that binds them together. With the addition of a second size of square, this pattern can be extended arbitrarily as well. The polygons we have identified thus far that work for ranges of (w, d) in offset twist tiles also work for centered twist tiles. Regular polygons and arbitrary triangles already worked for both; and since symmetric trapezoids are also cyclic (their vertices are concyclic), they, too, can be used for centered twist tiles. We can, therefore, use this exact same tiling of polygons to create a centered twist tessellation, as shown in Figure 4.60. Here I have chosen a twist angle of 30◦ , so as to make the triangle twists closed-back. Note, though, that only the triangle twists are closed-back at this twist angle. In centered twist tilings, all of the twist angles are the same. That means that the fewer-sided polygons are closer to their critical values, while larger polygons have holes in their twists. Conversely, with offset twist tiles, the twist angles vary from polygon to polygon and it is possible to have all of the polygon twists exist in the closed-back state.

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329

Figure 4.60.

A centered twist tessellation of M n and V n twists composed of squares, equilateral triangles, and trapezoids with base angles of 75◦ and centered twist parameters w = 0.2, α = 30◦ .

?? 4.6.4. Offset Twist Higher-Order Polygon Tiles We have found one type of quadrilateral that produces a flatfoldable offset twist tile for a range of (w, d) species parameters, the symmetric trapezoid. Such a polygon that works for a range of (w, d) parameters is said to be generic for offset twists. Are there other generic quadrilaterals? And what about higher-order polygons: are there generic pentagons, hexagons, and more? If we postulate the existence of such polygons, we can establish some of their properties by considering some limiting cases. Suppose, for example, we focus on closed-back offset twists, for which w = 2d (at least, in all the polygons we’ve seen so far). But now we consider what happens as the pleat width w becomes arbitrarily small. Then the central polygon will collapse toward a point, and the two lines of each pleat will collapse toward a single line, which is the perpendicular bisector of the edge from which each pleat emanates, as illustrated in Figure 4.61. Figure 4.61. Two offset twist pentagon tiles for a generic pentagon. Left: w = 0.25, d = 0.125. Right: w = 0.025, d = 0.0125.

330


In fact, for w = 2d, the mountain fold in each case is precisely the perpendicular bisector of its edge. So in the limit as w → 0, the perpendicular bisectors will all intersect at a point. And that means that the polygon must be cyclic, with the point of intersection the circumcenter of its vertices. We saw in the previous chapter that for a polygon to form a closed-back twist, it must be a Brocard polygon. And we have already seen that for an offset twist, the central polygon is geometrically similar to the outer polygon. So in the closed-back limit, the outer polygon must be a Brocard polygon. And thus, we have two requirements: the polygon must be both cyclic and a Brocard polygon. And it turns out that these two conditions are sufficient as well. Any cyclic Brocard polygon is generic for offset twist tiles, providing a flat-foldable tile for a range of (w, d) parameters. It turns out, though, that we don’t have a great deal of freedom in choosing the angles and sides of the polygon. Recall that for a given set of corner angles, the ratios between sides of a Brocard polygon are fixed, and there is no guarantee of concyclicity. It turns out that no matter how many sides we might choose, we can specify only two corner angles; then the dual requirements of Brocard-ness and vertex concyclicity completely specify the remaining angles and relative side lengths. Figure 4.62 shows three examples of offset twist polygon tiles where we have specified the first two angles for n = 4, 5, and 6 sides. In general, the Brocard angle for a convex n-gon must be solved for numerically, but we can establish some bounds on the Brocard angle and thus, the closed-back twist angle. In a paper by Besenyei describing earlier work by Dmitriev and Dynkin [12], it was shown that for any point P in a convex n-gon with vertices {Ak }, k = 1, . . . , n, min ∠P Ak Ak+1 ≤ k

π π − , 2 n

(4.61)

with equality if and only if P is the center of a regular n-gon. Thus, for a Brocard polygon (where all of the angles in Equation (4.61) are equal), this inequality applies to the Brocard angle and, thus, the maximum twist angle for the polygon.

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331

Figure 4.62. Crease pattern tiles (top row) and folded form tiles (bottom row) for generic offset twist polygons for species parameters w = 0.2, d = 0.1. Left: corner angles (85◦, 100◦, 95◦, 80◦ ). Middle: corner angles (85◦, 100◦, 124.35◦, 126.25◦, 104.40◦ ). Right: corner angles (85◦, 100◦, 134.07◦, 146.24◦, 140.93◦, 113.76◦ ).

? 4.6.5. Pathological Twist Tiles The construction algorithms for twist tiles give many useful patterns that can be used to create large, complex tessellations. But sometimes bad twists happen to good algorithms, and in this section, we look at a few of the things that can go wrong. For both centered twists and offset twists, one can choose a pair of species parameters and then find crease patterns that fold flat isometrically for • all regular polygons, • all triangles, • all cyclic polygons (for centered twists), • all cyclic Brocard polygons (for offset twists). Even within these collections, only certain crease assignments avoid self-intersection, but these are fairly limited sets. There

332


Figure 4.63. Three non-flat-foldable twist tiles. Left: a kite quadrilateral with corner angles (70◦, 100◦, 90◦, 100◦ ), centered twist parameters w = 0.12, τ = 75◦ . Middle: same kite quadrilateral, offset twist parameters w = 0.15, d = 0.1. Right: a rectangle with height 0.707 times its width, offset twist parameters w = 0.15, d = 0.1.

are other polygons that don’t work for tiles—even nicely symmetric polygons that are aesthetically desirable candidates for tilings. One prominent candidate is the generally symmetric kite quadrilateral, in which one pair of opposite angles are equal. This shape is neither cyclic nor closed-back for most sets of corner angles. The restrictions on offset twists are particularly severe: even a simple rectangle does not give a closed-back twist and thus cannot serve as an offset twist tile. Figure 4.63 shows examples of both polygons after applying the twist construction. It is clear that the angles between the pleats and central polygon—what would be the twist angle—are different as you go around the central polygons, and so none of the three patterns fold flat. There are other ways that a polygon can fail to support a flat-foldable twist tile if we try to construct the twist by propagating pleats in from the edges. For some polygons—notably parallelograms—both centered twist and offset twist constructions give rise to retrograde central polygons. Two examples are shown in Figure 4.64. If you travel around the outside of the tile polygon in a counterclockwise direction and trace the vertices of the central polygon, the latter circulate clockwise (and vice versa). Furthermore, the creases of the pleats cross the sides of the central polygon (and often, as in these two cases, each other). Even if the central polygon is not fully retrograde, it can still be malformed. Figure 4.65 shows two more examples. In one, the central polygon has a reflex vertex (with corner angle greater than 180◦ , and in the other, the central polygon crosses itself (so that half of it is retrograde).

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333

Figure 4.64. Retrograde central polygons formed in centered and offset twists in a parallelogram with vertices ((0, 0), (1, 0), (1.5, 1), (0.5, 1)). Left: centered twist with w = 0.2, τ = 70◦ . Right: offset twist with w = 0.2, d = 0.1.

Figure 4.65. Malformed central polygons in offset twists in a polygon with vertices ((0, 0), (1, 0), (1.2, 0.5), (0.7, 0.95)). Left: offset twist parameters w = 0.2, d = 0.05. Right: offset twist parameters w = 0.1, d = 0.05.

In general, whether a tile gives rise to a nonconvex, selfcrossing, or retrograde central polygon will depend upon the specific twist parameters being chosen, as well as the shape of the tile polygon, of course. Even if the central polygon is properly formed, as in Figure 4.63, it can fail flat-foldability due to the twist angles varying around the central polygon. That does not mean, however, that those twist parameters and the tile polygon are inherently incompatible; on the contrary, it is often possible to create more complex twist-like structures that allow one to rescue such problematic tile polygons, as we will now see. ? 4.6.6. Split-Twist Quadrilateral Tiles The non-flat-foldability of the tiles in Figure 4.63 is visible in the crease patterns by the varying twist angles around the central

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Figure 4.66. The three polygons and twist parameters from Figure 4.63, with additional creases that fold flat. Top row: crease pattern tiles. Bottom row: corresponding folded form tiles.

polygon. The incompatibility is also readily verified empirically by cutting out and attempting to fold each of the crease patterns; they will not fold flat. If you try to force the pleats and wedges to lie flat around the exterior of the folded tile, that will cause additional creases to form in the twist region. This is not necessarily a bad thing. In fact, forcing a non-flat-foldable pattern to fold flat while gently coercing into place additional creases that allow it to fold flat is an excellent way of discovering new flat-foldable structures. There are multiple ways to do this flattening with these three polygons and twist parameters; two distinct solutions are shown in Figures 4.66 and 4.67. For each polygon twist, there are two distinct ways of flattening it, but they result in very different appearances, both in the crease pattern and the folded form. In each pair, the pleats hit the edges at the same positions, widths, and angles, which are those specified by the species parameters, but we have a more complex structure Figure 4.67. Another way of adding creases for the three polygons and twist parameters from Figure 4.63 to make them fold flat. Top row: crease pattern tiles. Bottom row: corresponding folded form tiles.

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Figure 4.68. Construction of a flat-foldable split twist. Left: original non-flat-foldable twist. Right: modified for flat-foldability.

a3

q3

a4 a2

a4

q1

q3a q3b a1

a1

a4

q1

a1

in the center of the pattern: three polygons, rather than one, and if we look at the angles at the tops of each pleat—the analogs of the twist angles in simple flat twists—there are two different angles represented, two of each value. One way of looking at this structure is to think of it as splitting the original non-flat-foldable quadrilateral twist into two triangular twists with two different twist angles, and so we call this pattern a split twist. For each triangle, the pleats on two sides are simply ordinary twist pleats that make the same twist angle with the central polygon. On the third side, the pleats have coalesced into each other, forming the third triangle in the middle of the central region. Figure 4.68 illustrates this progression. In the original twist, shown on the left, we have four different twist angles, α1 –α4 . In the modified structure on the right, there are two triangles on either side of the shaded middle triangle. On one of them, twist angle α4 appears twice; on the other, twist angle α1 appears twice. We can also look at what happens to the vertices. Vertex q1 (black dot) and the two central polygon edges to either side are left unchanged in position. However, vertex q3 (gray dot) has been split into two vertices, each of which constitutes the third vertex of each of the two triangular twists. The role of the third pleat of each twist is taken up by the shaded triangle in the center. And we have added three new creases, completing the triangle of (q1, q3a, q3b ) and turning point q1 into a degree-6 vertex. It is, admittedly, not obvious that this new pattern folds flat, but it does—isometrically, at least, meaning that the KawasakiJustin Condition is satisfied at each vertex. It is not necessarily the case that such a pattern folds flat without self-intersection, and the question of whether a given pattern does so depends in a complex way on the pleat crease assignments and on the crease assignments chosen for the three newly added creases in the middle.

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b1c

b1b

b1a

b2

q1

Figure 4.69.

b4

Neighborhood of the degree-6 vertex q1 in a split twist.

b3

The geometric relationships in Figure 4.68 also suggest a geometric construction method for creating the split-twist crease pattern. Once we have chosen one corner to be the degree-6 vertex (q1 , in this case), twist angles α4 and α1 are fixed. We copy those two angles over to the other two pleats, which then define the two remaining vertices (q3a and q3b ). Connecting the three vertices (q1, q3a, q3b ) then completes the crease pattern. We cannot choose any of the four vertices to be the degree-6 vertex, however. Let us look more closely at vertex q1 , as shown in Figure 4.69. The original corner angle of the central polygon before the addition of the new creases was β1 = β1a + β1b + β1c . If, after subdivision, the vertex is flat-foldable, then it satisfies the Kawasaki-Justin Condition, which will be β1a − β1b + β1c − β2 + β3 − β4 = 0.

(4.62)

We define the alternating sum of sector angles around a vertex to be the Kawasaki excess κ for that vertex. Its sign depends on which angle you start from. For a flat-foldable vertex, the Kawasaki excess is zero. For the corner polygon of our original twist central polygon, the Kawasaki excess starting with that corner angle is going to be κ = β1 − β2 + β3 − β4,

(4.63)

which is generally not zero. But if, after constructing the split twist, the vertex is flat-foldable (so its Kawasaki excess is zero), then we can combine these two equations to find that κ = 2β1b .

(4.64)

The key thing about this is that angle β1b is a positive angle. So we can only create the split-twist construction if the Kawasaki

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337

excess at the desired degree-6 vertex is positive. In general, for a non-flat-foldable twist, the Kawasaki excesses of the central polygon corner vertices will contain both positive and negative values. Only the two positive-value vertices are eligible to receive the split-twist treatment. They give rise to the two constructions shown in Figures 4.66 and 4.67. We noted earlier that certain twist species parameters give rise to central polygons that may extend outside of the tile polygon, which requires special handling if we wish to use such polygons in a tiling-based tessellation. The same thing can happen with this construction. In addition, even if we succeed in finding a crease pattern in which the central polygons are self-contained and their creases are non-crossing, there is still the challenge of finding a non-self-intersecting crease assignment for the pleats and the three added creases. You may recall that for centered twist tiles, there was a simple, one-to-one relationship between the central polygon twist angle α and the pleat tilt angle τ that held for every tile, independent of polygon shape. It was tan α = w tan τ.

(4.65)

This simple relation no longer holds for split twist tiles. There are now two different twist angles (α1 and α4 in Figure 4.68), neither of which satisfies Equation (4.65). The tilt angle τ is, however, a constant on all four edges of the tile for centered twists, and the offset d is the same on all four edges for offset twists, and so this tile will still mate with any other twist tiles that have the same species parameters (w, τ) or (w, d), as applicable. Figure 4.70 shows an interesting phenomenon: not only are the twist angles different for the two triangle twists, but in one of them the twist directions are different: one twist is counterclockwise (α < 90◦ ), and the other is (barely) clockwise (α > 90◦ ). The change in twist direction enforces a change in the relationship between the type of the crease at the top of each pleat and the two creases of the pleat. Once we choose the pleat crease assignment, the crease assignment of the fold at the top is forced by the requirement that the smaller of the two angles at the top of each pleat have the anto crease assignment. On the topic of crease assignment, we do have some choice in how we assign the three new creases—but not much. We can choose the assignment on the crease between q3a and q3b , but once

338


Figure 4.70. Two mirror-symmetric quadrilateral tiles with corner angles (120◦, 90◦, 90◦, 60◦ ) and centered twist parameters w = 0.13, τ = 77.3◦ , completed as split twist tiles. Left: crease pattern tiles. Right: folded form tiles.

that is chosen, the remaining two creases have their assignments forced by the Maekawa-Justin Condition at both vertices. And there is no guarantee of non-self-intersecting flatfoldability. Once may choose a crease assignment that results in a self-intersection at vertex q1 or an adjacent-pleat self-intersection involving either of the partial triangular twists. In general, the relationships between possible crease assignments and possible modes of self-intersection are going to be complex and specific to each tile polygon and the particular twist parameters being tried out. Once one has found a split-twist configuration that folds flat without self-intersection, though, that tile can be added to the toolbox of tiles that work together, and they can be combined with the tilable polygons we have already identified to create new and interesting tessellations. The two centered twist quadrilaterals of Figure 4.70 were chosen for very specific reasons: two of each quadrilateral can be combined with two regular hexagon centered twist tiles to achieve a tiling that is a perfect square. This tiling, along with its crease pattern and folded form, is illustrated in Figure 4.71. In the same vein, four offset twist quadrilaterals can be combined to create a perfect square crease pattern that has a rotated square created by the pleats, as shown in Figure 4.72. Like all offset twist tessellations, the raw edges of the folded form are aligned with the outline of the folded form. In both of these tilings, the pleat crease assignments must be chosen to be consistent with their neighbors.

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339

Figure 4.71. A centered twist tiling consisting of hexagon centered twist tiles and mirror-symmetric quadrilaterals with split twist tiles for twist parameters w = 0.13, τ = 77.3◦ (which gives a twist angle in the hexagons of α = 30◦ ). Left: tiling. Middle: crease pattern. Right: folded form.

So while many regular polygons can’t combine with other regular polygons in sufficient ways to create large tilings containing them, when we throw arbitrary triangles, rectangles, symmetric trapezoids, and quadrilateral split twists into the mix, the possibilities expand enormously. The space of centered-twist-tiling-based tessellations is vast, and origami artists have just barely dipped their toes into the waters.

Figure 4.72.

A centered twist tiling consisting of four quadrilateral split twist tiles for twist parameters w = 0.14, d = 0.07. Left: tiling. Middle left: crease pattern. Middle right: folded form. Right: back side of the folded form.

340


Figure 4.73. “Double Spearhead” (2016), by Michał Kosmulski. Left: folded tessellation. Right: closeup of the same tessellation, backlit.

Both of these patches of tiles share a property; the crease assignments on the edges and shapes of the tilings are such that if you placed two copies of each tile side by side, the pleat creases, and hence the tiles, would mate (in the case of the offset tiling, you would place a copy side by side with its mirror image). Similarly, two such tiles, or even the mated pairs, could be placed with copies above and/or below it, and they would similarly mate. So these tilings have the special property that they could be replicated in two directions to make arbitrarily large tilings, and arbitrarily large tessellations. Although we have focused our attention on simple flat twists, the concept of tiles is not restricted to twists. Any polygon folding pattern containing pleats that cross edges in well-defined ways can be turned into tiles and combined to make tessellations. We can also manipulate the paper of a simple flat twist to create more complex tiles that still mate in the same way as a twist. There are hundreds of such variations in the origami world; I would like to present just one, shown in Figure 4.73, by Polish artist Michał Kosmulski. This pattern is based on a heavily manipulated square twist tile. I find its backlit form, seen on the right in the figure, especially striking. This tiling, like the combination of four offset twist quadrilaterals in Figure 4.72, uses a simple square tiling, which is one of the easiest patterns to plug tessellation tiles into. But the options for patterns are much greater. In the next chapter, we will look at the several ways that such patches of origami tessellations can be replicated into larger patterns of beauty.

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341

?

4.7. Terms Centered twist tile A tile containing a twist polygon in which the two creases of each pleat hit the tile edge at points equidistant from the center of the edge. Central angles The angles between consecutive lines from the circumcenter of a cyclic polygon to its vertices. Circumcenter The center of a circle that passes through a set of points. Concyclic A set of points is concyclic if they all lie upon a common circle. Cyclic polygon A polygon is cyclic if its vertices are concyclic. Diophantine equation An equation whose solution(s) must be integers. Fixed point A point that, after scaling, rotation, and offset, remains in its original location. Generic A geometric figure is generic if it has no special symmetries or properties. Genus A category of similar tiles that includes multiple species that are characterized by similar parameters. Join line A line along which two or more tiles join with each other. Kawasaki excess The alternating sum of sector angles around a vertex; has the value 0 for flat-foldable vertices, but is nonzero for non-flat-foldable vertices. Mating Two or more crease patterns mate if they join along a common border with no gaps. Offset twist tile A tile containing a twist polygon in which the two creases of each pleat hit the tile at 90◦ , with each pleat having the same width relative to the edge length and having the same relative offset with respect to the center of the edge. Pseudoinverse A matrix that behaves like an inverse of a singular matrix, in that its product with the singular matrix is as close as possible to the identity matrix.

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Self-contained tile A tile in which none of the crease lines (or fold lines, for folded form tiles) extends outside of the tile. Species A collection of tiles that can mate with one another. Split twist A twist pattern in a quadrilateral composed of two interacting triangular twists. Tilt angle The angle that, in a centered twist tile, each pleat makes with the tile edge. Vertex figure A sequential listing of the polygons around a vertex when those polygons are regular wherein each polygon is represented by the number of its sides.


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5

Tilings

?

5.1. Introduction to Tilings

In the previous chapter, we created many different types of twist tiles that mate: they can be assembled into larger structures that give flat-foldable patterns and beautiful folded forms. We can construct such patterns by creating a library of individual twist tiles, then putting them together so that they cover a planar region with no gaps. One way of doing this would be to simply start placing tiles, adding new tiles wherever there’s an angular gap, and then filling in the gaps that don’t fit any regular polygon tiles with triangle and/or trapezoid tiles, as needed. That’s a “bottom-up” way of creating origami tessellations. But there is also a “top-down” approach: we can start with an existing pattern of polygons, then “plug in” the relevant crease pattern (and/or folded form) tiles to realize an overall tessellation design. In this chapter, we’ll take this top-down approach. A salient characteristic of our origami crease patterns is that there be no gaps between tiles. In mathematics, a covering of the plane or of a region of the plane with no gaps is called a tiling. If a tiling is composed of polygons, it is a polygonal tiling. Such tilings have been decorative and artistic elements for centuries, appearing in Moorish mosaics, Escher lithographs, and in nature, from crystals to honeycombs. There is a vast mathematical and artistic literature devoted to tilings and their properties. For a comprehensive overview of their mathematical properties and classifications, see the classic book by Grünbaum and Shephard [40]. Polygonal tilings can be composed of a single type of polygon or several types, and individual tiles can be regular, semiregular, 345

Figure 5.1. Three examples of tilings created by arrangements of regular polygons.

or completely irregular polygons. Figure 5.1 shows a few simple examples of tilings with regular polygons. We have seen many more examples of tilings in the previous chapter. Some of those tilings were bespoke; others were members of well-known patterns. In this chapter, we’ll explore several families of tilings that lend themselves to origami tessellations. ? ???

5.2. Archimedean Tilings

? 5.2.1. Uniform Tilings Let us start by restricting our consideration to the regular polygons. These already offer great variety: we have identified two genera of tilings: centered twist and offset twist. Within each genus, we have many different species, each characterized by two parameters— (w, τ) or (w, α) for centered twist, and (w, d) for offset twist—and within each species, we can choose the number of sides n and the crease assignment around the central polygon. We must, of course, ensure that the crease assignments on adjacent tiles match up, but that still leaves quite a wide variety of choice. So now let’s turn back to the question of arrangement. How many different arrangements of unit regular polygons are there? As we saw in the previous chapter, there are 21 ways we can arrange regular polygons around a single vertex (see Figure 4.27).

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........CHAPTER 5. TILINGS

{3.3.3.3.3.3}

{3.3.3.3.6}

{3.3.3.4.4}

{3.3.4.3.4}

{3.3.4.12}

{3.4.3.12}

{3.3.6.6}

{3.6.3.6}

{3.4.4.6}

{3.4.6.4}

{3.12.12}

{4.4.4.4}

{4.6.12}

{4.8.8}

{6.6.6}

Figure 5.2. The 15 tileable vertex figures composed of regular polygons.

Any vertex of any tiling of the plane composed of regular polygons must match one of these 21 vertex figures. But we also saw that some of them cannot be extended arbitrarily by adding unit polygons. Six of them—{3.7.42}, {3.8.24}, {3.9.18}, {3.10.15}, {4.5.20}, and {5.5.10}—do not allow tilings of regular polygons that cover the plane. If you put down three polygons in these arrangements, eventually, you run into unfillable gaps. Taking away the six untilable vertex figures leaves fifteen vertex figures that allow plane tilings. These are shown in Figure 5.2. Eleven of these vertex figures have the nice property that it is possible to make a tiling of the plane in which every vertex is of the same type. The tilings thus formed are called the Archimedean, or semiregular, tilings. They are also sometimes called the uniform tilings. Each Archimedean tiling is said to be of the same type as its corresponding vertex figure: thus we can speak of the (3.3.3.4.4) tiling, which is composed of triangles and squares and all of

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347

whose vertices have vertex figure {3.3.3.4.4}. One of the 11 tiling types—(3.3.3.3.6)—comes in two forms that are mirror images of one another. Note that some of the tilings, like the (4.6.12) tiling, include both CW and CCW orderings of the specified polygons around different vertices. If we say that two tilings are the same if one can be transformed into the other by some combination of translation, rotation, and/or scaling (but not reflection), then there are a total of 12 distinct tiling patterns. The 12 patterns, along with their vertex figures, are shown in Figure 5.3. Any one of these 12 patterns can be used to construct a tessellation using the unit polygon twists described in the previous section. And what about the other four vertex figures? The vertex figures {3.3.4.12}, {3.4.3.12}, {3.3.6.6}, and {3.4.4.6} do not lead to tilings where all the vertices are alike—but they lead to interesting tilings nonetheless. A tiling composed of regular polygons that contains k types of vertices is called k-uniform. The Archimedean tilings are 1-uniform (or simply uniform). These other four vertex figures appear in k-uniform tilings for k > 1 (in fact, they all appear in 2-uniform tilings). We will examine higher-order uniform tilings a bit later, but at the moment, let us delve more deeply into the Archimedean tilings and examine how we can construct origami twist tessellations based upon them. ? 5.2.2. Constructing Archimedean Tilings One can construct a given Archimedean tiling by arranging polygons in one of the patterns shown in Figure 5.3. To make the pattern larger than what is shown, you simply keep adding tiles “in the same pattern.” Now what do we mean by that? The Archimedean tilings (as well as many others) have the property that they repeat themselves over and over as you travel in particular directions. They have translational symmetry. That means that if you place two copies of the tiling one on top of the other so that they line up, then shift one of them by a certain distance in a particular direction (the translation), they’ll still line up. As we saw in Chapter 2, a pattern or tiling is called periodic if it has translational symmetry. It is doubly periodic if it has translational symmetry in at least two different directions.1 1 Grünbaum and Shephard [40, p. 29] define periodic to require two different

348


(a) (3.3.3.3.3.3)

(b) (4.4.4.4)

(c) (6.6.6)

(d) (3.3.3.3.6)

(e) (3.3.3.3.6)

(f) (3.3.3.4.4)

(g) (3.3.4.3.4)

(h) (3.4.6.4)

(i) (3.6.3.6)

(j) (3.12.12)

(k) (4.6.12)

(l) (4.8.8)

Figure 5.3. The 12 Archimedean tilings.

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To be perfectly rigorous, only an infinite tiling is truly periodic. The patches shown in Figure 5.3 aren’t infinite, so if you translate a copy of the tiling with respect to the original, the edges of the two copies of the tiling don’t line up. But we’ll say that a finite tiling has translational symmetry if it is a piece of an infinite tiling with translational symmetry. Of course, any pattern of finite extent is part of the periodic tiling composed of repeated patches of that pattern, but we will always assume that any periodic pattern shows at least one full period (and usually more) within the image shown—which is the case in Figure 5.3. Now let’s do a little experiment. Suppose we mark a periodic tiling with a single dot, then translate a copy of the tiling in some direction so that all the edges of the shifted tiling line up with the edges of the original tiling. But we’ll only allow translation, not rotation, reflection, or glides (translation plus reflection). If it’s an infinite tiling, the only way you can tell that the tiling has moved is that the dot has moved to a new location. We’ll make a new dot in the new location, so now we have two dots. Then we’ll do this again in a different direction, again finding a new location for a new dot. We keep doing this until we’ve placed dots everywhere possible. If the tiling is doubly periodic, the dots will form a twodimensional lattice that is generally a lattice of parallelograms. The pattern of dots is called the lattice of the tiling. Figure 5.4 shows the lattice for one of the Archimedean tilings. The position of the lattice with respect to the tiling isn’t unique, by the way. Figure 5.4 shows two possible placements of the lattice on the 3.3.4.3.4 tiling. The position of the lattice points depends on where the first dot was placed. For a given tiling, however, the distances and angles between lattice points depend only upon the symmetries of the tiling itself; they are independent of the particular point chosen for the location of the first dot. What makes the lattice useful is that it can be used to extend the tiling indefinitely. We can associate with each lattice point a small group of tiles that can be duplicated and translated in two directions to extend the tiling to infinity. We call such a group a lattice patch. A lattice patch can be arranged with copies of itself in a 2D array to completely fill the plane with an Archimedean directions of translational symmetry. Other authors—and I—require only a single direction of periodicity to call something periodic, as was the usage in Chapter 2.

350


Figure 5.4. The lattice for the (3.3.4.3.4) Archimedean tiling. The position of the lattice with respect to the tiling isn’t unique; two different positionings are shown. The shaded region in the image on the right is one possible lattice patch.

tiling or, on a more practical note, to build as large a finite region of the tiling as we should like. Just as there is more than one way to position the lattice with respect to the tiling, for a given tiling, there are many possible lattice patches. Figure 5.5 shows two possible patches for the (3.3.4.3.4) tiling. You can choose the lattice patch for a tiling in many different ways. In fact, the tiles in a lattice patch need not even be contiguous. Any set of tiles will work if they can be used to completely cover the tiling. It is helpful, though, to pick tiles that roughly approximate the parallelogram formed by the lattice points themselves. If you are constructing tessellations, it is convenient to have lattice patches for the underlying tiling because they can be copied over and over in an orderly array to extend to arbitrarily large size. In Figure 5.6, we show lattice patches for each of the Archimedean tilings. Copies of each patch are aligned with their neighbors by aligning the black dots. You can use these patches to construct Archimedean tilings as large as you want. And if you fill each tile with tessellation creases, you can thus construct arbitrarily large tessellation crease patterns as well.

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Figure 5.5. Two possible lattice patches that tile to form the (3.3.4.3.4) tiling. Translated boundaries of the lattice patches are in black.

The (3.3.3.3.6) tiling comes in two versions that have the same vertex figure but are not the same tiling. One is the mirror image of the other. A pattern that is not the same as its mirror image is enantiomorphic; put another way, it comes in right- and lefthanded versions. As Figure 5.6 shows, the same lattice patch can be used to make both of the two enantiomorphic forms of the (3.3.3.3.6) tilings. They will simply be arranged differently with respect to each other, hence the different dot positions on the patches. That is, the lattice patch is the same, but the lattices are different. ? ? ? 5.2.3. Lattice Patches and Vectors By using lattices and lattice patches, we can describe an entire tiling of unit regular polygons quite concisely and can construct the tiling fairly easily as well. In such a tiling, all distances are unit length, and all angles are integral divisions of a full circle. Thus, the lattice, for example, can be described by the translation distances and directions from one lattice point to the next in each of the two directions of periodicity.

352


(3.3.3.3.3.3)

(4.4.4.4)

(6.6.6)

(3.3.3.3.6)

(3.3.3.3.6)

(3.3.3.4.4)

(3.3.4.3.4)

(3.4.6.4)

(3.6.3.6)

(3.12.12)

(4.6.12)

(4.8.8)

Figure 5.6. Lattice patches for the Archimedean tilings.

The lattice points form a two-dimensional array. We can describe each point by a vector vi, j , labeled by two indices that give its position in the grid of points. In general, the grid will not run precisely horizontally or vertically, but we can use the first index for the direction that is predominantly horizontal and the second for the direction that is predominantly vertical, as illustrated in Figure 5.7. We can also number the individual polygons in the patch, as is also shown in the figure.

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353

v1,2

v0,2

v0,1

Figure 5.7. Construction of the lattice vectors l1 and l2 and the specifications of the polygons of the lattice patch.

4 3

5

l2

6

v0,0

v1,1

2 1

l1

v1,0

Consider first the lattice. If we start from any point, such as v0,0 in Figure 5.7, we can get to the point v0,1 via a translation l1 , defined by l1 ≡ v1,0 − v0,0 . (5.1) Similarly, we can go from v0,0 to v0,1 via a translation l2 , defined by l2 ≡ v0,1 − v0,0 . (5.2) Once we have found these two lattice vectors for the lattice, every lattice point can be found by adding multiples of l1 and l2 , namely, vi, j = v0,0 + il1 + jl2 . (5.3) So, for example, the lattice point v1,1 shown in Figure 5.7 is v1,1 = v0,0 + l1 + l2 .

(5.4)

The lattice vectors themselves can be constructed by noting that each vector is a sum of a series of unit translations along edges of the tiling. We introduce for convenience the unit vector u(θ) ≡ (cos θ, sin θ).

(5.5)

For this tiling, we can see from inspection that l1 = u(0) + u( π6 ),

π l2 = u( 2π 3 ) + u( 2 ).

(5.6) (5.7)

Every lattice vector in any tiling composed exclusively of unit regular polygons can be written as a sum of such unit vectors,

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and we can provide a concise encoding of such a sum by simply giving a series of the angles, expressed as fractions of π. Thus, the lattice vector l1 would be described by the series (0, 1/6), and the lattice vector l2 would be described by the series (2/3, 1/2). Every other lattice vector for other regular polygon tilings can be similarly described. Once we have the lattice for a tiling, we need to fill it in with translated copies of the lattice patch, each copy translated by the same amount as its corresponding lattice point. We therefore only need a description of (a) a single patch and (b) the two lattice vectors in order to produce the complete tiling. Each lattice patch is made up of one or more unit regular polygons, each translated through some distance, and possibly rotated from its standard position (first vertex at (0, 0), second at (1, 0)). Thus, each polygon in the lattice patch can be fully described by a small set of numbers: • n, the number of sides in the polygon; • a sequence of fractions, which collectively gives the translation of the polygon from the lattice point (in the same format as the lattice vector series); • a rotation angle, which will also be given as a fraction relative to π. For the (3.3.4.3.4) tiling of Figure 5.7, the lattice patch contains six polygons: four triangles and three squares. Using the numbering shown in the figure, we can see that polygon #1 is a triangle (n = 3) that has its origin corner located at the reference lattice point, so its translation sequence is () (empty) and its rotation fraction is also 0. Polygon #2 is a square (n = 4), whose translation series is (0) (a unit translation at angle 0) and rotation fraction is 1/6. And so forth and so on, for each of the other four polygons. Clearly, the description of a lattice patch is not unique. We could number the polygons in any order; we could define our translation vectors for a given lattice patch relative to any nearby lattice point, and there is more than one way to describe each polygon by translation+rotation. Polygon #5, for example, could be translation (2/3), rotation 0, or could be translation (1/3), rotation 1/2. The actual choice is a matter of convenience.

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Once we have a full description of the lattice patch and lattice vectors, it is relatively straightforward to build the entire tiling. Table 5.1 gives the lattice vectors and lattice patch descriptions for the 12 Archimedean tilings whose lattice patches are shown in Figure 5.6: • l1 and l2 are the lattice vectors, specified by their translation fractions. • n is the number of sides of each unit polygon in the lattice patch. • offset is the translation from a lattice point to the origin vertex of the unit polygon (marked with a gray dot in Figure 5.7. Like the lattice vectors, these are given as a list of fractions, each of which specifies a unit vector at the given fraction of π in angle. • rotation is the angle (also expressed as a fraction of π) through which the polygon should be rotated about its origin vertex. With the descriptions given in Table 5.1, one can construct any of the 12 Archimedean tilings. Using the twist tiles from the previous chapter, one can fill in the crease patterns that give a flat-foldable tessellation for each tiling. There is still the problem of matching the crease assignments in adjacent crease pattern tiles. Of course, we could do this by brute force: start with tiles of all the same type (say, M n ), then go through and invert crease assignments as needed until everything matches up. However, we can address this issue at the tiling level, as we will see in the next section. ?

5.3. Edge-Oriented Tilings

? 5.3.1. Centered Twist Tiles Figure 5.8 shows two square twists that mate: an M 4 twist on the left and a V 4 twist on the right. They didn’t need to be this specific crease assignment, of course. All that was required was that the pleat on the left of the join line have the opposite assignment from the one on the right.

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Tiling (3.3.3.3.3.3)

l1 0

l2 1/3

(4.4.4.4) (6.6.6) (3.3.3.3.6)

0 0, 1/3 0, 0, 1/3

1/2 2/3, 1/3 1/3, 2/3, 1/3

(3.3.3.3.6)

(3.3.3.4.4)

(3.3.4.3.4)

(3.4.6.4)

(3.6.3.6)

(3.12.12)

(4.6.12)

(4.8.8)

0, −1/3, 0

0

0, 1/6

0, 1/6, −1/6

0, 0

0, −1/6, 0, 1/6

0, −1/6, 1/6, 1/2, 1/3, 1/6

0, −1/4, 0, 1/4

2/3, 2/3, 1/3

1/3, 1/2

2/3, 1/2

1/3, 1/6, 1/2

1/3, 1/3

1/3, 1/2, 1/3, 1/6

1/6, 1/3, 1/2, 2/3, 5/6

0, 1/2, 1/4

n 3 3 4 6 3 3 3 3 3 3 3 3 6 3 3 3 3 3 3 3 3 6 3 3 4 3 4 3 3 4 3 4 3 4 3 4 6 3 6 3 3 12 3 4 6 4 6 4 6 12 4 8

offset

rotation

0

1/3

0 0 0, 0 0, 1/3 0, 1/3, 0 0, 1/3, 1/3 0, 1/3, 1/3 1/3

1/3 1/3 1/3 1/3 2/3 1/3

0 1/3 0, 1/3 1/3,1/3 1/3, 1/3 2/3

1/3

0 1/3

1/3

0 1/3 1/3, 1/6 2/3

1/6 1/6 1/4

1/3 1/3

1/3 −1/3 −1/3 −1/3, −1/6 −1/3, −1/6, 0 0

1/3 −1/6 0 1/6 −1/6

0 0, 0, 1/3

1/3

0, −1/6 0, −1/6, 0, 1/6, 1/3, 1/2

1/3

0, −1/6 0, −1/6, 1/6, 1/2 0, −1/6, 1/6, 1/2, 1/3 1/2, 0, 1/6, 1/3, 1/2 1/2, 0, 1/6, 1/3, 1/2, 2/3 1/2

1/6 1/3 1/6 1/6 1/6

0, −1/4

Table 5.1. Characteristic description of the lattices and lattice patches for the Archimedean tilings.

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Figure 5.8. A mating pair of square twist tiles for w = 0.15, α = 30◦ .

Right along the edge, what matters is that the top pleat fold on each side is valley and the bottom pleat fold on each side is mountain. But since these pleats are anto pleats, the crease assignment of the pleats at the tile edge forces the assignment of the central polygon edge incident to the two pleats. So the central polygon edge on the left must be M, and the one on the right must be V. When we fold this pair of squares, the central polygon of the left twist lies on top of the the pleat in the middle, which, in turn, lies on top of the central polygon of the twist on the right. We could indicate this “left on top of right” relationship by drawing an arrow across the edge, pointing from the thing that’s on top to the thing underneath. The arrow signifies the direction of coverage, but it also signifies the crease assignment on the edge of each central polygon that faces the edge in the middle. The head of the arrow points toward a valley fold (albeit obliquely); the tail comes from a mountain fold (also obliquely). With an M 4 twist on the left, we know that whatever ends up mated to the other three sides of that tile, each mating tile will have to have a valley fold facing the tile. So we could indicate that relationship, and the forced crease assignment, by outgoing arrows on the other three sides. In a similar way, the other three sides of the V 4 tile on the right must have incoming arrows, indicating that the edges of the central polygons in its mating tiles must all be mountain folds. Figure 5.9 shows the same pair, now with arrows drawn on all of the tile edges. What is interesting about this type of labeling is that when we place an arrow on a tile edge, it unambiguously specifies the crease assignments on both of the central polygon edges that face it. If we start with a tiling and then place an arrow on every edge, that assignment (called an orientation of the edges of the tiling)

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Figure 5.9. A mating pair of square twist tiles for w = 0.15, α = 30◦ with stacking order arrows on the tile edges.

completely specifies the crease assignment of the centered twist tiles with which we might fill the tiling. We can now construct a fully assigned centered twist tessellation as follows: • Construct (or choose) a tiling. • Orient the edges with arrows. • Use the orientation to define the crease assignment for the individual tiles. Figure 5.10 illustrates this three-step process for a small regular polygon tiling.

Figure 5.10. A centered twist tessellation tiling with w = 0.15, α = 30◦ . Top left: oriented tiling. Top right: tiling with assigned crease pattern tiles. Bottom left: complete crease pattern. Bottom right: folded form.

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Figure 5.11. A centered twist tessellation tiling with w = 0.15, α = 30◦ and a different orientation of the tiling. Top left: oriented tiling. Top right: tiling with assigned crease pattern tiles. Bottom left: complete crease pattern. Bottom right: folded form.

A different choice of orientation of the original tiling leads to a very different appearance in the folded tessellation, as Figure 5.11 shows. For periodic tilings, such as the Archimedean tilings, one can create a large patch of tiling and then orient the edges individually, or one can create oriented lattice patches that tile in such a way that mating edges have the same orientation. In general, in a tiling, every edge can be independently oriented in either direction, so if there are Ne edges in the tiling, there would be 2Ne possible orientations and, thus, the same number of possible crease assignments (subject to limitations due to self-intersection). With a lattice patch, though, half of the edges’ orientations are fixed by the requirement that the patches themselves tile the plane, giving 2Ne −Nb /2 (5.8) possible assignments for each lattice patch, where Nb is the number of edges on the border of the patch. That is a considerable number, adding once more to the variety of possibilities. To illustrate just a few, Figure 5.12 shows a single oriented lattice patch with associated crease pattern tiles for each of the 12 Archimedean tilings. Figures 5.13–5.24 then show complete tessellations—both crease patterns and folded forms—making use of each lattice patch.

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Figure 5.12. Oriented lattice patch tilings and assigned crease pattern tiles for the 12 Archimedean lattice patches with w = 0.28, α = 28◦ .

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Figure 5.13.

Centered twist tessellation with w = 0.28, α = 28◦ for the (3.3.3.3.3.3) tiling. Left to right, tiling, crease pattern, and folded form.

Figure 5.14.

Centered twist tessellation with w = 0.28, α = 28◦ for the (4.4.4.4) tiling. Left to right, tiling, crease pattern, and folded form.

Figure 5.15.

Centered twist tessellation with w = 0.28, α = 28◦ for the (6.6.6) tiling. Left to right, tiling, crease pattern, and folded form.

362


Figure 5.16.

Centered twist tessellation with w = 0.28, α = 28◦ for the first (3.3.3.3.6) tiling. Left to right, tiling, crease pattern, and folded form.

Figure 5.17.

Centered twist tessellation with w = 0.28, α = 28◦ for the other (3.3.3.3.6) tiling. Left to right, tiling, crease pattern, and folded form.

Figure 5.18.

Centered twist tessellation with w = 0.28, α = 28◦ for the (3.3.3.4.4) tiling. Left to right, tiling, crease pattern, and folded form.

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Figure 5.19.

Centered twist tessellation with w = 0.28, α = 28◦ for the (3.3.4.3.4) tiling. Left to right, tiling, crease pattern, and folded form.

Figure 5.20.


Figure 5.21.


364


Figure 5.22.


Figure 5.23.


Figure 5.24.


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Figure 5.25. Lattice patches for the four two-colorable Archimedean tilings. Top left: (3.3.3.3.3.3). Top right: (4.4.4.4). Bottom left: (3.4.6.4). Bottom right: (3.6.3.6).

? 5.3.2. Offset Twist Tiles Naturally, we can use a similar approach to determine the crease assignment for offset twist tiles. Recall, though, that there are two types of matching conditions that apply to offset twist tilings. We must make the mountains and valleys agree on either side of a tile edge, similar to centered twist tiles. But we must also alternate the sign of the offset parameter d from one tile to its neighbor. That restricted offset twist tiles to only tilings where an even number of polygons meet at each vertex of the tiling. A tiling where an even number of polygons meet at a single vertex is called two-colorable, for the obvious reason that it can be colored using only two different colors so that no two tiles of the same color touch along a single edge. Only four of the 12 Archimedean tilings are two-colorable: the (3.3.3.3.3.3), (4.4.4.4), (3.4.6.4), and (3.6.3.6) tilings. Figure 5.25 shows twocolored lattice patches of the four tilings. Three of the patches are colored versions of the Archimedean tiling lattice patches, but we must double the square in the (4.4.4.4) tiling to make the patch create a two-colored tiling. Placing these patches according to their lattice vectors gives two-colored Archimedean tilings, as shown in Figure 5.26. With offset twists, given a two-colored tiling, there is an obvious mapping from the color to the offset parameter of the crease pattern and/or folded form tiles; we assign one color to +d and the other to −d. There is still the issue of crease assignment matching at the edges. With offset twist tiles, it is possible to give every tile the same cyclic crease assignment—either M n or V n —which have distinctive and aesthetically pleasing appearances. One may, though, wish to vary the crease assignment about a given central polygon,

366


Figure 5.26. Examples of the four two-colorable Archimedean tilings. Top left: (3.3.3.3.3.3). Top right: (4.4.4.4). Bottom left: (3.4.6.4). Bottom right: (3.6.3.6).

which will, necessarily, affect the assignments of its neighbors. We can, again, use edge arrows to ensure proper matching of the twist tiles placed into the polygon tiling. In contrast to centered twist tiles, with offset twist tiles, the central polygon edges facing the tile edge must match in crease assignment, rather than have opposite parity, and we indicate this matching by running arrows along the direction of the tile edges, rather than transversely. Figures 5.27–5.30 show offset twist tilings based on the four two-colorable Archimedean tilings. They all have cyclic crease assignments and closed-back twist parameters with the sign of the offset (and thus, twist direction) chosen according to the tile color. With offset twist tiles, if some of the twists in the tiling are closed-back, they all are. Conversely, we can open up the center of each tile by picking a pleat width w smaller than 2d. Figures 5.31– 5.34 show the same four tessellations as Figures 5.27–5.30, but with w = 0.14, |d| = 0.14.

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Figure 5.27. Offset twist tessellation with w = 0.2, |d| = 0.1 for the (3.3.3.3.3.3) tiling. Top left: tiling. Top right: crease pattern. Bottom left: folded form. Bottom right: folded form for the opposite crease assignment.

Figure 5.28. Offset twist tessellation with w = 0.2, |d| = 0.1 for the (4.4.4.4) tiling. Top left: tiling. Top right: crease pattern. Bottom left: folded form. Bottom right: folded form for the opposite crease assignment.

368




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369

Figure 5.31. Folded forms of offset twist tessellations with w = 0.14, |d| = 0.14 for the (3.3.3.3.3.3) two-colorable Archimedean tiling with both cyclic crease assignments.

Figure 5.32. Folded forms of offset twist tessellations with w = 0.14, |d| = 0.14 for the (4.4.4.4) two-colorable Archimedean tiling with both cyclic crease assignments.



370


?

5.4. k-Uniform Tilings

? 5.4.1. 2-Uniform Tilings The Archimedean tilings are called uniform because at each vertex, the same polygons occur in the same order around the vertex. For any two vertices, if you place one copy of the tiling on another so that the two vertices are aligned, you can rotate the copy so that the entire second tiling is aligned with the first. If this shiftrotation-alignment is possible for two vertices of a tiling, then the vertices are said to be in the same transitivity class. For a uniform tiling, every vertex is in the same transitivity class. There are tilings for which the vertices must be grouped into more than one transitivity class—for example, because different vertices have different vertex figures. A periodic tiling with k transitivity classes is said to be a k-uniform tiling. The k-uniform tilings are commonly labeled by the vertex figure for each transitivity class. The next level of the k-uniform tiling hierarchy beyond the uniform tilings consists of the 2-uniform tilings. There are 20 of these [40, p. 65]. Similarly to the Archimedean tilings, they consist of combinations of equilateral triangles, squares, hexagons, and dodecagons—but no octagons, interestingly. The 20 2-uniform tilings are illustrated in Figure 5.35. Each tiling can be characterized by the two vertex figures to be found within the tiling. For example, ((3.3.3.3.3.3); (3.3.3.3.6)) means that some vertices are of type (3.3.3.3.3.3) and the remaining ones are of type (3.3.3.3.6). Note, though, that the vertex figure pair is not a unique identifier of the tiling; two distinct tilings can be constructed of the same two vertex figures, and you can see several examples in the figure. Any of these can be transformed into a twist tessellation via the route we have already showed. Pick a genus, such as centered twists. Then choose a species parameter pair (w, τ) or (w, α). Orient the edges of the tiling, i.e., label each edge with an arrow, to specify the direction of the pleats in the tiles to either side. Then construct the twist in each tile according to the species parameters and assign the creases according to the edge arrows. Figure 5.36 shows an example for the ((3.3.4.3.4); (3.4.6.4)) tiling. There are 19 more 2-uniform tilings and hundreds of possible crease assignments for each. Enjoy!

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(a) ((3.3.3.3.3.3); (3.3.3.3.6))

(b) ((3.3.3.3.3.3); (3.3.3.3.6))

(c) ((3.3.3.3.3.3); (3.3.3.4.4))

(d) ((3.3.3.3.3.3); (3.3.3.4.4))

(e) ((3.3.3.3.3.3); (3.3.4.12))

(f) ((3.3.3.3.3.3); (3.3.4.3.4))

(g) ((3.3.3.3.3.3); (3.3.6.6))

(h) ((3.3.3.3.6); (3.3.6.6))

(i) ((3.3.3.4.4); (3.3.4.3.4))

Figure 5.35. The 2-uniform tilings.

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(j) ((3.3.3.4.4); (3.3.4.3.4))

(k) ((3.3.4.3.4); (3.4.6.4))

(l) ((3.3.3.4.4); (3.4.6.4))

(m) ((3.3.3.4.4); (4.4.4.4))

(n) ((3.3.3.4.4); (4.4.4.4))

(o) ((3.3.4.3.4); (3.4.6.4))

(p) ((3.3.6.6); (3.6.3.6))

(q) ((3.4.3.12); (3.12.12))

(r) ((3.4.4.6); (3.6.3.6))

Figure 5.35 (Continued). The 2-uniform tilings.

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Figure 5.35 (Continued). The 2-uniform tilings.

13

(s) ((3.4.4.6); (3.6.3.6))

14

(t) ((3.4.6.4); (4.6.12))

15 12

10

11

8

5

9

4

6

7 2

1

3

Figure 5.36.

A centered twist tessellation from the ((3.3.4.3.4); (3.4.6.4)) tiling with w = 0.18, α = 28◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

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? 5.4.2. Two-Colorable 2-Uniform Tilings As we saw with the Archimedean tilings, if we wish to create a tessellation entirely from offset twist tiles, we must base it upon a two-colorable tiling, so that some of them will get (w, d) twist tiles and the alternate tiles get (w, −d) tiles. Among the 20 2-uniform tilings, there are seven two-colorable tilings. All of them can be used to create closed-back offsettwist tessellations, but one, in particular, stands out: the ((3, 3, 3, 3, 3, 3), (3, 3, 4, 12)) tiling is the first two-colorable tiling we have encountered that includes dodecagons, which provide a particularly stunning accent when they appear in cyclic form in a closed-back tiling. I have constructed such a tessellation tiling, shown in Figure 5.37, as an example.

Figure 5.37.

An offset twist tessellation from the ((3.3.3.3.3.3); (3.3.4.12)) tiling with w = 0.2, d = 0.1. Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form. 375 CHAPTER 5. TILINGS

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Figure 5.38.

A centered twist tessellation from a ((3.4.4.6); (3.4.6.4); (4.4.4.4)) tiling with w = 0.2, α = 30◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

I find this to be one of the more striking tiling-based tessellations. Constructing offset twist tilings for the remaining six tilings will be left as an exercise for the ambitious reader. ? 5.4.3. Higher-Order Uniform Tilings There are higher-order k-uniform tilings, and the number of possible periodic tilings grows steadily as you allow more types of vertices. You can find many more tilings suitable for origami tessellations in the classic book Tilings and Patterns, by Grün-

376


Figure 5.39.

An offset twist tessellation from a ((3.4.4.6); (3.4.6.4); (4.4.4.4)) tiling with w = 0.2, d = 0.1. Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

baum and Shephard [40]. I will show just two of the possibilities here: centered twist and offset twist tessellations from a 3-uniform tiling, in Figures 5.38 and 5.39. As we add more and more types of vertices to tilings, the patterns grow richer and more varied. Really, though, there is nothing particularly significant about k-uniformity, other than its aesthetic mathematical appeal. For finite-sized tilings, one could toss k-uniformity out the window and construct a tiling by simply fitting together polygons, then converting that tiling to a twist tessellation. Figure 5.40 shows one such example. The set of equilateral triangles, squares, hexagons, and dodecagons form

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Figure 5.40. A centered twist tessellation from a tiling of triangles, squares, hexagons, and dodecagons, with w = 0.18, α = 30◦ . Left: oriented tiling. Middle: crease pattern. Right: folded form.

a rich set for assembling such tilings and their corresponding tessellations. One of the masters of creating such assemblies is the Italian artist Alessandro Beber, who developed his own tile-based methods and creates designs that make particular use of offset tiles (pleats perpendicular to the tile edges). Several of his works are shown in Figure 5.41. This first set makes use of a central dodecagon surrounded by triangles, squares, and/or hexagons, similarly to Figure 5.39.

Figure 5.41. Tessellations by Alessandro Beber with 12-fold rotational symmetry. Left: “Dodecagon2 .” Middle: “Dodecagon3 .” Right: “Dodecagon3/4 .”

378


Figure 5.42. Tessellations by Alessandro Beber with a square layout. Top left: “Square tessellation of dodecagons #0.” Top right: “Square tessellation of dodecagons #1.” Bottom left: “Square tessellation of dodecagons #3.” Bottom right: “Square tessellation of dodecagons #3 v.2.”

Many tilings lend themselves well to square layouts, and when folding from the traditional origami square, there is a natural place to truncate the tiling. When the repeating unit of a tiling contains many elements, it only takes two to four repetitions to strike an aesthetic balance: enough repetitions to establish the pattern, but not so many that it gets boringly repetitive. The four tessellations in Figure 5.42 strike that balance perfectly. The same concept is at play when folding from a hexagon: the tiling is naturally clipped by the boundaries of the paper, and

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Figure 5.43. Tessellations by Alessandro Beber. Top row: left to right, “Hexagonal tessellation of dodecagons # 1,” “Hexagonal tessellation of dodecagons # 2,” “Hexagonal tessellation of dodecagons # 3.” Bottom row: left to right, “4.6.12,” “4.36 .12,” “‘Rotated’ 3.4.6.4.”

a relatively small number of repetitions establishes the periodic pattern. Figure 5.43 shows six of Beber’s hexagonal tessellations. Note that Beber names the last three from their underlying tiling. Can you identify the tiling for the others? He tells us that the designs for “4.6.12” and “4.36 .12” had been previously (independently) discovered by Thomas Hull and Chris K. Palmer (albeit using different pleat widths), which highlights a property of origami tessellations and, for that matter, geometric origami in general. As artists, we often strive for pattern, beauty, and elegance in our designs. Mathematics provides a description of pattern and regularity, and it can sometimes precisely enumerate the small numbers of elegant regularities within a field. One of those elegant regularities is that there are precisely

380


12 Archimedean tilings. This is a universal truth: anyone who strives to use regular polygons and have every vertex the same is bound to arrive at one of those 12 tilings, whether designing now, in the past, or in the future. But we still have aesthetic choices that let us express individuality: in choices of pleat width, tile size, tile type, crease assignment, materials, overall artwork size, and even context and execution. We can create work that expresses our own individual vision, but if we choose to work within a strongly mathematical genre like tessellations, we should not be put off to find that others have (or will) find the same resonances within the artistic universe. ? 5.4.4. Periodic Tilings with Other Shapes The examples so far have been tilings composed of regular polygons, but we are not restricted to those. With centered twists, we can use any cyclic polygon; with offset twists, we can use any cyclic Brocard polygon as long as the tiling is two-colorable (all vertices of even degree). There are many ways to create periodic tilings composed of some of these other tiles. One very straightforward way is via subdivision. A square can be divided into four cyclic quadrilaterals, for example (with one of the quadrilateral angles as a free parameter). Figure 5.44 shows a centered twist tessellation based on such a subdivision. A hexagon can be divided into six cyclic quadrilaterals, which, when placed into the (6.6.6) tiling, give rise to the tessellation shown in Figure 5.45. This one is particularly interesting because of the unexpected appearances of squares and rectangles in the patterns of folded edges and pleats. Creating tilings with nonregular offset twist tiles is a bit more challenging; there are fewer polygons that form general-purpose offset twist tiles. As with centered twists, all regular polygons and all triangles work, but for other polygons, not only must they be cyclic, they must be Brocard as well. There’s more: the range of angles that give self-contained tiles is relatively small, and perhaps most limiting, the tilings themselves must be two-colorable. There is plenty of opportunity provided by subdividing tiles from existing tessellations, though. The (4.8.8) Archimedean tiling was not two-colorable, but if we subdivide half of the octagons into eight triangles each, we can obtain a two-colorable tiling and offset twist tessellation that is, arguably, more interesting visually, due to the

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Figure 5.44.

A centered twist tessellation from a subdivision of a square tiling, with w = 0.18, α = 37◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

range of different shapes represented. An offset twist tessellation based on this tiling is shown in Figure 5.46. In addition to creating tilings by subdividing tilings of regular polygons, there are also tilings that are composed of nonregular polygons and are not a dissection of a tiling of regular polygons. As you might expect, as we relax the restrictions on tilings, the possibilities explode in number. A challenge, perhaps, for the mathematicians who enumerate such things, but a boon to the artist, for this creates an immense richness of possibility. Even if we restrict ourselves to a single type of tile—what is called an isohedral tiling—there are 107 different types of tiling [40], each composed of a single type of polygon with three to six sides.

382


Figure 5.45.

A centered twist tessellation from a subdivision of a hexagonal tiling, with w = 0.18, α = 30◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

All of the tilings we have worked with so far have been edgeto-edge tilings, meaning that the vertices of each tile align with the vertices of its neighbors; nowhere does a vertex of one tile fall in the middle of the edge of another tile. This is not a restriction forced by origami tessellations in general; it is only a restriction imposed by the particular species of tiles that we have constructed. Our tiles mate only edge to edge, but one could imagine creating other twist tiles (or tiles containing other origami patterns) that allow non-edge-to-edge matings. Even so, there are, for example, 17 distinct edge-to-edge isohedral tilings of quadrilaterals, many of which lend themselves to tessellations. We have already seen a few—Figures 5.44 and 5.45 are two of them—but while these

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Figure 5.46.

An offset twist tessellation from a subdivision of the (4.8.8) tiling, with w = 0.14, d = 0.07. Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

were obtained by subdividing regular polygon tilings, there are others that do not come from such a subdivision. Figure 5.47 shows one such example, based on the P4 − 52 tiling from [40, p. 478]. There are 24 distinct isohedral tilings of pentagons. Of those, nine are edge-to-edge. Constructing origami tessellations from pentagon twists is a bit more challenging. From offset twist tiles alone, it is impossible, because all of the pentagon tilings have at least some degree-3 vertices and so are not two-colorable. Even if we restrict ourselves to centered twist tiles, there is the challenge

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Figure 5.47. An offset twist tessellation from Grünbaum and Shephard’s P4 − 52 tiling of quadrilaterals, with w = 0.18, d = 0.09. Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

that our tile polygons must be cyclic, having all five vertices on a common circle. Even so, isohedral tilings with cyclic pentagons may be found; Figure 5.48 shows an example. If two non-consecutive angles of an equilateral pentagon sum to 180◦ , as in Figure 5.49, then the pentagon and its mirror image can be assembled into a tiling in two different ways, as shown in Figure 5.50. This tiling is called a Cairo tiling and has been used as a floor tiling in the Arab world. Although we can vary the base angle δ of the tile, we cannot coerce it into a cyclic form. For the value δ = 98.7074◦ , however,

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Figure 5.48. A centered twist tessellation from Grünbaum and Shephard’s P5 − 21 isohedral tiling of pentagons, with w = 0.2, α = 30◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

Figure 5.49.

180¡Ð d

Schematic of a Cairo tile. All sides are equal length; two non-consecutive angles sum to 180◦ .

d

386


Figure 5.50. Two different forms of a Cairo tiling from an equilateral pentagon with corner angles of (105◦, 87.61◦, 143.33◦, 75◦, 129.06◦ ). Each tiling is composed of equal numbers of the pentagon and its mirror image.

the tile takes on a mirror-symmetric form that can then be dissected into an isosceles triangle and a symmetric trapezoid, both of which support compatible centered twist tiles, as in Figure 5.51. The subdivided pentagon can then be placed onto the Cairo tiling, giving rise to the tessellation in Figure 5.52. A comment about the parameters of the tiling: the pleats in this tiling are, I think, on the narrow side of aesthetically pleasing, but their width is driven by the requirement that the triangle tiles be self-contained. As an isosceles triangle approaches an apex angle of 90◦ , the central twist moves toward the base, forcing smaller pleats if we want to keep the folded form tile self-contained. (Of course, this is not strictly necessary, if we are willing to handle negative pleat lengths as we did in the previous chapter.) In addition, the requirement that the cyclic triangle tiles be non-selfintersecting limits the twist angle α to be less than the Brocard angle, which is about 28.1◦ for this triangle. Figure 5.51. 180¡Ð d 180¡Ð d/2 d

d

180¡Ð d d/2 d/2 180¡Ð d d

d

A mirror-symmetric Cairo tile. Left: the tile. Right: dissected into isosceles triangle and symmetric trapezoid.

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Figure 5.52.

A centered twist tessellation based on a subdivision of the Cairo pentagon, with w = 0.07, α = 28◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

Another way to create periodic tilings with pentagons is to combine them with other shapes. Regular pentagons can be combined with isosceles right triangles to make a periodic tiling, as shown in Figure 5.53. This tiling is two-colorable, which means that we can also construct an offset twist tessellation from the same tiling, shown in Figure 5.54. The pleat width, and thus the twists, in this tiling are quite small, their small size caused by keeping all of the tiles self-

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Figure 5.53. A centered twist tessellation based on a tiling of regular pentagons and isosceles right triangles, with w = 0.119, α = 26◦ . Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

contained in their folded forms. Once again, if we allow the central twists to expand outside of tile boundaries, we can obtain tessellations with a more pleasing balance between the sizes of the pleats and the sizes of the facets, and in subsequent chapters, we will explore other ways of constructing tessellations from these same tilings that overcome this limitation. ? 5.4.5. Grid Tessellations Once you have designed a set of mating twist tiles, they can be connected and combined to create an almost endless variety of patterns. However, there is a drawback to their generality: the CHAPTER 5. TILINGS

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Figure 5.54. An offset twist tessellation based on a tiling of regular pentagons and isosceles right triangles, with w = 0.09, d = 0.045. Top left: oriented lattice patch. Top right: oriented tiling. Bottom left: crease pattern. Bottom right: folded form.

resulting tessellations must be somehow transferred to the paper, either by careful drawing, or computer plotting, using a tool such as a sign cutter/scorer or laser cutter. Historically within the world of origami, the folding process is bootstrapped: one begins with an unmarked square (or other shape) and then sequentially builds up all of the reference points and lines needed purely by folding, with recent reference lines based on earlier ones, and so forth. In the world of tessellations, because the pattern of the tessellation is usually periodic, one can create a periodic set of reference lines by folding a grid. It is now common—in fact, typical—for

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origami tessellators to start by folding an N × N square grid, or an N × N × N equilateral-triangular grid—on the paper before starting, with N typically a power of 2: 16, 32, and 64 being the most common (and 128 is not unheard of). Once one has the prefolded grid, it is possible to construct the tessellation by using a subset of the folded lines of the grid directly, or, more commonly, by mixture of using existing folds and adding new folds that connect vertices of the grid. The tiling method of constructing tessellations is eminently adaptable to prefolded grids, but you will find that only particular combinations of tilt angles, pleat widths, and offsets put folds and/or vertices on tractably small grids. One of the masters of the gridded style of tessellation is Eric Gjerde, who it can fairly be said led the third wave of origami tessellations (the first two waves being those of Fujimoto and Momotani, and Chris K. Palmer, respectively). Gjerde’s book, Origami Tessellations: Awe-Inspiring Geometric Designs [37], is widely regarded as a definitive reference, and he regularly breaks new ground in genre and technique. A few of his creations are shown in Figure 5.55. You might see if you can identify the underlying tilings for each of these. ? ???

5.5. Non-Periodic Tilings Periodic tilings have the property that they can be extended forever on their lattice, which means that it’s easy to create a tessellation that is as large as you like. Mathematically, that’s a wonderful property. Aesthetically, though, it has its limitations. Tessellations that show simple repetition are just not very interesting. (In fact, I would say that the tessellation of square twists is downright boring.) The more interesting and visually intriguing tessellations contain a mix of twists in different sizes and orientations. Even if the lattice patch contains a mix of tiles, with more than a few repetitions in each direction, a certain uniformity begins to set in. The periodic tilings are only a subset of the vast, wide world of tilings, however, and many other tilings may be used as the basis of origami tessellations. Many of these tilings are still symmetric, but exhibit some other kind of symmetry: rotational symmetry or self-similarity.

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Figure 5.55. Tessellations by Eric Gjerde. Top row: left to right, “Square Weave” (2005), “Tiled Hexagons” (2005), “Pinwheels” (2006). Middle row: left to right, “Château-Chinon” (2006), design by Christiane Bettens; “Stacked Triangles” (2006); “Open-Back Hexagon Twist” (2006). Bottom row: left to right, “Arms of Shiva” (2006); “Field of Stars” (2006); “Double Triangle Sawtooth” (2006), design by Miguel Angel Blanco Muñoz.

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Figure 5.56.

Goldberg tilings for different rotational orders m with n = 3. Top left: m = 5. Top right: m = 6. Bottom left: m = 7. Bottom right: m = 8.

? 5.5.1. Goldberg Tiling In a 1955 paper, Michael Goldberg described and explored a set of tilings he called central tessellations [38], and which we will call Goldberg tilings. Each tiling is rotationally symmetric and is composed of isosceles triangles with their apex angles given by 2π/m, where m is the order of the rotational symmetry. There are two parameters that describe such a tiling: m, the rotational symmetry, and n, the number of rings about the center. Each tiling is composed of a single type of triangle, some pointing inward, others pointing outward. See the examples in Figure 5.56. We can realize centered twist tessellations using our triangle tiles from Section 4.5; for Goldberg tilings with m > 4, there are at least some self-contained tiles. An example with m = 7 is shown in Figure 5.57.

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Figure 5.57.

A centered twist tessellation on an m = 7 Goldberg tiling with w = 0.17, α = 30◦ . Left: oriented tiling. Middle: crease pattern. Right: folded form.

Some of the Goldberg tilings are also amenable to offset twist tessellations, which, you may recall, require two-colorable tilings. In these tilings, all of the interior vertices are of degree 6, except the vertex in the very middle, which is of degree m. So for even m, the tiling is two-colorable, and we can create an offset twist tessellation from the tiling, as in Figure 5.58. An interesting variation was noted by Goldberg. The evenorder tilings possess a “cleavage line,” a straight line that divides the entire tiling into two halves. The two halves can be shifted

Figure 5.58.

An offset twist tessellation on an m = 8 Goldberg tiling with w = 0.14, d = 0.07. Left: oriented tiling. Middle: crease pattern. Right: folded form.

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Figure 5.59. Offset Goldberg tilings for m = 10. Top left: no offset. Top right: offset = 1. Bottom left: offset = 2. Bottom right: offset = 3.

by a single tile edge length (or a multiple thereof), giving rise to a tiling with an unusual spiral pattern. Examples of these spiral tilings are shown in Figure 5.59. All of the offset forms can be used for centered twist tessellations. For offset twist tessellations, the two-colorability rule still applies when we split the central vertex, and only the tilings with m of the form 2(2k +1) (for integer k) turn out to be two-colorable. This includes m = 6, but for m = 6, we have equilateral triangles, which makes the tiling turn out to be the (3.3.3.3.3.3) tiling we have already seen. The smallest nontrivial offset-twist offset tiling is, then, m = 10. An example is shown in Figure 5.60. A folded version of an offset Goldberg tiling, by Alex Bateman, is shown in Figure 5.61. ? ? ? 5.5.2. Self-Similar Tilings The beauty of many tessellations arises from symmetry, and symmetry is, broadly speaking, simply an operation on an object that leaves the object unchanged. In periodic tessellations, the symmetry was translational; if we shift the pattern by some combination

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Figure 5.60.

An offset twist tessellation on an offset m = 10 Goldberg tiling with w = 0.088, d = 0.044. Left: oriented tiling. Middle: crease pattern. Right: folded form.

Figure 5.61. “Spiral Tessellation” (1997), by Alex Bateman.

of the lattice vectors, it remains the same (excluding border effects). The Goldberg tilings exhibit rotational symmetry; if we rotate the tessellation, it remains unchanged. There is a third type of symmetry that is useful and relevant in tessellations, and that is scaling symmetry, also called self-similarity. In that symmetry, if we magnify or demagnify the object, it remains unchanged, albeit possibly with an apparent rotation. One of the classic examples of self-similarity is the logarithmic spiral, an example of which is shown in Figure 5.62. A logarithmic spiral spirals outward from a point and has the property that for

396


Figure 5.62. Logarithmic spirals with winding angle ω. Left: the spiral cuts the circle at angle ω. Right: an arrangement of six equally-spaced spirals is also self-similar.

w

any circle centered on the point, the spiral cuts the circle at the same angle, independently of the size of the circle. This angle, called the winding angle of the spiral (which we will denote by ω), is a unique parameter that characterizes the spiral; every spiral with the same value of ω can be rotated relative to any other with the same central point, so that they are perfectly aligned. If we create an arrangement of m such spirals, each rotated by 2π/m relative to its neighbor, the resulting pattern of spirals will also be self-similar; changing the magnification gives the same pattern, albeit rotated through some angle. If we superimpose two such sets of spirals, as in Figure 5.63, then we get a pattern that is also self-similar and that is now composed of curved-edge quadrilaterals. Each quadrilateral is geometrically similar to every other quadrilateral in the pattern.

Figure 5.63. Two superimposed sets of logarithmic spirals. The red set consists of m L = 8 spirals with winding angle ω L = 50◦ . The green set consists of mR = 6 spirals with winding angle ωR = 30◦ .

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We can use such a pattern to create a self-similar tiling of straightedge quadrilaterals and then, potentially, turn that tiling into an origami tessellation. To create the straight-edge tiling of quadrilaterals, we will choose the vertices of the quadrilaterals to be the intersections of the curved logarithmic spirals. The logarithmic spirals can be parameterized as x 2πi (i) x p L (x) = e u − , (5.9) ωL mL x 2πi (i) x p R (x) = e u − + , (5.10) ωR mR

pi, j

where p(i) L (x) is a parameterized curve of the ith rotated spiral that bends left as it spirals outward (i = 1, . . . , mL ), p(i) R (x) is a parameterized curve of the ith rotated spiral that bends right as it spirals outward (i = 1, . . . , mR ), and u(φ) ≡ (cos φ, sin φ) is the directed unit vector function. We define pi, j as the intersection of the ith left-bending spiral and the jth right-bending spiral. It is given by 2πω L ω R i j 2π i j = exp + u ωR + ωL . (5.11) ωL + ωR mR mL ωL + ωR mR mL We take each quadrilateral consisting of vertices (pi, j , pi, j+1, pi+1, j+1, pi+1, j ) to be a quadrilateral tile, as illustrated in Figure 5.64. Each tile is geometrically similar to all of the other tiles, differing only in size and rotation. Now, in order to convert such a tiling into a tessellation, each tile polygon needs to be fillable with a crease pattern tile that can mate with copies of itself. For the crease pattern tiles we know of, that means the tile polygon must be a cyclic polygon for centered twists, a cyclic Brocard polygon for offset twists, or any quadrilateral for split twists. If the two sets of tiles have the same rotational orders, i.e., mL = mR ≡ m, then there is an overall m-fold rotational symmetry and all of the points {p k,−k } lie on a common circle. If, in addition, we give the two sets of spirals the same winding angles ω L = ω R ≡ ω, then the tiling takes on a particularly symmetric form and the quadrilaterals all become similar kite shapes, as shown in Figure 5.65.

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p2,0

p2,1 p2,2

p1,0 p0,0 p0,1

p1,1

Figure 5.64. p1,2

A portion of the double spiral tiling, for parameters m L = 8, mR = 7, ω L = 30◦, ωR = 35◦ .

p0,2

To make this tiling work with simple flat twists, the quadrilateral must be a cyclic polygon. Quadrilaterals are cyclic if and only if diagonally opposite angles sum to π, and because of the mirror symmetry of the kite shape, that condition is met if the two equal angles of the kite are both π/2. Using Equation (5.11) for the vertices of the tiling, we can work out the condition that ensures that the two incident edges meet at right angles: (p0,0 − p−1,0 ) · (p0,1 − p0,0 ) = 0,

(5.12)

Figure 5.65. A symmetric double spiral tiling, for parameters m L = mR = 7, ω L = ωR = 40◦ . Multiple vertices lie on the amber circle.

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which lets us solve for the symmetric winding angle ω: m π π ω = log sec + tan . (5.13) π m m Thus, for each value m of the rotational symmetry, there is a unique value of the winding angle ω that gives rise to a cyclic polygon that can be tiled with centered twists. There is still one problem, though: the theoretical tile becomes arbitrarily small in the center of the tiling. There is an infinite number of infinitesimally small tiles covered up by the black dot in Figure 5.65. That poses a slight practical problem when it comes to folding the pattern. For mL = mR , though, there is an overall rotational symmetry and each “generation” of points lies on a common circle, as shown in Figure 5.65. We can choose the quadrilaterals with vertices that lie on a particular circle and then cut those quadrilaterals in two, turning their outer halves into triangles, and then remove all of the tiling inside the resulting regular m-gon. We can then use the known regular m-gon centered twist tile for the central polygon and a set of m triangle tiles to glue together the finite inner tessellation and the arbitrarily extendable outer tessellation. The full process is illustrated in Figure 5.66. This construction is theoretical, of course, but the Chilean origami artist Nicolás Gajardo Henriquez has created a design very much like this (quite independently). A backlit version is shown in Figure 5.67. There are, of course, many other possible tessellations based on this general theme: one can vary the species parameters, crease assignments, rotational orders, and winding angles and, using split-twist tiles, construct offset twist tessellations as well. Given the wide variety of tilings that exist in the world of mathematics, it might seem that tilings alone provide a nearly inexhaustible set of possibilities for simple flat twist tessellations. However, tiling-based tessellations lean heavily on repetition and the ability to re-use a relatively small set of building blocks. There are, however, less regular patterns in which many more tiles might be needed. Indeed, we can conceive of patterns in which every single tile is unique. Designing tessellations based on such patterns requires a more holistic approach, where we consider the entire pattern to begin with, then work our way down to the individual twists. In the next chapter, we explore new strategies for twist tessellation construction.

400


Figure 5.66.

A centered twist tessellation, based on a symmetric double spiral with m = 12, w = 0.11, α = 30◦ . Top row: oriented tiling. Center row: three types of crease pattern tiles (not to the same scale). Bottom row: crease pattern and folded form.

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Figure 5.67. “Sunflower Mandala” (2017) by Nicolás Gajardo Henriquez, backlit.

?

5.6. Terms Archimedean tiling A polygonal tiling in which each vertex has the same vertex figure. See also semiregular tiling. Bootstrapped sequence A folding sequence in which all folds are defined based on previously made folds and/or the corners and edges of the paper. Cairo tiling A particular tiling of identical pentagons. Edge-to-edge tiling A tiling where the vertices of each tile align with the vertices of neighboring tiles (versus a vertex of one tile falling in the interior of an edge of its neighbor). Enantiomorphic Having both right- and left-handed forms that differ; a shape that is not the same as its mirror image. Goldberg tilings A class of tilings of triangles that are not translationally symmetric but have a particular central point of rotational symmetry. Isohedral tiling A tiling is isohedral if every tile can be translated/rotated to any other tile such that the entire tiling remains unchanged.

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k-uniform tiling A tiling is k-uniform if each every vertex is one of k classes, such that within each class, the same regular polygons occur in the same order around the vertex and if you translate the tiling from one vertex to another in the same class and rotate it so that the edges around the two vertices are aligned, the overall tiling remains unchanged. Lattice A collection of points that encapsulates the geometry of the tiling. The full tiling can be reconstructed by placing lattice patches on the lattice. Lattice patch A collection of polygons or shapes that, when copies are placed on the lattice, creates the complete tiling. Lattice vector A vector from one lattice point to an adjacent lattice point. A tiling lattice can be fully described by two independent lattice vectors. Orientation Assigning a specific direction to a tile edge or graph (e.g., drawing an arrow on each edge of the graph). Polygonal tiling A tiling composed entirely of (straight-sided) polygons. Scaling symmetry, self-similarity A pattern has scaling symmetry (or is self-similar) if a scaled (and possibly rotated) version of itself gives the same pattern. Semiregular tiling A polygonal tiling in which each vertex has the same vertex figure. See also Archimedean tiling. Tiling A partitioning of a region of the plane into individual tiles that completely cover the region with no overlaps and no gaps. Two-colorable tiling A tiling is two-colorable if only two colors are needed to color the tiling such that adjacent tiles that meet along a common edge are differently colored. Uniform tiling A tiling is uniform if at each vertex, the same regular polygons occur in the same order around the vertex.

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6

Primal-Dual Tessellations ?

6.1. Shrink and Rotate

In the previous chapter, we saw that we could construct a variety of tessellation patterns from building blocks that were individual polygonal tiles decorated with creases in such a way that • the creases within each tile fold flat, • the creases in adjacent tiles connect to each other (same point, same angle, same crease assignment). A common feature of these tessellation tile patterns is that each tile contains a polygon that is a shrunken, rotated version of the original tile. Many origami tessellations have this property: the tessellation has the symmetry of an existing tiling, and each polygon of the underlying tiling appears as a shrunken and rotated polygon in both the crease pattern and the folded form. This property has been noticed by many of the pioneers of tessellation, and it was used extensively by Chris K. Palmer in his works in the late 1990s and the next decade. And it suggests another possible way of creating tessellations: rather than trying to find individual tiles that fit together, could we instead take an entire tiling, shrink and rotate its polygons, and then add creases in such a way as to turn it into a valid tessellation? The allure of such an approach is clear. It gives us greater flexibility: instead of having to construct a tessellation tile for every polygon in the tiling, we could construct them all in one go. It would let us create tessellations from tilings that include a wider variety of shapes—as we saw, not all polygons can be turned into centered twists with arbitrary edge parameters (w, τ) 405

Figure 6.1. Progressive constructions of a shrink-rotate tessellation crease pattern. Top row: the original tiling; shrink the polygons by a factor ρ = 0.756; rotate them all through the same angle β = 19.10◦ . Bottom row: connect vertices with new creases; assign mountain and valley folds.

(and offset twists were even more limited). In fact, it would, if possible, allow us to transform literally any line drawing into a tessellation, opening up a vast new world of artistic expression. And indeed, this simple algorithm—shrink and rotate each polygon in the tiling—can be used to create a wide range of tessellations: those that arise from k-uniform tilings of regular polygons, and also tilings that are not periodic, such as the offset Goldberg spiral that we saw in Chapter 5 (see Figure 5.61). Bateman’s innovation in constructing origami tessellations was the realization that tessellations based on many diverse tilings could be constructed programmatically by this algorithm, which we call the shrink-rotate algorithm. In fact, it is a very straightforward algorithm, which he implemented in 2000 in the Perl programming language, in a program he called Tess [7]. I illustrate the shrink-rotate algorithm in Figure 6.1. We start with a tiling, shrink each polygon, rotate each polygon by the same angle, then connect shrunken-and-rotated vertex pairs that came from the same initial vertex of the polygon with lines. The result

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is a flat-foldable crease pattern (or at least, it can be, depending on the specifics of the shrinkage, rotation, and crease assignment). We call it a shrink-rotate tessellation. Finally, we assign the creases. And this is the algorithm that Bateman implemented in Tess. But even this simple algorithm raises several questions: • How do we characterize the family of possible tessellations for a given tiling? • What crease assignments are possible? • What does the folded form look like? • Does this algorithm work for every tiling? If not, where does it break down? In this chapter, we’ll address these questions. We’ll show, in fact, that the algorithm does not work for every possible tiling; but we’ll show how to tell if it does work. Along the way, we’ll explore a number of topics in geometry, computational geometry, and mathematics, and remarkably, we will find that the key principle for the whole field traces its lineage back over 100 years, to a work by physicist James Clerk Maxwell. And we’ll see and construct some beautiful tessellations along the way. ??

6.2. Properties

?? 6.2.1. Twist and Aspect Ratio The first question that arises if we say that we’re going to “shrink and rotate” each polygon is, “rotate about what point?” The obvious and logical answer to this question is, “the center of the polygon, of course.” But even this answer raises some questions: which center? Triangles, for example, have at least four points that can be considered the center: the centroid, incenter, circumcenter, and orthocenter. (For definitions of these terms, see any good text on geometry or, e.g., [127].) For the moment, let us consider only tilings composed of regular polygons, in which case symmetry removes much of the ambiguity; the obvious choice to use as the center of rotation, as well as the center of shrinkage, is the centroid, or center of mass,

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of the polygon (which, for a regular polygon, is also its center of rotational symmetry). We can characterize the shrinkage and rotation by two parameters, which we now introduce. We define the ratio between the shrunken polygon and the original as the scaling and denote it by ρ. And we define the rotation angle to be the angle by which the shrunken polygon is rotated relative to the original and denote it by β. We note, too, that it doesn’t matter in the construction whether you shrink first or rotate first; the end result will be the same. Now, rotation angle β is not the same as the twist angle α. But what is quite remarkable is that for any tiling with regular polygons, if you apply the same scaling factor and rotation angle to every polygon in the tiling, not only is the result a crease pattern that folds flat (which is itself somewhat unexpected), but every polygon becomes a twist with exactly the same twist angle. This is quite independent of the number of sides of each polygon. So whether you have a regular triangle next to another triangle, or a regular triangle next to a decagon, when both are shrunken and rotated, they each become the centers of simple flat twists with the same twist angle. This is sufficiently remarkable that it deserves closer scrutiny. Let us zoom in on one edge of the original tiling and the two centers of rotation, one for each tile, as shown in Figure 6.2. With regular polygons, the line between the rotation centers is perpendicular to the tile edge, so without loss of generality we can place the intersection of those two lines at the origin of our coordinate system with both rotation centers on the x-axis. Then the tile edge lies on the y-axis, and we can characterize the two vertices of the tile edge and the two centers of rotation by their distances from the origin. We denote these distances in the figure by the letters a–d. Now, with a little trigonometry, it’s possible to calculate the relationship between the scaling/rotation (ρ, β) and the resulting twist angle α, which turns out to be ! sin β α = cos−1 p . (6.1) ρ2 − 2ρ cos β + 1 This is truly remarkable because none of the distances a–d appear anywhere within the expression. The twist angle depends

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y

Figure 6.2.

b

a b

d

a c m

l

b

Configuration of a single edge, before (orange highlight) and after (tan highlight) the shrink-rotate construction.

x

a

only on the shrinkage and rotation, and not at all on the length of the tile edge (b + c) or the separation between the centers of rotation (a + d). What’s more, both angles labeled α in Figure 6.2 have this value, which means that not only are the two rotated versions of the tile edge parallel, the opposite pair of sides of the quadrilateral between them are parallel as well. So the quadrilateral between the two twisted polygons is, in fact, a parallelogram with two angles of α and the other two angles of 180◦ − α. And since every polygon is surrounded by such parallelograms, this means that each polygon is completely surrounded by pleats that have a twist angle of α (unless it is on the border of the pattern, of course). So each polygon is the center of a simple flat twist, and the twist angle is the same for every pair of adjacent twists—which means that the single twist angle α is common to the entire tessellation pattern. Now, the length of one edge of the parallelogram is simply the scaled length of the tile from which it came; if the tile edge had a length of (b + c), the corresponding pair of parallelogram edges that are shrunken/rotated versions of this edge must have length l = (b + c)ρ.

(6.2)

A bit more trigonometry gives the length of the other pair of parallelogram edges to be q m = (a + d) ρ2 − 2ρ cos β + 1, (6.3) which is also a nice result: it says that the length of the short side of the parallelogram depends only on the distance (a + d) between the centers of rotation and the scaling/rotation parameters, and not at all on the length (b + c) of the edge of the original tile.

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The ratio between the two adjacent sides of the parallelogram is thus l b+c ρ = ·p , (6.4) m a+d ρ2 − 2ρ cos β + 1 which is nice, too: the first multiplicative factor depends solely on the geometry of the original tiling, while the second depends solely on the scaling and rotation. We will call this second factor the aspect ratio parameter γ, since it controls the aspect ratio of the parallelogram, and indeed the aspect ratio of all such parallelograms, in a shrink-rotate tessellation: γ≡p

ρ ρ2 − 2ρ cos β + 1

.

(6.5)

We now have two new parameters by which we can characterize the tessellation: (γ, α), the parallelogram aspect ratio parameter and the twist angle. These are an alternative way of characterizing the twist, since given (ρ, β), we can calculate (γ, α). We can go the other direction, too: given a pair (γ, α) that characterizes the aspect ratio of the parallelogram and the twist angle, we can compute the scaling and rotation that gives rise to this pair, which are easily found to be ρ= p

β = ± cos

γ γ 2 + 2γ sin α + 1

,

γ + sin α

−1

p

γ 2 + 2γ sin α + 1

(6.6) ! ,

(6.7)

where the sign of β is positive for a CCW twist and negative for a CW twist. It turns out that while the parameter pair (ρ, β) is necessary to construct a shrink-rotate tessellation, the pair (γ, α) is a bit more useful to characterize such a tessellation, for a couple of reasons. First, as we saw in Chapter 3, the possible valid crease assignments for a simple flat twist depend on the twist angle. When it comes to crease-assigning a shrink-rotate tessellation crease pattern, we’ll need to know the twist angle α for each polygon in order to know whether a crease assignment is possible. But second, there is yet another surprise to be had in this algorithm.

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?? 6.2.2. Crease Pattern/Folded Form Duality Recall from our discussions of simple flat twists and tiling-based tessellations that the twist angle in the crease pattern falls within the range α ∈ (0◦, 180◦ ), with α ∈ (0◦, 90◦ ) giving a CCW twist and α ∈ (90◦, 180◦ ) giving a CW twist. And if we construct the pattern with a negative value for α, then the pattern lines overlap the central polygon; and, in fact, we obtain the geometrical pattern of the lines of the folded form of the twist. The same situation applies to a shrink-rotate tessellation. If we create a pattern with a constant negative twist angle α, then the pattern of lines, at least, will be the line pattern of the folded form of some shrink-rotate tessellation. The question is, which one? Consider the following: if we construct two shrink-rotate tessellations from a given tiling using a twist angle α for one of them and −α for the other, then all of the polygons in the former will be similar (in the strict geometric sense) to the corresponding polygon in the other, with a single scaling constant between every pair of polygons. What’s more, in the corresponding parallelograms, the angles that become twist angles are equal and opposite—which means the same relation must be true for the opposite pair of angles in each parallelogram. If, furthermore, the ratio between adjacent sides of each parallelogram in one tessellation is the same as it is in the other, then all parallelograms, and indeed all polygons in the first tessellation, are similar to the corresponding polygons in the other tessellation—except that the parallelograms in the −α tessellation are reflections of the parallelograms in the +α tessellation. Which is all to say that if this pair of conditions is satisfied—the twist angles are equal and opposite, and corresponding parallelograms have the same ratio between adjacent sides—then the −α tessellation pattern is similar—the same up to a scaling constant— to the folded form of the crease pattern given by the +α tessellation pattern. And here’s the clincher: the aspect ratio for each parallelogram is the product of a constant term (b + c)/(a + d) that depends only on the tiling and the coefficient γ, which depends only upon the shrinkage/rotation. The two aspect ratios will be the same for every pair of corresponding parallelograms if and only if both patterns are characterized by the same value γ. This brings us to the following theorem:

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Theorem 22 (Bateman CP-FF Duality Theorem). If a shrink-rotate tessellation crease pattern is constructed from a tiling using aspect ratio parameter and twist angle (γ, α), then the folded form of that tessellation is given to within a scaling constant by the same construction using (γ, −α). This symmetric result was identified by Alex Bateman, who used it within his Tess program; if one describes a tessellation crease pattern by the parameters (γ, α), rather than some other pair of parameters, then one can easily construct the line pattern of the folded form of any of the shrink-rotate tessellations that it constructed. That is, this gives the proper locations of the folded edges. It does not, of course, tell which edges are visible (which depends on the crease assignment—which, in turn, may or may not exist, depending on the twist angles and polygons involved). We can also determine the precise scaling constant between the crease pattern and folded form thereby constructed. Suppose a given edge of the tiling has length (b + c) = 1. Then in the crease pattern (+α), the scaled-and-rotated edge will have length ρ CP = p

γ γ 2 + 2γ sin α + 1

,

(6.8)

which is, in general, less than 1. Whereas, in the folded form (−α), the scaled-and-rotated edge will have length ρ FF = p

γ γ 2 − 2γ sin α + 1

,

(6.9)

which is, in general, greater than 1. But, of course, in the real folded form, the edge should have the same length as it did in the crease pattern, which means that the computed folded form pattern will be too large by a factor ρ FF /ρ CP . Thus, to construct the folded form at the same scale as a crease pattern constructed from parameters (γ, α), we should construct the same tessellation pattern for parameters (γ, −α), and then scale the result by a factor s ρ CP γ 2 + 2γ sin α + 1 . (6.10) = ρ FF γ 2 − 2γ sin α + 1 Now, if we construct the crease pattern by the shrink-rotate algorithm, the individual polygons are shrunken by the factor ρ CP

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Figure 6.3.

A shrink-rotate tessellation for ρ = 0.756, β = 19.10◦ (or, equivalently, γ = 2.0, α = 30◦ ). Left: crease pattern. Right: folded form.

and then rotated through an angle β CP = ± cos

γ + sin α

−1

p

γ 2 + 2γ sin α + 1

! ,

(6.11)

taking the (+) sign for a CCW twist and (−) for a CW twist. To construct the folded form, we make the substitution α → −α and use a shrinkage factor ρ FF with a rotation through an angle ! γ − sin α , (6.12) β FF = ± cos−1 p γ 2 − 2γ sin α + 1 taking the same sign as before. We then perform an overall scale by the factor ρ CP /ρ FF , and this will give the proper line pattern for the folded form. For the tiling and crease pattern of Figure 6.1, I show the crease pattern and folded form together in Figure 6.3. The folded form, of course, depends on the specific crease assignment. The outline of the folded form and the number of layers at each point (which is visible for translucent paper) is the same for any assignment, but the pattern of visible folded edges depends enormously on the specific crease assignment. In this pattern, all of the vertices satisfy the Kawasaki-Justin Condition and so can conceivably be folded flat for some assignment, but whether the creases can all be assigned so that the entire pattern can be folded flat without self-intersection depends on the angles of the twists. As we have seen earlier, for any given polygon, there are linked ranges of twist angles and crease assignments that affect global flat-foldability. By using the shrink-rotate algorithm, we ensure that every interior vertex satisfies the Kawasaki-Justin Condition, which is

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the first step of establishing flat-foldability. The next step is to establish local flat-foldability—that is, to ensure that each vertex, viewed in isolation, is flat-foldable. The last step is global flatfoldability, which ensures that the layer overlaps can be chosen in a consistent, non-intersecting way. The problem of global flatfoldability can be very hard to solve for the general case, but we will show how to establish some limits on the tessellation parameters that ensure a simple, flat-foldable crease assignment. ??

6.3. Nonregular Polygons

?? 6.3.1. A Broken Tessellation The shrink-rotate algorithm works for a wide range of tilings—all tilings composed of regular polygons, among others. But it doesn’t work for all tilings and, in fact, breaks down on a relatively simple tiling composed of rhombuses, as shown in Figure 6.4. This pattern, unfortunately, is not flat-foldable for any crease assignment, because it does not satisfy the Kawasaki-Justin Condition at its vertices. This is easily seen by considering a single rhombic twist from the pattern. Since the vertices are degree-4, in order to fold flat, opposite sector angles must add up to 180◦ , and this is clearly not the case here. Equivalently, the twist angles are not all the same around the central polygon. To emphasize the difference between the rhombic tiling of Figure 6.4 and the regular tiling of Figure 6.1, in Figure 6.5 I have explicitly drawn in the centers of shrinkage/rotation and the lines between the centers of rotation for adjacent polygons for the two Figure 6.4. A rhombus-tiling-based crease pattern using the shrink-rotate algorithm. Left: the initial rhombus tiling. Middle: the crease pattern constructed by shrinking/rotating about the rhombus centers. Right: a single twist from the crease pattern.

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Figure 6.5. Lines between the centers of rotation for the two tilings are shown in green. Left: rhombus tiling. Right: (3.4.6.4) tiling.

tilings. Note that every interior edge of the tiling is crossed by one of these lines in each tiling. A difference between the two patterns is immediately apparent: in the (3.4.6.4) tiling, the lines between the rotation centers (green) are perpendicular to the tiling edges (orange), whereas in the rhombus tiling, they cross at two different angles. This property, it turns out, is what makes the shrink-rotate algorithm fail for the rhombus tiling. In our previous analysis of how the parallelogram ratio γ and twist angle α depended upon the shrinkage factor ρ, the rotation angle β, and the dimensions of the tiling, we assumed that both rotation centers lay on the x-axis with the tiling edge on the y-axis; in other words, we assumed perpendicularity. This is not guaranteed, and so we should re-do our analysis for the general case. Figure 6.6 illustrates the general geometry, where we have now introduced an angle δ, which is the deviation of the line between the rotation centers from perpendicularity to the tile edge. Figure 6.6.

y

α b a β

d

β x

l

c m

α

δ

Configuration of a single edge, before (orange highlight) and after (tan highlight) the shrink-rotate construction, where the centers of rotation are rotated relative to the tiling edge.

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If we carry out an analysis similar to what we did in Section 6.2.1, we will find that we still obtain a parallelogram between the two shrunken tiles, and the twist angle α does not depend on the distance between the circle centers or length of the tile edge. But it does depend on the angle δ; in fact, we find that sin(β − δ) + ρ sin δ cos(β − δ) + ρ cos δ α = cos−1 p = sin−1 p , 1 + ρ2 − 2ρ cos β 1 + ρ2 − 2ρ cos β

(6.13)

and if, again, we define the ratio between the two sides to be

we find that

l b+c ≡ · γ, m a+d

(6.14)

ρ cos δ γ=p . 1 + ρ2 − 2ρ cos β

(6.15)

Comparing Equation (6.13) with Equation (6.1), we see that the twist angle of a parallelogram now depends on the angle between the tiling edge from which it comes and the line between the two centers of rotation. The problem with the rhombus tiling now becomes clear. It isn’t that the angle between these two lines deviates from perpendicularity; it’s that for any given rhombus, as you go around the rhombus, this angle alternates between 60◦ and 120◦ (when measured in a consistent way). And so the twist angle is not constant as you go around the rhombus; it alternates from one side to the next. And this variation is what kills flat-foldability. But this example also points the way to a possible solution. If we choose the centers of the rhombuses as the centers of shrinkage and rotation, that clearly doesn’t work. But perhaps there is a different choice of rotation/shrinkage centers that does. ?? 6.3.2. Dual Graphs and Interior Duals In Figure 6.5, the lines of the tiling, in orange, form a plane graph—one drawn on a flat surface with no edges crossing each other. For each polygon, there is a center of rotation, and for each interior edge of the tiling, there is a line between the centers of rotation (in green in Figure 6.5) that (in these two cases) crosses exactly one edge of the tiling. The pattern of green lines also forms a plane graph, which is called the interior dual graph of the tiling graph.

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Figure 6.7.

Left: a graph (orange) and its dual graph (green). Right: a straight-line embedding of its interior dual graph.

In general, for any plane graph, one can define a dual graph as a graph that has a vertex for every face of the plane graph and vice versa. The dual graph also has an edge for every edge of the original, such that for every edge of the original graph that separates two faces, the dual graph has an edge between the two vertices that correspond to the two faces. In a finite plane graph, the outside of the graph region is considered to be a face of the graph, so that the dual graph has a single vertex that corresponds to the exterior of the original graph; this vertex has edges that correspond to each of the border edges of the original graph. If we delete this vertex and its incident edges, then we have a dual graph whose edges have a one-to-one correspondence with the interior edges of the original graph, i.e., those edges not on the border; hence the interior dual qualifier in the name. Examples of a general dual graph and an interior dual graph are shown in Figure 6.7. You may recall from Chapter 2 that a graph is an abstract mathematical concept, but when we choose particular points for the vertices and lines for the edges (e.g., by drawing it on paper), the result is called an embedding of the graph. In general, there is no requirement that the edges of an embedding of a dual graph are straight lines. But for our purposes, we are interested in embeddings where the edges of the dual graph are straight lines, as on the right in Figure 6.7. We call such a graph a straight-line embedding of the dual graph. The concept of the dual graph relies on the original graph, called the primal graph, being a plane graph (and specifically, that it have well-defined faces). The dual and interior dual graphs are topological concepts: they exist without actually having to be drawn on a surface. However, it is easily shown that both the dual graph and interior dual graph are planar graphs—capable of being drawn on a plane or sphere with no crossings.

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Note the difference between planar graphs and plane graphs: a planar graph is an abstract concept, a graph that can in principle be drawn on a plane without crossings; a plane graph is an actual non-crossing embedding of a planar graph. It is clear, now, that when we choose a set of centers of rotation for a tiling, we are choosing an embedding of the vertices of the interior dual graph where the primal graph is itself the tiling graph. And the straight lines we draw between rotation centers are just the edges of the embedded interior dual graph. With these definitions in hand, we can now pose the problem of creating a shrink-rotate tessellation in a more abstract way: For a shrink-rotate tessellation, the centers of rotation of the tiling (the primal graph) should be the vertices of a straight-line embedding of the interior dual graph for which the angles are constant between the primal graph edges and their corresponding dual graph edges. So, how can we find such an embedding? ?? 6.3.3. A Valid Rhombus Tessellation Let us return to the rhombus tiling. We can think about this as a tiling of rhombuses, of course; but we can also think of it as a tiling of squares that has been stretched vertically by some factor m. If we start with both the square tiling and its interior dual graph and we stretch them both by the same amount, then we end up with the situation we have already seen, where the vertices of the interior dual graph wind up in the centers of the tiles but the edges of the dual graph no longer cross the edges of the original tiling at the same constant angle for every primal/dual pair. But what if we stretched the dual graph by a different amount? This strategy, in fact, works; as shown in Figure 6.8, if we compress the interior dual graph vertically by the same amount as we stretched the original tiling vertically, the edges of the dual graph are all perpendicular to their corresponding edges in the primal graph. If you carry out the full shrink-rotate algorithm, you will find that this procedure gives a tessellation that does, indeed, fold flat (at least locally, at each vertex, and by suitable choice of twist angle, can be made to fold flat globally as well). This techniques works for the rhombus tiling, but in fact, it can be readily shown that it works for any tiling; if you have a tiling

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Figure 6.8.

Left: a square tiling (orange) and its dual graph (green). Middle: the tiling is stretched vertically by a factor 1.25; the dual graph is compressed vertically by the same factor. Right: the crease pattern using the vertices of the dual graph as the centers of rotation.

and a straight-line embedding of the interior dual graph such that the edges of the primal and dual graphs are perpendicular, then if you stretch the primal graph in one direction and compress the dual graph in the same direction, the corresponding edges of the new tiling will remain perpendicular to each other, and so a shrinkrotate tessellation composed from the new tiling will be locally flat-foldable. An example by Alex Bateman based on a (4.8.8) tiling is shown in Figure 6.9. Now, observe that in the middle subfigure of Figure 6.8, the centers of rotation are no longer in the same place in each rhombus. In fact, since the periodicity of the dual graph is different from

Figure 6.9. “Skewed 4.8.8” (2009), by Alex Bateman.

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419

ρ

β

Figure 6.10. A square tiling (orange), its dual graph (green), and the resulting tessellation for a configuration where the dual graph is displaced from the primal graph. Each rhombus is shrunken and rotated relative to its corresponding vertex in the dual graph. One rhombus and its corresponding vertex are highlighted.

the periodicity of the original tiling, you can see that if we were to extend the tiling further in the vertical direction, the dual graph vertices would move closer and closer to the corners of their corresponding rhombuses, and each would, in fact, eventually fall outside of its corresponding rhombus. Does this cause a problem? Not at all! Recall that the twist angle α and aspect ratio parameter γ do not depend at all on the actual positions of the centers of rotation relative to the edges of the tilings. Although I have drawn the dual graph with its vertices enclosed by their corresponding tiling polygons, there is absolutely no requirement that this be the case. In fact, as you can quickly discover by trying to construct a few examples, one can shift the entire dual graph by an arbitrary amount in any direction; as long as the correspondence between dual graph vertices and tiling polygons is maintained—that is, you rotate each tiling polygon about its dual graph vertex—the result will still be a locally flat-foldable tessellation, merely shifted in space to lie somewhere between the original tiling and the interior dual graph, as in Figure 6.10. Now that we have an embedding of the interior dual graph, we can move it around as we like and use its vertices as the centers of

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rotation for a shrink-rotate tessellation; the result will, in fact, be the exact same tessellation, just shifted in position. What other transformations on the dual graph result in a valid tessellation? Instead of (or in addition to) translating the dual graph, we could rotate it. A rotation of the entire dual graph will rotate each edge by the same angle. If we started with a dual graph whose edges were perpendicular to their corresponding primal graph edges, then a constant rotation by some angle δ will give parallelograms with a constant twist angle α given by Equation (6.13) and with aspect ratio γ given by Equation (6.15). One of the side lengths of the parallelograms change. We still have l = ρ(b + c) (6.16) for the side that comes from the edge of the original tiling (from the primal graph), but q m = (a + d) sec δ 1 + ρ2 − 2ρ cos β (6.17) for the side that comes from the dual graph. Similarly, we could scale the dual graph instead of rotating it. Magnifying it uniformly in both directions would increase the lengths of all edges of the dual graph by the same amount. This doesn’t change the twist angle; it simply changes the aspect ratio of the parallelogram (not by changing the factor γ, but by changing the factor (a + d) in the preceding equation). Thus, once we have an embedding of the interior dual graph whose edges all make the same angle with the primal graph, we can construct a shrink-rotate tessellation that satisfies the KawasakiJustin Condition at every interior vertex; furthermore, any uniform translation, rotation, or scaling of that dual graph will work as well—at least, to ensure this condition. We note that Equation (6.16) for the primal graph edges of the twist parallelogram does not depend on the rotation angle. Equation (6.17) for the dual graph edges of the twist parallelogram does, and it reaches its minimum value for δ = 0, i.e., orthogonality between the edges of the interior dual graph and their corresponding edges of the primal graph. We call such a graph an orthogonal embedding of the interior dual graph. If any embedding of the interior dual graph has its edges at a constant angle to their corresponding edges in the primal graph, then it can be rotated into orthogonality. Thus:

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Theorem 23. A shrink-rotate tessellation satisfies the KawasakiJustin Condition at all vertices if and only if there exists an orthogonal embedding of its interior dual graph. ?? 6.3.4. Relation Between Primal and Dual Graphs There is another interesting observation to make. Each of the polygons of the primal graph, the original tiling, shows up in the tessellation pattern at reduced size with a rotation applied to it. But the same can be said for the polygons of the dual graph. In fact, you can see this in the rhombus tessellation; the dual graph contains horizontally oriented rhombuses, and those same polygons appear, reduced and rotated, in the tessellation, as highlighted in Figure 6.11. In fact, there is perfect duality between the primal graph and the dual graph. One can construct the tessellation by taking each edge of the primal graph, shrinking it, and rotating it toward/about a corresponding vertex of the dual graph. But one can construct the same tessellation by taking each edge of the dual graph, shrinking it, and rotating it about a corresponding vertex of the primal graph (though you will be missing some edges of the original tessellation border). In each case, the resulting tessellation will consist of shrunken/rotated versions of the primal graph polygons, shrunken/rotated versions of the dual graph polygons, and parallelograms that tie the whole thing together. The shrinkage and rotation factors for the dual graph are not the same as those for the primal graph, however. If, in Figure 6.2,

Figure 6.11. The primal graph, tessellation, and dual graph from Figure 6.10, with a polygon from each graph highlighted in is original form and in its shrunken-and-rotated form in the tessellation.

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we swap the primal graph and dual graph edge roles—that is, we let line (a + d) be the tile edge and let line (b + c) be the dual graph edge between the centers of rotation, then we can find the shrinkage and rotation (ρ0, β0) that would be applied to the dual graph and that gives the same twist angle α and parallelogram aspect ratio γ that we get from shrinkage/rotation (ρ, β). A bit of algebra reveals that the dual rotation and shrinkage factors are q 0 ρ = ρ2 + 1 − 2ρ cos β, (6.18) β0 = −sign(β) cos−1 p ? ??

1 − ρ cos β ρ2 + 1 − 2ρ cos β

! .

(6.19)

6.4. Maxwell’s Reciprocal Figures As we have seen in the previous section, we can construct a shrinkrotate tessellation that satisfies the Kawasaki-Justin Condition at all vertices from a plane graph by finding an embedding of the interior dual graph of the original tiling whose edges are all perpendicular to the edges of the original graph. For certain special classes of plane graphs, we can construct this orthogonal embedding of the interior dual graph in a straightforward way. But what about the general case? Is it always possible to construct an orthogonal embedding of the interior dual graph for a general plane graph? And if not, when is it possible? It turns out that this question has already been asked, and answered, in a field far removed from origami tessellations— removed by both topic and time. The topic of the orthogonal embedding of a (general) dual graph was considered, and addressed, in the field of mechanical engineering over a hundred years ago. And it was addressed by none other than one of the greatest physicists and mathematicians of the 19th century: James Clerk Maxwell. Maxwell is most famous for the four equations that bear his name that constituted the first unification of theories of electricity and magnetism—and which predicted and described the wave nature of light, and laid the basis for Einstein’s Special Theory of Relativity a half-century later. But Maxwell also made seminal contributions to thermodynamics, the kinetic theory of gases, the analysis of colors, and several other fields in mechanics and engineering. It was one of these “lesser” investigations where CHAPTER 6. PRIMAL-DUAL TESSELLATIONS

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he made a mark that ultimately finds its expression in the field of origami tessellations. The paper that provides the key to making shrink-rotate tessellations was titled “On Reciprocal Figures and Diagrams of Forces” and was published by Maxwell in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science in 1864. A reciprocal figure is defined as follows: Reciprocal figures are such that the properties of the first relative to the second are the same as those of the second relative to the first . . . . The kind of reciprocity which we have here to do with has reference to figures consisting of straight lines joining a system of points, and forming closed rectilinear figures; and it consists in the directions of all lines in the one figure having a constant relation to those of the lines in the other figure which correspond to them. In plane figures, corresponding lines may be either parallel, perpendicular, or at any constant angle. Lines meeting in a point in one figure form a closed polygon in the other. [78, p. 250] That, in a nutshell, describes the relationship between the primal graph and the dual graph that gives rise to a shrink-rotate tessellation satisfying the Kawasaki-Justin Condition. The vertices of the primal graph correspond to polygons of the dual graph and vice versa; and the edges of the one are perpendicular to the edges of the other or can have any constant angle we choose, as long as every edge makes the same angle with its opposite number. But why explore such reciprocal figures? Maxwell goes on to explain: The conditions of reciprocity may be considered from a purely geometrical point of view; but their chief importance arises from the fact that either of the figures being considered as a system of points acted on by forces along the lines of connection, the other figure is a diagram of forces, in which these lines are represented in plane figures by lines . . . . [78, p. 251]

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In the 19th century, there was a very practical application for reciprocal figures. If a planar figure represented a set of beams connected at their ends, such as a bridge truss, it was critical to know the distribution of tensile and compressive forces in the various members of this truss in order to be sure that no single element was stressed beyond its breaking point. What Maxwell knew and showed was that the distribution of forces could be described by a reciprocal figure. Draw the truss structure; then construct a reciprocal figure, a graph whose edges are perpendicular to the edges of the original figure. Then the relative edge lengths of the reciprocal figure would be proportional to the forces in the individual truss members. ? 6.4.1. Indeterminateness and Impossibility In a general reciprocal pair of figures, there is a one-to-one correspondence between each edge of the primal graph and the dual graph, so the two graphs have the same number of lines. Our situation is slightly different: our dual graph, which we called the interior dual graph, has fewer lines than the primal graph, since the dual graph does not contain edges that correspond to the border lines of the primal graph. If, however, we had a reciprocal figure whose edges were properly orthogonal, we could simply “snip off” the dual graph edges that correspond to the border edges of our primal graph, and we would have the desired orthogonal embedding of the interior dual graph. But does the reciprocal figure exist? Not always, as it turns out. Suppose, for example, that we have a primal graph consisting of V vertices, E edges, and F faces, with Eb edges on the border (and Ei = E − Eb edges in the interior; we also do not count the exterior as a face). These numbers are not entirely independent; they must satisfy the Euler relation, V + F − E = 1.

(6.20)

The number of border vertices Vb must be equal to the number of border edges, Vb = Eb , and so the number of interior vertices is given by Vi = V − Eb . Now, the interior dual graph has a vertex for every face of the primal graph, an edge for every non-border edge of the primal graph, and a face for every interior vertex of the primal graph. If we define the dual graph quantities by a primed variable, we find CHAPTER 6. PRIMAL-DUAL TESSELLATIONS

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425

that

V 0 = F, E 0 = E − Eb, F 0 = V − Eb .

(6.21)

And of course, you can see directly that the interior dual graph satisfies its own Euler relation V 0 + F 0 − E 0 = 1 as well. Now, let us count how many degrees of freedom there are in this system and how many equations must be satisfied. The degrees of freedom in the interior dual graph are the coordinates of its vertices, and so with two coordinates per vertex, there are 2V 0 degrees of freedom available. As we have seen, a uniform translation in any direction or a uniform scaling will preserve the orthogonality relation between the primal and dual graphs; so we can choose those three degrees of freedom arbitrarily: for example, the (x, y) coordinates of any one vertex, plus the length of one of the edges. Then there is a condition on each of the edges of the dual graph: each edge must be perpendicular to its corresponding edge in the primal graph. (In a later section, we will see what this condition is explicitly.) So there are E 0 conditions that must be satisfied for orthogonality, making E 0 + 3 conditions in total on the 2V 0 degrees of freedom available to us. If the number of conditions is exactly equal to the number of degrees of freedom, then there is precisely one solution for the dual graph embedding (apart from translation and scaling). If, however, the number of conditions is less than the number of degrees of freedom, then the system is indeterminate: there are multiple solutions for the dual graph embedding. And if the number of conditions exceeds the number of degrees of freedom, then the system may be impossible: it is not possible to construct a dual graph whose edges are all orthogonal to their corresponding primal graph edges. Let us define the “excess” degrees of freedom (DOF) as the difference between the number of degrees of freedom and the number of conditions. Then we have DOF ≡ 2V 0 − (E 0 + 3) = 2F − E + Eb − 3.

(6.22)

The condition on solvability is that the excess degrees of freedom is nonnegative, i.e., DOF ≥ 0. This places a condition on

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the dual graph:

2V 0 − E 0 ≥ 3.

(6.23)

This is a well-known requirement for the existence of a reciprocal figure, and it is commonly known as the Maxwell Condition for that reason. We can transform the Maxwell Condition into a condition on our primal graph using the relationships in Equation (6.22); the Maxwell Condition becomes, for the primal graph, 2F + Eb − E ≥ 3.

(6.24)

This, in turn, can be expressed in several equivalent ways, by substituting the Euler relation back into Equation (6.24), giving F + Eb − V ≥ 2,

(6.25)

E + Eb − 2V ≥ 1.

(6.26)

Equation (6.25) displays most clearly the requirements for existence of a solution: more faces than vertices and a manyedged border give solvability. Figure 6.12 show two examples: one that is perfectly determined and one that is insolvable. It must be noted, however, that Equation (6.25) does not mean that a solution is never possible if this condition isn’t satisfied; this is the condition that must be satisfied for there to be a solution for every embedding of the primal graph. There can be (and often are) particular embeddings that are sufficiently symmetric that orthogonality can be satisfied for multiple edges in ways that use F=3 Eb = 3 V=4 DOF = 0

F=4 Eb = 3 V=6 DOF = Ð1

Figure 6.12. Two primal graphs and their attempted reciprocal figures. Left: this figure satisfies the Maxwell Condition with no leftover degrees of freedom. Right: this figure does not satisfy the Maxwell Condition, and there is no way to close the outer triangle that preserves orthogonality.

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Figure 6.13. An embedding of the right-hand primal graph of Figure 6.12 that has an orthogonal interior dual graph.

up fewer degrees of freedom. An example of such a primal-dual combination is shown in Figure 6.13. We can make one further simplification: noting that V − Eb = V − Vb = Vi , Equation (6.25) can be written as F − Vi ≥ 2.

(6.27)

As the size of the graph becomes larger, the number of faces F grows quadratically, while the number of border edges Eb typically only grows linearly—at least, for those graphs that are most interesting for tessellations (roughly speaking, those that grow in both width and height, as opposed to a long linear chain). For a large graph, therefore, what matters is that the number of faces stays larger than the number of interior vertices. This, in turn, says something about the average size of a face. Let us denote the average size of a face in a graph (i.e., the average number of vertices per face) by DF . If we “split” each edge into two half-edges, then every face gets DF half-edges with Eb half-edges left over from the border; thus, the total number of half-edges is DF F = 2E − Eb . (6.28) Combining this with the Euler relation and Maxwell Condition gives that the average facial degree DF must satisfy DF − 4 ≤

1 (Eb − 6). F

(6.29)

Consequently, for very large (and interesting) graphs, the right side of Equation (6.29) is going to be a small value in the range (0, 1). This implies that, on average, the faces can have no more than four sides in order for an orthogonal interior dual graph to exist. So, in general, graphs primarily of small polygons—triangles and quadrilaterals—are required for us to be able to construct a

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shrink-rotate tessellation. Because this is a condition on the average face size, we can tolerate a few high-degree polygons studded like raisins within the overall pudding of quadrilaterals and triangles. And of course, as we saw in Figure 6.13, it is possible for highly symmetric patterns to have reciprocal diagrams even if they don’t strictly satisfy the Maxwell Condition. Conversely, it is possible for a small region of a graph to “poison” the the entire graph against Maxwell Condition satisfiability. If we construct a subgraph of the primal graph by taking some connected subset of its faces with no holes—a facial subgraph—in order to construct a shrink-rotate tessellation, the Maxwell Condition must apply to that subgraph. Every such subgraph has an interior dual graph that is a subgraph of the overall interior dual graph; for the overall interior dual graph to exist, every possible subgraph of the primal graph must have an orthogonal interior dual graph. Consequently, any such subgraph that violates the Maxwell Condition, no matter how small, ensures that there is no solution for the overall graph. ? 6.4.2. Positive and Negative Edge Lengths There is another issue, however, that can prevent the construction of a shrink-rotate tessellation, and it is illustrated in Figure 6.14. The primal graph (in orange) has F = 6, Eb = 4, V = 8, so that DOF = 0; the dual graph is uniquely defined (to within translation and scaling). For this particular embedding of the primal graph, F=6 Eb = 4 V=8 DOF = 0

Figure 6.14.

A primal graph with a crossing dual graph. Left: this figure satisfies the Maxwell Condition with no leftover degrees of freedom. Middle: the dual graph contains crossings. Right: a topologically proper (but non-orthogonal) embedding of the dual graph.

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however, the dual graph is not a plane graph: it has two crossings of its edges. That means that two of the polygons in the dual graph are improperly drawn. Figure 6.14 shows the dual graph drawn with non-crossing edges on the right; it is topologically correct, but it is no longer orthogonal to the primal graph. You should be able to convince yourself that there is no way to alter the dual graph that achieves orthogonality and avoids this crossing of the edges. This makes it impossible to turn this graph into a shrink-rotate tessellation crease pattern. Recall that the resized/rotated versions of the polygons of the dual graph show up in the tessellation pattern, so the crease pattern would also contain these crossing polygons. The crossings would be additional vertices in the crease pattern that would not satisfy the Maekawa-Justin Condition (they would have two mountains and two valleys or all four creases of one type). So the existence of crossing edges in the dual graph is sufficient to prevent turning the primal graph into a shrink-rotate tessellation. In fact, remember that the Maxwell Condition applies to any facial subgraph of the primal graph. Any primal graph that contains the graph shown in Figure 6.14 as a subgraph would have to contain the crossed dual graph as a subgraph of its dual graph. Thus, the presence of such a structure in any primal graph is enough to poison the entire tessellation. There is a physical interpretation of this situation. Suppose we pick a vertex of the primal graph and count the faces around it in CCW order, as shown in Figure 6.15. Each face of the primal graph corresponds to a vertex of the dual graph and vice versa; so cycling through faces around a vertex in the primal graph corresponds to cycling through vertices around a face of the dual graph. Thus, the marked vertex in the primal graph is the vertex that corresponds to the crossing polygon in the dual graph.

A

Figure 6.15.

Left: primal graph with four faces labeled A–D in cyclic order. Right: dual graph with corresponding vertices marked.

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A B

C D C D B


Because the primal graph and dual graph edges must be orthogonal, line CD in the dual graph must be orthogonal to the highlighted edge between faces C and D of the primal graph (on the left). If we fix the position of vertex C, vertex D must lie somewhere on the highlighted line through CD (on the right). For the polygon to be non-crossing, vertex D would have to lie above vertex C on this line. Any non-crossing rendering of the polygon would have vertex D somewhere above C. We normally talk about the length of an edge in a plane graph as the magnitude of the distance between the vertices—a number that is always positive. But we can associate a signed length with an edge by calling the signed length positive if the direction of circulation of the dual graph polygon is the same as the direction of circulation around the vertex (e.g., vertex D lies above vertex C on the orthogonal line) and calling the signed length negative if the direction of circulation goes the opposite direction. This definition corresponds exactly with the force model of Maxwell. Recall that if the primal graph is a planar beam construction under a series of stresses, the lengths of the edges of the dual graph give the stresses (compressive or tensile) in each corresponding member of the mechanical structure. In fact, this correspondence refers to the signed length as we have defined it here. If the primal graph describes a structure that is placed under tension, then the signed length of each edge of the dual graph gives the tension in each member corresponding to an edge of the primal graph. If the signed length is negative, the tension is negative; or in other words, the member is under compression, not tension. Thus, for a primal graph to be turned into a tessellation, it must have an orthogonal interior dual graph—i.e., a reciprocal figure— in which there are no edge crossings; all edges must have positive signed length. This means that there must be some application of tensile forces to the primal graph that makes the tension in each edge of the graph strictly positive. Such a graph has a name in the field of static analysis: it is called a spiderweb. The analogy is fairly direct. If you can create the graph from thread (or spider silk) and apply tensile stresses so that there are no slack threads, then the dual graph must be properly non-crossing and have edges that are orthogonal to the edges of the primal graph. Using the vertices of the dual graph, one can then shrink and rotate each of the polygons of the primal

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graph tiling to realize a crease pattern that satisfies the KawasakiJustin Condition. By choosing the shrinkage and rotation factors (ρ, β) to keep the twist angle α sufficiently small, it is possible to assign creases to every parallelogram, and thus every twist, to make them individually flat-foldable; and so, the entire crease pattern can be made flat-foldable. Or put succinctly: every spiderweb can be turned into a flatfoldable shrink-rotate tessellation. Now, while one could, in principle, start constructing tessellations from actual spiderwebs, what this property actually does is expand the possible tilings that can be transformed into twist tessellations. Recall from Chapter 4 that • in a tessellation composed of centered twist tiles, the tile polygons must be individually cyclic; • in a tessellation composed of offset twist tiles, the tile polygons must be individually cyclic and Brocard, and the tiling must be two-colorable. Using the shrink-rotate algorithm, we only require that a noncrossing reciprocal figure exists, and this is often possible (but of course not guaranteed) if the average facial degree is ≥ 4. As mentioned at the beginning of this chapter, the shrinkrotate algorithm was utilized as a concept by Chris K. Palmer, but it was implemented as an algorithm by Alex Bateman, who first observed the duality between crease pattern and folded form obtained by switching the sign of the twist angle [5]. Bateman and I subsequently described the connection with reciprocal diagrams [72]. Bateman has constructed quite a variety of tessellations using this algorithm (and his program Tess [7] can construct many more); a representative sampling is shown in Figure 6.16. As we have seen, we can construct shrink-rotate tessellations by algorithmically combining a primal graph, which provides the tiling, with an orthogonal interior dual graph, which provides the centers of shrinkage/rotation. Since the primal graph and dual graph are reciprocal figures of each other, we could also have shrunken/rotated the polygons of the dual graph about their corresponding vertices in the primal graph. We call a tessellation that uses information from both a primal graph and its interior orthogonal dual graph a primal-dual tessellation. Applying the shrink-rotate algorithm to a graph and

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Figure 6.16. A collection of shrink-rotate tessellations by Alex Bateman, folded from tracing paper.

its reciprocal figure gives a valid shrink-rotate tessellation. As we will presently see, though, the concept of a primal-dual tessellation has broader applicability that goes well beyond simple flat twists. ?? 6.4.3. Crease Assignment There is still the question of crease assignment. A shrink-rotate tessellation will be a composition of twists, and as we have

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seen, each twist will have its own conditions on crease assignment that determine whether it can be flat-folded without selfintersection. Recall, though, that for a general twist, there is a safe twist angle αsafe , given by Equation (3.28); for twist angles below this value, every twist is guaranteed to fold flat without selfintersection with anto pleats (where the creases at the acute angle of each parallelogram have opposite assignment). Although each twist parallelogram has four folds, there are only two possibilities for crease assignment. Parallel edges must have opposite assignment, and creases at the acute vertices must have opposite assignment. Thus, the crease assignment is fully determined simply by specifying, for each parallelogram, which of the two primal graph faces lies on top of the other. This specification can be simply encoded by orienting the dual graph, i.e., assigning a direction to each edge. Each vertex of the dual graph corresponds to a primal graph polygon; assigning a direction arrow to the edge means that the primal graph polygon at the tail of the arrow lies on top of the primal graph polygon at the head. From this orientation, the full crease assignment follows directly. Figure 6.17 shows a simple example. Figure 6.18 shows another somewhat more interesting example. Here I have placed the vertices of the primal graph on concentric ellipses. Observe that the reciprocal figure consists of similarly scaled polygonal ellipses.

Figure 6.17. Orientation of the dual graph fully specifies the crease assignment of a shrink-rotate tessellation. Top row: CCW twists. Bottom row: CW twists. Left: primal and oriented dual graphs. Middle: crease pattern. Right: folded form.

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Figure 6.18. A shrink-rotate tessellation based on concentric elliptical decagons. Left: primal and oriented dual graphs. Middle: crease pattern. Right: folded form.

Because the crease pattern is a simple vertical scaling of a circular pattern, we could construct the dual graph by starting with the circular pattern and constructing its dual graph (its vertices will be the circumcenters of the trapezoids of the primal graph), then scaling the primal and dual graphs oppositely, as we did to create the rhombus graph. For most primal graphs, though, the dual graph must be found numerically, as is the case for the tessellation shown in Figure 6.19. ?? 6.4.4. Triangle Graphs For primal graphs composed entirely of triangles, we can construct the reciprocal figure—or at least, a potential candidate—in a very straightforward way, relying on the geometric property that the perpendicular bisectors of the three sides of a triangle meet at a point, which is the circumcenter of the triangle. If we apply this construction to any two incident triangles, as illustrated in Figure 6.20, the perpendicular bisector segments join collinearly and are orthogonal to the edge between the triangles. It is clear that the circumcenters and the lines joining them form a dual graph that is a reciprocal figure; consequently, it can be used for the construction of a shrink-rotate tessellation.

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Figure 6.19. A primal-dual tessellation based on a rainbow of quadrilaterals. Top left: primal graph. Top right: oriented dual graph. Bottom left: crease pattern. Bottom right: folded form.

I have used this method to construct a tessellation based on a mapping of a (3.3.3.3.3.3.3) tiling in the hyperbolic plane1 into the Euclidean plane using a mapping called the Poincaré map. (Technically, the Poincaré map creates curved tile lines, but I have straightened them out for origami purposes.) Figure 6.21 shows the primal and dual graphs and the crease pattern. Figure 6.22 shows a folded example. If the triangles are all strictly acute (all corner angles less than 90◦ ), then every triangle contains its circumcenter, and the reciprocal figure is guaranteed to be non-self-crossing; this is the case in Figure 6.21. However, if there are any obtuse triangles, their circumcenters lie outside of the triangle. This can result in a circumcenter-constructed reciprocal figure being self-crossing, as in the example in Figure 6.23. Even if this happens, it may still be possible to perturb the dual graph to achieve a non-self-crossing reciprocal figure that 1

Yes, there are seven 3s in that tiling type. In the hyperbolic plane, there is a uniform tiling in which seven equilateral triangles meet at every vertex.

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Figure 6.20.

Left: the intersection of the perpendicular bisectors of a triangle gives the circumcenter of a triangle. Right: connecting adjacent circumcenters gives edges of a reciprocal figure.

maintains edge orthogonality between primal and dual graphs, as the figure shows. More generally, if the tessellation is composed of cyclic polygons—those that can be circumscribed by a circle—then the same construction gives a reciprocal figure from which, if it is non-self-crossing, a shrink-rotate tessellation can be constructed. Recall that the centered twist tiling method worked as well for any tiling composed of cyclic polygons. Thus, the shrink-rotate

Figure 6.21.

Left: primal and dual graphs for a tiling of triangles, based on a tiling in the hyperbolic plane. Right: the constructed crease pattern, with γ = 1.5, α = 20◦ .

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Figure 6.22. Folded form of the hyperbolic tiling tessellation.

Figure 6.23.

Left: obtuse triangles can give rise to a self-crossing reciprocal figure if the circumcenters are chosen as the centers of rotation. Right: in this case, another non-self-crossing reciprocal figure can be constructed.

algorithm is equivalent to centered twist tilings—just a different way of constructing the tessellation. ?? 6.4.5. Voronoi and Delaunay A primal graph composed of multiple quadrilaterals and/or triangles may well satisfy the Maxwell Condition for average facial degree, Equation (6.29), but it is still entirely possible (and in practice, quite common) to find quadrilateral arrangements that

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Figure 6.24.

Left: a Voronoi diagram (orange) of a set of points (green). Right: connecting points of cells incident to each other gives the Delaunay triangulation.

don’t satisfy the spiderweb condition. This is a bit frustrating. Wouldn’t it be nice to have a large category of graphs that is guaranteed to possess a reciprocal figure from which a shrink-rotate tessellation could always be constructed? There is such a class of graphs. An old concept in graph theory is the Voronoi diagram of a set of points [10, p. 160]. The Voronoi diagram is a subdivision of the plane into cells around a set of points such that • each point is enclosed by a single cell, • every point in any cell is closer to that cell’s point than to any other point, • points on the border of a cell are equidistant from the points of the incident cells. An example of a Voronoi diagram is shown in Figure 6.24. The Voronoi diagram fills the plane; in general, some of the lines will run off to infinity. If, for any pair of Voronoi cells that are incident to each other, we connect their corresponding points with a line, we get a second type of graph, shown in green in Figure 6.24, called the Delaunay triangulation. The Delaunay triangulation is the way of connecting all of the points to form triangles in such a way that it maximizes the minimum angle in any triangle. It has another nice property: the circumcircle of any triangle contains exactly one of the original points. What makes the Delaunay triangulation relevant to shrinkrotate tessellations is that the edges of the Delaunay triangulation correspond one-to-one with the edges of the Voronoi diagram,

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and corresponding edges are orthogonal to one another. Thus, for any set of points, the Delaunay triangulation is a valid reciprocal figure of the Voronoi diagram. We can use a Voronoi-Delaunay pair to generate a family of shrink-rotate tessellations. In the Delaunay triangulation, each vertex has, on average, six surrounding triangles. This means that each Voronoi face is, on average, of degree 6. So this provides a way to get around the Maxwell Condition and generate shrink-rotate tessellations for polygons of average degree higher than 4. Another property that makes the Delaunay triangulation desirable for tessellations is that it maximizes the minimum angle in any triangle. Since the safe angle for a triangle twist depends on the minimum angle of the triangle, the Delaunay triangulation is desirable to avoid being forced into very small twist angles by sliver triangles. So, a simple recipe for creating a Voronoi-Delaunay tessellation would be the following: 1. Choose a set of points to have some interesting pattern. 2. Compute the Voronoi diagram and Delaunay triangulation. 3. Orient the Delaunay triangulation (which will determine the crease assignment). 4. Apply the shrink-rotate algorithm (subject to limitations on twist angle). Since the Voronoi diagram extends to infinity, we will need to clip its polygons to some shape. A square is convenient. As an example, Figure 6.25 shows the Voronoi-Delaunay pair and the resulting shrink-rotate tessellation for a set of points that is a randomly perturbed 4 × 4 grid of points. By varying the aspect ratio parameter γ, it is possible to “tune” the tessellation to emphasize the higher-order polygons of the Voronoi diagram (for large γ) or the triangles of the Delaunay triangulation (for small γ). Figure 6.26 shows crease patterns and folded forms for the same primal-dual pair as in Figure 6.25, but with larger and smaller values of γ. I chose the points in Figure 6.24 and 6.25 randomly, to demonstrate the generality—the algorithm works for any set of points— but the result illustrates one potential weakness of this method.

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Figure 6.25.

Left: a clipped Voronoi diagram for a set of points (orange) and its Delaunay triangulation (green) for aspect ratio γ = 1 and twist angle α = 20◦ . Middle: crease pattern. Right: folded form.

Figure 6.26. The Voronoi-Delaunay tessellation of Figure 6.25 for different values of γ. Top row: γ = 2.5. Bottom row: γ = 0.4. Left: crease patterns. Right: folded forms.

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Figure 6.27.

Left: a clipped Voronoi diagram for a set of points (orange) and its Delaunay triangulation (green) based on a phyllotactic spiral. Middle: crease pattern. Right: folded form.

Although the Delaunay triangulation maximizes the minimum angle of any triangle, with a large number of random points, one can still end up with some fairly small angles, and this, in turn, sets a fairly small limit on the twist angle. The crease pattern in Figure 6.25 has a twist angle of 15◦ , which is pretty small (and correspondingly challenging to fold). Using randomly chosen points shows off the versatility of the Voronoi-Delaunay tessellation algorithm, but it has an aesthetic weakness: the pattern is “too random.” An aesthetically pleasing tessellation is neither too symmetric nor too irregular. In Figure 6.27 I show an example of a Voronoi-Delaunay tessellation based on a phyllotactic spiral, a pattern often seen in the growth patterns of plants [101]. In this pattern, every hexagonal region is slightly different from every other hexagon, but the overall pattern displays a pleasant mixture of alternating-direction spirals and, in principle, could be extended forever without ever precisely repeating a hexagon. ?

6.5. Flagstone Tessellations The origami tessellations we have considered thus far are composed of simple flat twists. In such tessellations, some polygons are on top of all of their neighbors; others may be wholly or partially covered by all of their neighbors; but usually, most are somewhere in between, partially covered by some of its neighbors and partially covering others.

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There is another class of origami tessellations in which every visible polygon lies entirely on top of all of its neighbors and no other polygon covers any visible polygon. In such tessellations, the visible polygons must necessarily butt-join (with no overlaps) at their folded edges. Such tessellations are called flagstone tessellations, because the pattern of polygons separated by recessed linear gaps resembles the flagstones of a walkway. Flagstone tessellations have a special appeal in the origami tessellation world, and there are many examples (see, e.g., Gjerde [37]). We may define a flagstone tessellation formally as follows: we assume a crease pattern P is partitioned into vertices, creases, and facets, all of which have both position and layer ordering defined in the folded form. A flagstone tessellation is a flat-foldable origami crease pattern P containing a subset F of facets— the flagstone facets—that satisfies the following: 1. In the folded form, the closure of F is simply connected. 2. In the folded form, no facet of P overlaps and covers any part of any facet in F . In layman’s terms, item 1 says that there are no gaps between flagstone facets; item 2 says that they don’t overlap each other. This definition is quite broad and supports many possible schemes for constructing flagstone tessellations. Many, particularly those of Gjerde [37], incorporate twisted pleats behind the flagstones. Here I present a conceptually simple method for constructing flagstone tessellations, which has surprising connections to the shrink-rotate algorithm and Maxwell’s reciprocal figure. ? 6.5.1. Spiderwebs Revisited In most of this book, vectors, which were introduced in Section 1.6, are ? ? ? material, but in this section, we don’t need their full definition and properties. If you haven’t seen a vector before (and have been skipping the ? ? ? sections), you can probably follow this section with just the following: • A vector behaves like an arrow that has a specified length and direction.

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p

p

q

q

−q

p−q

p+q

−q p

Figure 6.28. An illustration of vectors, represented by arrows with specified length and direction. Left: a first vector p. Middle left: a second vector q. Middle: a vector (blue) that is the sum p + q. Middle right: the negative of vector q. Right: a difference vector p − q is the sum of p and −q.

• You can add vectors together to get a vector sum by joining their arrows head to tail. The sum is the arrow that connects the unpaired tail to the unpaired head, as illustrated in Figure 6.28. • The negation of a vector is an arrow going the opposite direction. An important property used in flagstone tessellations can be seen in a simple flat twist. Figure 6.29 shows the crease pattern and folded form for a simple flat twist created from an irregular quadrilateral. In the crease pattern of Figure 6.29, I have drawn red arrows to represent vectors that are perpendicular to the valley folds of the pleats and that connect a point on the mountain fold to the point that it maps to on the opposite side of the valley fold. These four vectors are then connected by blue vectors, head to tail, forming a closed loop. We call the red arrows pleat vectors and the blue arrows wedge vectors. In Figure 6.29, we sum the pleat vectors and wedge Figure 6.29. A simple flat twist. Left: crease pattern. Pleat vectors are red; wedge vectors are blue. The shaded regions are regions that are not visible in the folded form. Right: folded form.

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b

b a

a

c d d


c

b

b b

a

b

a

c d d

c

a

a

c

c

d d

Figure 6.30.

Left: illustration that the pleat vectors independently must sum to zero. Right: rotating each of the four vectors by 90◦ aligns each vector with its corresponding line.

vectors by joining them head to tail, forming a closed loop. Since the head-to-tail vectors sum to a closed loop, the vector sum of all eight vectors must be zero, i.e., an arrow of zero length. In the folded form in Figure 6.29, the wedge vectors connect with each other to form a closed loop. So they, too, must independently vectorially sum to zero. If the sum of the pleat vectors and wedge vectors together is zero, and the sum of the wedge vectors alone is zero, then the sum of the pleat vectors must also be zero. This is illustrated in Figure 6.30, in which the four pleat vectors have been translated together to connect head to tail. And indeed, they do form a closed loop, as the figure shows. If we rotate all four vectors by 90◦ , they will be parallel to their respective pleats, but the rotation will not change the fact that they vectorially sum to zero. But if we translate them so that their tails touch the common point, we can give a different interpretation to this diagram: the vectors can be thought of as lines of force along their four corresponding lines, and since they sum to zero, we can say that these are the forces that would be experienced by a static network of four lines in tension at a common point. This condition is the mechanical spiderweb condition of Maxwell; thus, we can see that there is a direct correspondence between the forces on a spiderweb under static tension and the pleat widths (the lengths of the red vectors) needed to realize a simple flat twist in origami. But this has a broader interpretation: if we somehow “remove” parallel strips of paper along the boundaries of a tiling to bring the edges of the tiling together, the widths of the removed strips must obey the spiderweb condition. We can use this property to create other types of tessellations.

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Figure 6.31.

Left: side view of a single pleat, as in a simple flat twist. Right: a double pleat, which can be used to realize flagstone tessellations.

? 6.5.2. The Flagstone Geometry A simple flat twist tessellation effectively removes parallel-edged strips of paper from visibility; in Figure 6.29, the shaded regions are completely hidden from view. But a simple flat twist isn’t compatible with the flagstone concept because the paper on one side of the removed region lies on top of the paper on the other side. We can imagine, though, that some other arrangement of folds and layers could hide the removed region and also give rise to the flagstone configuration. One such example is a double pleat, as illustrated in Figure 6.31. If the shaded region were folded with two oppositepolarity pleats, as on the right in Figure 6.31, rather than with a single pleat as on the left, then the top surfaces would meet along their folded edges, realizing the flagstone condition. In fact, if we were allowed to cut away regions of the paper, a very simple algorithm could be used to realize a flagstone tessellation. As with the simple flat twist tessellation, we construct the reciprocal figure and use those vertices as the centers of shrinkage of each polygon—but this time, we don’t employ any rotation. This has the effect of inserting rectangles of paper between the edges of formerly touching tiles; these rectangles are our parallel-edged strips. We cut away the polygon outlined by the rectangles—the dark quadrilateral in the center—then add two side-by-side pleats to each rectangle. The construction is illustrated in Figure 6.32, where we have created each pleat by dividing the rectangles evenly into fourths. In the folded form, the wedges of the crease pattern all come together at a point (the black dot in the center), which means that the vectors that define the pleat widths must form a closed loop, as they did on the crease pattern. We already know how wide the pleats must be. Their width vectors must sum to zero, and so the original tiling must satisfy the spiderweb condition,

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b

b

a

Figure 6.32.

a

d

c

c

d

Left: geometry for a flagstone construction. The dark region is cut away. Right: folded form. Dotted lines indicate hidden layers.

exactly the same way as a tiling must to create a shrink-rotate tessellation. So, satisfaction of the spiderweb condition is a necessary condition for the realization of this type of flagstone tessellation as well. And by solving for the edge lengths of the reciprocal figure, we can have pleat widths for all of the pleats of the flagstone tessellation (to within an arbitrary constant of proportionality). However, we cannot simply cut away the polygons at the corners where the rectangles come together—at least, not if we’re going to call this origami. Instead, we must find folds that cleanly continue the pleats between the tiles and interact in such a way that the corner structure lies flat and has no self-intersections. There is also the question of how wide to make the pairs of pleats. In Figure 6.32, we divided each pleat region evenly, but that division is not required; we could have as easily devoted most of the paper to one side or the other of each pair. There is, as it turns out, some freedom in choosing the folding pattern for the pleats and the junctions. One such folding pattern has some elegant symmetry properties with respect to the underlying tiling. ? 6.5.3. Flagstone Vertex Construction For many tiling patterns that satisfy the spiderweb condition, an embedding of the reciprocal figure may be found such that each vertex of the dual graph is enclosed by its corresponding primal graph polygon. Such graphs permit a particularly elegant choice of flagstone pleat vertex construction that permits a purely geometric (as opposed to numerical) construction, allowing the entire construction to be carried out within a computer drawing program.2 2

Or for those of a more traditional bent, using compass and straightedge.

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Figure 6.33. Geometry for a flagstone vertex construction. Left: the primal graph (orange) is the original tiling. Its polygons are shrunken about the vertices of the reciprocal figure (green). Right: construction of pleat pair valley folds. Dotted lines connect corresponding vertices in adjacent polygons.

Once we have shrunken each polygon about its corresponding dual graph vertex, each edge of the primal graph lies parallel to and strictly between the corresponding edges of its shrunken polygon, and each vertex of the primal graph lies within the “cut out” polygon, which is, in fact, a shrunken polygon of the dual graph. This situation is illustrated in Figure 6.33. When the pleats are made that bring two polygon edges together, those pleats must bring the polygon edges onto a common line that lies somewhere in the interior region of the pleat rectangle. We make the choice that this line is the original line of the tiling. The valley folds of the pleats then each lie halfway between the mountain folds (the edges of the shrunken polygons) and the lines of the original tiling, as shown on the right in Figure 6.33. Similarly, when the pattern is folded, the corners of the shrunken polygons around a vertex must all come together at a point somewhere in the cut-out polygon; we choose that point to be the original vertex of the primal graph tiling. And with those two choices, the remaining creases are forced and can be constructed geometrically—in fact, they may be constructed using only reflection and translation, as follows and illustrated in Figure 6.34 for the central polygon of Figure 6.33. These operations are typically built into standard computer drawing programs. Applying this construction to each vertex of the tiling gives a flagstone tessellation for the entire tiling. Note that the procedure

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1. Begin with the central polygon; construct its surrounding mountain and valley folds as shown here.

2. Construct perpendicular bisectors of the (dotted) segments between each of the shrunken polygon corners and the original corner, outlining a new polygon with valley folds, which we will call the central polygon.

3. Connect each vertex of the central polygon with the two adjacent shrunkenpolygon corners using mountain folds.

4. Translate copies of the central polygon by the vector from the original vertex to each of the shrunken vertices.

5. For each of two corners of each copy, reflect the corner across both the mountain and valley (in that order) of the adjacent pleat, resulting in one pair of coincident points for each pleat.

6. Construct perpendicular bisectors of the segments between each of the new points and the corresponding corner of the central polygon using mountain folds, extending each bisector to the valley folds within each pleat.

7. Connect the new vertices with those of the central polygon and the surrounding shrunken polygons using valley folds.

8. Remove the original tiling, leaving the completed crease pattern.

Figure 6.34. Construction of the folds around a single vertex of the flagstone tessellation.

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applies even for vertices on the border of the original tiling. By construction, each of the added vertices satisfies the KawasakiJustin Condition and the Maekawa-Justin Condition, and thus the entire crease pattern is flat-foldable—with two significant caveats. The first, which is relatively minor, is that since the pleats extend in both directions away from their mutual lines right up to the point where they all come together, it is quite possible—in fact, downright likely—to incur collisions between pleats around a vertex. This is easily dealt with, simply by adding some extra folds to one or the other of the colliding layers. The second is more serious. It is also possible for the construction of the central polygon in step 2 of Figure 6.34 to result in a self-crossing polygon. This is not allowed in a crease pattern, and any such occurrence at any vertex effectively kills the algorithm for that particular tiling. Note, though, that the choice of using the original tiling lines and vertices as the alignment features was an aesthetic choice, not driven by mathematical necessity. In fact, the lines along which the shrunken polygon edges align may be chosen arbitrarily between said edges, and the point where the shrunken polygon corners come together may be chosen arbitrarily in the vicinity of the tiling vertex. So, if a self-crossing central polygon arises, it may be removable by choosing a different meeting point for surrounding corners in the central polygon. ? 6.5.4. Examples I show here two examples of flagstone tessellations computed using this algorithm. The first is a section of (3.4.6.4) tiling (the same section, in fact, that we started with in Figure 6.1). Figure 6.35 shows the primal and dual graph and the crease pattern of the resulting tessellation. Figure 6.36 shows photographs of the folded form. As noted already, this algorithm only works for a tiling (a) that satisfies the spiderweb condition and (b) within which each polygon encloses the corresponding vertex of its reciprocal figure. This includes many tilings of aesthetic interest, including all tilings composed of regular polygons and/or arbitrary acute triangles. For such tilings, the dual graph vertices are simply the circumcenters of each polygon. The restriction to acute triangles is necessary to ensure that the circumcenter lies in the interior of every triangle. Since the dual graph may be translated relative to the primal

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Figure 6.35.

Left: primal and dual graph tiling of a piece of a (3.4.6.4) Archimedean tiling. Right: computed crease pattern.

Figure 6.36.

Left: the flagstone tessellation. Right: the reverse.

graph, even if there are non-acute triangles, there may still be a flagstone tiling, if the graph can be translated so that all dual graph vertices lie within their corresponding polygons (and/or exceptions lie outside the border of the polygon). An interesting property of flagstone tessellations is that the completed tessellation is a reduced-size copy of the original tiling. This is in contrast to ordinary shrink-rotate tilings, for which the tiles are offset relative to one another. Because of these offsets, in shrink-rotate tessellations, the silhouette of the folded shape can differ significantly from that of the original paper. In this form of

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Figure 6.37.

Left: primal and dual graph tiling for a Penrose rhomb tiling. Right: computed crease pattern.

Figure 6.38.

Left: the flagstone Penrose rhomb tessellation. Right: the reverse.

flagstone tessellation, the silhouette of the tiles in the folded form is an identical smaller copy of the original tiling. As a final example, I have created a flagstone version of the well-known Penrose rhomb tiling [19] shown in Figures 6.37 and 6.38. The Penrose rhomb tiling is interesting because its reciprocal figure takes the form of a series of sets of intersecting and parallel lines, called Ammann bars. It is potentially problematic because the vertices of the dual graph do not reside within their correspond-

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Figure 6.39.

Left: “Cairo Tessellation” (2011), a flagstone tessellation by Eric Gjerde, based on the Cairo tiling. Right: the reverse, showing simple folds at the vertices.

ing tiles (the fat rhomb tiles contain their dual graph vertices, but the dual graph vertices corresponding to skinny rhomb tiles are offset to one side of the tile). Nevertheless, for this portion of a rhomb tiling, it is possible to construct the flagstone tessellation. Folding it, though, is quite the challenge. There are multiple ways to realize flagstone tessellations, and while the construction in this section is quite general, for specific patterns, there are often simpler treatments of the folds around the vertices. A particularly elegant one is the flagstone tessellation by Eric Gjerde shown in Figure 6.39. This is based on the Cairo tiling that we met back in Chapter 5; fitting, perhaps, because the Cairo tiling is itself often used for flagstone paving. Both shrink-rotate tessellations and the flagstone tessellations of this section make use of the reciprocal diagram of the original tiling to determine how much paper is inserted between shrunken tiles. Thus, they are both examples of primal-dual tessellations. There are undoubtedly other families of tessellations that can be constructed using a primal-dual approach. ? ???

6.6. Woven Tessellations The primal-dual tessellations shown in this chapter illustrate a generalizable concept: take an existing mathematical tiling and then transform it mathematically to realize an origami tessellation in which folds recreate some aspect of the original tiling. Both

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twist and parallel-pleat flagstone tessellations make use of the reciprocal figure to realize a flat-foldable tessellation, but there are other transformation algorithms to create tessellations that can act directly upon the original tiling. One such algorithm that we will now address is an algorithm for woven tessellations [70, 71]. ? 6.6.1. Woven Concepts One of the origami tessellations we have already seen is the alternating simple flat twist tessellation based on the (4.4.4.4) tiling with offset twist tiles, which we saw in Chapter 4. It is composed of twisted squares in which adjacent squares rotate in opposite directions. In such a tessellation, it is possible to assign creases so that all of the squares are surrounded by mountain folds, i.e., they are all in cyclic form. An example of such a tessellation is shown in Figure 6.40. If you invert the crease assignment in such a tessellation (or turn it over), there is a surprise: the pattern of edges looks uncannily like a set of woven strips, as shown in Figure 6.41. This is a pleasing illusion, because origami tessellations are, by definition, folded from a single sheet, but this structure looks like it was constructed from many separate strips of paper, plus a background field. This pattern suggests that there might be a family of such patterns as we start to consider generalizations, and indeed, several

Figure 6.40.

Left: crease pattern for an alternating simple flat twist tessellation consisting of square twists with cyclic mountain crease assignment. Right: folded form of this tessellation.

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Figure 6.41.

Left: crease pattern for an alternating simple flat twist tessellation consisting of square twists with cyclic valley crease assignment. Right: folded form of this tessellation. This crease assignment displays the appearance of continuous woven strips.

people have done so; see [6, 8] for some examples. One of the artists who explored woven tessellations was Alex Bateman, who we have already met, and it was a conversation with Bateman, in which he showed me some of his examples of woven tessellations, that inspired me to look into the genre. In the square woven tessellation, the strips run up-and-down and side-to-side and are evenly spaced. But one could envision patterns in which the strips run at other angles, or other spacings. For example, there is a strong woven effect in the work “Pinwheels” by Eric Gjerde, shown in Figure 6.42.

Figure 6.42.

Left: “Pinwheels” (2005), by Eric Gjerde. Right: the reverse.

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We could imagine more complex patterns with woven strips running at still other sets of angles—or even at no particular angle, with every strip running in a different direction. There is a lot that is possible; if we think of such patterns as “patterns formed by woven strips,” we not only have the spacing and angles of the strips to choose from: we can also choose, at every intersection, which strip appears to be on top at the crossing. We can also choose the width of the apparent strips, and we can, in principle, choose them all independently. These variables outline a vast space of potential patterns, any of which may or may not be realizable as a single-sheet origami tessellation. I call such patterns woven tessellations. The descriptions of the patterns possible with woven tessellations parallel the patterns possible in textile weaving: in both cases, one can vary the strips (analogous to the warp and weft threads in textiles), their widths, their angles, and the pattern of crossings—how the various strips and threads cross over one another. The simplest woven pattern is called a plain weave, or a simple over-and-under weave. Any given strip, followed along its length, alternately goes over and under the strips that it crosses. If we further stipulate that • no more than two strips cross at a given point, • every strip travels in a straight line, then we can narrow the field of possible woven tessellations considerably. We will call any tessellation that uses a simple overand-under weave and that satisfies these two conditions a simple woven tessellation. The question then arises: what are the possible simple woven tessellations, and how might they be folded? ? 6.6.2. Simple Woven Patterns To simplify matters, let us assume that all strips are the same width. Then we can construct a simple woven pattern by the following prescription, illustrated in Figure 6.43: 1. Choose some pattern of straight lines such that no more than two intersect at any vertex. 2. Fatten each line to the desired width of the strips.

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Figure 6.43. Construction sequence for a simple weave. Top left: begin with a pattern of lines for which no more than two intersect at any given point. Top right: thicken each line. Bottom left: add a border to the field. Bottom right: selectively erase crossings based on a two-coloring of the polygons between the woven edges.

3. Add a border to the pattern to define the background field. 4. At each crossing, erase two of the four lines at the crossing to create a simple over-and-under weave. How to do this last step is, perhaps, not entirely obvious, but there is a simple procedure. Note that all interior vertices of the line pattern have degree 4, which means that the pattern can be two-colored as shown in Figure 6.43. Each background polygon is surrounded on all interior sides by partial strips. If we give each polygon a counterclockwise (CCW) circulation, each side of the polygon has a head, which is the end of the strip segment in the CCW direction, and a tail, which is the end of the strip segment at the other end. We can use these definitions, plus the two-coloring, to create the over-and-under pattern as follows: • In colored polygons, the head of one strip segment goes under the tail of the next strip segment. • In white polygons, the head of one strip segment goes over the tail of the next.

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Figure 6.44. Crease pattern and folded form with the key trapezoidal structure highlighted. Left: crease pattern. Right: folded form.

So that defines the woven pattern. But how do we turn it into an origami figure? We can get an idea of this by looking at the square pattern again, but this time let’s give the paper some translucency so that we can see the hidden layers of paper—which is where all the action is—and we’ll zoom in a bit. Figure 6.44 shows the crease pattern and folded form. Pay particular attention to the highlighted region in Figure 6.44. We see that this shaded trapezoid reappears throughout the pattern; in fact, every crossing of two strips has one of these trapezoids. Every trapezoid has two obtuse-angle vertices, which are the (hidden) ends of a going-under strip, and two acute-angle vertices, which are the endpoints of a covering-up strip. And then, of course, each of the acute-angle vertices of a trapezoid is incident to an obtuse-angle vertex of an adjacent trapezoid. When the pattern is unfolded, each trapezoid appears explicitly in the crease pattern. In principle, one could imagine that any woven tessellation pattern might be realized by placing some version of this same trapezoidal structure at every strip crossing in a woven tessellation. Using our test pattern of strips from Figure 6.43, a hypothetical example is shown in Figure 6.45. We can start by designing a folded form, as illustrated in Figure 6.46, based on the desired pattern of lines. Once we’ve constructed a folded form, we simply unfold it to get the desired crease pattern. The challenge is, can we make the folded form unfold to a flat sheet of paper, i.e., ensure that it is developable? We have some freedom in choosing the dimensions of the trapezoids; with any

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Figure 6.45. A hypothetical folded form of the woven-strip pattern from Figure 6.43.

Figure 6.46. Sequence for constructing a simple woven tessellation. Left: begin with the pattern of lines, including the border. Middle: create a folded form consisting of strips connected by hidden trapezoids, where the strips are obtained by thickening the lines. Right: unfold the folded form to get the desired crease pattern.

luck, we can choose their dimensions to ensure developability of the unfolded form. ? ? ? 6.6.3. Woven Algorithm The first step in creating the woven tessellation is to create an image of what the desired folded form would be. A simple process is shown in Figure 6.47. We start with the pattern of lines, which defines our desired pattern of strips and choose a strip width 2h. We can then construct the pattern of strips by insetting each of the polygons in the array by a distance h from the tile lines and then connecting pairs of vertices to establish a continuous border and the pattern of overlaps among the strips. Next, we add trapezoids (or partial trapezoids along the border), as illustrated in Figure 6.48. We don’t know yet the precise shape of the trapezoids. We do know that every point of a trapezoid lies somewhere along a line that is an extension of an edge of one of the inset polygons. For

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Figure 6.47. First steps for constructing a simple woven tessellation. Left: begin with the pattern of lines, including the border. Inset each polygon by the same perpendicular distance h. Right: connect pairs of points to establish the border and the pattern of overlapping strips.

example, in Figure 6.48, point pi is a corner of an inset polygon, whose edges correspond to folded edges of the desired folded form. Point qi , a corner of one of the hidden trapezoids, must lie on the extension of the inset polygon edge. In fact, if we define di as the distance along that edge underneath the covering strip, we can write an explicit expression for qi : qi = pi + di ri, (6.30) where ri is the unit vector in the direction of the edge. This makes it clear what freedom we have in choosing the folded form: for each trapezoid vertex qi (green dots in Figure 6.48), the quantities pi and ri are determined from the desired form of the folded form, but we can choose the value of each di

ri

pi

qi di

Figure 6.48.

Left: add trapezoids, which are layers hidden behind the visible strips. Each trapezoid point is an extension of an existing point along an existing edge. Right: all trapezoid points and the inset polygon points of which they are extensions.

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independently. In Figure 6.48, there are 20 such vertices, so 20 possible values of di . It’s good that we have some freedom in choosing the values of di , because we have conditions to satisfy. Each of the vertices in the interior of the pattern is a flat-folded degree-4 vertex, and the angles around the vertex must satisfy a condition. Recall that for a crease pattern to be flat-foldable, i.e., to go from unfolded to flat-folded, the angles around the vertex must satisfy the Kawasaki-Justin Condition, a flat-foldability condition. Here we have something new: we require that a known flat-folded vertex must unfold to a flat sheet of paper. This is a flat-unfoldability condition. ? 6.6.4. Flat Unfoldability The question just posed is a specific example of a more general problem: if we are given a description of a flat-folded form (complete or partial), what are the conditions that ensure that it can be folded from a simple flat sheet of paper? This question is an inverse problem, and its answer would be the opposite of the more well-known flat-foldability conditions that determine whether a crease pattern can be folded into a flat-folded form. Those conditions are the Kawasaki-Justin Condition (that the alternating sum of the angles at any interior vertex equals zero) and the Justin Non-Crossing Conditions (that disallow self-intersection). The Kawasaki-Justin Condition is a metric condition; it ensures that the folded form lies flat when constructed from non-stretchy (isometric) material. Justin’s Non-Crossing Conditions are combinatorial, and because they describe the layer ordering of the folded form, they are, implicitly, a condition on the folded form. If a folded form satisfies Justin’s conditions—we have chosen a layer ordering that avoids self-intersection—then the only condition still to be met is some isometric condition, and the particular isometric condition would be that the sector angles around each interior vertex, when the paper is unfolded, must sum to 360◦ , so that the vertex lies flat. Given a folded vertex and the information about each layer, we can define the sector angles of the vertex, in order, as the rotational angle from each folded edge to the next within each separate layer. If we adopt the usual convention that CCW rotation is a positive angle, then we will find that successive sector angles in the folded form alternate in sign: first positive (CCW), then negative (CW),

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4 4

6 a5

3

2

2

a3

3

b5

a1

1

b4

a2

b3 b2

5 a6

6

a4

1

b1

5

Figure 6.49.

Left: crease pattern for a vertex. Middle: the measured angles of the folded form from one vertex to the next. CCW angles are positive; CW angles are negative. Right: angle βi is the angular extent from the ith fold to the next.

then positive, and so forth, until we reach the folded edge at which we started. If we label these folded sector angles α1, α2, . . ., as shown in the middle of Figure 6.49, then the condition that we end up where we started is simple: α1 + α2 + α3 + α4 . . . = 0.

(6.31)

But if this folded vertex arose from a flat sheet of paper, when we unfold the vertex, we should get 360◦ of angle around the unfolded vertex, and so the condition that the folded form sector angles must satisfy will be α1 − α2 + α3 − α4 . . . = ±360◦,

(6.32)

where the sign of the result depends on whether we started with a positive or negative angle. This equation is the flat-unfoldability condition, analogous to the Kawasaki-Justin Condition for flatfoldability. To apply Equation (6.32), one must be able to identify the layers incident to each folded edge in order to construct the cyclic ordering of the sector angles. However, sorting the order of the folded edges is not necessary. If we examine the folds that emanate from the vertex, they will always lie strictly within a 180◦ arc. Starting at one end, we take βi to be the angular extent of the ith arc, as on the right in Figure 6.49, and we denote by ni the number of layers of paper that are incident to the vertex (i.e., we

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don’t count layers that are not part of the vertex figure). Then, since every layer incident to the vertex must form part of the flat, unfolded vertex, it must be the case that Õ ni βi = 360◦ . (6.33) i

For the commonly encountered degree-4 vertex, the pattern of fold lines falls into one of three possible patterns, and it is possible to construct special cases of Equation (6.33) that apply to the line pattern of the folded form. Every flat-foldable degree-4 vertex falls into one of the following cases: 1. All four sector angles are distinct. 2. The sector angles come in two pairs of equal angles. 3. All four sector angles are equal to 90◦ . If all four sector angles are distinct, then there are four distinct lines in the line pattern of the folded form and the largest angle in the line pattern < 180◦ will be one of the four sector angles, as shown on the left in Figure 6.50. If we number the three visible angles within this angle by {β1, β2, β3 }, then the four sector angles in the crease pattern will be, respectively, (β2 ), (β1 + β2 ), (β2 + β3 ), and (β1 + β2 + β3 ),

(6.34)

b2

b3

Figure 6.50. b1

b2 b1

b1

The three distinct configurations of a degree-4 vertex. Top: folded forms. Bottom: crease patterns. Left: all four sector angles are distinct. Middle: sector angles form two pairs of equal angles. Right: all four sector angles are equal to 90◦ .

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and so the flat-unfoldability condition applicable to the visible angles in the folded form is 2β1 + 4β2 + 2β3 = 360◦

(6.35)

β1 + 2β2 + β3 = 180◦ .

(6.36)

or, equivalently,

If the sector angles come in two pairs of equal angles, then there will be three distinct lines in the line pattern of the folded form and the largest angle in the line pattern < 180◦ will be equal to two of the four sector angles, as shown in the middle of Figure 6.50. Numbering the two visible angles within this angle by {β1, β2 }, the four sector angles in the crease pattern will be, respectively, (β1 + β2 ), (β1 + β2 ), (β2 ), and (β2 ),

(6.37)

and so the flat-unfoldability condition applicable to the visible angles in the folded form is 2β1 + 4β2 = 360◦ .

(6.38)

And finally, if all four sector angles are equal, there are two distinct lines in the line pattern of the folded form; the angle in this pattern < 180◦ will be equal to each of the sector angles of the crease pattern, as shown on the right in Figure 6.50, and so the relationship between the sector angles of the crease pattern and the corresponding condition on the visible angle {β1 } of the folded form will be α1 = α2 = α3 = α4 = β1 = 90◦ .

(6.39)

Thus, given the line pattern of a folded form composed of degree-4 vertices and a valid layer ordering, the folded form can be unfolded to a flat sheet of paper if and only if for every interior vertex of the line pattern, one of Equations (6.35)–(6.39) (as appropriate) is satisfied (and there is no self-intersection).3 3

There is no guarantee, of course, that any such pattern can be rigidly unfolded.

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q1

q1 A

A p3

q2

q0 q4

q2

p3

q4

Figure 6.51.

Left: each interior vertex of the folded form pattern is a flat-folded degree-4 vertex, whose surrounding vertices are all parameterized on the variables {di }. Right: each vertex should unfold to a developable vertex (sum of angles is 360◦ ).

? ? ? 6.6.5. Woven Algorithm, Continued So, how does the flat-unfoldability condition apply for a woven tessellation? In our case, the vertices are only degree-4, which simplifies things considerably. Consider an interior vertex such as the one marked q0 in Figure 6.51. That vertex is a flat-foldable degree-4 vertex, and so it should unfold into a developable vertex that will look something like the pattern on the right in the figure. (Note that facet A has the same orientation in both the folded form and crease pattern.) Of course, if you have indexed every pi and qi vertex in the pattern, the numbering is unlikely to match what I’ve shown here. I’ve numbered the surrounding vertices so that the vertices of the crease pattern are in numerical order counterclockwise. We don’t know yet the positions of the vertices in the crease pattern, but we can infer their relative ordering in the crease pattern from their ordering in the folded form, as shown in Figure 6.52. We can compute expressions for each of the four angles {αi } from the vertex coordinates {pi } and {qi } (and remember, the {qi } coordinates depend on the variables {di }). The flat-unfoldability condition is then α1 + α2 + α3 + α4 = 360◦ .

(6.40)

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a4 q1

A p3

a2

Figure 6.52. Sector angles of the vertex, as derived from the desired folded form.

q2

q0

a1 q4 a3

There is a similar flat-unfoldability condition for each of the interior vertices in the pattern—eight such vertices, in this example. And that’s all that’s needed to generate a developable crease pattern. In this example, we have 8 conditions on 20 unknowns, and in general, there will be fewer conditions than variables, since there is a variable for each qi but a condition only for those qi in the interior of the pattern. That tells us that there are multiple solutions for any given simple woven tessellation, and we need some way to distinguish among them. A good way to do so is to frame the problem as a constrained optimization: a class of problems in which one minimizes (or maximizes) a merit function subject to a set of equality and possibly inequality constraints over a set of variables. For this problem, the equality constraints are, clearly, the flatunfoldability conditions on each of the interior vertices. What about inequality constraints? Well, an implicit assumption in all of the figures has been that the distances {di } were positive. So for every variable, we need an inequality constraint of the form di ≥ 0. Actually, we need more than that. What happens if di is positive, but very, very tiny? In that case, the trapezoid becomes exceedingly thin and difficult to fold. So for purposes of ease of foldability, we should establish some minimum amount of overlap dmin . Thus, there will be a set of inequality constraints of the form di ≥ dmin for each i. (We may want to choose different dmin for interior and border vertices.) Last, there is the question of the figure of merit: the thing to be minimized. In general, this should be something related to the quality of the pattern: for instance, we don’t want the overlaps to be too small, but neither do we want them to be too large. In

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particular, if both trapezoids around a strip are wider than the strip itself, we will run into a violation of the Three Facet Theorem, and the pattern will not be flat-foldable without self-intersection (or the creation of additional folds). So, we would like the overlaps to be as small as possible while respecting that minimum. That suggests a figure of merit of the form Õ FOM = di2, (6.41) i

i.e., we minimize the least-square sum of all of the overlap distances {di }. This turns the origami problem into a well-formed mathematical problem: minimize FOM subject to the inequality and equality constraints. Finding a solution gives a valid folded form; unfolding that folded form to a flat sheet gives the desired crease pattern. ? 6.6.6. Woven Examples I implemented the algorithm described above using the Mathematica programming environment. The algorithm takes as input a pattern of lines that defines a woven pattern and a desired strip width (which must be small enough that there are no points where three or more woven strips overlap) and computes a valid developable crease pattern. Another Mathematica function takes the embedded graph of the folded form and algorithmically unfolds it to realize the crease pattern that gives rise to the desired folded form. Using this program, I computed and folded a variety of simple woven tessellations. Figure 6.53 shows an example of this series: the computed crease pattern and a folded example. Figure 6.54 shows another example, this time with sevenfold symmetry, but two parallel stripes along each direction of symmetry. When I first started exploring this family, I chose patterns of lines that exhibited geometric regularity and a high degree of symmetry, such as those in Figure 6.53 and 6.54. But then I started wondering about ways to break symmetry, using the fact that any pattern of overlapping lines was possible. An example of this exploration is “Dislocation,” shown in Figure 6.55. It takes the form of a regular grid of overlapping stripes, which we could

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Figure 6.53. “13-Fold Star” (2009), by the author. Left: crease pattern. Right: folded artwork.

Figure 6.54. “Two-Layer Seven-Fold Weave” (2009), by the author. Left: computed crease pattern for a woven pattern with seven-fold symmetry. Right: folded artwork.

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Figure 6.55. “Dislocation” (2009), by the author. Left: crease pattern. Right: folded artwork.

obtain from the regular tiling of offset twist tiles. But then there’s that one extra thread making its way through the pattern. And, taking this idea to its extreme, for “Courteous Anarchy,” I picked a random pattern of lines (that’s the “anarchy” part). The algorithm requires that there be no place where three lines come together—or even come close together, as that will limit the width of the strips and the trapezoids—so I selected random lines that maintained a reasonable distance between line intersections (that’s the “courteous” part). The resulting pattern is in Figure 6.56. One can also envision various generalizations and variations of this notion. Readers might enjoy the challenge of constructing algorithms to address some of these generalization: • Instead of using an unconstrained border, one could implement periodic boundary conditions to realize a woven tessellation pattern that tiles the plane. • Many origami tessellations are gridded, i.e., they have their vertices (and/or at least some of their folds) on a regular grid. Given an arbitrary line pattern and grid, can we find a woven tessellation with all vertices on the grid? • Is it possible to create woven tessellations for more complex woven patterns, e.g., herringbone or twill?

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Figure 6.56. “Courteous Anarchy” (2009), by the author.

Even beyond woven tessellations, the technique used here of applying flat-unfoldability conditions to a partially defined folded form pattern is a broadly useful tool for the design of geometric origami figures. I close this section with a somewhat more complicated woven pattern, a “double-weave” pattern that appears to be composed of pairs of woven strips, shown in Figure 6.57. This pattern is more complicated than the simple woven pattern: there are both degree4 and degree-6 vertices in the line pattern of the folded form (and therefore, of course, in the crease pattern). Nevertheless, it, and others like it, can be constructed in the same way, by applying flat-unfoldability conditions to a partially defined desired crease pattern. One of the interesting experiences of folding these woven tessellations is that they are rather hard to assemble, even when the patterns are fully precreased. They almost seem to “fight back” against being folded: as you bring one group of creases together, the paper attempts to unfold other creases, and as the pattern is folded, the paper forms a progressively more rumpled form. Just when you think there’s no more progress to be made, things suddenly start to relax; folds start to fall into place, and suddenly the pattern settles happily into its final, flat-folded form. This type of behavior is well known among origami tessellators, but it’s not universal among folding patterns. Some patterns

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Figure 6.57.

Left: crease pattern for a double-strip woven tessellation. Right: a folded example.

fall nicely into shape from the get-go. This dichotomy suggests that there’s some new phenomenon going on. There’s something about the intermediate, partially folded state that either permits or resists smooth motion. It’s time now to move out of the realm of purely flat folds and look more closely at 3D states, at partially folded vertices and fold patterns, and at the mechanisms that underly the movement of folded paper. ?

6.7. Terms Centroid The “center of mass” of a polygon. Delaunay triangulation A triangulation of a set of points such that no point is inside the circumcircle of any of the triangles; instead, each point is on the boundary of the circumcircles of the triangles that include the point. Dual graph A graph associated with another (the primal graph) in which for every face of the primal graph, there is a vertex of the dual graph and vice versa; for every edge of the primal graph there is an edge of the dual graph; and the vertex and face incidences of the primal graph edges are the same (but switched) for the corresponding dual graph edges. Embedding An assignment of coordinates to the vertices of a graph that fixes the vertices in space, often in the plane. CHAPTER 6. PRIMAL-DUAL TESSELLATIONS

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Facial subgraph A plane graph composed of some subset of faces of a graph that has no holes (missing faces) in its interior. Interior dual graph A dual graph of a primal graph, but not including the vertex corresponding to the exterior face of the primal graph or the connecting edges. Maxwell Condition A condition on the number of vertices and edges of a dual graph that ensures that there are sufficient degrees of freedom to allow for orthogonality between the primal graph and its reciprocal figure. Orthogonal embedding (of an interior dual graph) An embedding of the interior dual graph such that the edges of the interior dual graph are perpendicular to their corresponding edges in the primal graph. Plain weave A woven pattern of strips in which each strip alternately goes over and under the strips it crosses. Planar graph A graph that can be drawn in the plane in such a way that its edges intersect only at their endpoints (though the edges do not necessarily have to be drawn as straight lines). Plane graph A planar graph together with a specific embedding of the graph such that the edges intersect only at their endpoints. Pleat vector A vector transverse to a pleat whose length and direction gives the translation accomplished by the two folds of the pleat. Primal graph A graph for which we have constructed (or will construct) a dual graph. Primal-dual tessellation A tessellation that is constructed by manipulating polygons of a primal graph according to information embodied in a dual graph, such as the reciprocal figure. Reciprocal figure An embedding of a dual graph in which the edges are perpendicular to the corresponding edges of the primal graph.

472


Shrink-rotate algorithm A method of constructing tessellations from a tiling by shrinking the polygons of the tiling, rotating them, then reconnecting vertices that were originally in contact with new edges. Shrink-rotate tessellation A tessellation constructed using the shrinkrotate algorithm. Simple woven tessellation A woven tessellation in which the pattern is a plain weave, no more than two strips cross at a given point, and every strip travels in a straight line. Spiderweb A graph and embedding that has a reciprocal figure whose edges are straight and non-crossing. Straight-line embedding An embedding of a graph whose edges are straight lines. Voronoi diagram A partitioning of the plane into polygons around a set of points, one polygon for each point, such that every point within each polygon is closer to that polygon’s point than to any other point. Points on polygon edges are equidistant between the two points on either side. Wedge vector A vector that crosses a wedge around a simple flat twist. Woven tessellation An origami tessellation whose appearance resembles a pattern of woven strips.


........

473

7

Rigid Foldability ?

7.1. The Easy Way or the Hard Way

Normally, one folds an origami tessellation by pre-creasing all of the creases and then assembling the entire tessellation at once—a process usually called a collapse. After folding several tessellation patterns, you will observe that some tessellations are a lot easier to fold than others when it comes time to bring all the folds together. In some tessellations, if you put some of the creases in place, the neighboring creases also tend to fall into place, whereas in others, as you form some of the creases, the paper seems to try to unfold others. Even worse, sometimes bringing creases together forces layers to bend, buckle, and/or interfere with each other. In these recalcitrant patterns, even though the paper lies flat in the unfolded state and the folded tessellation lies flat in the fully folded state, there is no intermediate state that exists without bending or buckling of the facets of the tessellation. Whether a tessellation is easy or hard to assemble depends on both the geometry of the crease pattern and the specific crease assignment. Exactly the same crease pattern can be easy or hard, depending on the crease assignment. Two assignments for the same tessellation are shown in Figure 7.1—one easy, one hard. You will find if you try them that the crease pattern on the right cannot be assembled without bending several of the facets of the pattern during assembly. The patterns that are easy to assemble have the property that they can vary smoothly between the folded and unfolded state without bending or buckling any of the facets of the pattern. An origami tessellation with this property is said to be rigidly fold475

Figure 7.1.

Left: a crease assignment that is easy to assemble. Right: a crease assignment that is hard to assemble.

able.1 Origami tessellations that don’t vary smoothly between their folded and unfolded state are non-rigidly foldable. We encountered this concept in Chapters 2, where some periodic patterns, such as the Miura-ori and Yoshimura patterns, were rigidly foldable, and again in Chapter 3, when we were exploring the different crease assignments of the square twist. Rigidly foldable origami tessellations are interesting not only because they’re easier to assemble than inflexible tessellations, but also because in their half-folded state, they become threedimensional corrugations, and the interplay of light and shadow off of the three-dimensional structure makes them visually appealing. While the question of whether a pattern is rigidly foldable can affect how easy or hard a tessellation is to fold, with time, effort, and a little care, even the most recalcitrant non-rigidly foldable pattern can usually be coaxed into shape. However, as we move away from paper to other materials, we may well encounter materials that are truly rigid. Paper is remarkably forgiving: it can be bent and distorted, creases buckled, and crease locations slightly stretched out of position. All of these allow us to create origami forms from non-rigidly foldable materials. But when we use truly rigid materials—metal, wood, thick plastic—many of the 1

OK, “rigidly foldable” sounds like an oxymoron: if it’s foldable, then it isn’t rigid, right? Nevertheless, that’s the generally accepted term.

476

........CHAPTER 7. RIGID FOLDABILITY

ways of cheating rigidity become no longer tenable, and we must pay careful attention to whether the fold pattern is theoretically rigidly foldable. The analysis of which and whether origami patterns are rigidly foldable became of great interest in the first two decades of the 21st century and considerable theoretical work was carried out: some by myself and collaborators at Brigham Young University [24, 25, 26, 74], but most notably and comprehensively by University of Tokyo professor Tomohiro Tachi [111, 115, 114, 112, 88, 119, 116, 117]. Another signficant work was by Yan Chen et al. [15], who tied the problem of rigidly foldable origami back to the classical theory of mechanisms from the field of mechanical engineering. Much of the analysis of rigid foldability calls upon mathematics more advanced than what we have used thus far, but a few key relationships are readily accessible with high school mathematics (trigonometry) and are enough to design, construct, and fold some quite interesting origami patterns. In this chapter, we will analyze some of the conditions that make a rigidly foldable origami tessellation.

??

7.2. Half-Open Vertices Thus far, our analysis of origami tessellation patterns has been two-dimensional. All geometric figures were restricted to a plane. However, partially folded tessellation patterns are threedimensional, so if we are going to analyze half-folded tessellations, we must use three-dimensional geometry. We’ll start by establishing some tools and terminology and by studying the simplest possible origami tessellation: a single vertex. Let us draw a unit circle centered on the vertex. We will examine the outline of the circle as the paper moves between the unfolded and folded states. Figure 7.2 shows a representative vertex, the folded form, and a partially unfolded (and therefore three-dimensional) version. We will pay particular attention to the angles between adjacent facets around the vertex, the fold angles between the facets. The labeling of the fold angles {γi } is shown in Figure 7.3. Recall that the fold angle is the “deviation from straightness” from one facet to the next, as shown in the figure; that is, an unfolded crease has

........

CHAPTER 7. RIGID FOLDABILITY

477

a2

a1

a3

a4 a4

a4

Figure 7.2.

Left: a crease pattern for a vertex. Middle: the folded form. Right: a partially folded form.

a fold angle of 0, while a folded crease has a fold angle of π (for valley folds) or −π (for mountain folds).2 Also, by convention, we measure the fold angle as positive moving in the counterclockwise direction (and moving counterclockwise around the vertex, again, as shown in the figure). Thus, in Figure 7.3, γ1 , γ2 , and γ4 have positive values, and γ3 , since the arrow goes in the opposite direction, has a negative value. For consistency in numbering, we will generally number sector angles and fold angles in counterclockwise order (when viewed from the white side of the paper), and we will adopt the convention that the ith sector angle αi lies between fold angles γi and γi+1 (with cyclic wrap-around numbering, of course).

g2

g1 a2

Figure 7.3. Labeling and measurement of the fold angles.

g3

a1 a3

a4 g2

g3

g4

g1

a4

g4 2

In previous chapters, I have given angles in degrees. In this chapter (and other places that use trigonometry), we’ll use the mathematician’s convention of giving angles in radians.

478


g3 g2

a2

a3

a1 a4

g1

g4

Figure 7.4. The vertex placed inside a unit sphere.

Prior to this point, we have assumed that all folds are mountain, valley, or otherwise flat, and therefore have fold angles of −π, π, or 0, respectively. Now that we are allowing angles to vary continuously, we can characterize any partially folded crease whose fold angle is γ as mountain if 0 < γ ≤ π, valley if −π ≤ γ < 0, and of course, still flat or unfolded if γ = 0. If γ = ±π, then the fold is flat-folded (mountain or valley, as applicable); if it is less than flat-folded, it is partially folded. The convention of using “deviation from straightness” as the angular measure gives the sensible result that an unfolded crease has a fold angle of 0. If we place our partially folded circle into a unit sphere with the vertex at the center of the sphere, then the edges of the circle lie on the surface of the sphere and in fact are segments of great circles on the sphere, as shown in Figure 7.4. A great circle is any circle on a sphere whose center is also the center of the sphere. (For example, on the Earth, the equator and meridians of longitude are great circles, but most circles of constant latitude are not because their centers are not the center of the earth.) In the vertex embedded within the sphere, the edges of the original paper circle become arcs of great circles on the sphere. Since our sphere is a unit sphere, the length of each circular arc is equal to the corresponding sector angle at the vertex measured in radians. The fold angles between the facets are simply the exterior angles between the lines on the surface of the sphere; the dihedral angles simply become the interior angles of the arcs on the surface of the sphere. The arcs generated by all of the angular sectors created a closed polygon on the surface of the sphere.

........


479

A polygon on a unit sphere whose sides are composed of arcs of great circles is called a spherical polygon. There is an entire branch of geometry devoted to figures drawn on spheres, called spherical non-Euclidean geometry, or spherical geometry, for short. We can use formulas from spherical geometry to derive relationships between the sector angles {αi } and the fold angles {γi } of the partially folded vertex. Now both the unfolded vertex and the flat-folded vertex are just special cases of a partially folded vertex. Therefore, any relationship we derive for a partially folded vertex should be equally applicable to the unfolded paper, where all the fold angles are equal to 0, as well as to the fully folded form, where all the fold angles are π (for valley folds) or −π (for mountain folds) as well. ??

7.3. Spherical Geometry The mathematics in this section makes substantial use of spherical trigonometry, the trigonometry of angular relationships in 3D, as opposed to the planar trigonometry that is often the sole focus of current-day high school mathematics. Spherical trigonometry used to be a regular part of mathematics education; it is necessary for astronomy and, in the days before GPS, it was necessary for navigation on the high seas, when the skies and a good chronometer were the only means of finding one’s position. Knowing a little bit of spherical trigonometry (or having someone with that knowledge on board) made the difference between knowing where you were and dying of scurvy or being eaten by your shipmates. Nowadays, spherical trigonometry is not as widespread, though is still used by amateur astronomers. The functions themselves—sine, cosine, tangent, and the like—are the same as for planar trigonometry, but we will use them in new ways. To aid the unfamiliar, I have included a short collection of basic spherical trigonometric relationships in this section. A spherical triangle is a triangle formed on the surface of a sphere whose sides are arcs of great circles, as shown in Figure 7.5. If A, B, and C are the angles of a spherical triangle and a, b, and c are the arc lengths of their respective opposite sides, then the following relations hold: sin a sin b sin c = = ; sin A sin B sin C

480


(7.1)

B a c C A

b

Figure 7.5. A spherical triangle with sides a, b, c and corner angles A, B, C.

Equation (7.1) is called the Law of Sines. Similarly, we have the Law of Cosines: cos a = cos b cos c + sin b sin c cos A, cos b = cos c cos a + sin c sin a cos B, cos c = cos a cos b + sin a sin b cos C;

(7.2)

cos A = − cos B cos C + sin B sin C cos a, cos B = − cos C cos A + sin C sin A cos b, cos C = − cos A cos B + sin A sin B cos c.

(7.3)

When talking about geometric figures on the surface of a sphere, it is a convenient shorthand to use the same letter both for the corner angle at a point and to identify the point on the sphere. Thus, with respect to Figure 7.5, A is the angle at point A, B is the angle at point B, and so forth. We can thus identify a spherical triangle by its corners—triangle ABC in Figure 7.5, for example. Usually context will make it clear whether we are referring to a point or an angle. In three dimensions, we have two types of angles: the usual angle measured between two intersecting lines, and the solid angle, which is the area of a patch outlined by circular arcs on the surface of a unit sphere, like triangle ABC in Figure 7.5. Solid angle is measured in square radians, or steradians. The solid angle Ω subtended by the spherical triangle ABC is given by its area on a unit sphere, which is Ω = (A + B + C) − π.

(7.4)

Equation (7.4) generalizes to an arbitrary spherical polygon; the solid angle of a spherical polygon with n sides and corner

........


481

angles (β1, . . . , βn ) at the vertices is given by ! Õ Ω= βi − (n − 2)π.

(7.5)

i

Equation (7.5) is known as Girard’s Theorem, and the solid angle area Ω is called the spherical excess (often denoted by the letter E). If we place a folded vertex at the center of a unit sphere as in Figure 7.4, the dihedral angles of the folds become the interior angles of the corners of the spherical polygon on the surface of the sphere; the fold angles of the folds become the exterior angles of the spherical polygon. Expressed in terms of the exterior fold angles γi = π − βi , Girard’s Theorem becomes Õ Ω = 2π + γi . (7.6) i

If the corner angles of a spherical triangle are all π (i.e., consecutive sides are collinear), the spherical triangle becomes a great circle on the unit sphere, and so its area according to Equation (7.6) is 2π, i.e., the area of a unit hemisphere. Girard’s Theorem provides a two-line derivation3 of the Maekawa-Justin Theorem. If each of M mountain folds has a fold angle of −π, and each of V valley folds has a fold angle of π, then the summation in Equation (7.6) can be rearranged to become Õ Ω − 2π = γi = M(−π) + V(π) = π(V − M). (7.7) i

For a flat-folded vertex, if the white side is on the inside, then its enclosed solid angle is Ω = 0; if the colored side is on the inside, then the enclosed solid angle is the entire sphere, so that Ω = 4π. Plugging these two values into the preceding and dividing both sides by π gives the familiar result: V − M = ±2. 3

(7.8)

A derivation, not quite a proof. The Maekawa-Justin Theorem relies on non-self-intersection of the paper but with flat-folded layers. To get there from Girard’s Theorem, we need to assume a non-self-intersecting 3D vertex, then take its limit as we approach flat-foldability while preserving non-selfintersection.

482


Girard’s Theorem gives the area of a spherical triangle in terms of its angles; another formula gives the same result in terms of its sides. Let s be the semiperimeter, s = (a + b + c)/2. Then s 1 1 1 1 1 tan Ω = tan s tan (s − a) tan (s − b) tan (s − c) . 4 2 2 2 2

(7.9)

Equation (7.9) is known as L’Huilier’s Theorem [122]. An alternative formula for the area, used by Huffman (see unmarked equation preceding Equation (1) in [47]), gives the area in terms of an angle and two adjacent sides: tan

tan a2 tan 2b sin C Ω = , 2 1 + tan a2 tan 2b cos C

(7.10)

and its simpler equivalent, Ω a b = cot C + cot cot csc C, (7.11) 2 2 2 and, of course, the equivalents of Equations (7.10) and (7.11) under cyclic permutation of (a, b, c) and (A, B, C). There are a number of identities that make use of the haversine function hav(x) [124], defined by 1 2 x hav x ≡ (1 − cos x) = sin , (7.12) 2 2 which also gives the identities cot

and

hav(π − x) = 2 − hav x

(7.13)

hav(2π − x) = hav x.

(7.14)

In a spherical triangle, the haversine formula for sides, relating a side to the opposite angle and its two adjacent sides, is hava = hav(b − c) + sin b sin c hav A.

(7.15)

We can also put this relationship in terms of the exterior angle opposite the given side: hava = hav(b − c) + sin b sin c hav(π − Aext ),

(7.16)


........

483

which, after some manipulation, can be shown to be equivalent to hava = hav(b + c) − sin b sin c hav Aext,

(7.17)

which we will use in Section 8.3.5. There is similarly a haversine formula for an angle, which can be expressed in terms of the semiperimeter and two adjacent sides, hav A =

sin(s − b) sin(s − c) ; sin b sin c

(7.18)

hava − hav(b − c) ; sin b sin c

(7.19)

in terms of all three sides, hav A =

or in terms of the opposite side and its adjacent angles, hav A = hav [π − (B + C)] + sin B sin C hava.

(7.20)

The haversine is not usually taught in a conventional course of trigonometry, but it turns out to be quite useful in spherical trigonometry. Two additional identities of the haversine are easily derived and are occasionally useful: 1 ( hav(x + y) − hav(x − y)) , 2

(7.21)

1 (1 − hav(x + y) − hav(x − y)) . 2

(7.22)

sin x sin y = and cos x cos y =

Because of the identity in Equation (7.12), formulas involving the haversine function quite naturally lead to relations between half-angles, and there are a number of useful spherical trigonometric identities in which half-angles enter explicitly. First, we have Gauss’s formulas: sin a−b sin A−B 2 2 = (7.23) , sin 2c cos C2 sin sin

484


a+b 2 c 2

=

cos sin

A−B 2 C 2

,

(7.24)

cos

a−b 2

A+B 2 cos C2

A+B 2 C 2

a−b 2

sin

= cos 2c sin a+b cos 2 = cos 2c sin

,

(7.25)

.

(7.26)

,

(7.27)

Then we have Napier’s analogies: A−B 2 A+B 2

sin sin

A−B 2 A+B 2

cos cos sin

a+b 2

a−b 2

sin cos cos

a−b 2

a+b 2

=

tan tan

=

= =

tan tan

c 2

a+b 2 c 2

,

A−B 2 cot 2c

A+B 2 cot 2c

tan

tan

(7.28)

,

(7.29)

.

(7.30)

For a summary of these (and other) formulas and citations for their sources, see Weisstein [124]. While we’re on the topic of identities involving half-angles, we introduce an exceedingly useful formula: the Weierstrass substitution. For any angle ξ, we define ξ x ≡ tan , 2

(7.31)

which lets us turn trigonometric functions of ξ into algebraic functions of x, using the following identities: 2x , 1 + x2 1 − x2 cos ξ = , 1 + x2 2x tan ξ = . 1 − x2 sin ξ =

(7.32)

........


485

The Weierstrass substitution (named for Karl Weierstrass, a 19th-century German mathematician) is commonly used in integral calculus as a change of variable to turn integrals involving trigonometric expressions into integrals involving polynomials. It can serve the same purpose in the simplification and analysis of fold angles and/or sector angles in origami design, and in fact, when you see a formula that involves tangents of half-angles, it is a good bet that “Weierstrass was here.” Or at least, his substitution. ??

7.4. A Degree-4 Vertex in Spherical Geometry Let us now look at the spherical polygon defined by the partially folded degree-4 vertex shown in Figure 7.4. We redraw it in Figure 7.6, drawn as if we were looking down onto the vertex from the “North Pole” of the unit sphere. Figure 7.6 shows the spherical quadrilateral that corresponds to the partially folded vertex. We cut the quadrilateral into two triangles by connecting diagonally opposite vertices by an arc of a great circle. We denote the arc length of the line by ζ. The new arc divides the interior angle at γ3 into two parts, which we denote by φ and ψ. We will now work out some useful relationships between fold angles and sector angles.

?? 7.4.1. Opposite Fold Angles All of the lines in Figure 7.6, including the ζ line, are curved arcs of great circles. Their lengths and the angles inside the triangle are related by formulas from spherical geometry. Using the cosine law, we can solve for cos ζ in two different ways in the two triangles in Figure 7.6: cos ζ = cos α1 cos α2 + sin α1 sin α2 cos(π − γ2 ), Figure 7.6.

Left: the spherical quadrilateral corresponding to the partially folded vertex. Right: the quadrilateral cut into two triangles.

486

a1

g2 a2

g4

a2 a3

a3 a4


a1

p-g2 g1

g3

(7.33)

f y p-g4

z

a4

g1

for the upper triangle, and cos ζ = cos α3 cos α4 + sin α3 sin α4 cos(π − γ4 ),

(7.34)

for the lower triangle. Since we started with a flat-foldable vertex, the angles around the vertex must satisfy the Kawasaki-Justin Condition, that is, α3 = π − α1 and α4 = π − α2 . Substituting these into Equation (7.34) and then taking the difference of the two equations gives sin α1 sin α2 (cos γ2 − cos γ4 ) = 0.

(7.35)

Since neither of the sector angles α1 or α2 is zero, the term in parentheses must be zero; therefore, cos γ2 = cos γ4 .

(7.36)

Exactly the same analysis may be carried out for the other two vertices, giving cos γ1 = cos γ3 . (7.37) In other words, in a partially folded flat-foldable degree-4 vertex, the opposite fold angles have the same angle cosines. Now, just because two angles have the same cosines doesn’t mean the angles themselves are equal, since cos(γ) = cos(−γ)

(7.38)

for any angle γ. But there are only two choices for any value of the angle cosine: if the ith fold is a valley fold, then γ > 0 and we take positive angle value; otherwise, if the fold is a mountain fold, we take the negative angle value. So, based on the crease assignment, we can choose the proper sign of the fold angle to assign to the crease. In Figure 7.6, both γ2 and γ4 are valley folds, so they both must be equal to the same value, which we will call γ+ ; that is, γ2 = γ4 ≡ γ+,

with 0 ≤ γ+ ≤ π.

(7.39)

We will refer to these two angles as the major fold angles of the degree-4 vertex and the corresponding folds as the major folds of the vertex. For the other pair of angles, one is a valley fold and the other is a mountain fold. We can define a second angle γ− for that pair, with γ1 = γ−, γ3 = −γ−, with 0 ≤ γ− ≤ π. (7.40)

........


487

We will refer to these two angles as the minor fold angles of the degree-4 vertex and the corresponding folds as the minor folds of the vertex. Why these particular names and symbols? We will see, presently. But the thing to remember now is that for any flatfoldable degree-4 vertex, the following hold: • The major folds at a degree-4 vertex are opposite one another at the vertex, have the same sign, and have the same fold angle. • The minor folds are opposite one another at the vertex, have the opposite sign, and have the same fold angle but with opposite sign. And yes, this is the same labeling of major and minor that we applied to the folds of a Miura-ori back in Chapter 2. This nomenclature applies equally to a degree-4 vertex formed from three mountain folds and one valley fold; opposite mountain folds have the same fold angle, while the opposite mountain and valley folds have the same fold angle going in opposite directions. Let us define the fold angle magnitude as the magnitude of the fold angle, independent of mountain or valley status. The fold angle magnitude is always between 0 (unfolded) and π (fully folded). Then we can summarize Equations (7.39) and (7.40) as follows: Theorem 24 (Equal and Opposite Angles Theorem). In a partially folded degree-4 flat-foldable vertex, opposite fold angle magnitudes are equal. This result implies some remarkable long-range structure to a partially folded degree-4 flat-foldable origami form. If you pick any fold line and draw a line along it and every time you come to a vertex, you continue with the line on the opposite side of the vertex, the fold angle remains constant all along that line. Fold lines can run from one side of a tessellation completely across the tessellation, as shown in the {4.4.4.4} tessellation in Figure 7.7. These networks (lines or loops) of angle constancy are formed by joining folds opposite each other at every vertex and are based on the equality of opposite folds. There is also a relationship between adjacent fold angles at a degree-4 vertex. We’ll solve for this in the next subsection.

488


Figure 7.7. A {4.4.4.4} tessellation. The highlighted lines are two representative lines of constant fold angle.

An interesting side note: substituting Equations (7.39) and (7.40) into Girard’s Theorem for a spherical quadrilateral gives a corollary relating the two major fold angles to the solid angle subtended by the folded vertex: Ω = 2(π − γ+ ),

(7.41)

where, as before, we have taken γ2 = γ4 ≡ γ+ . Noting that the corresponding dihedral angles are given by βi = π − γi , this reduces simply to Ω = 2βi . (7.42) Thus, the solid angle subtended by a flat-foldable degree-4 vertex is simply twice the dihedral angle of the two opposing valley folds. ?? 7.4.2. Adjacent Fold Angles Now, let us seek a condition relating adjacent fold angles in a flat-foldable degree-4 vertex. We begin with an expression for sin γ3 : sin γ3 = sin (π − (φ + ψ)) = sin φ cos ψ + cos φ sin ψ.

(7.43)

We can use the Laws of Sines and Cosines for spherical geometry to get expressions for each of the 4 quantities on the right: sin α1 sin γ2 , sin ζ sin α4 sin γ4 sin ψ = , sin ζ sin φ =

(7.44) (7.45)


........

489

cos α1 − cos α2 cos ζ , sin α2 sin ζ cos α4 − cos α3 cos ζ cos ψ = . sin α3 sin ζ cos φ =

(7.46) (7.47)

Using our definitions γ2 = γ4 ≡ γ+ and γ1 = −γ3 ≡ γ− , and the relations α4 = π − α2 and α3 = π − α1 , these become sin α1 sin γ+ , sin ζ sin α2 sin γ+ sin ψ = , sin ζ cos α1 − cos α2 cos ζ cos φ = , sin α2 sin ζ cos α2 − cos α1 cos ζ cos ψ = − . sin α1 sin ζ sin φ =

(7.48) (7.49) (7.50) (7.51)

Substituting these into Equation (7.43) and carrying out some simplifications yields an expression relating major and minor angles: sin γ− =

cos α2 − cos α1 sin γ+ . 1 − cos ζ

(7.52)

Now, we can substitute in a relation from Equation (7.33), cos ζ = cos α1 cos α2 − sin α1 sin α2 cos γ+, which gives sin γ− =

cos α2 − cos α1 sin γ+ . (7.53) 1 − (cos α1 cos α2 − sin α1 sin α2 cos γ+ )

This relates the major and minor angles, but it’s a bit messy. Time for a Weierstrass substitution! We define g+ ≡ tan 12 γ+,

(7.54)

g− ≡ tan 12 γ−,

(7.55)

substitute, and solve for one of {g+, g− } in terms of the other. Remarkably, all the complexity collapses, giving g+ = g− 490


sin 12 (α1 + α2 ) sin 12 (α1 − α2 )

,

(7.56)

or putting it back in terms of angles, tan 12 γ+ tan 12 γ−

=

sin 12 (α1 + α2 ) sin 12 (α1 − α2 )

.

(7.57)

I think this result is one of the most beautiful results in all of mathematical origami due to its unexpected simplicity and symmetry. But it is especially important because of its meaning, first identified by Tachi [111]. The expression on the right involves only sector angles, so is a fixed property of the crease pattern. The expression on the left describes a relationship between the fold angles that must hold at all possible folded states. So, once we have defined a crease pattern and fixed the sector angles, the right side of Equation (7.57) is constant; thus, adjacent fold angles are constrained to have their half-angle tangents vary proportionally to that constant. In terms of the individual fold angles, tan 12 γ2 tan 12 γ1

=

tan 12 γ4 tan 12 γ1

=−


=−


=

sin 12 (α1 + α2 ) sin 12 (α1 − α2 )

So now, all four angles are related. Equations (7.36), (7.37), and (7.58) imply that once we’ve picked a particular crease assignment of a vertex, as we vary any one fold between 0 and π, the other three folds are completely determined, and so the folds around a single vertex can all be characterized by a single parameter. In a degree-4 tessellation, though, every vertex is connected to every other vertex by some chain of folds; this means that every fold in the entire degree-4 tessellation is determined by a single parameter. If you know (or choose) the fold angle for a single crease in the tessellation, then by using Equations (7.36) and (7.37) to relate opposite folds and Equation (7.58) to relate adjacent folds, you can calculate the fold angle for every fold in the tessellation. If you were to make a degree-4 tessellation with faces from perfectly rigid material and flexible hinges at each of the fold lines, you could flex a single fold back and forth and the entire tessellation would fold and unfold in sync. But, as we have observed empirically, not all degree-4 tessellations have the property of rigid foldability, and we are about to see why.

.

........


(7.58)

491

??

7.5. Conditions on Rigid Foldability Observe that in Equation (7.57) the quantity on the right always has magnitude greater than 1, at least, for sector angles α1 and α2 that permit the crease assignment we assumed in Figure 7.2. Actually, it’s more general than that: Equation (7.57) holds for any set of sector angles that allow γ2 and γ4 to have the same sign (and thus γ1 and γ3 have opposite sign). Since the tangent functions are strictly monotonic, this means that for all possible folded states, |γ− | < |γ+ | for γ ∈ (0, π), |γ− | = |γ+ | for γ = 0, π.

(7.59)

Thus, Equation (7.59) tells us that at any partially folded flatfoldable degree-4 vertex, the fold angle magnitude of the major pair is strictly greater than the fold angle magnitude of the minor pair at all non-flat states. Equality is reached only at a flat state— unfolded or flat-folded. The inequality between major and minor fold angles is the reason for their names: the major fold angle magnitude is larger than the minor fold angle magnitude in any non-flat state. We call this relationship the flat-foldable vertex inequality. The flat-foldable vertex inequality lets us examine whether a given crease assignment permits a rigidly foldable origami tessellation. To see how, let’s examine the crease patterns for three different crease assignments for a unit square centered twist with three different assignments: M 4 , M 2V 2 , and (MV)2 , which are shown in Figure 7.8. At each vertex, the two minor (dissimilar-assignment) fold lines must have smaller fold angle magnitudes than the two major (same-assignment) fold lines in accordance with Equation (7.59). For each sector, one of the crease pairs has larger fold angle magnitude than the other. We’ll indicate these relationships by

Figure 7.8. Unit square centered twist tiles. Left: M 4 . Middle: M 2V 2 . Right: (MV)2 .

492


(minor)

(major)

Figure 7.9.

g1

g2

The fold angle graph for a single vertex. Inequality arrows point from the major folds (larger fold angle magnitude) to minor folds (smaller fold angle magnitude).

g3

(minor)

g4

(major)

putting a dot on each crease and drawing an arrow pointing from the larger fold angle toward the smaller fold angle, as illustrated for a single vertex in Figure 7.9. When we draw all such arrows for a multi-vertex crease pattern, each crease with two interior vertices has two inequality arrows connected to it, and we get a potentially complex directed graph (a digraph) that we call the fold angle graph. The fold angle graphs for our three square twists are shown in Figure 7.10. At this point, we can ignore whether the creases are mountain and valley folds; the relevant information is carried by the graph consisting of the arrows that display the relationships between fold angle magnitudes. We can think of this graph as a network through which fold angle magnitude “flows downhill,” from larger magnitude to smaller, according to the directions of the arrows.

gc

gc

gc

gb gd

gb gd

ga

gb gd

ga

ga

Figure 7.10.

The same twists. The fold angles of the highlighted central square are γa –γd . The arrows indicate the direction of inequality of the fold angles around the central square. Left: M 4 . Middle: M 2V 2 . Right: (MV)2 .

........


493

Now let’s look at the nodes of this graph that correspond to the creases of the central polygon with fold angles labeled γa –γd in the first twist pattern, M 4 , on the left in Figure 7.10, and specifically, the loop of arrows in the center of the highlighted twist polygon. The arrows point from larger to smaller fold angle magnitude. If this M 4 twist is partially folded, then following the arrows around the highlighted path in the figure, we have that γa > γb > γc > γd > γa,

(7.60)

which means that the fold angle for crease a is larger than itself! This cannot be. We have arrived at a contradiction, and the only questionable assumption in our analysis was the assumption that the M 4 crease pattern was partially folded. Therefore, the M 4 twist cannot ever be partially folded and retain flat facets. And indeed, if you try it—cut out and fold the cyclic unit square twist— you will find that although it folds flat, in the partially folded state, some of the facets must be bent or creased in order to form the pattern.4 This also explains why inflexible tessellations are more difficult to assemble than rigidly foldable ones. Paper is springy. If the tessellation cannot be folded without curving some of the facets, then when the paper is partially folded you will have the springiness of the paper working against you, trying to unfold what you’re trying to fold. By the same token the figure on the right in Figure 7.8—the (MV)2 unit twist—also has a circular chain of inequalities. In this case, we find that γa < γb < γc < γd < γa,

(7.61)

which is equally impossible. Therefore, the flat-foldable vertex inequality says that this unit square twist cannot be partially folded with flat facets. And if you fold this unit square twist (which is iso-area), you will find that when it is partially folded, the central square must bend along one of its diagonals unless the paper is pressed flat, which suggests a possible way to rescue rigid foldability; we will return to this pattern. 4

Thus, for example, for the square twist in Chapter 3 with instructions on pages 196–197, it was necessary to add a valley fold across the middle of the square.

494


Recall that the flat-foldable vertex inequality becomes an equality for both the unfolded and fully folded forms of the tessellation. So it does not say that the pattern can’t be flat-folded; it just says that you have to bend some of the faces along the way in order to make it flat-folded. The middle form in Figure 7.8 , M 2V 2 , does not have a loop of inequalities like the others. For this crease assignment, we have γa, γc > γb, γd ;

(7.62)

therefore, the flat-foldable vertex inequality is satisfied at every vertex. The graph at least suggests that this assignment and its complement, V 2 M 2 , (and rotated versions of both) are both rigidly foldable. But are they really? Is the simple absence of a closed loop sufficient to guarantee rigid foldability? The answer is no, it is not sufficient. But with a small addition to the fold angle graph, we will be able to determine rigid foldability for twists and, indeed, any crease pattern composed of flat-foldable degree-4 vertices. ?? 7.5.1. The Weighted Fold Angle Graph The flat-foldable vertex inequality establishes a qualitative relationship between the major and minor creases of a degree-4 vertex, but Equation (7.58) quantifies it. This relationship between adjacent fold angles, coupled with the fact that opposite fold angles are pairwise equal, implies that there is a simple proportionality between the half-angle tangents for any pair among all four angles around a flat-foldable degree-4 vertex. There is a single parameter that describes all possible pair angle relationships. For a general flat-foldable degree-4 vertex with major crease fold angles γ2, γ4 and minor crease fold angles γ1, γ3 , we define the fold angle multiplier by µ≡

sin[ 12 (α1 + α2 )] sin[ 12 (α1 − α2 )]

,

(7.63)

where α1, α2 are the adjacent sector angles as in Figure 7.6. Then the half-angle tangents are all proportional to a common factor, independent of the degree of openness of the vertex: tan 12 γ2 tan 12 γ1

= µ,


= −1,


= µ.

(7.64)

........


495

Figure 7.11. The weighted fold angle graph for a single vertex. Half-angle tangents of the fold angles are proportional in the ratio µ or −µ, with the arrow pointing from larger to smaller fold angle magnitude.

(minor)

(major) g2

g1

m -m

m -m

g3

(minor)

g4

(major)

From these, all other pairwise relationships may be similarly derived. Note that in this definition, the magnitude of µ is always greater than 1, reaching equality only for the degenerate cases α1 = 0 or α2 = 0 (and remember, this is based on the assumption that γ2 and γ4 are the major pair). If µ is positive, then γ2 and γ4 have the same sign as γ1 ; if µ is negative, they have the same sign as γ3 . So this relationship holds for any crease assignment of the degree-4 vertex, as long as the major and minor crease pairs are properly identified. We can thus augment the fold angle graph by adding weights to the directed edges of the graph, as shown in Figure 7.11, creating a weighted fold angle graph. In the weighted graph, the weight µ on an edge gives the ratio between the half-angle tangents of the creases at each node of the graph. And it is quantitative, with the values of µ around each vertex given by Equation (7.63) for each vertex with the appropriate sector angles. Let us now look again at the two inflexible square twists. Figure 7.12 shows the M 4 and (MV)2 twists with the weighted fold angle graph. Here I have labeled the vertices, rather than the creases, with letters a through d and have chosen the crease numbering about the vertex so that µi > 0 for each vertex. Because the multipliers express coefficients of proportionality between half-angle tangents, the self-consistency condition can be written as a product going around the central loop. For the cyclic (M 4 ) twist, the self-consistency condition is µa µb µc µd = 1.

(7.65)

And since we know that for all non-degenerate cases (positive

496


c d

c

mc

md

mb

ma

d b

-md

-mc

-ma

a

-mb

Figure 7.12. The two inflexible twists and their weighted fold angle graphs. Left: M 4 twist. Right: (MV)2 twist.

b

a

sector angles) µ always has magnitude greater than 1, this cannot possibly be satisfied. For the alternating ((MV)2 ) twist, the arrows point the other direction, and so the proportionality coefficient going against the arrow is its inverse: −1 −1 −1 (−µ−1 a )(−µb )(−µc )(−µd ) = 1.

(7.66)

And in this case, since the left side of the equation must have magnitude less than 1, it, too, cannot possibly be satisfied. Now, let’s look at the third twist, the M 2V 2 twist, as well as a new one, the M 3V, as illustrated in Figure 7.13. Consider first the M 2V 2 twist. We can write down its loop consistency condition by inspection: −1 (−µ−1 a )µb (−µc )µd = 1.

(7.67)

For this M 2V 2 square twist, all of the vertices are geometrically similar (same angles); furthermore, which creases are major and minor are also the same. (This is not always the case for geometrically similar vertices; recall from Chapter 1, Figure 1.27,

c d

md

-mc

-ma a

c

mb

d b

-md

-ma a

Figure 7.13.

mc mb

b

Two rigidly foldable twists and their weighted fold angle graphs. Left: M 2V 2 twist. Right: M 3V twist.

........


497

that among the four possible crease assignments, any given crease could be either major or minor.) So all of the multipliers are equal: µ1 = µ2 = µ3 = µ4 ≡ µ. And our loop consistency condition reduces to µ−1 µµ−1 µ = 1, (7.68) which is clearly always satisfied no matter what the value of µ is. So this tells us that any flat-foldable M 2V 2 square twist is rigidly foldable as well, no matter what the twist angle is. In the same fashion, we see that for the M 3V twist, the loop consistency condition is −1 (−µ−1 a )µb µc (−µd ) = 1,

(7.69)

which works out to the same thing. So again, both twists are rigidly foldable. The M 2V 2 twist is well known; it has featured in tessellations by Paulo Taborda Barreto and Chris K. Palmer, among others, and in fact, the Miura-ori also consists of M 2V 2 twists with parallelogram central polygons. The M 3V twist, though, is less common, possibly because of an interesting bit of additional complexity that we will now explore. ?? 7.5.2. Distinctness of Fold Angle Once again, let us consider the two rigidly foldable square twists, which I have redrawn in Figure 7.14 with some different highlighting colors. Recall that in a flat-foldable vertex, the two major creases have the same fold angle as each other, and the two minor creases have the same fold angle magnitude as each other. So in the M 2V 2 twist, Figure 7.14. Two rigidly foldable twists and their weighted fold angle graphs along with colored highlighting of constant-foldangle-magnitude chains of creases. Left: M 2V 2 twist. Right: M 3V twist.

498

R

S

c

Q d

P

R

-m

m

c

Q

-m

m

S

d b

a


P

-m -m a

m m

b

all three red-highlighted creases in the top half must have the same fold angle magnitude. So, too, must the three red-highlighted creases in the bottom half, and the two triplets of vertical creases must also have the same fold angle magnitude, at least within each triplet. I have labeled each triplet of creases of constant fold angle magnitude with a letter: P, Q, R, and S. Observe that along a given line, some of the fold angles are positive (valleys) and others are negative (mountains), but the fold angle magnitude is constant. This, in turn, implies that the magnitude of the half-angle tangent is also constant, and so we can use the weighted fold angle graph to determine proportionalities between half-angle tangents of any two crease lines in the pattern, as long as they are connected by a path of edges in the weighted fold angle graph. For simplicity, let us define gP ≡ | tan 12 γP |,

(7.70)

where γP is any fold angle along the line of constant fold angle magnitude labeled P, and similarly for Q, R, and S. We can use the weighted fold angle graph to determine the relationship between fold angles for different creases simply by tracing a path in the fold angle graph and multiplying by the µvalues encountered along the way (or dividing, depending upon the direction of the arrow). Thus, for example, in the M 2V 2 twist, we can see that gQ = | µ|gR, (7.71) which establishes that crease chain R has a smaller fold angle than crease chain Q. We already knew that, of course. But we can also see that gS = | µ|gR, (7.72) which, together with Equation (7.71), establishes something new:

and

g R = gS ,

(7.73)

gP = gQ .

(7.74)

So the P-chain and Q-chain have the same fold angle and the R-chain and S-chain have the same (smaller) fold angle, and that is why I used the same color for their highlighting: each color conveys a distinct fold angle.

........


499

The second twist is a bit more interesting, because the same chain of reasoning establishes that there are three distinct fold angles in this pattern, satisfying gP > gR = gS > gQ .

(7.75)

For three distinct fold angles, we use three distinct colors. (For the optical physicists in the audience, the ordering of the wavelengths of the colors—red > green > blue—matches the relative ordering of fold angle magnitudes.) The utility of coloring the fold angles arises when we start to consider splicing together tiles of crease patterns to make larger patterns. The fold angle coloring establishes matching rules on fold angle that must apply for rigid foldability to persist. Let’s try putting some of these tiles together into larger patterns and see if the result will be rigidly foldable. ?? 7.5.3. Matching Fold Angle Figure 7.15 shows the rigidly foldable square centered twist tiles: M 2V 2 (two versions), M 3V, and MV 3 . And, of course, we should include their cyclic permutations, equivalent to rotating a tile by some multiple of 90◦ . We can assemble these tiles along with copies of themselves, and rotated copies of themselves, to make rigidly foldable tessellations, but we must respect (a) position, angle, and assignment of the creases (as was the case in flat-foldable twist tessellations) and also (b) fold angle magnitude, represented by distinct colors.

Figure 7.15. Square centered twist tiles with fold angles colored. Top: two crease assignments for an M 2V 2 rigidly foldable square centered twist tile. Bottom: an M 3V and MV 3 rigidly foldable square twist tile.

500


Figure 7.16.

Left: two side-by-side M 2V 2 rigidly foldable square twist tiles. Right: same twist tiles, aligned top to bottom.

Figure 7.16 shows two M 2V 2 twists aligned side to side and top to bottom. Each individual twist is rigidly foldable, and because the fold angles match (same color), the combination of two tiles into a single crease pattern is also rigidly foldable. I encourage you to cut one out, precrease from stiff paper, and verify this for yourself. What if we rotate one of the tiles by 90◦ ? Can we still join them up in a stacked arrangement? Figure 7.17 shows this configuration. We can see on the left that the two tiles don’t match in either fold angle or fold direction. We can get the fold angles right by inverting the direction of all of the folds (converting mountain

Figure 7.17.

Left: two stacked M 2V 2 rigidly foldable square twists. The tiles are mismatched in fold parity. Middle: same thing, with the fold parity swapped on the upper tile. Still mismatched in fold angle. Right: same as middle, with coloring verifying rigid flexibility.

........


501

Figure 7.18. A 4 × 4 array of M 2V 2 centered twist tiles with three fold angle magnitudes represented. Left: crease pattern. Right: folded form.

to valley and vice versa), getting the arrangement shown in the middle subfigure. This looks like we don’t have a match of the fold angle, since we have red mating to green, but remember, the coloring of fold angles is not something absolutely defined; it’s only relative within a tile. So we can choose the coloring of the vertical crease chains in the upper tile to continue the green folds of the lower one, whose fold angles are related by gred > ggreen .

(7.76)

The same inequality exists in the upper tile, so we can’t just swap the colors; we need to introduce a new color (as in the right subfigure), representing a new fold angle magnitude, that maintains the same relative fold angle inequality, i.e., ggreen > gblue .

(7.77)

This twosome now folds rigidly and can, in fact, be used as a larger building block that may be assembled into a larger tile with three fold angles represented. Figure 7.18 shows a tessellation built from a 4 × 4 array of M 2V 2 tiles (or, equivalently, a 4 × 2 array of this two-tile building block). Note, though, that there is no compelling need to butt-join tiles; we can extend their folds arbitrarily, creating arrays of threedimensional pleats in which the individual twist tiles are simply “gadgets” that allow pleats to interact. For example, we could join two rows of centered twist tiles that are angled with respect to one another, connecting them by their pleats, as shown in Figure 7.19. As long as we respect fold angle magnitudes, the results are guaranteed to be rigidly foldable.

502


Figure 7.19. A crease pattern constructed by tilting twist tiles so that they join up with their mirror image.

If we’re going to use twist tiles as gadgets within rigidly foldable tessellations (something that can be done with flat-foldable tessellations as well), then we don’t need to tilt them with respect to their boundaries; we can use the offset twist form for each tile, as in Figure 7.20. A rigidly-foldable tessellation based on this concept is shown in Figure 7.21. Even with just a few rigidly foldable square twists to work from, there are many possible patterns that can be built by plugging twists into rectilinear arrangements of pleats, and as long as we respect both crease assignment and fold angle color, the full pattern will be rigidly foldable. The imposition of the requirement of rigid foldability limits the range of patterns, but it opens up the range of material. We can now use rigid materials other than paper: wood, metal, composites. (Of course, we still must find a way to make compliant—flexible— hinges where the folds take place.) In particular, rigidly foldable mechanisms can be used to realize artworks that combine the beauty of wood with the visual

Figure 7.20. Rigidly foldable offset twist tiles. Left: M 2V 2 twist. Right: M 3V twist.

........


503

Figure 7.21. A crease pattern constructed from offset twist tiles. Left: crease pattern. Right: folded form.

interest of folded patterns, by using wood panels joined by flexible hinges. One of my first wooden mechanism origami sculptures was based on the M 3V square twist constructed by matching fold angles. As in Figure 7.17, there are three different fold angles at play, which gives rise to a mechanism with a deeply incised pattern reminiscent of complex millwork. The artwork and its crease pattern are shown in Figure 7.22. For this design (and others I will show later), the material is a laminate of wood veneer, thin aluminum foil, and a paper backing.

Figure 7.22.

Left: “M 3V Pillars” (2015), by the author, folded from stained maple veneer laminate. Right: its crease pattern. Mountains are blue, valleys are red, and line widths are proportional to the fold angle magnitude.

504


The creases are laser-cut into the laminate; the laser cuts through the wood but is blocked by the metal layer, and then the uncut paper backing provides a compliant hinge. Even with only two fold angles and a single set of vertex angles, the possibilities are great. Thomas Crain is a tessellation artist and woodworker who has, for many years, combined folded tessellations with stunning wood joinery for mounting and incorporating folded tessellations into wooden tables, boxes, and other furniture items. More recently, he has taken to creating the origami tessellations directly from wood. As he describes it, Origami artists create within a framework of selfimposed constraints. The usual one being using only a single sheet of paper with no cuts or glue. I further constrain myself by designing using only twist folds. Because paper is flexible, another layer of constraints may be added by using rigid materials. Thus, for me, origami is a design challenge with the goal being the creation of an esthetically pleasing object while operating in the realm of self-imposed constraints. His rigidly foldable origami tessellations are composed primarily of M 2V 2 twists, but with wooden panels—sometimes, quite thick ones—and brass or monofilament hinges. Examples of Crain’s work are shown in Figures 7.23–7.25. Quite complex rigidly foldable patterns may be constructed with nothing more than square twist tiles. Another nice example of the genre is shown in Figure 7.26, by Polly Verity, whose work we will see more of in a bit.

Figure 7.23. A rigidly foldable tessellation in oak and brass, by Thomas Crain. Left: partially folded. Right: unfolded.

........


505

Figure 7.24. “Oak and Brass” (2015), by Thomas Crain. Oak panels and brass hinges with wooden pins.

Figure 7.25.

Left: a rigidly foldable tessellation by Thomas Crain with monofilament hinges. Right: a rigidly foldable paper pattern, destined for wood.

??

7.6. General Twists Square twists are only a small part of what’s possible in the area of single-degree-of-freedom rigidly foldable origami patterns. Some of the other possibilities we can consider are • quadrilateral twists other than squares, such as kites, trapezoids, and completely irregular polygons; • twists based on polygons with fewer or more sides;

506


Figure 7.26.

Left: “Mayan Steps” (2012), by Polly Verity. Right: detail from the crease pattern of “Mayan Steps.”

• structures that are not twist-based (a polygon surrounded by parallel pleats). All of these are potential avenues for exploration, and we’ll come back to some of them. The possibilities raise the issue of what general statements we can make about classes of rigidly foldable structures. For the moment, we will stick to twists but broaden our consideration. Consider a general simple flat twist in normal form (each acute twist angle is anto). For any two consecutive creases around the polygon, there are four possible crease assignments. Let us assume also that the twist angle is the smallest angle at each vertex (usually, but not always, the case). Then for each of the four crease assignments, we can construct the directed edge of the fold angle graph, as shown in Figure 7.27. For both M M and VV, the fold angle graph edge points CCW around the vertex. For both MV and V M, it points CW. As we have already seen for the square twist, a complete directed loop (either CCW or CW) implies a contradiction in the assumption of existence of a partially folded state. So we can say something stronger than the nonexistence of a rigidly foldable square twist with cyclic crease assignment: there is no all-anto twist, for any polygon, for which a cyclic crease assignment gives rigid foldability.

........


507

M

M

M

V

V

M

V

V

Figure 7.27. The four possible crease assignments for two consecutive edges of a general twist.

The same argument establishes that there are no normal alternating form twists that are rigidly foldable, since those, too, would result in a directed loop (going in the opposite direction). Of course, alternating-form twists are only possible for even-order polygons. As for other crease assignments, any other crease assignment on the central polygon can potentially result in a rigidly foldable twist. There is still a loop condition to be satisfied, however, and it may well turn out that there could still be no solution, as we will now see. ?? 7.6.1. Triangle Twists The simplest rigidly foldable twist is not the square twist, of course; it would be a triangle twist. And if we wished to create a rigidly foldable tessellation based on Archimedean tilings, we would most likely need to include triangle twists, since many of the Archimedean tilings contain triangles. Figure 7.28 shows a schematic of a hypothetical rigidly foldable triangle twist with three interior angles βi, i = 1, 2, 3, that must sum to π. Consider first the possible crease assignments. We already know that this twist can’t have a cyclic crease assignment, so it must be either M 2V or MV 2 , which are equivalent under parity swap. So let us choose the M 2V assignment for definiteness, and we make the base of the triangle the valley fold. We seek a set of interior angles {βi } and a twist angle α that gives rigid foldability (and flat-foldability as well). Now, for this crease assignment, the BLBA Theorem already forces some constraints on the interior angles and sector angles:

508


p Ð a p Ð b3 b a 3

m3 a p Ð b1

b1

Ð m1

Ð m2 a

pÐa

Figure 7.28.

b2 p Ð a

Schematic of a hypothetical rigidly foldable triangle twist.

p Ð b2

• Either α or β1 must be the smallest angle at vertex 1. • Either α or β2 must be the smallest angle at vertex 2. • Either α or π − β3 must be the smallest angle at vertex 3. Then the fold angle multipliers {µi } are given by µ1 =

sin[ 12 ((π − α) + β1 )] sin[ 12 ((π − α) − β1 )]

,

µ2 =

sin[ 12 ((π − α) + β2 )] sin[ 12 ((π − α) − β2 )]

,

µ3 =

sin[ 12 (β3 + α)] sin[ 12 (β3 + α)]

. (7.78)

The loop consistency condition is

or equivalently,

(−µ1 )(−µ2 ) = µ3,

(7.79)

µ1 µ2 − µ3 = 0.

(7.80)

We can immediately eliminate β3 from the variables; since the sum of the interior angles of a triangle is π, we can substitute β3 = π − (β1 + β2 ). Making this substitution and simplifying the resulting expression, we find that the loop condition factors into a product as follows: µ1 µ2 − µ3 = cos( 12 β1 + β2 ) − cos( 12 β1 ) · sec( 12 (α + β1 )) · sec( 12 (α + β2 )) · sec( 12 (α + β1 + β2 )) · sin( 12 α) · sin( 12 β1 ). A rigidly foldable triangle twist would be some combination of β1 , β2 , and α that makes this expression go to zero. For that to happen, at least one of the product terms must vanish. But

........


(7.81)

509

• none of the secant terms can vanish because the secant has no real zeroes; • neither of the last two terms can vanish unless α or β1 go to degenerate values (0 or π or 2π). That leaves the first term. But since cosine monotonically decreases for its argument between 0 and π, the second cosine is always smaller than the first (for valid values of β1 and β2 ). This leads to a strong conclusion: There is no nontrivial rigidly foldable triangle twist. What about higher-order or nonregular polygon twists? It turns out that some exist, but only under fairly specialized geometries. Thomas Evans, a student of Larry Howell at Brigham Young University, carried out a detailed analysis of rigidly foldable origami twists and gadgets [25, 26] and found (among other things) the following: • There are no rigidly foldable triangle twists (as we have just seen). • Rectangles, parallelograms, isosceles trapezoids, and rhombus twists are rigidly foldable for the same crease assignments as the square twist. • Kite twists exist for particular combinations of vertex angles. • Higher-order regular polygon twists are rigidly foldable only for special combinations of crease assignment and twist angle. This result has broad implications for creating rigidly foldable twist tessellations based on Archimedean tilings (or really, on any tiling). Suppose we created a twist tessellation based on a tiling (replacing each polygon of the tiling with a simple flat polygonal twist). If the original tiling contained any triangles, then the tessellation would contain triangular twists—and we now know that a rigidly foldable triangular twist does not exist. So the full tessellation could not be rigidly foldable since at least some portion of it is not. But what about tilings that don’t include triangles, such as the {4.4.4.4}, {6.6.6}, or {8.8.4} tilings? Well, we’ve already seen

510


that the {4.4.4.4} tiling of square twists does give rise to a rigidly foldable twist tessellation (more than one, as it turns out). But what about the others? One of the attributes we saw in twist tessellations is that the original polygons show up as twist polygons (sometimes distorted, as in alternating-twist-direction tessellations). But we also see twist polygons for every vertex of the tiling where the center polygon has the same number of sides as the degree of the vertex. So in any twist tessellation based on a tiling with degree-3 vertices, each of those degree-3 vertices will give rise to a triangle twist in the crease pattern. And so again, since there is no rigidly foldable triangle twist, there could be no rigidly foldable tessellation from such a pattern. The only Archimedean tiling with no triangles or degree-3 vertices is the {4.4.4.4} tiling. Of the Archimedean tilings, therefore, the only possible rigidly foldable twist tilings are tilings composed solely of quadrilaterals. Indeed, the two best-known rigidly foldable periodic patterns—the Miura-ori and Barreto’s Mars tessellation—are based on this type of tiling. More broadly, any periodic tiling other than a tiling of quadrilaterals with all vertices of degree 4 must contain either a triangle or a degree-3 vertex within its unit cell. So the lack of a rigidly foldable triangle twist has quite far-ranging consequences for rigidly foldable twist tessellations. All is not entirely lost, however; there are non-periodic patterns that can potentially and in reality give rise to rigidly foldable mechanisms (for example, patterns based on individual twists). And the limitations imposed by the weighted fold angle graph apply only to mechanisms composed of flat-foldable degree-4 vertices; as we will see, selectively increasing the degree of certain vertices—effectively, adding more folds—can change a locked mechanism into one that is rigidly foldable. The relative paucity of rigidly foldable twists may seem at first disheartening, until we realize that even the few rigidly foldable twists we know about enable an enormous variety of rigidly foldable mechanisms. All of the varieties of Miura-ori and Barreto’s Mars patterns are single-degree-of-freedom rigidly foldable mechanisms, and indeed, many of the applications of the Miura-ori, such as space deployables, rely on that very property when they are implemented in aluminum, polymer, and silicon.

........


511

g2

g2

g3 g1

g3 g3

a2

a1

a3

a4

g4

g1

g4

g2

a2

a1

a3 a2

a3

a4

a4

a1

g1

g4

Figure 7.29.

Left: a 90◦ vertex with a singular multiplier. Middle: a 120◦ vertex with a singular multiplier. Right: a different crease assignment that gives a well-behaved multiplier. For all three vertices, the fold and sector angles are numbered so that γ2 and γ4 are the major creases.

?? 7.6.2. Mechanical Advantage In the preceding discussion, I have assumed without comment a certain reasonable behavior of the degree-4 vertices, captured in the assumption that the fold angle multiplier µ remains finite. This doesn’t have to be the case; since the denominator of µ contains the factor sin( 12 (α1 − α2 )), µ → ∞ when α1 → α2 . If µ goes to either 0 or ∞, we say that both µ and the vertex are singular. Two examples of this situation are illustrated in Figure 7.29. In the left and middle figures, we have α1 = α2 , and therefore the denominator of µ goes to zero, implying that µ goes to infinity. To be precise, at the point of equality, µ becomes undefined, but in the limit, lim µ = ∞. (7.82) α1 →α2

What does this limit physically correspond to? If you cut out and fold the first vertex in Figure 7.29, you will also find an apparent violation of an earlier claim, that all four creases had to fold together at once. In fact, to fold that first vertex, you need to fold the major folds completely to flatness; only then can you fold the minor folds. Then again, this behavior is qualitatively consistent with the singular fold angle multiplier; if the minor fold angle must remain flat (zero) while the major fold angle changes, then the ratio between the two half-angle tangents is indeed (something)/(zero).

512


g2 g2 a1=170¡

g2

a2

a4

a1=135¡

g1

a2

a4

a3

g1

g3

a2

a1=100¡

a3

a4

g1

g3

a3 g3 g4

g4

g4

Figure 7.30.

Three vertices whose sector angles are successively closer to the singular state.

To see more clearly what’s really going on, let’s consider three patterns that approach the limiting case of a crossed-right-angle vertex. For all three cases, we’ll choose α2 = α4 = 90◦ , but we’ll vary α1 to approach 90◦ from above, as shown in Figure 7.30. We can rearrange Equation (7.57) to plot the relationship between the two angles, which we do in Figure 7.31. Far from the singular state (α1 = 170◦ ), the major and minor angles change at almost the same rate. But as we approach the singular state, the motion changes. First, the major angle moves with relatively little motion of the minor angle; then the situation switches and most of the motion happens in the minor angle within the last several degrees of motion of the major angle. From this plot, it is easy to see the evolution from equal motions for α1 very 180°

Minor Angle

135°

α1 =170°

90°

α1 =135° α1 =100°

45°

Figure 7.31.

0° 0°

45°

90° Major Angle

135°

180°

Minor angle versus major angle for the three vertices.

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different from α2 , to first one moving, then the other, as the two sector angles approach equality. As we approach the singular state, we have the situation where one angle moves a lot while the other only moves a little (with the roles switching throughout the course of the motion). This behavior can be seen in many origami mechanisms such as action figures that operate over a wide range of motion: as the controller region is actuated, relatively little motion happens, but then toward the end of the range the actuated motion “darts” into position. The asymmetry of the motion is strongest when the vertex is closest to the singular case. As with many other mechanical linkages, a degree-4 vertex can be considered a motion amplifier or reducer, depending on the range of motion over which one is measuring. The ratio between the relative motions of two moving objects is called the mechanical advantage of the mechanism. In the case of a degree-4 vertex, there is a varying non-unity mechanical advantage between the rotations of the major and minor folds. This can be either advantageous or disadvantageous, depending on the particular application and other forces and mechanisms at play. When a small motion in one component creates a large motion in the other, we get an amplification of distance moved (or in this case, of the rotational angle). Conversely, when a large motion in one component creates a small motion in the other, we get an amplification of force (or with folds, of torque). The mechanical advantage between the major angle γ+ and minor angle γ− is given by m(γ+ ) ≡ dγ− /dγ+ . We plot the mechanical advantage for our three cases in Figure 7.32 over the full range of motion. Near the singular state, there is a significant amplification and reduction between the two coupled motions. In fact, it is easy to see from Equation (7.57) that near γ+ = 0, we have m(γ+ )| γ+ →0 = µ−1, (7.83) where µ is the fold angle multiplier for the vertex, while near γ+ = 180◦ , m(γ+ )| γ+ →180◦ = µ. (7.84) The mechanical advantage function in Figure 7.32 is monotonic, which means that for every flat-foldable degree-4 vertex, the mechanical advantage varies smoothly from µ−1 to µ over the

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Mechanical Advantage (dγ'/dγ)

10 5 2 α1 =170°

1

α1 =135° α1 =100°

0.5 0.2 0.1

Figure 7.32. 0°

45°

90°

135°

180°

Major Angle

Mechanical advantage for the three vertices.

full range of motion. That, in turn, implies that the torque transfer varies over a factor µ2 over the full range of motion. As we approach the singular state with α1 = α2 , we will have µ → ∞. We will also have the major-minor transfer function graphed in Figure 7.31 approach an L shape, consisting of a horizontal line from γ+ = 0◦ to 180◦ , followed by a vertical line at 180◦ . That is precisely the motion that we see in action; first the two major angles fold, and then only when they are completely folded can the pair of minor angles fold. So the motion breaks down into two separate motions. But the true significance is this: the mechanical advantage goes to 0 and/or ∞ (depending on where you are in the motion). That means that an infinite amount of force would be needed to actuate the mechanism over the full range of motion. Effectively, then, the singular state is not actuable over its full range from a single fold. Any folding mechanism with the goal of actuating all four folds from a single fold must stay well away from the singular state in its design in order to keep the forces and mechanical advantages within a tractable range. ??

7.7. Non-Twist Folds

?? 7.7.1. General Meshes Now, we’ve focused a lot on rigidly foldable twist folds, and one can certainly do a lot with quadrilateral twists. But twists— polygons surrounded by parallel-edge pleats—are still a pretty specialized collection of creases. If we cast our net more broadly,

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Figure 7.33. Two examples of general bird’s-foot vertex patterns. Left: a valley fold tripod plus another mountain fold is rigidly foldable. Right: a valley fold cross plus another mountain fold is rigidly foldable.

we can find many more crease patterns that can be constructed to be rigidly foldable. In fact, by way of example, most patterns consisting entirely of triangles are not just rigidly foldable; they have multiple degrees of freedom. Put colloquially, they are somewhat floppy. That is not to say that all triangulated patterns are flexible. In a recent analysis, Abel et al. established conditions for a singlevertex crease pattern of degree > 2 to be able to fold rigidly [1]. Every rigidly foldable single-vertex pattern must contain a general bird’s-foot pattern: a set of three creases of one fold assignment (e.g., mountain), separated sequentially by angles strictly between 0 and π (possibly with additional creases between them), plus one additional crease of the opposite assignment (e.g., valley). Examples of general bird’s-foot vertices are shown in Figure 7.33. (Note that neither of these folds fully flat, but they both fold rigidly from the unfolded state over some finite range). For a general crease pattern composed of multiple vertices, assuming that each vertex satisfies the bird’s-foot condition, the number of degrees of freedom of the overall pattern was derived by Tachi [115], as Õ DOF = B − 3H + S − 3 − Pk (k − 3), (7.85) k=4

where • B is the number of edges on the border, • H is the number of holes in the pattern, • S is the number of redundant constraints (typically geometry-specific), • Pk is the number of k-gon facets.

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If we start with a triangulated crease pattern, the last sum term in Equation (7.85) goes away. Joining triangles together into polygons with larger numbers of sides will reduce the flexibility of a pattern; conversely, the greatest flexibility is attained in a fully triangulated pattern. Note, though, the importance of the border term B in what’s left. If there are no holes (H = 0) and no special geometry (S = 0), the number of degrees of freedom is B − 3, no matter how many triangles might be in the surface. Thus, for example, a generic triangulated surface with four border edges will, in general, have one degree of freedom, no matter how many vertices it contains. If a triangulated surface is constructed with three edges on the border, Equation (7.85) gives DOF = S = 0 for generic (nonspecial-geometry) configurations, implying rigidity. Indeed, since the polygon composed of those three edges will be a triangle and must necessarily be rigid, adding a triangular facet to that closed hole will give a closed polyhedron. A famous theorem by Cauchy [22] established that a closed convex polyhedron was rigid, and indeed, any generic closed polyhedron, and thus any generic noholes crease pattern with three edges, is rigid. “Generic” means “non-special,” though; special geometries can give S = 1 and thus one degree of freedom even with B = 3. For polyhedra, Connelly constructed nonconvex closed flexible polyhedra, and it was subsequently shown that all such polyhedra must have constant volume as they flex. See [17, pp. 239–247] for a fuller discussion. As crease patterns consisting of ever larger numbers of facets are created, unless care is taken to wrap them in ever longer edge polygons to force rigidity, as the size of the array grows, the number of facets on the border will grow as well. For crease patterns composed entirely of triangles, as they scale up to larger arrays with increasing border, they gain progressively more degrees of freedom; informally, they get floppier. If, however, we build patterns from quadrilaterals, the scaling behavior changes dramatically. As they scale up to larger arrays with increasing border, the last term in Equation (7.85) dominates; they lose degrees of freedom. Thus sufficiently large arrays of generic quadrilateral (or higher-degree) facets will be rigid. The key term here, though, is generic—meaning no special geometric properties. By careful choice of geometry, it is possible to find patterned arrays that scale arbitrarily. In particular, if a

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pattern is composed of rigidly-foldable repeated building blocks where the fold angle consistency is maintained from one block to the next, then arbitrarily large arrays may be created with nonzero degrees of freedom. An example of this is the Miura-ori pattern that has only one degree of freedom, as well as all of the rigidly foldable twist tessellations we have seen thus far. ?? 7.7.2. Quadrilateral Meshes I’ve mentioned Tomohiro Tachi quite a lot in the context of rigid foldability; much of the recent work in the field can be traced back to his analytic/computational work investigating the kinematics of origami mechanisms in the first decade of the 2000s. In a seminal paper [111], he analyzed the design and kinematics of quadrilateral meshes—crease patterns consisting entirely of degree-4 vertices and quadrilateral facets—and (fully) generalized Miuraori: adopting the crease assignment of Miura-ori, but letting all of the facets become arbitrary quadrilaterals. Now, in order to be developable (foldable from a flat sheet of paper), all of the facets cannot be truly arbitrary; the sector angles at each vertex must add up to 360◦ . If that condition is satisfied, then quite a wide array of three-dimensional structures can be designed by distortions of the underlying Miura-ori. However, the vast majority of those patterns are indeed structures, not mechanisms: they are rigid in their folded form and so cannot be smoothly folded from a flat sheet while maintaining flat facets during the folding motion. A subset of those patterns is actually rigidly foldable, though. Tachi showed that if the vertices are flat-foldable, then because the half-angle tangents of all fold angles around any vertex are linearly proportional to each other (as above), then every pair of half-angle tangents would be proportional to a common quantity. Thus, if you can find a single partially folded state, the entire pattern would flex from unfolded to flat-folded with a single degree of freedom. This was demonstrated dramatically by a kinetic sculpture designed and built by Tachi for an exhibition in 2014, shown in Figure 7.34. The connection between flat-foldability and rigid foldability is a remarkably powerful result that could, in principle, allow for a much wider variety of rigidly foldable patterns than those that can be assembled from the relatively small number of building blocks we have constructed from twists and their kin. And in practice, it leads to exactly that [73].

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Figure 7.34. Tomohiro Tachi demonstrating his rigidly foldable kinetic sculpture.

If we consider a quadrilateral mesh consisting of flat-foldable degree-4 vertices, then the folds can be grouped into vertical and horizontal chains of folds, where, regardless of crease assignment, every fold in each chain has a constant fold angle magnitude, as illustrated in Figure 7.35. In a rigidly foldable quadrilateral mesh whose vertices are all developable and flat-foldable degree-4 vertices, since the magnitude of the fold angle is constant across each vertex, the folds can be organized into continuous chains running roughly vertically and horizontally in which the fold angle magnitude is constant for each entire chain. However, the sign of the fold angle may or may

Figure 7.35. Schematic of a quadrilateral mesh composed of flat-foldable vertices. The folds can be grouped into chains of constant fold angle running horizontally (red) and vertically (green).

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g4

Figure 7.36.

Schematic of a single vertex tile in a rigidly foldable quadrilateral mesh.

g1

g3

a3

a4 a1

a2

g2

not switch at any given vertex, depending on whether the chain is major or minor at that particular vertex. A chain of fold angles could be major at one vertex and minor at the next, depending on the relative magnitudes of the fold angles of the two chains intersecting at the vertex. We can construct such a pattern by building an assembly of tiles—the orange squares in Figure 7.35—connected by creases. But in this case, rather than a tile containing an entire twist, each tile consists of an individual vertex, whose configuration is determined by the fold angles and relative orientations of the creases entering and exiting the tile. Let us look again at a single degree-4 vertex, but now thinking of it as a tile, as illustrated in Figure 7.36. We introduce a variation on the notion of the fold angle multiplier: define µi, j as the ratio of half-angle-tangents between the ith and jth folds around the vertex, e.g., µ1,2 ≡


=

sin 12 (α1 + α2 ) sin 12 (α1 − α2 )

.

(7.86)

This is for γ2 and γ4 being the major fold angles and has | µ1,2 | > 1 for all sector angles and crease assignments. Since γ2 and γ4 are major, we must have that γ2 = γ4, γ1 = −γ3, |γ{2 or 4} | > |γ{1 or 3} |.

(7.87)

It is straightforward to rearrange vertex indices to obtain the equivalent ratio for γ1 and γ3 being major, in which case | µ1,2 | ≤ 1 and sin 12 (α1 − α4 ) µ1,2 = , (7.88) sin 12 (α1 + α4 )

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and thus,

γ1 = γ3, γ2 = −γ4, |γ{1 or 3} | > |γ{2 or 4} |.

(7.89)

Now, rather than starting with the sector angles, we could, instead, start with the fold angles at each vertex, which strongly constrains the possible sector angles. Because of Equations (7.86) and (7.88), the fold angles and sector angles at a vertex cannot be chosen entirely independently. In the conventional way of looking at crease patterns, the sector angles {αi } are a given, setting the value of µi, j ; then by choosing one fold angle, Equation (7.86) or (7.88) (as applicable) implicitly determines the adjacent fold angles. However, there is another way to look at this: we could instead take the two fold angles as the given. In this case, we can choose only one of the sector angles; the other three must be set to satisfy the Kawasaki-Justin Condition, α1 + α3 = π,

α2 + α4 = π,

and Equations (7.86) and (7.88). If we carry out this procedure for the vertex of Figure 7.36 with γ2 and γ4 major, the fold angle multiplier µ1,2 is given by Equation (7.86) and the four sector angles can be shown to be α1 = arbitrary, α2 = 2 cos−1 q

1 + µ1,2 cos 1 α1 2

1+

µ21,2

+ 2µ1,2 cos α1

, (7.90)

α3 = π − α1, α4 = π − α2 . If, however, γ1 and γ3 are major, then we will have α1 = arbitrary, α4 = 2 cos−1 q α3 = π − α1, α2 = π − α4 .

1 + µ1,2 cos 1 α1 2

1+

µ21,2

+ 2µ1,2 cos α1

, (7.91)

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521

g4 f4 f3

Figure 7.37. Schematic of a single vertex tile in a rigidly foldable quadrilateral mesh.

g1

f1

a4 a1

g3

a3 a2 f2 g2

We can use these relationships to define the four sector angles of the tile from any one of them. That, in turn, places constraints on the angles at which the four folds hit the edges of the tile. We define φi as the fold direction angle, the angle between the ith fold and the horizontal, as in Figure 7.37. We could choose φ1 and φ2 arbitrarily, or at least with broad latitude, but the other two fold direction angles are fully determined from the requirement of flat-foldability and the fold angles. Suppose φ1 and φ2 are given. That determines the first sector angle: α1 = φ2 − φ1 .

(7.92)

Then, depending on which fold angle pair is major, either Equation (7.90) or (7.91) gives the other three sector angles. Those, in turn, define the two remaining fold direction angles: φ3 = φ1 + (α1 + α2 − π), φ4 = φ2 + (α2 + α3 − π).

(7.93)

And that completes the design of the vertex tile. So, to recap, we can choose the two fold angles {γ1, γ2 }, the two fold direction angles {φ1, φ2 }, and the crease assignments for those two folds. From those, all four sector angles, the other two fold angles, fold directions, and crease assignments are fully determined. Crease assignments too? Yes: • If |γ1 | < |γ2 |, then γ2 and γ4 are major, and γ3 = −γ1 , γ4 = γ2 . • If |γ1 | > |γ2 |, then γ1 and γ3 are major, and γ3 = γ1 , γ4 = −γ2 .

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γ2 = 15°

γ2 = 30°

γ2 = 45°

γ2 = 60°

γ2 = 75°

γ1 = 15°

γ1 = 30°

γ1 = 45°

Figure 7.38.

γ1 = 60°

A grid of rigidly foldable single vertex tiles with φ1 = 0◦ , φ2 = 90◦ , with varying fold angles γ1, γ2 .

γ1 = 75°

What if the two fold angles are equal? If they are, then we will have | µ1,2 | = 1, which will make one of the sector angles zero. This is not allowed, so at any interior vertex, no two adjacent folds can have the same fold angle magnitude. Choosing the fold angles or fold direction angles of two of the folds sets the geometry of the vertex. Choosing the assignment of those two folds doesn’t affect the geometry; it just affects the assignments of the other two folds. Figure 7.38 shows a grid of the possible vertices with the first two fold direction angles fixed at right angles to one another and varying the fold angles in those two directions. The diagonal is empty because, as noted, it’s not allowed to have the fold angles be the same. In principle, using these rigidly foldable vertex tiles, it is possible to build up quite complex rigidly foldable mechanisms. We can construct them manually, by sliding tiles around, extending their fold lines, and ensuring that we only mate tiles where fold angles match, in both magnitude and sign. Here, as before, using distinct colors for different fold angle magnitudes can help keep track of matching fold angles. When working with individual vertices, though, it is challenging to build up patterns that are large enough to be interesting. We

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γ2 = 15°

γ2 = 30°

γ2 = 45°

γ2 = 60°

γ2 = 75°

γ1 = 15°

γ1 = 30°

Figure 7.39. A grid of rigidly foldable four-vertex tiles with φ1 = 0◦ , φ2 = 90◦ , with varying fold angles γ1, γ2 . In each tile, the fold angle pairs along the left and bottom of the tile are the same magnitude, opposite assignment.

γ1 = 45°

γ1 = 60°

γ1 = 75°

can, though, construct small collections of vertices as building blocks. If, for example, we connect four tiles so that side-by-side creases have equal fold angles but opposite assignment, then we get a set of tiles as shown in Figure 7.39, which all turn out to be some version of a square twist. Of course, the distances between the fold pairs can be scaled (which will have the effect of changing the central polygons from squares to rectangles). And of course, we can change the fold direction angles, both individually and pairwise. Figure 7.40 shows a set of 2 × 2 tiles with equal-fold-angle-magnitude pairs along the left and bottom of the tile, but with the bottom pair tilted to 60◦ . We can combine vertices in many more ways, and as long as we respect matching of fold direction, angle, and assignment, we will be guaranteed of getting a rigidly foldable mechanism. In [73], I described a computational algorithm that combines such vertices in a programmatic way. Figures 7.41–7.44 show several examples of rigidly foldable quadrilateral meshes constructed by this method. Figure 7.43 shows something a bit unusual: degree-6 vertices. How does this arise? By allowing two vertices to approach one

524


γ2 = 15°

γ2 = 30°

γ2 = 45°

γ2 = 60°

γ2 = 75°

γ1 = 15°

Figure 7.40.

γ1 = 30°

A grid of rigidly foldable four-vertex tiles with φ1 = 0◦ , φ2 = 60◦ , with varying fold angles γ1, γ2 . In each tile, the fold angle pairs along the left and bottom of the tile are the same magnitude, opposite assignment.

γ1 = 45°

γ1 = 60°

γ1 = 75°

Figure 7.41. “RFQM #2” (2017), by the author. Left: crease pattern. Right: folded form, from cherry wood veneer laminate.

another—basically, taking the intervening edge length to zero— we can collapse the two degree-4 vertices into a single degree-6 vertex, as we did in Chapter 2 in moving from the Huffman grid to the Yoshimura pattern. This doesn’t change the compatibility between the fold angles passing through the vertices, but such a vertex in isolation would have additional degrees of freedom. Most of the patterns I’ve generated using this approach have made use of a small number of vertex tiles, which simplifies the process of fitting tiles together and creates aesthetically appealing

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Figure 7.42. “RFQM #4” (2017), by the author. Left: crease pattern. Right: folded form, from cherry wood veneer laminate.

Figure 7.43. “RFQM #5” (2017), by the author. Left: crease pattern. Right: folded form, from walnut wood veneer laminate.

regularities in the finished form. In Figure 7.44, however, I decided to exercise the computational tool that I used to design these, by making every direction along the left edge and bottom edge slightly different. This gives rise to every vertex in the pattern being geometrically unique and adds an overall flow to the finished artwork. There’s an important point I’ve glossed over but need to come back to now. I’ve mentioned that when we build rigidly foldable

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Figure 7.44. “RFQM #6” (2017), by the author. Left: crease pattern. Right: folded form, from camphor wood veneer laminate.

patterns from flat-foldable degree-4 vertices, if an intermediate state exists, the pattern will flex all the way from the unfolded state to flat-folded, but this is limited by self-intersection: collision between layers. Most of the patterns that we’ve seen don’t experience such collisions, but it is relatively easy to construct a pattern for which self-intersection limits the range of motion. Figure 7.45 shows a simple pattern of four twists that cannot fold rigidly to the flat-folded state (although if you allow some dis-

Figure 7.45. A rigidly foldable pattern that does not fold all the way to flat-folded because of self-intersection. Left: crease pattern. Middle: partially folded. Right: at the limit of motion. Note that the right and left sides are about to meet.

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tortions during the folding process, it can be coerced past the collision to fold fully flat). Even if we don’t run into layer collisions limiting the range of motion, it is possible that the full range of collision-free motion may not be reachable from the unfolded state. Abel et al. [2] demonstrated a triangulated pattern that was rigidly foldable, but the space of attainable configurations was discontinuous; it needed to be “popped” (distorted) to get from one region of continuous motion to another (similar to the way a triangulated cylinder must be popped between its two stable states). It seems plausible that there are many more such patterns waiting to be discovered in the genre of rigidly foldable mechanisms.

?

7.8. Non-Quadrilateral Meshes

? 7.8.1. Forced Rigid Foldability The M 2V 2 and M 3V square twists are rigidly foldable; the M 4 and (MV)2 twists are not. Sometimes, though, if we are presented with a non-rigidly foldable pattern, we can transform it into a rigidly foldable pattern by a simple strategem: force it to fold, look for facets that are bending, and then add a fold along the bend. Sometimes this works; sometimes it doesn’t. In any fully triangulated pattern, like the triangulated cylinder, there are no facets that can bend along diagonals and so the distortions that enable motion are going to be more complicated, commonly buckling and/or rolling of creases continuously through the layers of the paper. One pattern where this forcing method works very neatly is the (MV)2 square twist. Forcing it to open and close very clearly induces a bend along one or the other diagonal of the central square; by adding this diagonal fold explicitly, as in Figure 7.46, we regain rigid foldability, and the motion has a single degree of freedom. The extra crease can run either vertically or horizontally; its assignment must match the creases to which it is collinearly connected. The additional crease also creates a singularity in the flat state by creating a second potential motion. Since there is now a continuous chain of collinear mountain (or valley) folds across the middle of the square, the entire pattern could simply fold in half along that chain.

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Figure 7.46. Crease pattern for an (MV)2 twist, which is ordinarily not rigidly foldable. Adding a diagonal crease (in either direction) makes it singledegree-of-freedom rigidly foldable.

Figure 7.47 shows an application of this split-(MV)2 twist, created from wood composite with Tyvek™ hinges. There are thickness-accommodating offsets between adjacent panels that allow the layers to stack neatly when flat-folded, designed according to the technique of Edmonson et al. [24]. My colleagues at Brigham Young University used the same pattern to develop their own large-scale kinetic sculpture displayed at the BYU Museum of Art in 2015, shown in Figure 7.48. A nice feature of this pattern is that it has an intermediate state that is stable when sitting on a flat surface, shown in the middle image of Figure 7.47. This property makes the small size a nice display stand for other artwork. David Morgan’s students at BYU have also demonstrated a deployable desk using this same pattern.

Figure 7.47. An (MV)2 square twist, made rigidly foldable by adding a fold along the center diagonal, from laser-cut wood composite.

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Figure 7.48. A kinetic sculpture based on the split-diagonal (MV)2 square twist. Left: collapsed. Right: expanded.

? 7.8.2. Non-Flat-Foldable Vertices In the split-diagonal (MV)2 twist, the vertices connected by the added crease become degree-5, and they are then no longer individually flat-foldable—at least, not including all creases. (In the rigidly foldable mechanism, the diagonal folds are unfolded in the state where all of the other folds are flat-folded). There are, in fact, many rigidly foldable mechanisms that make use of non-flat-foldable vertices. When we go to non-flat-foldable vertices, or vertices of degree higher than 4, we lose the absolute guarantee of continuous rigid foldability that came with flatfoldable degree-4 vertices, but in many cases, either symmetric considerations, a high degree of triangulation, or both, can bring about rigid foldability. One example of a rigidly foldable mechanism based on nonflat-foldable vertices was the modified Miura-ori pattern explored by Yves Klett and colleagues that we saw back in Section 2.4.9. This pattern (like so many others) has been discovered independently several times. A 1991 patent by Czaplicki [18] describes this pattern for cellular-core structures; a later patent by Gale [34] describes still more variations. Conceptually, it can be seen as an evolution of the Miura-ori. In the Miura-ori, because the vertices are all symmetric bird’sfoot vertices, the minor folds in each row are collinear in the crease pattern and coplanar in the folded form, as shown in Figure 7.49. We could split the crease pattern along that minor fold chain, insert a strip of paper between the halves, and then connect the

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Figure 7.49. A Miura-ori. Note that each row of minor folds is collinear in the crease pattern and coplanar in the folded form. This allows the crease pattern to be split along each zigzag chain of minor folds.

Figure 7.50. Splitting the Miura-ori and adding spacers creates a new 3D structure but preserves the continuous motion and rigid foldability of the original Miura-ori pattern.

split vertices across the strip with folds that continue the major fold chains, as shown in Figure 7.50. This bit of surgery transforms the split vertices: they are no longer flat-foldable; in fact, they exhibit a hard stop as the layers are pressed together, a phenomenon that Klett and colleagues call a Blockfaltung. This is, as the saying goes, a feature, not a bug, at least in the application of folded cores. In artistic applications, it is useful, too, because it can introduce an inevitable—in fact, unavoidable—three-dimensionality into a folded crease pattern and creates surfaces at roughly normal incidence to the plane that contains the folded form. Since the vertices have been transformed away from flatfoldability, the formulas we have derived that relate the fold angles are no longer applicable. There are more general formulas that describe the fold angle relationships, and we will explore them

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Figure 7.51.

Top left: “Aztec Iterated” (2008), by Polly Verity. Top right: detail of the crease pattern of “Aztec Iterated.” Bottom left: “Pixelstrip” (2012), by Polly Verity. Bottom right: detail of the crease pattern of “Pixelstrip.”

in the next chapter. But for simply constructing artistic shapes, there is no need to compute the fold angles; we can simply start with flat-foldable forms, then “graft in” strips of paper by splitting chains of collinear folds. This idea of splitting and grafting is quite versatile and applies to much more than just Miura-ori patterns. Two beautiful examples of the concept are shown in Figure 7.51 by artist Polly Verity, who we have already met. Verity’s work extends across many geometric genres and categories (and much of her work defies easy categorization), but these two examples are lovely illustra-

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tions of the grafting concept and, I hope, might inspire further explorations of rigidly foldable concepts. Many rigidly foldable origami patterns can be constructed with minimal use of mathematics; we can take vertices, twists, or periodic patterns, such as Yoshimura, Miura-ori, and their kin, and pull them open, graft in strips, and otherwise modify them using simple geometric manipulations. If, however, we wish to quantitatively model these patterns, then we need to compute relationships between fold and sector angles, between edges and vertex coordinates, and between 2D and 3D representations of crease patterns and folded forms. To do that, we need to bump up the level of mathematics used to describe origami. Over the next two chapters, we will do exactly that. ?

7.9. Terms Blockfaltung A limitation on the motion of a rigidly foldable origami mechanism where the range of motion is constrained by self-intersection avoidance. Digraph (directed graph) A graph together with an orientation, an assignment of direction to each edge. Fold angle graph A digraph whose vertices correspond to folds of a crease pattern with directed edges connecting pairs of vertices that are adjacent in the crease pattern, the edge directions corresponding to the relative fold angle magnitudes of their corresponding edges. Fold angle multiplier The ratio between half-angle tangents of pairs of folds at a flat-foldable degree-4 vertex. Forcing method A technique for turning origami structures into rigidly foldable mechanisms, by forcing the folding and adding creases where facets are forced to bend. General bird’s-foot pattern A set of three creases of one fold assignment separated sequentially by angles strictly between 0 and π, plus one additional crease of the opposite assignment. Great circle A circle on a sphere whose center is also the center of the sphere. CHAPTER 7. RIGID FOLDABILITY

........

533

Major fold (vertex) In a degree-4 vertex, the two folds opposite one another that have the same sign fold angle. Mechanical advantage The ratio between the relative motions of two moving objects. Mechanism An object that is intended to be flexible. See also structure. Minor fold (vertex) In a degree-4 vertex, the two folds opposite one another that have the opposite sign fold angle. Quadrilateral mesh A crease pattern in which every facet is a quadrilateral. Singular multiplier The fold angle multiplier of a degree-4 vertex whose value is 0 or ∞. Singular vertex A degree-4 vertex where the fold angle multiplier goes to 0 or ∞. Spherical polygon A polygon on a sphere whose sides are composed of arcs of great circles. Structure An object that is intended to be inflexible. See also mechanism. Weighted fold angle graph A fold angle graph with weights assigned to each edge that correspond to the fold angle multiplier between the pairs of folds corresponding to the vertices of the fold angle graph.

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8

Spherical Vertices ?

8.1. Generalizing Vertices

In the previous chapter, we began analyzing flat-foldable degree-4 vertices in their partially folded state—that is, in the intermediate configuration between (flat) unfolded and (fully) flat-folded. We learned three important properties of such vertices: • The fold angles of the major folds (opposite creases of the same type) are equal to one another. • The fold angles of the minor folds (opposite creases of differing type) are also equal to one another. • The half-angle tangents of the major and minor folds are proportional to one another, with the constant of proportionality being the fold angle multiplier µ. The analysis of that last relationship was not terribly complicated, but it did rely on algebraic simplification that arose from flat-foldability and the sector angle relationships implied by the Kawasaki-Justin Theorem. For degree-4 vertices that are not flatfoldable—and vertices of higher degree—these simplifications no longer apply. We can, though, still learn quite a bit about general non-flatfoldable degree-4 vertices and the patterns folded from them, using further tools from spherical geometry. To do so, we now introduce a new concept, the Gaussian sphere, that will allow us to analyze vertices—and indeed, entire crease patterns—and can ultimately be a tool for the construction of quite sophisticated and beautiful 3D origami structures. In this chapter, I will follow the approach 535

of David Huffman [47], who pioneered much of the mathematical analysis of 3D folded shapes and for whom the Gaussian sphere was a familiar friend and a valuable design tool. ??

8.2. The Gaussian Sphere

?? 8.2.1. Plane Orientation To introduce the Gaussian sphere, let us consider a general smoothly curved surface, such as that illustrated in Figure 8.1. At every smooth point of the surface, there is a unique plane that is tangent to the surface. As one slides the plane around on the surface, keeping it in contact, it will, in general, change its orientation, tilting back and forth as it moves. For each point of contact, there is a unique orientation of the plane that preserves tangency. How does one represent that orientation? There is a straightforward way of representing a direction in space using Cartesian coordinates and vectors, which we will make full use of in the next chapter, but there is also a purely geometric representation, which is to represent the orientation of the plane by the point on a unit sphere where a plane in the same orientation would touch the a a

C

C¢

Figure 8.1. Representing a direction as a point on the Gaussian sphere via the tangent plane. Left: for any smooth surface, at any point, there is a unique plane that is tangent to the surface. Right: if we bring the plane into contact with a unit sphere without changing its orientation, we can represent the orientation of the plane by the point on a unit sphere where the plane contacts the sphere.

536

........CHAPTER 8. SPHERICAL VERTICES

sphere, as illustrated on the right in Figure 8.1. A plane tangent to a sphere at a point is called an osculating plane of the sphere. Every point on a sphere has a unique osculating plane with its own orientation in space,1 and every possible orientation corresponds to some point on a sphere. Thus, we can use the position of the point of tangency of an osculating plane as a way of representing the orientation of any plane that is translated (while preserving its orientation) to be tangent to a sphere. Another equivalent way of representing the orientation of a plane is by the unique direction that is perpendicular to the plane,2 known as the normal to the plane—the purple arrow in Figure 8.1. If we draw a line from the center of the sphere to the point of tangency of the plane, that arrow is perpendicular to the plane. One way of picturing the relationship between the plane of tangency and its perpendicular direction is to think of the plane as the flat head of a nail and its direction as the sharp end of the nail; as we slide the head of the nail around on the surface, its sharp end changes its direction as the head tilts and turns. If we preserve the orientation of the nail as we move along the path, but translate its head to the center of this unit sphere, then the point of tangency— the point where the head touches the sphere—will move around on the sphere and trace out a path thereupon. So, the point on the sphere where the osculating plane contacts it is equivalent to the direction of the normal to the plane, and both are equivalent ways of representing the same information: the orientation of the plane, or a particular direction in space (which is perpendicular to the plane). We will call that direction a direction vector, and while we can think of that direction vector as an arrow that has a preferred direction in space—like a compass needle that always wants to point north—we can give that direction a concrete geometric representation as a point on a unit sphere. Although I’ve drawn the arrow as if it started at the point of tangency, if we move the direction arrow so that its tail is at the center of the sphere, its head points to the spot on the sphere that represents it. So another way to think of the unit sphere 1

A reasonable question to ask is, aren’t there two such points on a sphere, at opposite poles? Yes. But if we give our plane a “top side” and a “bottom side,” then we can say that we use the point where the bottom side of the plane contacts the sphere, and that gives a unique point for every orientation of the plane. 2 And points out of the top side of the plane.

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CHAPTER 8. SPHERICAL VERTICES

537

representation of a point on a surface is to translate the direction vector directly to the center of the sphere, and then its tip lies on the surface of the sphere. A unit sphere that is used in this way to map out orientations of direction vectors (or, equivalently, tangent planes to a surface), is called the Gaussian sphere. The image of the point of the nail on the sphere, or equivalently, the point of tangency of the osculating plane, is called, variously, the spherical mapping [45], spherical image [66, 3], or Gauss map [100, 96] of the original path or surface. ?? 8.2.2. The Trace Consider now a surface that could be any combination of flat and smoothly curved regions as well as sharply folded creases and corners. Pick a point P on that surface and draw a closed contour C around it in the counterclockwise direction. At each smooth point of the surface, there is a well-defined direction vector, which is perpendicular to the plane of tangency. As we follow the path around the contour, the direction of the vector varies in space along the path. Now, think about how, as the direction vector changes along the path, its position on the Gaussian sphere changes. If we think about a direction vector as an arrow whose tail is fixed to the center of the sphere, as the direction vector changes orientation, its tip will trace out a different path on the surface of these sphere. This new contour on the Gaussian sphere is called the trace of the original path, and we denote it by C 0. An example of this, with a circular contour on a bulging surface, is shown in Figure 8.2. In this simple example the trace is similar in shape to the original contour. This is not usually the case, however. Figure 8.3 a

b c

Figure 8.2.

Left: a contour on a surface with unit normals. Right: its trace on the Gaussian sphere.

538

d

a

b

e

C


c

d

C¢ e

C′

Figure 8.3.

C

Left: a contour on a cube corner. Right: its trace on the Gaussian sphere.

shows a contour on a corner of a cube and its trace, where a new situation arises: a sharp fold. (Three of them, in fact). On the face of a flat surface, the direction vector is unchanged as you travel along the point, and so every point along that path maps to the same point on the Gaussian sphere. For surfaces with sharp edges, (i.e., folds in origami vertices, or polyhedral vertices in general), though, at each edge, the direction vector needs to abruptly rotate about the edge as it moves from one facet to the next. So, the direction vector sweeps out a fan of direction vectors rooted at the edge and smoothly rotating around the corner, thus connecting the surface normals on one side with the surface normals on the other. This rotation creates an arc in the trace on the Gaussian sphere. The arc is a segment of a great circle of the sphere: a circle whose center is also the center of the sphere. This process is illustrated in Figure 8.3. Each sharp edge of the cube gives rise to a great circle arc of the trace (shown in green). Conversely, for each flat face of the cube, all of the normals along the contour C point in the same direction so that along each face, all of the direction vectors map to the same point of the trace (shown in orange), a corner where two great circle arcs come together. There are three faces to the the vertex of the cube, so there are three corners (and three great circle arcs) in the trace. But there is a duality to the structure: each arc of the path on a flat face gives rise to a point on the Gaussian sphere, whereas each point of sharp bend on the path gives rise to a full arc of the trace. Our nail analogy still works here. We take a flat-headed nail, place its head on the surface on the contour, and then slide the nail along the contour, keeping the head flush with the surface and

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539

C′

C′

C′

Figure 8.4. Traces on the Gaussian sphere. Left: A(C 0) > 0. Middle: A(C 0) < 0. Right: A(C 0) = 0.

pivoting it around each sharp edge. Translate the head of the nail to the center of the sphere and see what path its sharp end traces out on the surface of the sphere. Remember, it is the orientation of the nail in 3D that defines the trace; its position along the path is entirely irrelevant (other than to determine whether the path runs clockwise or counterclockwise on the surface). Whether the original surface was smooth or angled, the trace C 0 always forms a closed loop that encloses some figure on the surface of the Gaussian sphere, which has a well-defined spherical area A(C 0) (in units of steradians). However, we also assign a sign to this area: if the trace encloses a region in the counterclockwise direction, we give that a positive sign; if it encloses it in the clockwise direction, the sign is negative. If the trace includes areas with both clockwise and counterclockwise circulation, the net area is the sum of all the signed areas, so that, for example, a symmetric figure-8 will enclose zero net area (see Figure 8.4). The contour C, too, encompasses an area A(C) (which is a “real-space” area, having units of (distance)2 ). In general, as the size of the contour C is decreased, both the area of the trace A(C 0) and the area of the original contour A(C) will decrease as well. The Gaussian curvature K is defined as the limit of their ratio: K(P) ≡ lim

C→P

A(C 0) . A(C)

(8.1)

The Gaussian curvature, therefore, has units of (distance)−2 . On a smooth surface, the Gaussian curvature K is given by the

540


product of the two principle curvatures at that point, which are the reciprocals of the radii of curvature in two particular orthogonal directions. In fact, there are several equivalent ways of describing the Gaussian curvature using the tools of differential geometry [39]. Many of these definitions require smoothness, i.e., differentiability of the surface at the given point. This contour-based definition is useful because it works well for surfaces containing sharp edges and corners—exactly the surfaces we encounter within mathematical folding. And the relevance to mathematical folding is this: paper (or any other developable surface) has the property that the Gaussian curvature is everywhere zero. This is true on the flat planes of an origami polyhedral surface (obviously); it is true along folded edges; and it is true most especially at folded vertices, which is where we now turn our attention. ?? 8.2.3. Polyhedral Vertices Now we consider specifically polyhedral vertices, i.e., vertices at the junctions of connected flat facets. If we use our “sliding nail” analogy for the construction of the trace, we can see that there are two types of correspondence between the contour around the vertex and its trace: • As the nail slides across a flat facet, its orientation never changes; thus all points on the contour along a single facet map to a single point on the Gaussian sphere, which is located on the sphere at the surface normal to the corresponding facet. • When the nail goes around an edge from one facet to the next, it rotates about the edge; this creates a great circle arc on the Gaussian sphere, connecting the two points that correspond to the two facet normal vectors. This leads to a relatively straightforward method for constructing the trace of a polyhedral vertex: mark the surface normals of all of the facets on the Gaussian sphere; then connect them with great circle arcs in the same order that they are encountered in a counterclockwise tour around the vertex. An example of a real-space contour and its trace on the Gaussian sphere is shown in Figure 8.5. The purple arrows are the surface normals on the

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541

B

c D d C

c b

B

e

b a

A

E

A

C E D

d e

a

Figure 8.5.

Left: a polyhedral vertex with facets A through E and folds a through e. Right: its trace on the Gaussian sphere. Note how arcs on facets of the original vertex become vertices of the trace while points on edges give rise to arcs of the trace.

polyhedral facets and the same set of unit vectors form the vertices of the spherical polygon on the trace. But that is not the only correspondence: we can also identify correspondences between angles. On the real-space vertex, at each folded edge, the surface normal changes by an amount equal to the fold angle of the edge. Thus, on the Gaussian sphere, the great circle arc must have an arc length equal to the fold angle of the polyhedral vertex. Conversely, after the sliding nail has crossed a facet, the next great circle arc will launch in a different direction from its predecessor, and it is readily shown that the change in direction is the sector angle of the facet of the polyhedral vertex. If we embed a unit sphere centered on the corner of the polyhedral vertex, then the intersection of its surface with the polyhedral vertex is a spherical polygon whose sides are the sector angles and whose vertices have exterior angles given by the fold angles of the polyhedral vertex. There is a pleasing duality between this spherical polygon in real space and its trace on the Gaussian sphere; the sides of the real-space spherical polygon (the sector angles) become the exterior angles of the trace spherical polygon, and the sides of the trace spherical polygon become the exterior angles (fold angles) of the real-space spherical polygon. (This duality will become clearer in a moment.) With this correspondence under our belt, we now turn our attention to the simplest polyhedral vertex of a developable surface: the degree-4 vertex.

542


?? 8.2.4. A Degree-4 Vertex It is worth pausing, just for a moment, to consider why the degree4 vertex must be the simplest polyhedral (flat-facet) vertex for a developable surface (like paper). Why not degree-3? As we saw earlier, a degree-3 vertex has a trace that is a spherical triangle, which must have some finite area if its corner angles are nonzero. But a developable surface must have a trace that encloses zero net area. Thus, the trace of any nontrivial folded vertex must include regions of both positive and negative area, i.e., both counterclockwise and clockwise polygons, as in Figure 8.5 (which isn’t developable, but does have a trace with regions of both positive and negative area). Figure 8.5 describes a vertex of degree 5, so it isn’t minimal; but it should be clear that the simplest trace with regions of both positive and negative area would be two triangles connected at a point, as in a bow-tie, formed from four crossing arcs; such would be the trace of a degree-4 vertex. Now let us reintroduce the degree-4 vertex considered in the previous chapter, with sector angles {αi } and fold angles {γi }. Figure 8.6 shows the vertex embedded within a sphere in real space, along with its 2D crease pattern and its trace on the Gaussian sphere. We will assume that it is folded from a developable surface, but we will not assume flat-foldability in what follows. Now, you might wonder: where is the contour in Figure 8.6? And the answer is: it doesn’t matter, as long as it encloses the vertex. You should be able to convince yourself that any path on γ2 γ2

α2

α4

γ3 γ3

α2

α2 γ1

γ4

γ1

α3 α4

γ4

α1

α3

α1

γ2

γ4

α2

−γ3

γ1

α1

α3

α4

Figure 8.6.

Left: the 2D crease pattern for a degree-4 vertex. Middle: the 3D degree-4 vertex in a real-space sphere. Right: its trace on the Gaussian sphere.

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543

the vertex, as long as it encloses the vertex and crosses every edge exactly once, will give exactly the same trace. Because of the duality between the vertex contour and the trace, sector angles on the vertex become angles between arcs of the trace, while fold angles of the vertex become the spherical polygon arcs of the trace. However, we must be careful of fold angle signs, which affect the corner angles on the trace. As we follow each arc of the trace, we turn counterclockwise in the amount of the sector angle. Since in this vertex angle γ3 has a negative fold angle, after we make our turn, the next arc runs backward; the net effect is of making a clockwise turn of π − α2 and then proceeding along an arc length |γ3 |. A similar thing happens when we get to the surface normal of sector α3 . The net effect of γ3 being negative is that while sector angles α1 and α4 are the exterior angles of the left spherical triangle in the trace, positive sector angles α2 and α3 become the interior angles of the other spherical triangle in the trace, as shown on the right in Figure 8.6. Note that in order to form a bow-tie with one region of negative area, it is essential that as one traces out the figure, exactly one of the four arcs must go in the negative direction after making a turn. This means that of the four arcs of the trace, one must be negative and three positive, or vice versa. Since arcs of the trace correspond to fold angles of the vertex, this is equivalent to telling us that of the four creases, there must be three mountains and one valley or vice versa; that is, M −V = ±2. We already knew that this must be true for any flat-foldable vertex (from Girard’s Theorem, or equivalently, the Maekawa-Justin Theorem), but for a degree-4 vertex, it must be the case even if the vertex is not flat-foldable. (It is not necessarily true for vertices of higher degree.) We will, over the coming sections, spend considerable time and analysis talking about the relationships between the vertex in real space and the trace on the Gaussian sphere. As a bit of shorthand, we will refer to the geometric objects, the creases of the vertex and the arcs of the trace, by the angles associated with both: {γi } are the fold angles of the creases and the arc lengths of the corresponding arcs on the trace. Similarly, we will refer to the sector angles of the vertex and the corner angles of the trace by their angles: {αi } are the former on the vertex and the latter on the trace. It should be clear from context to which geometric object (and which space, real or Gaussian sphere) we are referring.

544


??

8.3. Sector and Fold Angles There are two general ways to look at the analysis of a degree-4 vertex (or any other): • We can assume the sector angles are specified plus some measure of the degree of openness and then solve for the fold angles and other parameters of the vertex. • We can assume various other parameters of the vertex and then solve for the sector angles. The first way is the forward problem: given a crease pattern, describe its 3D shape. The second is the inverse problem: given a desired 3D shape, what is the crease pattern that gives rise to it? The first is easier to treat, and so we begin with this descriptive exercise. As already noted, for fixed sector angles, there is only one degree of freedom left; we require only one other parameter to fully specify the fold angles. Possible choices include two quantities from the previous chapter: • the solid angle Ω of the polyhedral vertex; • the arc length ζ between creases γ1 and γ3 . We could also characterize a partially folded degree-4 vertex by parameters of the trace, e.g., • the spherical area of one of the two triangles in the trace; • the angle between the two crossing arcs in the middle of the bow-tie. As we will see, all of these have their uses.

?? 8.3.1. Osculating Plane Every degree-4 vertex from a developable surface will have a fundamentally bow-tie-shaped trace such as in Figure 8.6 (it may, of course, be squashed flat or become vanishingly small as the paper opens flat). A common characteristic of all such traces is the crossing point, labeled q in Figure 8.7. Just as the trace corners

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545

g2

Figure 8.7.

Left: vertex in real space with osculating plane (lavender), which contains creases γ2 and γ4 . Right: trace on the Gaussian sphere with osculating normal (purple).

q

g3

a4

a2

a1

g1

a3 g4

a4

g1

a1

g4

q

q q

g2

a2

-g3

a3

{αi } are the normal vectors to planes in the vertex (specifically, the sector planes), the point q must also be a normal to a plane in real space; and so it is, and we can identify it by the correspondences between the vertex and the trace. Since point q is the intersection of arcs γ2 and γ4 , its associated plane in real space is the plane containing creases γ2 and γ4 , as shown on the left in Figure 8.7. We call this plane the osculating plane of the vertex. A line from the origin to point q on the Gaussian sphere would be perpendicular to this plane, and so we call point q the osculating normal of the vertex. Point q on the trace defines two spherical triangles, which we identify by their corners: 4(q, α4, α1 ) and 4(q, α2, α3 ). These, too, are identified in Figure 8.7, highlighted in yellow and orange. They are, respectively, the traces of two spherical triangles of the vertex, each formed by two sectors of the vertex and a sector of the osculating plane: 4(γ1, γ2, γ4 ) and 4(γ3, γ2, γ4 ). These, too, are highlighted correspondingly in the figure. A key angle of the trace is the angle θ between the two branches of the “bow-tie.” Angle θ is clearly the exterior angle of both triangles 4(q, α4, α1 ) and 4(q, α2, α3 ) at point q, so that the interior angles of both are π − θ. From this relation, we can compute the area of both triangles in terms of its corner angles from the excess angle of both triangles: area(4(q, α4, α1 )) = 2π − (α1 + α4 ) − θ, |area(4(q, α2, α3 ))| = (α2 + α3 ) − θ.

(8.2) (8.3)

(We use | · · · | in Equation (8.3) to make clear that we are talking about unsigned area in this equation.)

546


For a developable vertex, the total area enclosed by the trace must be zero and so the two (signed) triangle areas must be equal and opposite. We define the unsigned area as E, and so we must have that E = 2π − (α1 + α4 ) − θ = (α2 + α3 ) − θ.

(8.4)

If the four sector angles {αi } are specified, then there is only one free parameter required to define all four fold angles; that parameter could be E, or could be θ. They may be taken as measures of the “openness” of the vertex, although, clearly from the equations, they go in opposite directions as the vertex is opened or closed. As Equations (8.2) and (8.3) show, they could be used interchangeably (with appropriate offsets) to characterize a vertex. It is worth examining how these two quantities vary as a vertex moves from its fully open (flat) state to its fully-closed state (or, in the case of a non-flat-foldable vertex, as closed as it can get). Of the two values E and θ, the latter has a more immediate physical representation in real space; Since θ is the exterior angle of a corner of a spherical triangle in the trace, it must be a sector angle in real space. And indeed, angle θ is the sector angle of the segment of the osculating plane contained between creases γ2 and γ4 in real space, and so it provides a clear measure of the “openness” of the vertex. In the fully open (flat) state, the angle between γ4 and γ2 is simply α2 + α3 . In this case the area E goes to zero. (That E must vanish follows also from the observation that in a flat vertex, all of the sector normals are aligned and so the vertices of the trace must overlap.) In the open state, however, the situation is a bit more complicated, as we will see. ?? 8.3.2. Binding Condition As we know, if a vertex satisfies the Kawasaki-Justin Condition, it is flat-foldable for some crease assignment. If it doesn’t, then there is a limit to how far the vertex can be closed up with all creases folded; that limit is reached when the paper reaches a point that further closure would cause self-intersection or, for a degree-4 vertex, when one or more of the fold angles reaches π. This situation can arise in two different ways, illustrated in Figure 8.8. For the degree-4 vertex of our current discussion, as it is closed up, either fold angle γ2 or γ4 will reach an angle of π (fully folded).

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547

g2

a2

g3 a1

a3

Figure 8.8.

Left: a binding degree-4 vertex with αK > 0. Right: a binding degree-4 vertex with αK < 0.

g2

g4 g4

g1

a4

a1

a2 a3

a4

g3

g1

We call this state the binding state for the polyhedral vertex and call the crease whose fold angle is π the binding crease. Which fold angle reaches this condition first depends on the values of the sector angles. Imagine cutting the vertex along fold angle γ1 and then flattening it out, as shown in Figure 8.9. If γ2 had been the binding crease (γ2 = π), then the right side of sector α1 lies to the right of its mating point on sector α4 and fold angle γ1 , as shown on the left in Figure 8.9. On the other hand, if γ4 had been the binding crease, the right side of sector α1 would lie to the left of its mating point in the flattened figure. (And of course, if the vertex is flat-foldable, the two points are perfectly aligned.) The separation between the two ends is given by the alternating sum of the sector angles. We define αK ≡ α1 − α2 + α3 − α4 .

(8.5)

We call αK the Kawasaki excess (see Chapter 4), naming it for Kawasaki for the obvious reason: we are closing in on the Kawasaki-Justin Theorem again. Clearly, γ2 binds first if αK > 0, while γ4 binds first if αK < 0. And if αK = 0, then both creases reach fold angles of π simultaneously and the vertex is

g2

a2 g4 a3

g3

a1 a4

g1

Figure 8.9.

g4

a2 g2 a3

g3

a1 a4

Left: a cut and flattened degree-4 vertex for which γ2 binds first. Right: a cut and flattened degree-4 vertex for which γ4 binds first.

548


g1

flat-foldable, which is, of course, KJT. Thus: αK > 0 ⇐⇒ γ2 binds first, αK = 0 ⇐⇒ the vertex is flat-foldable, αK < 0 ⇐⇒ γ4 binds first.

(8.6)

?? 8.3.3. Ruling Plane There is one more plane of interest here. Just as the osculating plane contained creases γ2 and γ4 , we can also identify the plane that contains creases γ1 and γ3 , which we call the ruling plane of the vertex. This plane is represented on the Gaussian sphere by its normal vector z, which is not part of the trace—at least, not so far. However, if we extend arcs γ1 and γ3 to form great circles, then z lies on the intersection of the great circles (the same way that q lay on the intersection of arcs γ2 and γ4 . Of course, there are two such intersection points; it is only a matter of convention which one we choose to represent the ruling plane. If we assume that the surface is right-handed rotating from γ1 toward γ2 , then the point z should be the lower one in Figure 8.10. The ruling plane divides the vertex up into two spherical triangles, 4(γ1, γ2, γ3 ) and 4(γ1, γ3, γ4 ), with sides (α1, α2, ζ) and (ζ, α3, α4 ), respectively. These two triangles have spherical triangles as their traces on the Gaussian sphere, and by identifying the correspondences between sector angles and fold angles in the vertex and corner angles and arc lengths in the Gaussian sphere, we can see that newly-created sector angle ζ shows up as a corner angle at point z on the Gaussian sphere, specifically, as the exterior angle of the corner corresponding to its arc on the vertex. g2

g3

Figure 8.10.

a4

a2 z

a1 g1

a3

g4

q q

g4 a4

a1

g1 z

a3

g2 -g3

a2

Left: vertex in real space with ruling plane (lavender), which contains creases γ1 and γ3 . Right: trace on the Gaussian sphere with ruling normal (purple).

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549

So, the trace for a developable degree-4 vertex will always be some variant of a bow-tie, but it can be rather distorted, depending on the sector angles, the degree of folding, and how far it is wrapped around the Gaussian sphere. Figure 8.11 shows several examples of vertices in different levels of openness and for different sets of sector angles along with their corresponding traces. At this point, our geometric figures are getting wrapped rather far around the sphere, and so to make them easier to see, we will distort and unwrap them as we did in the previous chapter. Figure 8.12 shows the spherical polyhedra identified so far in both the vertex and the Gaussian sphere. We are now in a position to work out the properties of a degree4 vertex and the relationships between them. For some properties, it will be easier to work in real space; for others, to work on the Gaussian sphere; and for some, to switch back and forth. ?? 8.3.4. Real Space Solid Angle Let us now look at the solid angle in real space subtended by the polyhedral vertex. We begin by determining its range. In the unfolded state, the answer is simple; the flat sheet bisects the unit sphere and the area enclosed is half the sphere. Thus, the maximum solid angle subtended by the vertex is Ωmax = 2π.

(8.7)

At the other end of the range, as the vertex is closed, the solid angle decreases until it reaches some minimum value Ωmin . For a flat-foldable vertex, that minimum value is simply zero. For a non-flat-foldable vertex, however, the minimum solid angle is reached when one of the two angles γ2 or γ4 binds. Consider first the case when αK > 0 (the vertex on the left in Figure 8.8). In this case, the solid angle of the vertex is the solid angle of a triangle with sides (α1 −α2, α3, α4 ). Applying L’Huilier’s Theorem [122] from the previous chapter (Equation (7.9)), the area of this triangle is given by h α − α − α − α Ωmin α1 − α2 + α3 − α4 1 2 3 4 tan = tan tan 4 4 4 α − α + α + α −α + α + α − α i 1/2 1 2 3 4 1 2 3 4 · tan tan . 4 4

550


(8.8)

a4 a1

g2

g3

a2

a2

a3

a1

a3 g1

a4

g4

g3

g2

a1

g3

a4

g1

a2

a2

g2

g4

g1 -g3

a1

g4

a3

g2 a2

a4

a1

a2

g1 g3

a4

g4

-g3 g1

g2 a1 a3

Figure 8.11. a4

g2

a2

a1

a3 g4

a2

g3

-g3

g1

g4 g2

g1

a4

a3 a1 a4

g2

g4

a1

a2 a3 g3

a4

g1

a2

g4

g1

g2 a1

-g3 a3

Left: polyhedral vertices on the Gaussian sphere. Right: their corresponding traces. Top row: a nearly flat fully opened vertex. Second row: a nearly closed vertex with a small anto sector angle. Third row: a nearly closed vertex with a large anto sector angle. Fourth row: a vertex binding at γ2 . Bottom row: a vertex binding at γ4 .


........

551

g2

a2 q

Figure 8.12.

Left: schematic view of a polyhedra vertex. Right: schematic view of its trace.

z

p-g3 a3

a4

a1

p-g2

-g3

p-g4

g1

p-a4

g4 p-q

g1

g2

q q q

p-q

p-a1

-g3 a3

a1

p-a3

a4

g4

a2

p-z

z z

p-z

z

Using the definition of αK and the fact that the sector angles sum to 2π, this simplifies to

h i 1/2 Ωmin tan = − cot 12 αK cot 12 α1 cot 12 α2 cot 12 (α2 + α3 ) 4 h i 1/2 = − cot 12 αK cot 12 α1 cot 12 α2 cot 12 θ .

(8.9)

The equivalent formulas for αK < 0 are the same with indices swapped, 1 ↔ 4, 2 ↔ 3, and the sign of αK reversed. Before moving on to intermediate cases, it is worth considering: what is the largest possible value of the minimum solid angle in the binding condition, and what sector angles give rise to this? In terms of the area of the spherical triangle, for αK > 0, clearly, all that sector α2 does is “take away angle” from sector α1 , so, to maximize the solid angle, we should set α2 = 0. In which case, the question becomes that of how best to partition 2π of vertex angle among the remaining three sectors so as to maximize the area of the resulting spherical triangle. As is the case for planar triangles, the maximum area is obtained for constant perimeter when the three sides are equal. Therefore, the maximum area is found when π α1 = α3 = α4 = , α2 = 0, (8.10) 2 and this, in turn, makes the triangle equal to one octant of the sphere; thus, π max {Ωmin } = . (8.11) 2 Now, for an intermediate case, that is, Ω ∈ (Ωmin, Ωmax ), we have several potential ways of computing Ω. First, there is Girard’s

552


Theorem: Ω = 2π + (γ1 + γ2 + γ3 + γ4 ) .

(8.12)

To use this, we’d need the fold angles. We could also compute Ω by adding the areas of the two spherical triangles that make up the vertex: Ω = area (4(γ1, γ2, γ3 )) + area (4(γ1, γ3, γ4 )) .

(8.13)

For each of these triangles, we can use L’Huilier’s Theorem, which gives 1 1 −1 area (4(γ1, γ2, γ3 )) = 4 tan tan (ζ + α1 + α2 ) tan (−ζ + α1 + α2 ) 4 4 1/2 1 1 · tan (ζ − α1 + α2 ) tan (ζ + α1 − α2 ) 4 4

(8.14)

and area (4(γ1, γ3, γ4 )) = 4 tan

−1

1 1 tan (ζ + α3 + α4 ) tan (−ζ + α3 + α4 ) 4 4 1/2 1 1 · tan (ζ − α3 + α4 ) tan (ζ + α3 − α4 ) . 4 4

(8.15)

These three equations give an expression for Ω in terms of just the sector angles and the arc ζ, which we will discuss in a moment, and can be taken as a measure of the openness of the vertex. We could also compute Ω by subtracting the area of two spherical triangles: Ω = area (4(γ1, γ2, γ4 )) − area (4(γ3, γ2, γ4 )) .

(8.16)

Again, L’Huilier’s Theorem supplies the two areas: 1 1 −1 area (4(γ1, γ2, γ4 )) = 4 tan tan (θ + α1 + α4 ) tan (−θ + α1 + α4 ) 4 4 1/2 1 1 · tan (θ − α1 + α4 ) tan (θ + α1 − α4 ) 4 4

........


(8.17)

553

and area (4(γ3, γ2, γ4 )) = 4 tan

−1

1 1 tan (θ + α2 + α3 ) tan (−θ + α2 + α3 ) 4 4 1/2 1 1 · tan (θ − α2 + α3 ) tan (θ + α2 − α3 ) . 4 4

(8.18)

Here, we have an expression for Ω in terms of just the sector angles and the arc θ; this, too, can be taken as a measure of the openness of the vertex. Both θ and ζ can be used to characterize the openness of the vertex; it is worth looking at them both a bit more closely. ?? 8.3.5. Ruling Angle Angle ζ in Figure 8.12 has some physical significance of its own. It is the angle between the two minor creases of the vertex (γ1 and γ3 ); we met it back in Chapter 2 (see Section 2.4.3). It can be used as a measure of the openness of the vertex, and its range is fairly easily identified. When the paper is fully flat, we have that ζmax = min(α1 + α2, α3 + α4 ),

(8.19)

where we have taken the minimum value to enforce continuity with the limit for a nearly-flat sheet. The minimum value occurs at binding and depends on which side binds first (which, recall, depends on the Kawasaki angle αK ). For αK > 0, examination of Figure 8.8 shows that ζmin = α1 − α2,

(8.20)

ζmin = α4 − α3 .

(8.21)

while for αK < 0, These can be combined simply, giving ζmin = max(α1 − α2, α4 − α3 ).

(8.22)

So, if we take the ruling angle as a measure of openness, we can derive two of the fold angles very simply by applying the haversine formula, Equation (7.15), to the two triangles that lie on either side of the ζ arc.

554


First, working with the upper triangle in Figure 8.12, we have that havζ = hav(α1 − α2 ) + sin α1 sin α2 hav(π − γ2 ),

(8.23)

or equivalently, using Equation (7.17), havζ = hav(α1 + α2 ) − sin α1 sin α2 havγ2 .

(8.24)

Using the substitution hav x → 12 (1 − cos x), we can find that cos ζ = cos α1 cos α2 − sin α1 sin α2 cos γ2,

(8.25)

and thus, ζ = cos−1 [cos α1 cos α2 − sin α1 sin α2 cos γ2 ]

(8.26)

Note that there are, in general, two solutions to Equation (8.26), one in [0, π] and the other in [π, 2π]. For a valley-like vertex (γ2 > 0), the first root applies; for a mountain-like vertex (γ2 < 0), the second applies. The analysis is the same for the lower triangle in Figure 8.12 with the substitutions α1 → α3, α2 → α4, γ2 → γ4, so that

or

havζ = hav(α3 − α4 ) + sin α3 sin α4 hav(π − γ4 ),

(8.27)

havζ = hav(α3 + α4 ) − sin α3 sin α4 havγ4,

(8.28)

which gives cos ζ = cos α3 cos α4 − sin α3 sin α4 cos γ4 .

(8.29)

Note that we can eliminate ζ by combining Equations (8.24) and (8.28) to get a relation between the fold angles directly: hav(α1 + α2 ) − sin α1 sin α2 havγ2 = hav(α3 + α4 ) − sin α3 sin α4 havγ4 .

(8.30)

For a developable vertex, though, hav(α1 +α2 ) = hav(α3 +α4 ) (see Equation (7.14)), so that sin α1 sin α2 hav(γ2 ) = sin α3 sin α4 hav(γ4 ),

(8.31)

........


555

or sin2

sin2

1 2 γ2 1 2 γ4

=

sin α3 sin α4 . sin α1 sin α2

(8.32)

Equation (8.32) was one of two elegant identities first derived by Huffman [47], and I will call this one Huffman’s Major-Angle Identity. (We will encounter the other shortly.) This is the generalization of the relation derived in the previous chapter for the major creases of a flat-foldable degree-4 vertex, which told us that the major crease angles are equal. In general, they are not, and Equation (8.32) applies. While Equation (8.32) is the most elegant form of this relationship, we can also express it in terms of cotangents of the half-angles, which will presently come in useful: cot2 ( 12 γ2 ) = csc2 ( 12 γ4 )

sin α1 sin α2 sin α1 sin α2 − 1 = (1 + cot2 ( 12 γ4 )) − 1, sin α3 sin α4 sin α3 sin α4

(8.33)

cot2 ( 12 γ4 ) = csc2 ( 12 γ2 )

sin α3 sin α4 sin α3 sin α4 − 1 = (1 + cot2 ( 12 γ2 )) − 1. sin α1 sin α2 sin α1 sin α2

(8.34)

If a vertex is flat-foldable, then the right side of Equation (8.32) is unity. Is the converse true? That is, if we have γ2 = γ4 , does that mean that the vertex is flat-foldable? We subtract the denominator of the right side of Equation (8.32) from the numerator and simplify, taking the assumption of developability—that the four sector angles sum to 2π—so that α4 = 2π − (α1 + α2 + α3 ). We find that sin α3 sin α4 − sin α1 sin α2 = − sin(α1 + α3 ) sin(α2 + α3 ). (8.35) This expression goes to zero if either α1 + α3 = π, which we recognize as the flat-foldability condition, or α2 + α3 = π. In this latter case, the pattern is not necessarily flat-foldable, but the two major fold lines are collinear. We therefore call this the straight-major condition. So: Theorem 25 (Major Angle Equality Theorem). If the major fold angles of a degree-4 vertex are equal, then the vertex is flat-foldable or straight-major or both. As we will see, a straight-major vertex is rather uninteresting when it comes to considering what the minor creases might be doing.

556


?? 8.3.6. Osculating Angle Angle θ in Figure 8.12 has some physical significance of its own. It is the angle between the two major creases of the vertex (γ2 and γ4 ). We met this, too, back in Chapter 2. It, like the ruling angle, can be used as a measure of the openness of the vertex. When the paper is fully flat, we have that θ max = α2 + α3 .

(8.36)

The minimum value θ min depends on the binding configuration, and it is a bit harder to analyze, so we will defer it for now. As we did above with the ruling angle, we can relate two of the fold angles rather simply to the osculating angle. Working with the two spherical triangles that share arc θ, we have, first, from the exterior-angle version of the haversine formula, Equation (7.17), that havθ = hav(α1 + α4 ) − sin α1 sin α4 havγ1 .

(8.37)

Using the substitution hav x → 12 (1 − cos x), we get cos θ = cos α1 cos α4 − sin α1 sin α4 cos γ1,

(8.38)

θ = cos−1 [cos α1 cos α4 − sin α1 sin α4 cos γ1 ] .

(8.39)

so that

For the other triangle, we find that havθ = hav(α2 + α3 ) − sin α2 sin α3 havγ3,

(8.40)

and thus, cos θ = cos α2 cos α3 − sin α2 sin α3 cos γ3 .

(8.41)

Similarly to what we did in the previous subsection, we can eliminate θ by combining Equations (8.37) and (8.40) to get a relation between the fold angles directly: hav(α1 + α4 ) − sin α1 sin α4 hav(γ1 ) = hav(α2 + α3 ) − sin α2 sin α3 hav(γ3 ).

(8.42)

For a developable vertex, as before, hav(α1 + α4 ) = hav(α2 + α3 ) (see Equation (7.14)), so that sin α1 sin α4 hav(γ1 ) = sin α2 sin α3 hav(γ3 ),

(8.43)

........


557

or sin2

sin2

1 2 γ1 1 2 γ3

=


(8.44)

Equation (8.44) is another identity first derived by Huffman [47], and I will call this one Huffman’s Minor-Angle Identity. This is the generalization of the relation derived in the previous chapter for the minor creases of a flat-foldable degree-4 vertex, which told us that the minor crease angles are equal and opposite. In general, they are not, and Equation (8.44) describes their relationship. As with the Major-Angle Identity, we can write this in terms of the half-angle cotangents: cot2 ( 12 γ1 ) = csc2 ( 12 γ3 )

sin α1 sin α4 sin α1 sin α4 − 1 = (1 + cot2 ( 12 γ3 )) − 1, sin α2 sin α3 sin α2 sin α3

(8.45)

cot2 ( 12 γ3 ) = csc2 ( 12 γ1 )

sin α2 sin α3 sin α2 sin α3 − 1 = (1 + cot2 ( 12 γ1 )) − 1. sin α1 sin α4 sin α1 sin α4

(8.46)

As we did with the major-angle identity, we can ask under what conditions the right side of Equation (8.44) goes to unity and the minor angles are equal and opposite. Subtracting the denominator from the numerator, we find that sin α2 sin α3 − sin α1 sin α4 = − sin(α1 + α2 ) sin(α1 + α3 ). (8.47) This goes to zero if and only if either α1 + α3 = π, which is, again, the flat-foldability condition, or α1 +α2 = π, in which case the two minor creases form a straight line. We call this, naturally enough, the straight-minor condition and thus, can say Theorem 26 (Minor Angle Equality Theorem). If the minor fold angles of a degree-4 vertex are equal, then the vertex is flat-foldable or straight-minor or both. ?? 8.3.7. Adjacent Fold Angles For flat-foldable vertices, the relationship between adjacent fold angles was surprisingly simple and elegant, but that made use of the considerable simplifications afforded by the condition of flatfoldability. For the case of a general vertex, that relationship is more complex. Huffman [47] gives a relationship between adjacent fold angles that is rather forbidding (and which he describes

558


a4 g1

p-a4 E

p-a1 a1

F

a2 g2

g4

E

Figure 8.13.

-g3

Trace of a degree-4 vertex. E, F, and G denote areas of spherical triangles.

a3

G

as finding after “a difficult derivation”): v u   u u sin α3 sin α4 r 1 2  u t1 − sin α1 sin α2 sin 2 γ4   1 ∓ sin α1 sin α3    sin α sin α 1 2 4 2   1 − sin γ 4 2     v u   u u sin α2 sin α3 r 2 1γ   u t 1 − sin  sin α1 sin α4 2 3  sin α sin α sin α sin α 1 3 1 3  = 1− · 1 ± . (8.48)  sin α4 sin α2 sin α sin α 1 2 2 4   1 − sin 2 γ3     We can, however, find a relatively straightforward relationship between any fold angle and that of one or the other (or both) of its neighbors. We begin again on the Gaussian sphere, as illustrated in Figure 8.13, where we have added two great-circle arcs connecting point pairs (α4, α2 ) and (α1, α3 ). We now consider the four triangles 4(α4, α1, α2 ), 4(α4, α3, α2 ), 4(α1, α4, α3 ), and 4(α1, α2, α3 ). Because the first two share the triangle F and the two halves of the bow-tie of the trace have equal area, the first pair of triangles have equal area, as do the second pair: area(4(α4, α1, α2 )) = area(4(α4, α3, α2 )) = E + F, area(4(α1, α4, α3 )) = area(4(α1, α2, α3 )) = E + G.

(8.49) (8.50)

And so, using the cotangent formula for triangle areas from Equation (7.11), we can establish an equality for each pair of triangles: cot( 12 γ1 ) cot( 12 γ2 ) csc(π − α1 ) + cot(π − α1 ) = cot( 12 γ4 ) cot(− 12 γ3 ) csc α3 + cot α3, (8.51) cot( 12 γ1 ) cot( 12 γ4 ) csc(π − α4 ) + cot(π − α4 ) = cot( 12 γ2 ) cot(− 12 γ3 ) csc α2 + cot α2 . (8.52)

........


559

Now, we can eliminate one of the half-angle cotangents from these two equations and solve for any one of the four in terms of the two remaining. For example, cot( 12 γ2 )

=

cot( 12 γ3 )(cot α2 + cot α4 ) csc α3 − cot( 12 γ1 )(cot α1 + cot α3 ) csc α4 cot2 ( 12 γ3 ) csc α2 csc α3 − cot2 ( 12 γ1 ) csc α1 csc α4

.

(8.53)

But this can be simplified; by using Huffman’s Minor-Angle Identity (Equation (8.44)), the denominator simplifies to cot2 ( 12 γ3 ) csc α2 csc α3 − cot2 ( 12 γ1 ) csc α1 csc α4 = csc α1 csc α4 − csc α2 csc α3 . (8.54) Applying this and corresponding relations for all four halfangle cotangents, we find linear relationships among the four: cot( 12 γ1 ) =

cot( 12 γ2 )(cot α1 + cot α3 ) csc α2 − cot( 12 γ4 )(cot α2 + cot α4 ) csc α3 , csc α3 csc α4 − csc α1 csc α2

(8.55)

cot( 21 γ2 ) =

cot( 12 γ3 )(cot α2 + cot α4 ) csc α3 − cot( 12 γ1 )(cot α1 + cot α3 ) csc α4 , csc α1 csc α4 − csc α2 csc α3

(8.56)

cot( 21 γ3 )

cot( 12 γ2 )(cot α2 + cot α4 ) csc α1 − cot( 12 γ4 )(cot α1 + cot α3 ) csc α4 = , csc α3 csc α4 − csc α1 csc α2

(8.57)

cot( 12 γ4 )

cot( 12 γ3 )(cot α1 + cot α3 ) csc α2 − cot( 12 γ1 )(cot α2 + cot α4 ) csc α1 = . csc α1 csc α4 − csc α2 csc α3

(8.58)

We have four linear equations in four variables (each variable being cot( 12 γi )), and so it is tempting to simply try to solve the linear system of four equations. This is, however, an ill-determined system; we need to bring in another equation to find one-to-one relations truly between the half-angle cotangents. We can obtain this additional information from the cotangent forms of the two Huffman opposite-angle identities in the two preceding subsections, which provide expressions for each the terms cot( 12 γi ) in terms of its opposite number. Recall that for the two pairs of opposite angles, one pair must have the same sign and the other must have the opposite sign. When we take square roots of Equations (8.33) and (8.45), we must choose signs that respect this convention. We can denote the possible choices by introducing operations ±i , where the sign of each is chosen to match the desired crease assignment. Then we

560


have that cot( 12 γ2 ) cot( 12 γ4 ) cot( 12 γ1 )

r

= ±2 (1 + cot2 ( 12 γ4 )) r

= ±4 (1 + cot2 ( 12 γ2 )) r

= ±1 (1 + cot2 ( 12 γ3 )) r

cot( 12 γ3 ) = ±3 (1 + cot2 ( 12 γ1 ))

sin α1 sin α2 − 1, sin α3 sin α4

(8.59)


(8.60)


(8.61)


(8.62)

and each of these can be substituted into the proceeding four equations to give a one-to-one relationship between any fold angle and either of its immediately adjacent angles. ?? 8.3.8. Flat-Foldable and Straight-Major/Minor Vertices In the previous chapter, we analyzed the relationship between adjacent fold angles for flat-foldable vertices. We should be able to find an equivalent expression using this analysis. Recall that for a flat-foldable vertex, γ2 = γ4 , γ1 = −γ3 , and opposite sector angles sum to π, that is, α1 + α3 = π, α2 + α4 = π. In principle, we could substitute these relationships into Equations (8.55)–(8.58), but the expressions become undetermined (zero in both numerator and denominator). Instead, we make the substitution α3 = π − α1 + , α4 = π − α2 + ,

(8.63)

and then take the limit as → 0 using L’Hôpital’s rule. This gives the beautiful result that we have already seen in Equation (7.58): tan 12 γ1 tan 12 γ2

=

sin 12 (α1 − α2 ) sin 12 (α1 + α2 )

.

(8.64)

Recall that this equation implied an inequality: |γ2,4 | ≥ |γ1,3 |, with equality only at either flat state. For a general vertex, the relationship between adjacent fold angles is more complex, and there is not such a strict inequality between individual pairs of angles. This can be seen from the example in Figure 8.14, which has sector angles α1 = 150◦ , α2 = 30◦ , α3 = 90◦ , and α4 = 90◦ ;

........


561

g2 g2

g3 a2

g4

a1 a2

g3

g1 a3

g1

a3

a4

a2

a4

a1

g2

-g3

a3

g1 a1

a4

g4 g4

Figure 8.14.

Left: a non-flat-foldable vertex. Middle: the 3D configuration for γ1 = 90. Right: its trace.

for γ1 = 90◦ , the other three fold angles are γ2 = 180◦ (larger), γ3 = −90◦ , and γ4 = 60◦ (smaller). Clearly, in this case, both minor angles (γ1,3 ) are not smaller than both major angles (γ2,4 ). However, we can identify a similar inequality between sums of angles. We divide fold angles γ2 and γ4 into two parts each, as shown in Figure 8.15. γ2 ≡ γ2a + γ2b, γ4 ≡ γ4a + γ4b .

(8.65)

Because the surface of the Gaussian sphere is a metric space, the triangle inequality holds, so that |γ1 | ≤ |γ2a | + |γ4a |, |γ3 | ≤ |γ2b | + |γ4b |.

(8.66)

Summing these two inequalities gives the following: |γ1 | + |γ3 | ≤ |γ2 | + |γ4 |,

(8.67)

with, again, strict inequality holding at all configurations except for the fully open and fully closed states, where the two sides are indeed equal. a4

Figure 8.15. Schematic of a degree-4 vertex. We divide fold angles γ2 and γ4 into two parts.

562

p-a4

g1

g4a

g2a p-a1 a1


g2b

a2 -g3

g4b a3

There is another special case for which Equations (8.55)– (8.58) become undetermined, which is when α1 +α2 = π, α3 +α4 = π. We already saw this: this is the straight-minor configuration, in which the two minor creases are collinear. Figure 8.14 is, in fact, a straight-minor configuration since γ1 and γ3 are collinear. We can analyze this configuration in the same way as the flat-foldable case. We substitute α2 = π − α1 + ,

α4 = π − α3 + ,

(8.68)

and again take the limit as → 0 using L’Hôpital’s rule. The results are not quite as simple as Equation (8.64) but are still tractable; the results are summarized in the next subsection. There is yet one more configuration to consider: the straightmajor configuration, when the two major creases are collinear and α1 + α4 = π, α2 + α3 = π. In this case, the two opposing pairs of fold angles are not monotonically coupled, and, as we saw in Chapter 7, the solution has two parts. If α2 = α3 , then γ2 = γ4 ∈ (−π, π) γ2 = γ4 = ±π

and or and

γ1 = γ3 = 0, γ1 = −γ3 ∈ [−π, π].

(8.69)

In this type of straight-major vertex, the two minor creases come into alignment as the major creases are folded, and once the vertex is folded in half, the two minor creases can fold together. If, however, we have a straight-major configuration with α2 , α3 , then the two minor creases never come into alignment and cannot be folded at all. There is only one solution: γ2 = γ4 ∈ [−π, π] and

γ1 = γ3 = 0.

(8.70)

?? 8.3.9. Sector Angle/Fold Angle Relations Via the various relations derived thus far, it is possible from the four sector angles and any one fold angle to compute the remaining three fold angles. We must know (or decide), though, whether the initial fold angle is part of a minor or major pair. Consider first a minor pair, where γ1 is the known angle. Then, from Huffman’s Minor-Angle Identity (Equation (8.44)), the other minor angle is given by " # r sin α sin α 1 4 γ3 = −2 sin−1 sin( 12 γ1 ) , (8.71) sin α2 sin α3 which is valid for either sign of γ1 .

........


563

The two major angles are then given by "

# cot( 12 γ3 )(cot α2 + cot α4 ) csc α3 − cot( 12 γ1 )(cot α1 + cot α3 ) csc α2 , (8.72) csc α1 csc α4 − csc α2 csc α3

"

# cot( 12 γ3 )(cot α1 + cot α3 ) csc α2 − cot( 12 γ1 )(cot α2 + cot α4 ) csc α1 . (8.73) csc α1 csc α4 − csc α2 csc α3

γ2 = 2 cot−1

γ4 = 2 cot−1

These two expressions give the proper signs for γ2 and γ4 for all possible sets of valid sector angles and given angle γ1 , except for the special cases where the two expressions are undetermined: flat-foldable, straight-minor, and straight-major. For the flat-foldable special case, where α1 +α3 = π, α2 +α4 = π, we have γ3 = −γ1, (8.74) " # 1 sin 2 (α1 − α2 ) γ2 = γ4 = 2 cot−1 cot( 12 γ1 ) = 2 cot−1 µ−1 cot( 12 γ1 ) , (8.75) 1 sin 2 (α1 + α2 ) where we have included the fold angle multiplier µ that we introduced in the previous chapter. These equations, too, give the proper sign for all possible sets of sector angles and γ1 . For the straight-minor special case, where α1 + α2 = π, α3 + α4 = π, we have γ3 = −γ1, (8.76) " # 1 1 (1 − cot2 ( 2 γ1 )) cot α1 + (1 + cot2 ( 2 γ1 )) cot α3 γ2 = 2 cot−1 , (8.77) 2 cot( 12 γ1 ) csc α1 " # 2 ( 1 γ )) cot α + (1 + cot2 ( 1 γ )) cot α (1 − cot 1 3 1 1 2 2 γ4 = −2 cot−1 . (8.78) 1 2 cot( 2 γ1 ) csc α3 And finally, for the straight-major special case, where α1 +α4 = π, α2 + α3 = π, we have γ3 = −γ1, γ2 = γ4 =

±π unspecified

(8.79) if γ1 , 0, if γ1 = 0.

(8.80)

Instead of being given a minor fold angle, we might instead be given a major fold angle, e.g., γ4 . From this we can calculate the other three fold angles. For the general case, the other major angle

564


comes from Huffman’s Major-Angle Identity (Equation (8.32)), given by " # r sin α sin α 3 4 γ2 = 2 sin−1 sin( 12 γ4 ) . (8.81) sin α1 sin α2 The two minor angles are then given by " # 1 1 cot( γ )(cot α + cot α ) csc α − cot( γ )(cot α + cot α ) csc α 2 1 3 2 4 2 4 3 2 2 γ1 = 2 cot−1 , (8.82) csc α3 csc α4 − csc α1 csc α2 " γ3 = 2 cot−1

# cot( 12 γ2 )(cot α2 + cot α4 ) csc α1 − cot( 12 γ4 )(cot α1 + cot α3 ) csc α4 . (8.83) csc α3 csc α4 − csc α1 csc α2

For the flat-foldable special case, where α1 +α3 = π, α2 +α4 = π, we have γ2 = γ4, (8.84) " # sin 12 (α1 + α2 ) −1 −1 1 1 γ1 = −γ3 = 2 cot cot( 2 γ4 ) = 2 cot µ cot( γ ) . 4 2 sin 12 (α1 − α2 )

(8.85)

For the straight-minor special case, where α1 + α2 = π, α3 + α4 = π, we have sin α3 −1 1 γ2 = 2 sin sin( 2 γ4 ) , (8.86) sin α1    cot( 12 γ2 ) csc α1 + cot( 12 γ4 ) csc α4 (cot α1 + cot α3 )    γ1 = −γ3 = 2 cot−1  . 2 2   csc α3 − csc α1    

(8.87)

And last, for the straight-major special case, where α1 +α4 = π, α2 + α3 = π, we have γ2 = γ4, (8.88) 0 if γ4 , 0, γ1 = −γ3 = (8.89) unspecified if γ4 = ±π. Examples of each of these are shown in Figure 8.16. The formulas above have somewhat broader applicability than they might appear. In general in this section, we have assumed that γ1 , γ2 , and γ4 all have the same sign and γ3 has the opposite

........


565

γ2

γ3 α2

γ3

α1

α3

γ2

γ1

α4

γ1

γ4

γ4 γ2

γ2 α1

α2

α3

Figure 8.16. Four degree-4 vertices as crease patterns and folded forms. In all cases γ1 = 60◦ . Left: crease pattern. Right: folded form. Top row: a general vertex with sector angles (120◦, 50◦, 70◦, 120◦ ). Second row: a straight-major vertex with sector angles (120◦, 120◦, 60◦, 60◦ ). Third row: a straight-minor vertex with sector angles (120◦, 60◦, 90◦, 90◦ ). Bottom row: a flat-foldable vertex with sector angles (120◦, 90◦, 60◦, 90◦ ).

566

γ1

α4

γ3

γ3

γ1

γ4

γ4 γ2

γ2

γ3 α2

γ3

α1

α3 α4

γ1 γ1

γ4 γ4 γ2

γ2

α1

α2 γ3

α3

α4

γ3

γ1

γ4


γ4

γ1

sign. However, it turns out that the formulas above are valid no matter which of γ1 and γ3 matches the sign of the other two. What is still necessary is that the major versus minor pair be respected: {γ1, γ3 } must be the minor pair (opposite sign), while {γ2, γ4 } must be the major pair (same sign). That is not to say that there are solutions for all possible input angles, though. The presence of sin−1 (x) in some of the formulas is a tip-off, because there is no real solution for values of its argument with magnitude greater than 1. This situation can arise in any non-flat-foldable vertex configuration, such as the one shown in Figure 8.14. Earlier in this chapter, we introduced the notion of the binding condition, which imposes a limit on the range of motion due to self-intersection of the paper. The limitation is stronger than mere self-intersection, though; even if we were to allow ghost paper that could pass through itself, there are still limits on the reachable fold angles that are imposed by the values of the sector angles. These limits are embodied in the fold angle relations above. Let’s go back to the relationship between minor angles expressed by Equation (8.71). There is no real solution for γ3 when the argument of sin−1 is larger than 1; this sets an upper limit on the value of γ1 , which is r sin α2 sin α3 −1 γ1,max = 2 sin . (8.90) sin α1 sin α4 Similarly, there is an upper limit on the magnitude of γ3 , which is r sin α1 sin α4 −γ3,max = 2 sin−1 . (8.91) sin α2 sin α3 These limits define the binding condition on the minor fold angles. Since the terms inside the sin−1 are the inverse of one another, in general, either γ1 or γ3 will reach some maximum angle, unless that term is exactly 1—in which case, as we have seen, the vertex is flat-foldable or straight-minor. In the same way, from the relationship between major angles expression by Equation (8.81), we can see that there is no real solution for γ2 when the argument of sin−1 is larger than 1. This sets an upper limit on the value of γ4 , which is r sin α1 sin α2 γ4,max = 2 sin−1 . (8.92) sin α3 sin α4

........


567

Similarly, there is an upper limit on the magnitude of γ2 , which is r sin α3 sin α4 −γ2,max = 2 sin−1 . (8.93) sin α1 sin α2 And again, one or the other will reach its limit, unless the term under the square root is exactly 1, in which case the vertex is either flat-foldable or straight-major. ??

8.4. More Angles and Planes The sector angles {αi } and fold angles {γ j } are coupled, and if we know enough of them, we can solve for the remaining angles. If we start with the four sector angles, then we are effectively starting with the crease pattern, and when we solve for the four fold angles, we are solving for the folded form. We can, however, address the inverse problem: suppose we wish to specify the folded form. What is the crease pattern that gives rise to a desired 3D shape? The simplest version of this problem is still a single vertex, but now we will put the vertex in a new light: instead of taking the sector angles as a given, we will specify other things and solve for the sector angles. Consider the system shown in Figure 8.17: a strip of paper, folded into a V shape in such a way that it makes a long trough with a single turn along it. How might we specify such a shape, and given that specification, what are the sector angles of the creases at the single vertex that give rise to this shape? g1 g2

E

s4

s1

q q

q

g3

a3

a1

a4

a2 a4

s3

q a1

s2

g1

E a3

a2 -g3

g4

Figure 8.17.

Left: two creases with a vertex between them. Right: trace of the vertex. The highlighted fold directions and sector elevation angles are specified.

568


One way of describing this shape would be to specify the locations of all of the points along the base of the V. A sequence of such points would form a polyhedral space curve; if we wanted to fold a given 3D shape, we might very well wish to describe the space curve of the folded edge. And so we would give the positions of the three points along the bottom of the fold, as well as the fold angles themselves; these would define the two creases γ4 and γ2 of the vertex at the position of the bend. We would like to perform a full analysis of such a vertex, defining all of its relevant angles and, ultimately, the sector angles of the crease pattern. This much already defines one point on the trace; the osculating plane contains creases γ4 and γ2 , and so its normal vector q is perpendicular to them both. Knowing the direction of q, we can find its location on the Gaussian sphere; this point anchors the trace, so to speak. These two creases also define the parameter θ, which, recall, is the angle between the two creases γ4 and γ2 on the vertex and is also the angle between the corresponding two arcs on the trace. Now, if we imagine ourselves traveling along the bottom of the valley, one degree of freedom to be set at the very beginning is the angle of each valley wall—specifically, the angles that the walls make with the already-established direction of the normal vector to the osculating plane. Thus, we might also wish to specify the fold angles that the incoming and outgoing sector planes make with the osculating plane. Those angles turn out to be important and useful; we turn our attention to them now. ?? 8.4.1. Sector Elevation Angles As we have seen, point q on the trace divides arcs γ2 and γ4 into two separate arcs each, which we will call sector elevation arcs going forward. We label these four sector elevation arcs σ1, . . . , σ4 , giving each arc the index of the sector angle it is incident to, as shown in Figure 8.18. Since arcs in the trace correspond to fold angles in the vertex, each of these sector elevation arcs must be a fold angle; and indeed, each sector elevation arc specifies the angle between the normals to the respective sector planes and the normal to the osculating plane. So each sector elevation angle is the angle between its sector plane and the osculating plane. The osculating plane is the plane that contains the two major creases, so if we were to set a folded vertex on a desk resting upon its major

........


569

g3

s2

a4

g1 a2 a3

s3

a1

g2 s1

s4

s1

p-a4

s3

g1 a 1 s2 a2

p-q

g1

a3 -g3

s2

s4

p-a1

q q

a2

p-q s3

s1

-g3

a3

q

s4 g4

z

z z

Figure 8.18. Sector elevation arcs on the trace. Left: a folded vertex. The sector elevation angles σ1 –σ4 are the angles between the osculating plane and sector planes. Middle: the trace of this vertex on the Gaussian sphere. Right: a schematic of the trace of a degree-4 vertex with sector elevation arcs and angles.

folds, the sector elevation angles would be the angles between each sector plane and the desk. Once again, we use the formula that relates the area of a spherical triangle to an angle and the two adjacent sides (Equation (7.10)). Applying this to the spherical excess E of the two triangles of the trace individually, we find that tan 12 σ1 tan 12 σ4 sin(π − θ) tan 12 σ1 tan 12 σ4 sin θ tan( 12 E) = = , (8.94) 1 + tan 12 σ1 tan 12 σ4 cos(π − θ) 1 − tan 12 σ1 tan 12 σ4 cos θ and similarly, tan 21 σ2 tan 12 σ3 sin(π − θ) tan 12 σ2 tan 12 σ3 sin θ tan 12 (E) = = . (8.95) 1 1 1 1 1 + tan 2 σ2 tan 2 σ3 cos(π − θ) 1 − tan 2 σ2 tan 2 σ3 cos θ Both relations are, again, due to Huffman [47], as is the following: by eliminating the area from the two equations, we have the symmetric relation, tan 12 σ1 tan 12 σ4 = tan 12 σ2 tan 12 σ3 . (8.96) We will call Equation (8.96) Huffman’s Sector Elevation Identity.

570


This identity implies the existence of a constant ρ such that tan 12 σ1 tan 12 σ2 = , ρ= (8.97) 1 1 tan 2 σ3 tan 2 σ4 as well as a similar relation for the other way of comparing ratios of sector elevation angles. We will call this parameter ρ the fold angle expansion, for reasons that will become clearer in a moment. Given any two sector elevation angles, the other two are fully determined by the value of ρ. So, for example, given σ3 and σ4 , the other two sector elevation angles are defined by 1 1 tan 2 σ1 = ρ tan 2 σ3 , (8.98) tan 12 σ2 = ρ tan 12 σ4 , so that h i σ1 = 2 tan−1 ρ tan 12 σ3 , h i −1 1 σ2 = 2 tan ρ tan 2 σ4 .

(8.99)

Now, since γ2 = σ1 +σ2 , γ4 = σ3 +σ4 , and tan x is a monotonic function, we have that σ1 < σ3, σ2 < σ4, γ2 < γ4 ⇐⇒ ρ < 1, σ1 > σ3, σ2 > σ4, γ2 > γ4 ⇐⇒ ρ > 1, σ1 = σ3, σ2 = σ4, γ2 = γ4 ⇐⇒ ρ = 1.

(8.100)

Thus, the parameter ρ specifies whether the major fold angle and its two constituent sector elevation angles grow, shrink, or remain constant across the vertex. These three cases are illustrated in Figure 8.19. Observe that in Figure 8.19, the fold angle γ4 = σ3 + σ4 is constant for the three examples. In the case of ρ = 1, we have equality between the incoming and outgoing sector elevation angles (taken in opposite pairs) and between the incoming and outgoing fold angles. These are precisely the conditions that are satisfied for a flat-foldable vertex, and indeed, ρ = 1 is equivalent to specifying that the degree-4 vertex is flat-foldable and all that that implies. So any one of the following implies all of the others:

........


571

g3

g1

a4 s4 a3

s2

a2

a2

r = 0.5

s3 -g3

g2

a1

a4 g4

a1 s1

a3

g1

g3 g1

a4 s4 a2

a3

a1

g4

Figure 8.19. Vertex and trace for three values of ρ. Top: vertex with ρ < 1. Middle: vertex with ρ = 1. Bottom: vertex with ρ > 1.

s3

s2

r = 1.0 a4

a1 s1

a2

g2

-g3

a3

g1

g3

g1

a4 s4 a2 a3

s3

r = 2.0

g1 a4

a1 s1

a1

s2

a3

g2

g4

-g3 a2

• The vertex is flat-foldable. • The fold angle expansion parameter satisfies ρ = 1. • Two nonzero opposing sector elevation angles satisfy σ1 = σ3 . • Two nonzero opposing sector elevation angles satisfy σ2 = σ4 . Recall also, from Theorems 25 and 26, that flat-foldability may come from fold angle equalities:

572


• The two nonzero major fold angles satisfy γ2 = γ4 if and only if the vertex is flat-foldable or is straightmajor. • The two nonzero minor fold angles satisfy γ1 = −γ3 if and only if the vertex is flat-foldable or is straightminor. The flat-foldability of a vertex can be significant even if the vertex never folds fully flat. As we have seen in rigidly foldable quadrilateral meshes, flat-foldability brings with it desirable properties in larger networks of vertices. We note a few more relations between the sector elevation angles. If σ3 and σ4 are given, we can express the tangent formulas for the other two sector elevation angles in terms of sines and cosines of σ3 and σ4 as s 1 − cos σ3 tan 12 σ1 = ρ tan 12 σ3 = ρ , 1 + cos σ3 (8.101) r 1 − cos σ4 tan 12 σ2 = ρ tan 12 σ4 = ρ . 1 + cos σ4 From these, we can construct expressions for the other four trigonometric functions of σ1 and σ2 from their counterpart trigonometric functions of σ3 and σ4 : sin σ1 =

(ρ2

2ρ sin σ3 , + 1) − (ρ2 − 1) cos σ3

(ρ2 + 1) cos σ3 + (ρ2 − 1) , (ρ2 − 1) cos σ3 − (ρ2 + 1) 2ρ sin σ4 sin σ2 = 2 , (ρ + 1) − (ρ2 − 1) cos σ4

cos σ1 =

cos σ2 =

(8.102)

(ρ2 + 1) cos σ4 + (ρ2 − 1) . (ρ2 − 1) cos σ4 − (ρ2 + 1)

Formulas going the other direction between {σ1, σ2 } and {σ3, σ4 } may be derived in a similar fashion. ?? 8.4.2. Sector Angles The sector elevation angles are fairly closely related to the sector angles and, given the former and a few other parameters, we can

........


573

compute the latter. Applying the Law of Sines to the left triangle of the trace in Figure 8.18, we have that sin α1 sin θ sin α4 = = , sin σ4 sin γ1 sin σ1

(8.103)

from which we can get the two sector angles: sin α1 =

sin σ4 sin θ sin γ1

and

sin α4 =

sin σ1 sin θ . sin γ1

(8.104)

Similarly, applying the Law of Sines to the right triangle yields sin α2 sin θ sin α3 = = , sin σ3 sin |γ3 | sin σ2

(8.105)

from which we can get the other two sector angles: sin α2 =

sin σ3 sin θ sin |γ3 |

and

sin α3 =

sin σ2 sin θ . sin |γ3 |

(8.106)

This also leads to a symmetric relation between the sector angles and sector elevation angles: sin α1 sin σ4 = sin α4 sin σ1

and

sin α2 sin σ3 = . sin α3 sin σ2

(8.107)

For a flat-foldable vertex, γ1 = −γ3 , and therefore all four sector angle sines are proportional to a corresponding sector elevation angle sine with the same constant of proportionality for all four pairs of angles. This result points toward a strategy for designing a crease pattern: by specifying properties of the pattern in terms of its trace, we can solve for sector elevation angles and then from there, find the sector angles that give rise to the desired 3D vertex. We can solve for the two minor fold angles from the Law of Cosines, finding that γ1 = + cos−1 [cos σ1 cos σ4 − sin σ1 sin σ4 cos θ] , γ3 = − cos−1 [cos σ2 cos σ3 − sin σ2 sin σ3 cos θ] ,

(8.108)

where here we have chosen the sign explicitly to match the crease assignment of the vertex shown in Figure 8.18. The other two fold angles, of course, are simply the sums of their respective sector elevation angles: γ2 = σ1 + σ2, γ4 = σ3 + σ4 .

574


(8.109)

We then can solve for the sector angles from the Law of Sines: −1 sin σ4 sin θ α1 = sin , sin γ1 −1 sin σ3 sin θ α2 = sin , sin |γ3 | (8.110) −1 sin σ2 sin θ α3 = sin , sin |γ3 | −1 sin σ1 sin θ α4 = sin . sin γ1 Since the sector angles are in the range (0, π), use of sin−1 yields an ambiguity (since sin x = sin(π − x)), but the cosine function has the proper domain to give an unambiguous result, and further use of the Law of Cosines gives explicit expressions for all four sector angles in terms of values previously found: −1 cos σ1 cos γ1 − cos σ4 α1 = cos , sin σ1 sin γ1 −1 cos σ2 − cos σ4 cos γ3 α2 = cos , sin σ4 sin |γ3 | (8.111) −1 cos σ4 − cos σ3 cos γ3 α2 = cos , sin σ3 sin |γ3 | −1 cos σ4 cos γ1 − cos σ1 α4 = cos . sin σ4 sin γ1 These equations provide a prescription for calculating the sector angles given a specification on the two major creases, their relative angle, one of the other major fold angles, and the expansion parameter ρ. Equations (8.103) and (8.105) also give rise to another elegant identity relating the sector elevation angles to the minor crease angles. Taking the appropriate product of two equalities from each set gives sin2 γ1 sin α2 sin α3 sin σ1 sin σ4 = · . (8.112) sin2 γ3 sin α1 sin α4 sin σ2 sin σ3 We recognize the first fraction as one that appears in Huffman’s Minor-Angle Identity. Substituting gives the following: 1 sin2 γ1 sin2 ( 2 γ1 ) sin σ1 sin σ4 = · . sin2 γ3 sin2 ( 12 γ3 ) sin σ2 sin σ3

(8.113)

........


575

g2

k

g2

k/ q

g3

a2

Figure 8.20.

a3

Left: bend angle κ in the crease pattern. Right: bend angle κ 0 in the folded form.

a1 a4

g3 a1

a3

g1 g4

a4

g1

g4

?? 8.4.3. Bend Angle In the crease pattern of Figure 8.17, the two major creases γ2, γ4 are defined by the shape of the specified curve, while the two minor creases γ1, γ3 are somewhat incidental, defined by requirements of developability and the incoming fold angles. A fundamental (and rather interesting) question then relates to the angle between the two major creases, which occurs in two places: in the crease pattern, and in the 3D folded form. It is worth considering the relationship between these two angles in 2D and 3D. To do so, we will consider specfically the bend angle, that is, the deviation from straightness, which we denote by κ and κ0 in the crease pattern and folded form, respectively, as shown in Figure 8.20.3 In the crease pattern, κ is related to the sum of two sector angles, specifically, κ = α1 + α4 − π,

(8.114)

while, in the folded form, κ0 is related to the angle θ: κ0 = π − θ.

(8.115)

To find a relation between these two, we turn to two of Gauss’s formulas as applied to 4(q, α4, α1 ) in the trace: 3

Here we introduce a notational convention for quantities that have the same meaning but different values between the 2D crease pattern and the 3D folded form: crease pattern variables (e.g., κ) will be unprimed; folded form variables (κ 0) will be primed. The notational mnemonic to keep track of which is 2D and which is 3D is “extra dimension” = “extra marking” (at least, in this section).

576


1 1 (σ − σ ) sin ((π − α ) + (π − α )) sin (α + α ) 1 4 4 1 1 4 2 2 2 = =− , 1 1 cos 2 γ1 cos 2 (π − θ) sin 12 θ cos 12 (σ1 + σ4 ) cos 12 ((π − α4 ) + (π − α1 )) cos 12 (α1 + α4 ) = = . cos 12 γ1 sin 12 (π − θ) cos 12 θ

cos

1

Taking the ratio of these two equations gives 1 1 cos 12 (σ1 − σ4 ) tan 2 (α1 + α4 ) = − tan 2 θ , cos 12 (σ1 + σ4 ) and, putting this in terms of the two bend angles, cos 1 (σ1 + σ4 ) tan 12 κ = tan 12 κ0 · 21 . cos 2 (σ1 − σ4 )

(8.116)

(8.117)

(8.118)

(8.119)

Now, in the fully symmetric case σ1 = σ2 = σ3 = σ4 ≡ σ,

(8.120)

the incoming and outgoing fold angles are γ2 = γ4 = 2σ ≡ γ, and Equation (8.119) becomes tan 12 κ = tan 12 κ0 · cos σ 1 0 = tan 2 κ cos 12 γ .

(8.121)

(8.122)

And thus, the bend angle in the crease pattern and the bend angle in the 3D folded form are related by the cosine of half of the fold angle. This formula has an analog in the properties of curved creases: the curvature of a fold in the crease pattern and folded form are also related by the cosine of half of the fold angle. In the general case, the coefficient of proportionality, cos 12 (σ1 + σ4 ) , cos 12 (σ1 − σ4 ) has a more complex relationship with the fold angle (and note that it includes sector elevation angles from both the “incoming” and

........


577

“outgoing” sides of the vertex). However, it is readily shown that the proportionality factor satisfies cos 1 (σ + σ ) 1 4 21 = 1 if σ1,4 = 0 or π, cos (σ1 − σ4 ) (8.123) 2 < 1 otherwise. Thus, for any partially folded vertex, the bend angle in the crease pattern is less than the bend angle in the folded form; or, put differently, the act of creating a fold along a bent segment in the crease pattern always increases the angular bend in the folded form. ?? 8.4.4. Edge Torsion Angle Now let us consider what happens when two vertices are connected along their major creases. Figure 8.21 shows a folded form consisting of three creases and two vertices, with the two vertices connected by one of their major creases, along with the traces of the two vertices. To distinguish the sector and fold angles of the two vertices, I have given each angle a superscript: thus {αi(1) } and {γi(1) } are the angles for the first vertex, while {αi(2) } and {γi(2) } are the angles for the second. In this first example, all three major creases lie in the same plane. Since the osculating plane of each vertex is the unique g2(2) a1(1) g3(2)

a2(2)

a1(2) a3(2) a 2) 4 g3(1) a2(1) a3(1) a (1) 1 a4(1)

g4(1)

g4(2) g2(1)

g1(1)

a4(1)

s1(1)

a4(2)

s4(1)

g1(2)

s2(1)

g1(2) s3(1) a3(1)

a2(1) -g3(1)

a1(2)

s4(2) s1(2)

s2(2) s3(2)

a2(2) -g3(2) a3(2)

g1(1)

Figure 8.21.

Left: three creases connected along the major creases of the two intervening vertices. Middle: trace of the first vertex. Right: trace of the second vertex. Note that the highlighted arcs are the same for the two vertices.

578


plane containing the two major creases, the two vertices have the same osculating plane, and therefore the same osculating vector, shown in purple in the figure. And so, the center of the trace is the same point on the Gaussian sphere. In fact, since sector planes α2(1) and α1(1) are part of the same planar surfaces as α3(2) and α4(2) , respectively, their corresponding points on the Gaussian sphere are the same as well, along with the arc between them, which is fold angle γ2(1) = γ4(2) . Similarly, two sector elevation angles are common between the two vertices: σ4(2) = σ1(1),

(8.124)

σ3(2) = σ2(1) .

This relationship points to a simple method for solving for the angles of every vertex along a chain: at each vertex, two of the sector elevation angles are identical to the sector elevation angles of the preceding vertex, while the other two may then be calculated from Equations (8.99) and the value of ρ chosen for each vertex. If the creases do not all lie in the same plane, however, then things get a bit more complicated. Figure 8.22 shows a set of three creases with two vertices in which the three creases do not lie in the same plane, so that the osculating planes for vertices 1 and 2 are different and the osculating vectors point in different directions. Since each osculating vector is perpendicular to the

g2(2)

-t a1(1)

g3(1)

g3(2)

a2(2) a3(2) g4(2)

g2(1) a2(1) a1(1) (1) a4

a1(2)

a42)

g1(1)

-t

a4(1)

s1(1)

a4

s4(1) g1(2)

s2(1)

g1(2) s3(1) a3(1)

a2(1) -g3(1)

2)

s4(2)

s2(2)

s1(2)

a2(2) s3(2)

-g3(2) a3(2)

a1(2)

a3(1) g4(1)

g1(1)

Figure 8.22.

Left: three non-coplanar creases connected along the major creases of the two intervening vertices. Middle: trace of the first vertex. Right: trace of the second vertex. Note that the highlighted arcs are the same for the two vertices, but the osculating normal for the second vertex is shifted along the arc.

........


579

two creases on each side of it, they both are perpendicular to crease γ2(1) = γ4(2) , and so they are related by a rotation about this crease through some angle. We call this angle the edge torsion angle of the crease, and denote it by τ. The edge torsion angle is defined as a right-handed rotation about a direction, so the edge torsion angle in the figure has a negative value. Of course, going the opposite direction (from vertex 2 to vertex 1), it would be positive. Because the sector planes between the two vertices are common to each vertex, their traces share a common arc (highlighted in the figure). However, the position of the osculating vectors along this arc are different, as can be seen in Figure 8.22. In fact, the position of the osculating vector is shifted by exactly the amount of the edge torsion angle, so that the sector elevation angles at the second vertex are now given by σ4(2) = σ1(1) + τ, σ3(2) = σ2(1) − τ.

(8.125)

In general, then, in a chain of vertices connected along major creases, the sector elevation angles will be related along the chain, but they will be potentially altered by edge torsion τ along each segment and expansion ρ at each vertex. We note that this relation implies a limit on the value of the edge torsion angle. If the edge torsion angle ever exceeds the sector elevation angle in magnitude, then the desired new osculating vector will fall outside of the fold angle arc and it will not be possible to construct the next vertex in the chain. For any chain of vertices with nonzero edge torsion, there will be some lower bound on the sector elevation angle, i.e., on the fold angle, that allows the realization of the given folded edge. If the edge torsion angle is zero along a chain of creases, then all creases share a common osculating plane, and their traces will all intersect in the same place on the Gaussian sphere; that is, the center of each “bow-tie” will be the same point. If we now imagine flexing one of the vertices of this zerotorsion structure, then its trace will open and close accordingly, as will the traces of all of the connecting vertices. If we position any one vertex in space so that as it is flexed, its osculating vector remains fixed in space, then that same point on the Gaussian sphere will be the osculating vector of each of the connected vertices as well. So no matter what the degree of openness is, all vertices

580


share a common osculating plane, which, therefore, must contain all of the major creases of all of the vertices. This leads to the following remarkable result: Theorem 27 (Zero-Torsion Planarity Theorem). A folded chain of major-connected creases with no edge torsion remains planar as it is folded and unfolded. The simplest example of such a chain of creases would be a chain of identical creases and vertices, i.e., with σ1 = σ2 = σ3 = σ4 ≡ σ,

ρ = 1,

τ=0

(8.126)

at each vertex. Each vertex would be fully symmetric and could be characterized as to its degree of openness by the angle θ, which, of course, also specifies the 3D bend κ0. For such a chain, as you open and close one vertex of the chain, because consecutive vertices share major creases with their neighbors, all would open and close in unison, and so the values of σ, θ, and κ0 would vary as you opened and closed the mechanism. The sector angles of the crease pattern {αi } are, of course, fixed for a given mechanism. We can choose particular values for the kinematic quantities, then solve for the invariant quantities—namely, the sector angles—that define the underlying crease pattern. Let us choose the 3D bend angle κ0 = κ00 and major fold angles γ2 = γ4 ≡ γ0 as the design point for a chain of identical zero-torsion vertices. At the design point, we then have σ = 12 γ0 ≡ σ0, θ = π − κ00 ≡ θ 0 .

(8.127)

Because the vertex is flat-foldable, Equation (8.122) relates the crease pattern bend to the folded bend: κ0 = 2 tan−1 tan( 12 κ00 ) cos σ0 . (8.128) The vertex is zero-torsion, so α1 = α4 and α2 = α3 . But the crease pattern bend is given by κ0 = α1 + α4 − π,

(8.129)

so we now have the four sector angles: α1 = α4 = 12 (π + κ0 ), α2 = α3 = 12 (π − κ0 ).

(8.130)


........

581

Figure 8.23.

Left: crease pattern of a chain of identical vertices. Right: folded form of the chain of identical vertices.

To determine the kinematic behavior, we can now take these sector angles along with a single parameter that describes the state of openness of the vertex (e.g., a major fold angle), and then use the relations from Section 8.3.9 to compute all of the other relevant angles. Both the crease pattern and folded form of a chain of such vertices form a circle—or rather, a polygonal approximation of a circle. Figure 8.23 shows the crease pattern and folded form for such a chain with a bend angle of κ00 = 11.52◦ and sector elevation angles of σ0 = 60◦ on both sides. This gives a bend angle in the crease pattern of κ0 = 5.77◦ . The folded form and crease pattern both form polygonal arcs of a circle, but the circle of the crease pattern, having a smaller bend angle at each vertex, has less in-plane curvature. And as you can see, the chain of major (valley) folds is, indeed, planar. The remarkable property is “once planar, always planar”—that is, if the folded edge is planar for one value of the fold angle, then it will be planar for all other fold angles (as long as the same angle is applied uniformly along its length). If we now allow a nonzero edge torsion angle τ, we can also form a chain of identical vertices, but we will no longer have identical sector elevation angles. Instead, we must choose the sector elevation angles so that Equation (8.99) is satisfied for each consecutive pair of vertices. In this case, the fold angle at each vertex can be characterized by a value σ (each major fold angle will be 2σ), but the individual sector elevation angles must be

582


chosen to accommodate the desired edge torsion; they will be given by σ1 = σ3 = σ − τ/2, σ2 = σ4 = σ + τ/2.

(8.131)

Each vertex is no longer zero-torsion, but since the two major fold angles match, they are still individually flat-foldable, and we can follow a similar procedure as the above to compute the sector angles of the crease pattern. We again take the 3D bend angle κ0 = κ00 and major fold angles γ2 = γ4 ≡ γ0 as the design point, though we now take τ = τ0 as the design torsion angle. We then have σ = 12 γ0 ≡ σ0, θ = π − κ00 ≡ θ 0 .

(8.132)

The crease pattern bend κ0 = α1 + α4 − π is now given by the more general Equation (8.119): " 1 # cos (σ + σ ) 1 4 2 −1 −1 1 0 1 0 cos σ0 κ0 = 2 tan tan( 2 κ ) · tan( 2 κ0 ) . = 2 tan cos τ0 cos 12 (σ1 − σ4 )

(8.133)

That gives us (α1 + α4 ). But how do we divide this angle (unevenly) between the two angles? In this case, we can use Equations (8.110) or (8.111) to give explicit expressions for the four sector angles. Once we’ve solved for the crease pattern, we can compute the kinematic form at any stage of folding or unfolding. The folded form takes the shape of a polygonal approximation of a helix, as shown in Figure 8.24. For this pattern, we have the following parameters: κ0 = 17.22◦, κ = 8.67◦, σ = 60◦, τ = 5.2◦ .

(8.134)

One can, of course, specify the osculating angle θ, expansion factor ρ, and edge torsion angle τ individually for every vertex or segment along a path. More generally, one can specify the positions of the vertices arbitrarily in 3-space as a polygonal space

........


583

Figure 8.24.

Left: crease pattern of a chain of identical vertices with constant edge torsion. Right: folded form of the chain of identical vertices.

curve, which defines the osculating angle θ and edge torsion angle τ while still leaving one free to specify the expansion factor ρ at each vertex. In Section 9.3, we will present a complete algorithm to do exactly that. ?? 8.4.5. Midfold Angles and Planes There is another significant set of planes and angles that can be identified with a degree-4 vertex and that rely on an 18th-century construction by the mathematician Anders Johan Lexell, who was a student of Leonhard Euler. The construction that now bears his name is illustrated in Figure 8.25. N

B*

A* C

D p

A

B

Figure 8.25.

D

C

A

E q r

s B

Left: the Lexell circle is the small circle passing through A∗, B∗, and C. Every spherical triangle ∆ABD with base AB and vertex D has the same area as ∆ABC if and only if D lies on the Lexell circle. Right: the small circles passing through A and B, and C and D, are lines of latitude if the great circle passing through midpoints p and r of edges AC and BC of triangle ∆ABC is the equator.

584


Given a spherical triangle ABC, define the antipode (point diametrically opposite) of A as A∗ and the antipode of B as B∗. There is a unique circle on the sphere that passes through the points A∗, B∗, and C, called the Lexell circle. (This is a small circle, not a great circle, of the sphere; that is, the center of the circle is offset from the center of the sphere.) Lexell’s Theorem states that any spherical triangle ∆ABD has the same area as ∆ABC if and only if point D lies on the Lexell circle. This result seems surprising and unexpected, but some years after Lexell proved it, Euler gave a proof that makes it seem almost intuitive, based on the second construction in Figure 8.25 [99]. Construct the (unique) great circle through the midpoints of arcs AC and BC. If we think of that great circle as the equator of a globe, Euler showed that points A and B lie on a line of latitude south of the equator, while C must lie on a line of latitude the same distance north of the equator—and that line is precisely the Lexell circle. From there, he showed that every spherical parallelogram with base AB whose upper vertices lay on the Lexell circle (north latitude line) had the same area, no matter how far around the globe the top was shifted from the bottom. From this result Lexell’s Theorem follows directly. A useful byproduct of his proof was that it established that for every triangle ∆ABD, the midpoints of sides AD and BD (q and s in the figure), as well as those of sides AC and BC, all lie on the great circle that is the equator of our sphere. That result has relevance to our developable degree-4 vertex trace. It is easy to see that triangles ∆ABC and ∆ABD have the same area if and only if triangles ∆AEC and ∆E BD have the same area, which would be the case for spherical quadrilateral ADBC being the trace of a developable degree-4 vertex. So the midpoints of all four arcs of the trace lie on a single great circle of the Gaussian sphere. This is a remarkable result that was among the many relationships identified by Huffman [47]. Each arc of the trace comes from the rotation of a direction vector about a fold angle as it rotates from one sector plane to the next. As the normal to the plane rotates, it sweeps out an arc on the surface of the Gaussian sphere. And so we define four new directions and associated planes that correspond to the midpoints of each of the four arcs of the trace. We call these points on the

........


585

a2 a2 a1

a3

a1

a3

a4 m1 a1

a4 a4

m2 m4

q a3

a2 m3

Figure 8.26.

Left: a vertex crease pattern. Middle: vertex with midfold plane identified. Right: trace on the Gaussian sphere with midfold points {mi } identified. The midfold plane passes through all four midfold points.

Gaussian sphere the midfold points {mi }. The midfold points for a vertex trace are illustrated in Figure 8.26, which is drawn as if looking straight down the osculating point. The midfold points lie on a great circle; that great circle defines a plane, the midfold plane, that contains those four points; and so we can also identify a point on the Gaussian sphere that characterizes the normal of this plane. It is the tip of the orange arrow in Figure 8.26. Now, since the osculating normal q is the crossing of the bowtie, it is tempting to think that q would be the center of both arcs γ2 and γ4 . But this is not necessarily the case. And so since q, m2 , and m4 do not coincide, neither do their associated planes. The normal direction of the midfold plane is roughly aligned with the two major creases of the vertex. It is not parallel to either of them; rather, it roughly “splits the difference” between the two fold directions. There is a physical interpretation of this vector and its plane. If a polyhedral chain of degree-4 vertices approximates a continuous space curve, then the midfold direction approximates the tangent to the space curve. But we will see that the midfold vector has additional significance in two-dimensional arrays of creases. ?? 8.4.6. Infinitesimal Trace As already noted, as a folded vertex approaches fully flat, the normals of the sector planes converge toward one another and the

586


trace collapses down to a point. It does so by shrinking in size, but it retains its general bow-tie shape. Recall that the angle between the two triangles, θ, varies with the degree of openness, but takes on the value α2 + α3 as the paper approaches flatness, which is, generally, not a straight line; and so, as the bow-tie of the trace shrinks into oblivion, it remains a bow-tie, albeit one vanishingly small. It is worthwhile, then, to consider what happens to the trace and to the various angles and directions associated with it as the paper approaches flatness, and we will work out several important relations in this section. In general, as the configuration approaches flatness, some quantities will vary but will approach constant values; some will remain constant (like the fixed sector angles); and some will approach zero, like the sector elevation and fold angles. In the limit as these quantities approach zero, we can make use of several asymptotic relations: sin x ∼ x as x → 0, tan x ∼ x as x → 0, cos x ∼ 1 as x → 0.

(8.135)

Taking these limits will allow us to simplify several of the trigonometric relations between the various angles in the trace. In order to distinguish between relationships that are only valid as we approach flatness, we will add a tilde ( x) ˜ to any quantity for which we have made such an approximation. We can also take advantage of the fact that as the trace becomes infinitesimally small, the curvature of the Gaussian sphere becomes negligible and the trace effectively becomes a planar figure, with the arcs replaced by straight lines. The limit plane of the trace is perpendicular to the osculating vector, which is, itself, perpendicular to the limit plane of the paper, and so the plane of the trace becomes parallel to the plane of the paper. If we view both the trace and paper from the direction of the osculating vector, then both planes will be perpendicular to the line of sight, and we can superimpose the trace over the vertex with the osculating point coinciding with the center of the vertex. This allows us to examine and work out the orientation of the trace relative to the orientation of the vertex, with all angles given as rotations within their respective parallel planes, as shown in Figure 8.27. The infinitesimal quantities of interest are the sector elevation angles σ ˜ 1, . . . , σ ˜ 4 , the fold angles γ˜1, . . . , γ˜4 , and the fold angle

........


587

s1

s4 g3 a3

Figure 8.27.

Left: trace and vertex for a nearly-flat vertex. Right: schematic of an infinitesimal trace superimposed on a crease pattern.

g4

a2

g2

s2 s 3

g3

~ m

g2 a1

s1 g1

a4

s4

s3

g1

s2

g4

expansion factor ρ. ˜ In the limit, several of the identities we have encountered already simplify significantly. First, Huffman’s Minor-Angle Identity (Equation (8.44)) becomes γ˜1 = γ˜3

r


(8.136)

Huffman’s Major-Angle Identity (Equation (8.32)) becomes γ˜2 = γ˜4

r

sin α2 sin α3 σ ˜1 + σ ˜2 = . sin α1 sin α4 σ ˜3 + σ ˜4

(8.137)

Huffman’s Sector Elevation Identity (Equation (8.96)) becomes σ ˜ 1 = ρ˜ σ ˜ 3,

σ ˜ 2 = ρ˜ σ ˜ 2.

(8.138)

And, applying the Law of Sines (of the Euclidean plane) to the sides of the two triangles of the trace, we have σ ˜2 σ ˜3 | γ˜3 | = = , sin α3 sin α2 κ σ ˜4 σ ˜1 γ˜1 = = , sin α1 sin α4 κ

(8.139)

where κ = π − (α2 + α3 ) is the bend angle of the crease pattern. Solving all of these simplified equations together allows us to express both the sector elevation angles and the fold angles in

588


terms of a single common factor σ ˜ (no subscript) as follows: r sin α4 σ ˜1 = σ ˜ , sin α1 r sin α2 σ ˜2 = σ ˜ , sin α3 (8.140) r sin α3 σ ˜3 = σ ˜ , sin α2 r sin α1 σ ˜4 = σ ˜ . sin α4 And the infinitesimal fold angle expansion parameter is r sin α3 sin α4 ρ˜ = . (8.141) sin α1 sin α2 We can use these values to construct the infinitesimal trace by superimposing it over the crease pattern of the vertex. Each of the fold angle arcs γ˜i are now straight lines and will be perpendicular to their respective creases. “Arcs” γ˜2 and γ˜4 will intersect at the osculating point, which will be exactly above the vertex. And the positions of the sector plane normal vectors will be located at the endpoints of the “arcs,” at distances given by sector elevation angles σ ˜ i. We can also construct the midpoints of the four fold angle arcs, to obtain the four infinitesimal midfold points; the line through them all will be the great circle of the midfold plane and a line perpendicular to this plane will be perpendicular to the midfold plane. Observe that the orientation of the midfold plane approaches a fixed value as σ ˜ goes to zero and the vertex approaches flatness; thus, we can identify the midfold plane direction even for a fully flat crease pattern. Let us define by µ˜ the angular displacement of the midfold plane direction from crease γ1 as in Figure 8.27. Some algebraic manipulations reveal that this angle is given by √ √ sin α sin α sin α + sin α sin α1 sin α2 1 3 4 4 −1 −1 ρ˜ sin α1 + sin α4 µ˜ = tan = tan . √ √ ρ˜ cos α1 − cos α4 cos α1 sin α3 sin α4 − cos α4 sin α1 sin α2 (8.142) In the case of a flat-foldable vertex, ρ˜ = 1 and this formula simplifies, and the rotation of the tangent vector relative to γ1 becomes µ˜ = 12 (π + α1 − α4 ). (8.143)

........


589

These relations will turn out to be useful when we consider a 2D array of identical vertices in a following section. ?? 8.4.7. What Specifies a Vertex? With all of the various angles, directions, and relationships of this section, it is fairly easy to get lost in the forest of trigonometric functions and relationships. We now have a great deal of trigonometric machinery for analyzing 3D vertices. The question is, which tool(s) do we use? And in particular, what do we need to fully specify a single folded vertex? While there are many combinations of quantities we could choose to specify a vertex and then derive all of the others, there are two sets that are particularly useful. The first is the crease pattern centric definition. Suppose we have a crease pattern. What additional information do we need to fully specify the folded form of the vertex, in addition to the crease pattern? We can fully specify everything from the following: • the designation of which creases are major/minor, • the four sector angles {αi }, • any one fold angle γi . If we have this information, then we can use the formulas from Section 8.3.9 to determine the remaining three fold angles, and then from there, everything else is determined. An alternative is the folded form centric definition. Suppose we have some specification of the folded form. What do we need to know in order to fully determine the crease pattern? In this case, we could fully define the crease pattern from the following information: • the designation of which creases are major/minor, • the four sector elevation angles {σi }, • the folded form bend angle κ0 (which, recall, defines the osculating angle θ = π − κ0). From the sector elevation angles we can compute the two major fold angles γ2 and γ4 ; then from the Law of Cosines, compute γ1 and γ3 as in Equations (8.108); and then from the Law of Sines, compute the four sector angles as in Equations (8.110).

590


Of course, these are not the only possible ways to specify the folded form of a degree-4 vertex, but this will turn out to be a particularly useful way when it comes to solving the inverse problem of origami design: given a desired folded form, find the crease pattern (and fold angles) that fold into that shape. ??

8.5. Networks of Vertices

?? 8.5.1. Huffman Grid You might remember from Section 2.3.1 that it was possible to create a crease pattern consisting of a 2D array of identical vertices for any degree-4 vertex, as shown in Figure 8.28. If we consistently number the creases of each vertex so that corresponding creases around each vertex are numbered the same, we can see that in the array, the ith crease of each vertex meets up with the corresponding ith crease of its mate, and so forth, all the the way across the array, forming an array we met in Chapter 2, called a Huffman grid [47]. Back in Chapter 2, I mentioned that no matter what angles you choose for the generating vertex, whether flat-foldable or not, it always curled up into a cylinder as it was flexed. We are now equipped to understand why. Constructing a Huffman grid is straightforward. Each facet of such an array is a quadrilateral with four corners; each corner is

γ2

γ3 α2 α3

γ3

α1 γ1 α4 γ4

γ3

γ2

γ3

α2 α α3 α1 α4 3 α4 α α α γ1 1 3 α1 α3 4 α2 γ4 α 2

γ2

γ3

γ2

γ3

Figure 8.28.

Left: crease pattern consisting of a 2D repeated array of vertices. Right: close-up of the array with sector and fold angles labeled.

........


591

one of the sector angles of the vertex, so that all four sector angles of the vertex form the four interior angles of a quadrilateral facet. If we trace around the border of such a facet in the counterclockwise direction, then the path will make four turns; the total turn angle will be the sum of the interior angles, which is equal to the sum of the sector angles, which is 2π, so the path will end running at the same angle that it started. To ensure that it truly closes on itself, we must choose the side lengths appropriately. It is relatively easy to show that one can choose any two of the four side lengths arbitrarily; then the other two are fixed by the requirement that the path close upon itself. (There is some limit on the arbitrary choice of lengths, as some choices give negative side lengths.) Once we have chosen the side lengths so that a single quadrilateral closes, copies of this quadrilateral can be arranged into a 2D array that tiles the plane, which gives the crease pattern shown in Figure 8.28. Now, this same arrangement of vertices can be created in 3D from a partially folded vertex. Since each quadrilateral in 3D has the same side lengths and interior angles as the crease pattern, each quadrilateral in the 3D arrangement remains locally flat, and so the 3D arrangement of quadrilaterals also forms a closed tiling. But the total 3D arrangement will, in general, not be flat; in fact, since each vertex is 3D, the folded form of this 2D array will also, in general, be 3D. Locally, each vertex is surrounded by sector planes whose normals point in many different directions. But globally, there is a clear long-range order to such a pattern. For the crease pattern shown in Figure 8.28, the macroscopic form is roughly cylindrical, as shown in Figure 8.29. In fact, it remains cylindrical as one opens or closes the underlying vertex, as can be seen for two different degrees of openness in the figure. This behavior is interesting, perhaps a bit surprising, but what is most surprising is that it is ubiquitous; every structure of this type will exhibit this cylindrical behavior, curling and uncurling in one direction, but not the other, as the pattern is flexed. And what is also surprising is that the axis of symmetry of the cylinder bears no obvious relationship to any particular crease angle of the crease pattern or to either direction of periodicity of the crease pattern. So, there are two questions: (1) why is the form cylindrical (and is it really always so), and (2) what determines the axis of cylindrical symmetry?

592


Figure 8.29.

Left: crease pattern of a 10 × 10 array of vertices. Middle: folded form for a fold angle γ1 = −40◦ . Right: folded form for γ1 = −20◦ .

(Note that since each vertex is degree-4, each vertex has exactly one degree of freedom, and since the vertices are connected to one another, the entire structure has exactly one degree of freedom; if constructed from perfectly rigid material, it would all flex uniformly as any one vertex is opened or closed.) The Gaussian sphere, as it turns out, holds the key to understanding this structure (for us, as it did for Huffman). ?? 8.5.2. Gauss Map A single polyhedral vertex in real space gives rise to a trace on the Gaussian sphere, in which • facets give rise to points (whose positions are determined by the facet normals); • folds give rise to arcs (whose lengths are given by the fold angles); • vertices give rise to spherical polygons (which are, in general, self-crossing). If we have a fold pattern containing multiple facets, folds, and vertices, each interior vertex creates its own trace; each trace is composed of arcs, one for each interior folded edge. (Edges that are on the border of the paper are not folds and so do not give rise

........


593

to arcs on the Gaussian sphere.) The collection of all such arcs, one for each interior edge of the fold pattern, constitutes a more complicated mesh on the Gaussian sphere: the Gauss map of the folded surface. If the set of correspondences facet ↔ point, fold ↔ edge, vertex ↔ polygon sounds familiar, it should: the Gauss map sounds very much like the interior dual graph that we encountered in Chapter 6. And indeed, the Gauss map is a particular embedding of the interior dual graph of the crease pattern. But now, instead of both the primal and dual graphs residing in the Euclidean plane, the embedding of the primal graph is a 3D folded form, and the embedding of the dual graph lives on the Gaussian sphere. Another difference from before is that when we used reciprocal figures to construct twist tessellations, we took care to ensure that the dual graph embedding had no self-crossings. But the Gauss map is jam-packed with self-crossings. Indeed, even a single flat-foldable vertex creates a self-crossing spherical polygon: its bow-tie trace. The structure of the folded form is reflected in structure of the Gauss map. Each vertex in a larger folded pattern contributes its own bow-tie trace to the Gauss map. But since any two adjacent vertices share a common edge, the traces corresponding to those two vertices must also share a common edge in the Gauss map. Similarly, since the vertices around a facet share a common sector plane, the traces of those vertices will share a common corner in the Gauss map, whose position is given by the normal vector to the facet shared by the vertices. And so, turning back to the Huffman grid, just as the 2D array of identical vertices forms an interconnected network of identical quadrilaterals in real space, the array of traces will form an interconnected network of identical bow-tie traces in the Gauss map. While the array of quadrilaterals in a Huffman grid forms a 2D array that grows steadily large in real space as we add quadrilaterals to the array, the array of traces that constitute the Gauss map do not extend to cover the Gaussian sphere. In fact, they appear to arrange themselves into a narrow ring that runs along a great circle on the sphere, as shown in Figure 8.30. Adding more quadrilaterals to the Huffman grid simply adds more bow-tie traces to this band of traces wrapping around the Gaussian sphere.

594


Figure 8.30.

Top: a single vertex of the Huffman grid. Bottom: an 8 × 5 Huffman grid. Left: crease patterns. Middle: folded forms. Right: Gauss maps. Note that the Gauss map of all of the vertices forms a band around the Gaussian sphere.

Let us look specifically at the four traces of four vertices around a single quadrilateral of the array, as illustrated in Figure 8.31. I have drawn each trace in a separate color and displaced them slightly so they can be distinguished. I have also labeled each facet around the quadrilateral with a letter, A–I. Each facet has its own normal vector, which maps to a point on the Gaussian sphere. Since each facet has four vertices, each facet normal vector will be a sector plane normal for each of its four vertices, meaning that point on the Gauss map will be shared by the four traces. The sector plane normals on the Gaussian sphere are also labeled A–I in Figure 8.31. Since two vertices that share a common incident edge must share the two adjacent sector planes in real space and must share the same fold angle arc in the Gauss map, one could build up

........


595

G

G

α3

α1 α2

Figure 8.31.

Left: four vertices around a quadrilateral. Sector planes are labeled A–I. Right: the four traces on the Gaussian sphere (schematic view). Note that fold angles are shared between consecutive creases around the quadrilateral.

γ4

α4

γ1

E

H

F

α3 γ4

α1 α4

α4

γ1

α3

α1

α3

α2

B γ2

H

I

E α2

I

γ2

A

γ3

F

A

γ3

γ2

γ1

D

C

γ3

γ4

B

D γ2

C

γ4

γ1

γ3

both the real-space folded shape and its Gauss map iteratively, by sequentially adding edges to the former and arcs to the latter. Folded edges around a vertex can be grouped into vertex traces; traces from adjacent vertices are grouped in such a way that mating traces share the same fold angle arc. The specific way that two mating traces share the same fold angle arc in the Huffman grid is that one is the 180◦ rotation of the other about the midpoint of the shared arc. Recall from Section 8.4.5 that the midpoints of all four arcs of a degree-4 vertex trace lie on a common great circle, which is the great circle of the midfold plane of the vertex. This arc is shown in orange in Figure 8.31. Because of this property, the midpoints of all four arcs of both traces that share any one common arc will lie on the same great circle. And so, since all of the vertices are connected to one another by sharing a fold angle, all of the traces must be connected to others in this way, and thus: The midpoints of all arcs of all traces of all vertices of a Huffman grid of identical degree-4 vertices lie on the same great circle, which is the common midfold plane of all of the vertices. This result explains the cylindrical symmetry of the folded form. All of the vertex traces in the Gauss map are arranged around a single great circle of the Gaussian sphere, and so all of the sector plane normals are confined to a narrow band around

596


Figure 8.32. An 8 × 5 Huffman grid at three different major fold angles. Top: γ+ = −5◦ . Middle: γ+ = −20◦ . Bottom: γ+ = −35◦ .

this great circle. Thus, the macroscopic shape will curve in the same direction that the band of traces curves and will be uniform in the direction perpendicular to the band. This direction defines the cylindrical axis of the folded form, and we can see now that the direction of cylindrical symmetry of the folded array will be the direction of the normal to the midfold plane of any given vertex. The way the vertex traces mate with each other on the Gaussian sphere also explains why the pattern curls ever tighter with increasing fold angle. As the fold angles increase, the arc lengths in the Gauss map increase, and the sizes of the individual bow-tie traces increase. Although they remain stacked on the midfold great circle, their increasing widths extend the Gauss map along the circle, wrapping ever farther around it, as shown in Figure 8.32. ?? 8.5.3. Miura-ori and Mars Now that we know how to construct Gauss maps, let’s look at another fold pattern: the venerable Miura-ori. A 4 × 4 Miura-ori is shown in Figure 8.33, along with its Gauss map.

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597

Figure 8.33. A 4 × 4 Miura-ori. Left: crease pattern. Middle: folded form. Right: Gauss map.

This is interesting. The crease pattern has 3 × 3 = 9 internal vertices, but the Gauss map looks like the trace of a single vertex. In fact, in the Gauss map of a Miura-ori, all of the vertex traces are precisely superimposed, one atop the other. This symmetry gives the Miura-ori a distinctly different kinematic behavior from that of the Huffman grid. As we saw in Figure 8.32, as the individual vertex traces of the Huffman grid expand, the Gauss map wraps farther and farther around the Gaussian sphere, imparting ever-greater cylindrical curvature to the real-space folded shape. But in the Miura-ori, since the traces remain superimposed on one another, while the individual traces grow and shrink in size, they remain co-aligned, and so the overall shape of a Miura-ori undergoing flexure remains planar. And just as the midfold plane of the Huffman grid vertex defined the axis of cylindrical curvature, in the Miura-ori, we can also identify the plane of expansion with a feature of the trace. The fixed point of all of the traces is the crossing point of the bow-tie. This, recall, corresponds to the normal vector of the osculating plane, the plane that contains both major folds in a vertex. And indeed, in the real-space Miura-ori, each chain of major folds lies within a fixed plane as the pattern expands and contracts. The same behavior occurs in a Mars pattern, as shown in Figure 8.34; the Gauss map consists of traces of all of the vertices, each superimposed on the others. And in fact, this behavior exists not just for the perfectly periodic Miura-ori and Mars pattern, but for all of the variations in which we varied the distances between vertices. The Gauss map

598


Figure 8.34. A 6 × 4 Mars pattern. Left: crease pattern. Middle: folded form. Right: Gauss map.

is completely independent of any distances in the folded form: it is affected only by fold angles and orientations. We see similar behavior even in patterns of non-flat-foldable vertices. In the Miura-ori with spacers, the individual vertices are not flat-foldable, but they can be grouped into four types and orientations, which, when we tile their traces together into a Gauss map of the surface, once again, all of the osculating points of the trace line up as shown in Figure 8.35, implying a planar flexing mechanism. This pattern looks like a bisected bow-tie, but it is, in fact, the overlap of two different bow-ties that align along common edge arcs. A single vertex of the pattern is shown in Figure 8.36.

Figure 8.35. A 6 × 7 Miura-ori with spacers. Left: crease pattern. Middle: folded form. Right: Gauss map.

........


599

Figure 8.36. A single non-flat-foldable vertex of a Miura-ori with spacers. Left: crease pattern. Middle: folded form. Right: Gauss map.

In a flat origami tessellation, we saw that tiles of creases must match in position, direction, and assignment. A 3D origami tessellation can be similarly built from tiles, but the creases that cross tile edges must match in position, direction, and fold angle (which is, of course, just the extension of assignment to continuous angle). A similar matching takes place on the Gaussian sphere; each tile has its own Gauss map, and the Gauss maps of adjacent tiles must match up on the Gaussian sphere. Figure 8.37 shows the tile for the Huffman grid; each tile contains two vertices and has six fold angles emanating from it. The figure shows that the tile’s Gauss map is a double bow-tie. To transform the tile into a periodic array, we create repeating copies of the tile in two different directions in real space. On the Gaussian sphere, this translates into creating repetitions of the base tile unit while matching folds of the same fold angle and orientation. So, for example, in the Gauss map tile, two tiles could mate by aligning edge CD of one tile with AF of the next, or CB with E F of the next, or AB of one with E D of the next. But not along AD: that edge appears only internally in the Gauss map tile. Figure 8.38 shows these three different ways of mating pairs of tiles. Note that in all three ways of mating, the tiles are spaced out along the great circle that passes through the midpoints of all of the fold angle arcs on the Gaussian sphere. Thus, every Gauss map tile in the periodic array will lie on the same great circle. The Miura-ori is built from a four-vertex tile; each vertex gives rise to the same bow-tie trace. Significantly, though, all four traces are superimposed on one another. When we start putting together tiles on the Gaussian sphere, we need to match folded edges in fold angle, and so we need to be a bit careful with how the tiles

600


γ4

α4

γ1

α3

α1 α2

E

γ2 γ3

F

γ2

A

γ3

E α2

α3 γ4

F

A

α1

α4

γ1

α4

B

α3

α1

I

γ1

α3

α2

γ4

C

D

B

C γ2

γ3

γ4

D

γ2

γ3 γ1

Figure 8.37.

Left: the Huffman grid is a periodic pattern with a two-vertex tile. Tiles must mate with matching fold angles along the mating edges. Right: the Gauss map of the two-vertex tile. Arcs are colored to match their corresponding fold angles in the tile. Colored dots indicate the bow-tie trace contributed by each vertex in the tile.

mate. In Figure 8.39, I have colored the four arcs of the Gauss map and, correspondingly, the fold angles that give rise to those arcs. Note that tiles mate with their neighbors only along the yellow and orange fold angles. That will force the Gauss maps for adjacent tiles to be similarly superimposed, and so no matter how large we make the 3D tiling, all of the vertex traces will be superimposed. So, for some periodic 3D fold patterns, like the Miura-ori, when they are tiled with 2D periodicity in the crease pattern, their tiling on the Gaussian sphere is stationary, and this gives a planar shape as we expand the tiling. For others, like the Huffman grid, when they are tiled with 2D periodicity in the crease pattern, the tiling on the Gaussian sphere is distributed along a great circle. That leads to a question: suppose we created a much more complex tile, with more vertices, but that still tiled periodically in the crease pattern (including periodic fold angles). Could it assume any other form on the Gaussian sphere?

........


601

γ3

γ2

γ1

γ1

γ2

γ3

γ3 γ2

Three ways to mate Huffman grid tiles and their corresponding Gauss map tiles. Blue/yellow lines indicate the mating edges. Top: mating along a γ1 fold. Middle: mating along a γ2 fold. Bottom: mating along a γ3 fold.

γ1

γ2

γ3

γ2

γ3

γ1 γ3

Figure 8.39. A tile of the Miura-ori. Left: crease pattern and a tile of four vertices. Right: Gauss map of the tile. The traces of all four tiles are superimposed.

602

γ1

γ3

γ2

γ1

γ2

γ3

γ2

γ1

Figure 8.38.

γ1


γ2

γ1

γ3

Surprisingly, perhaps, the answer is no, for a very simple reason. We can have Gauss map tiles that distribute periodically around a great circle, even with multiple periodicities, but there is no way to create a pattern with periodicity in two different directions on the surface of a sphere. Thus, we can conclude a very general fact about periodic 3D origami patterns. If we create a 2D periodic crease pattern where the periodicity applies to fold angles as well as vertices and edges, then, no matter what the crease pattern is, the resulting 3D shape either grows in a planar fashion or grows in a cylindrical form as the array is extended. Similarly, for rigidly foldable periodic crease patterns, the only possible periodic expansion motions are planar or cylindrical. There are, of course, spherical forms that start from a periodic pattern, the Waterbomb tessellation of Section 2.4.7 being the classic example. But in order to take on a spherical form, each tile of the pattern needs to distort in a somewhat different way, and indeed, this spatially varying distortion can be seen in, for example, Figure 2.100. There is much we can learn about the kinematics of 3D origami by studying the geometry of the trace, Gaussian sphere, and Gauss map. As you might imagine, though, because the traces and Gauss maps are all self-crossing, they can become horribly complex for all but the simplest and most symmetric patterns. When we wish to study larger and/or less regular patterns, we need a methodology for describing crease patterns and folded forms in 3D. For that, we need to add some mathematical machinery: vectors and linear algebra. ?

8.6. Terms Antipode A point on a sphere diametrically opposite a given point. Binding crease The fold that becomes flat-folded at the binding state of a non-flat-foldable vertex. Binding state The folded state for a non-flat-foldable vertex where motion can no longer proceed due to self-intersection avoidance. See also Blockfaltung (Chapter 7). Direction vector A representation of a particular direction in space. Edge torsion angle The rotation angle of the osculating plane from one vertex to the next in a major-connected chain of vertices.

........


603

Gauss map The collection of points and arcs on the Gaussian sphere that correspond to the facets and folds, respectively, of a polyhedral folded surface. Gaussian sphere A unit sphere that is used to represent directions in space. A point on the sphere represents the direction from the center of the sphere to that point. Equivalently, that direction is perpendicular to an osculating plane that touches the sphere at that point. Osculating normal (of a vertex) The point on a Gaussian sphere that corresponds to the perpendicular direction of the osculating plane of a degree-4 vertex. Osculating plane (of a sphere) A plane that is tangent to a sphere (touches it at exactly one point). Osculating plane (of a vertex) At a partially folded degree-4 vertex, the plane that contains the two major folds. Polyhedral vertex A vertex in 3D whose surrounding facets are all flat and planar. Ruling plane (of a vertex) At a partially folded degree-4 vertex, the plane that contains the two minor folds. Sector elevation angle The angle from the osculating plane of a degree-4 vertex to each of the sector planes. Small circle A circle on a sphere that is not a great circle, i.e., the center of the circle is not the same as the center of the sphere. Spherical mapping, spherical image, Gauss map A mapping of a particular direction in space to a point on the Gaussian sphere. Straight-major A degree-4 vertex in which the two major folds are collinear. Straight-minor A degree-4 vertex in which the two minor folds are collinear. Trace (on a Gaussian sphere) The mapping of a moving direction vector from some path in real space to a path on the Gaussian sphere. 604


9

3D Analysis ???

9.1. 3D Vectors

The computational design of a 3D origami figure involves the construction of two surfaces: the flat (2D) crease pattern, and the polyhedral (3D) folded form. This is most naturally carried out using vectors, similarly to what we did in the ? ? ? sections of previous chapters, but we will now use 3-vectors for the vertices of the folded form. The mathematics of 3D vector spaces are described well in any basic course on computer graphics and rendering. Tools and formulas for manipulating and analyzing the related data structures are built into many computer libraries and software packages. You will find many 3D vector identities in the literature of fields ranging from electricity and magnetism to computer graphics, as well as in many online resources. Here, for completeness (and commonality of notation), I will give a few of the more useful formulas. A point p in 3-space is represented by the ordered triple p = (p x, p y, pz ). As before, I will usually use the same letter for a vertex in the folded form as I used for the corresponding vertex in the crease pattern, but I will add a prime to one or the other to distinguish the two, as in Figure 9.1.1 A line segment pq is fully specified by its endpoints p and q, and the points along the segment are given by the parametric 1

I usually try to use unprimed variables for crease patterns and primed variables for the folded form, but this is not an absolute rule. Most of the time, I make the choice that means I have to write the fewest primes. So if most of my expressions involve vertices of the folded form, I’ll let those be the unprimed variables.

605

p5

p6

p7′

p7

p4′

p4 p3′

p3

p1

p2

p6′ p5′

p1′

p2′

Figure 9.1. A crease pattern can be specified by its vertices as 2-vectors; its folded form is specified by its vertices as 3-vectors. Left: crease pattern, with coordinates {pi }. Right: folded form, with coordinates {pi0 }.

equation

r(t) = (1 − t)p + (t)q,

t ∈ [0, 1].

(9.1)

For t ∈ (−∞, ∞), we have a parameterization of the line through p and q. A direction r in 3-space is also represented by an ordered triple (r x, ry, rz ) and is typically normalized: r x2 +ry2 +rz2 = 1. In general, vectors can be normalized or not; if its magnitude is 1, then it is a unit vector. We define three orthonormal unit vectors as xˆ ≡ (1, 0, 0), yˆ ≡ (0, 1, 0), zˆ ≡ (0, 0, 1).

(9.2)

We note further the well-known relations between any two vectors p, q and the angle θ between them: the scalar (dot) product p · q, which satisfies p · q = kpkkqk cos θ,

(9.3)

and the vector (cross) product p × q, which satisfies kp × qk = kpkkqkk sin θ k.

(9.4)

And we define the scalar triple product for three vectors:

606

........CHAPTER 9. 3D ANALYSIS

[p, q, r] ≡ p · (q × r) = (p × q) · r p x p y pz = qx qy qz r x r y rz

.

(9.5)

The triple product is invariant under cyclic permutation: [p, q, r] = [q, r, p] = [r, p, q].

(9.6)

In 3D, a fold consists of a rotation of some region of the paper through a specified angle about a fold line. Such rotations can be expressed as left-multiplication by a rotation matrix (possibly preceded and/or followed by a translation). Let w = (w x, w y, wz ) be a unit direction vector defining an axis of rotation that passes through the origin. The result of a rotation through angle φ about axis w can be written in two equivalent ways: q = p cos φ + (w × p) sin φ + w(w · p)(1 − cos φ) = R(φ, w) · p,

(9.7)

where the rotation matrix R(φ, w) is efficiently computed with Rodrigues’s Formula [123]: R(φ, w) = I + W(w) sin φ + W(w)2 (1 − cos φ),

(9.8)

where I is the 3 × 3 identity matrix and 0 −wz w y © ª 0 −w x ® . W(w) ≡ wz 0 ¬ « −w y w x

(9.9)

It is useful to define rotations about the three axes xˆ, yˆ, zˆ in the global coordinate system, namely 1 0 0 © Rx (φ) ≡ R(φ, xˆ ) = 0 cos φ − sin φ « 0 sin φ cos φ cos φ 0 sin φ © 0 1 0 Ry (φ) ≡ R(φ, yˆ ) = − sin φ 0 cos φ « cos φ − sin φ 0 © ˆ Rz (φ) ≡ R(φ, z) = sin φ cos φ 0 0 1 « 0

ª ®, ¬ ª ®, ¬

(9.10)

ª ®. ¬

We can define a 3D coordinate system by a 3 × 3 matrix whose columns are the three orthonormal basis vectors of the system. If î = (i x, i y, iz ), ˆj = ( j x, j y, jz ), kˆ = (k x, k y, k z ) are three orthonormal vectors defining a coordinate system, we can write that system as

........

CHAPTER 9. 3D ANALYSIS

607

the matrix

i j k © x x x ª T ˆ ˆ ˆ M = (i, j, k) = i y j y k y ® . (9.11) i j k z ¬ « z z For example, the coordinate system consisting of the unit vectors xˆ = (1, 0, 0), yˆ = (0, 1, 0), zˆ = (0, 0, 1) is represented by the identity matrix. From such a coordinate system, we can obtain a new, rotated coordinate system by left-multiplying the original by R(φ, w), where w is the axis of rotation in the global coordinate system. So, for example, if M is a local coordinate system, then M0 = R(φ, w) · M

(9.12)

gives the new coordinate system M0, whose columns are the rotated versions of the unit vectors from which M was assembled. Often, though, we would like to transform a coordinate system by rotating it about an axis that is expressed within the local ˆ system: for example, rotating M about its own î-, ˆj-, or k-axis through some angle φ. In this case, the transformed coordinate system is obtained by right-multiplying by the transposed rotation matrix; that is, M0 = M · R(φ, w0)T , (9.13) where the rotation axis w0 is now defined in the local coordinate system of M. In practice, we will commonly wish to rotate a local coordinate matrix about one of its three local coordinate axes, and for those three cases, R(φ, w0) will be one of Rx (φ), Ry (φ), or Rz (φ). Given two vectors a and b perpendicular to a third vector c, we define the 3D directed angle function ∠(a, b, c) as the right-handed rotation angle from a to b about (not necessarily normalized) axis c. It is given by ∠(a, b, c) ≡ tan−1 ((a · b)kck, (a × b) · c) = tan−1 ((a · b)kck, [a, b, c]),

(9.14)

where we use the four-quadrant arctangent function tan−1 (x, y), whose range covers 2π, e.g., (−π, π]. So ∠(a, b, c) satisfies the identity c b = R ∠(a, b, c), · a, (9.15) kck as illustrated in Figure 9.2.

608


a

b

Figure 9.2. c

∠(a, b, c)

Geometry of R(∠(a, b, c), c) and ∠(a, b, c).

When we need a 3D rotation angle from one plane to another, we are not always lucky enough to have perpendicular vectors a, b, c. It is convenient to have formulas based on arbitrary 3vectors. Given three 3-vectors {p, q, r} as shown in Figure 9.3, the area of the triangle formed by the three points {p, q, r} is ∆pqr =

1 k(q − p) × (r − p)k, 2

(9.16)

or its equivalent under any permutation of {p, q, r}. The rotation angle α from line pq to line pr satisfies | sin α| =

k(q − p) × (r − p)k . k(q − p)kk(r − p)k

(9.17)

and

(q − p) · (r − p) . (9.18) k(q − p)kk(r − p)k The right side of the expression for sin α is always positive; we cannot determine the sign of sin α because the direction of rotation is undetermined. The sign of the rotation angle depends on which side of the triangle is taken as the direction of the axis of rotation. Once we choose an axis of rotation, we can assign the sign of sin α. This relation is often useful when α is the sector angle around a vertex, in which case α is going to lie in the range (0, π) (as a sector angle is supposed to); in this case we know that sin α is positive and can drop the | . . . | from the left side of Equation (9.17). For cosine, the sign is determined; there is no ambiguity. cos α =

r p

a q

r

a Figure 9.3.

p q

Left: a triangle with vertices {p, q, r}. Right: same triangle but with α > π.

........


609

s

g

Figure 9.4.

r

p

A tetrahedron with vertices {p, q, r, s}.

q

If the vertices p, q, r form a triangle and α is taken to be the interior angle of that triangle, then we know that α ∈ [0, π], sin α ≥ 0, and so α = tan−1 ((q − p) · (r − p), k(q − p) × (r − p)k) .

(9.19)

Four vectors {p, q, r, s} define a tetrahedron with four triangular faces, as in Figure 9.4. Any edge of a tetrahedron can define an axis of rotation for its two incident faces; each face defines a plane, and we can ask what the signed rotation angle is from one plane to the other. For the tetrahedron with ordered vertices (p, q, r, s), we define angle γ as the rotation angle from the plane of triangle pqr counterclockwise about axis pq to the plane of triangle pqs. That angle satisfies two trigonometric identities: sin γ =

k(q − p)k ((s − r) · [(p − r) × (q − r)]) , k(q − p) × (r − p)kk(q − p) × (s − p)k

(9.20)

(p − s) × (q − s) (p − r) × (q − r) · . (9.21) kp − skkq − sk kp − rkkq − rk These (and many other related and useful identities) may be found in [126, 125]. The tetrahedron angle identities arise when line pq defines a fold and points {r, s} are points in two adjacent facets. Then Equations (9.20) and (9.21) can be used to compute the fold angle γ between the two facets. With those relations and notation now defined, we can turn to computing relations between the sector angles, fold angles, and directions around a vertex in 3D. cos γ =

???

9.2. 3D Vertices In the previous chapter, we characterized 3D vertices by various geometric quantities, aided by points on the Gaussian sphere to represent directions and planes. While there is much that can be

610


said about vertices using this geometric model, in many cases calculations involving vertices are simplified considerably if we use their vector representation. Here we develop vector descriptions of the points, concepts, and angles of general 3D vertices. ? ? ? 9.2.1. Fold Direction Vectors If we place the vertex at the origin, then each of the folds defines a vector pointing outward along the fold line. For the 3D folded form, we call the unit normal vectors giving these directions the fold vectors {gi } for the vertex. The paper surrounding the vertex consists of the facets of paper between the folds. Each facet has a unit normal vector (defined to point toward the “white side” of the paper by convention); these normal vectors are called the sector vectors {ai }. The sector vectors point toward the corners of the trace on the Gaussian sphere, as shown in Figure 9.5. If we order the facets around the vertex—by convention, we will order them counterclockwise when viewed from the white side of the paper—then we can number both the fold vectors and sector vectors to go along with the fold angles and sector angles already introduced. Each point on the Gaussian sphere defines an important direction in the folded form of the vertex, and we can define a direction vector for each of the points we identified in the previous chapter. There is one that turns out to be particularly significant in degree-4 vertices, which is the osculating plane normal q. This point is the crossing of the bow-tie trace; it is the normal to the plane that contains the major folds γ2 and γ4 , and so is perpendicular to both of g2 and g4 . g2 g2

g3

g3 a2

a1

a3 g4

g4

a3

a4 a4

a2

a4 a4

a1

g1 g1 g1

a1

a1

g4 a2 q a3

-g3

Figure 9.5. Fold vectors gi giving fold directions and ai giving sector normal directions on the vertex and Gaussian sphere.

........


611

g2

g2¢ a2 g3

Figure 9.6.

g3¢

a1

g1¢

g1

a4

a3

g4¢

Vectors gi0 giving crease directions in the crease pattern.

g4

Vector notation permits both a compact expression for directions and angles and an efficient computational procedure for moving among the various parameters of a vertex. The particular labels we have chosen reflect the greek letters used for related angles: • gi is the direction of the fold with fold angle γi . • ai is the direction of the normal to facet with sector angle αi . • q is the direction of the normal to the osculating plane where θ was the osculating angle. We can similarly construct crease vectors giving the directions of creases in the crease pattern; these will be 2-vectors and are related by the 2D rotation matrix R(θ) introduced in Section 1.6.4, as illustrated in Figure 9.6. To distinguish the 2D direction vectors from their 3D counterparts, I will prime the 2D variables here. We can choose any one of the 2D direction vectors arbitrarily: say, g01 . Then the others are simply given by rotations in the plane of the sector angles, i.e., g0i+1 = R(αi ) · g0i,

(9.22)

where the 2D rotation matrix R(φ) is given by Equation (1.23). ? ? ? 9.2.2. Vertex from Fold Directions We first provide relationships between the vectors gi , ai and the angles γi, αi for an arbitrary 3D vertex. For a general vertex of arbitrary degree, given a set of fold vectors {gi } in counterclockwise order as viewed from the white side of the paper, the vertex

612


is fully specified and the other sets of angles and vectors can be calculated. The sector vectors {ai } can be calculated as follows: ai =

gi × gi+1 , kgi × gi+1 k

(9.23)

where subscripts are to be interpreted cyclically. The fold angles at each crease are then given by γi = ∠(ai−1, ai, gi ).

(9.24)

The assignment of each crease is then given simply by the sign of the fold angle. If the vertex comes from a developable surface and is nondegenerate (none of the sector angles or fold angles are zero), then the sector angles must each lie in the range (0, π) and so are given by αi = cos−1 (gi · gi+1 ). (9.25) For degree-4 vertices, we also have the osculating normal vector q and the sector elevation angles {σi } to deal with. To compute the osculating and ruling planes, we must identify the major and minor folds of the vertex. So far in this chapter, we have consistently assumed that γ2 and γ4 were the two major folds and γ1 and γ3 were the two minor folds, but a general degree-4 vertex might well arrive without the folds numbered this way, and so we need an algorithm to identify these folds and assign a set of numbers—at least for purposes of computing the parameters unique to degree-4 vertices. We note that, as mentioned earlier, for every non-degenerate degree-4 vertex, whether flat-foldable or not, there must be three vertices of one type and one of the other—either V 3 M or M 3V. Thus, there will only be one pair of opposite folds of the same parity; those will be the two major folds, with the remaining two being minor. Of the two remaining folds, which are the minor folds, one will match the major folds; we will call it the matching minor fold, while the other will be the mismatched minor fold. There is still the possibility that the major and matching minor folds could all be either valley or mountain. In our definition of sector elevation angles earlier, we only showed a valley-like vertex, i.e., vertex configuration in which those three folds were valley. For this configuration, each of the sector elevation angles CHAPTER 9. 3D ANALYSIS

........

613

g2

s1

q

a4

a2 a2

a1

g3

a3

g1

a4

a3

s3

g1

g3

g2

g4

s3

g1 a1 s2

a1

a3 a2

-g3

s4

g4 g2

a2

a1

g3

q

g1

a3 g3

a4

a3

-s3

a3 g2

g4 a4

a2

g1 a1

g3

-s2 a2

-s4 -s1 a1

a4 -g1

-s4

g4

Figure 9.7.

Top row: a V 3 M vertex (positive sector elevation angles). Bottom row: an M 3V vertex (negative sector elevation angles). Left: crease pattern. Middle: folded form. Right: trace.

fell within the range (0, π/2), where an angle of 0 corresponded to a fully flat unfolded vertex and an angle of π/2 corresponded to a fully folded vertex. If we start with a fully folded valley-like vertex, unfold it to flat, and then keep going until it is fully folded but mountain-like, then the sector elevation angles will go from π/2 to 0 to −π/2, as shown in Figure 9.7. This has the effect that the trace will become inverted about the osculating vector. Note that it always remains within the hemisphere centered on the osculating vector. So, for the general degree-4 vertex, each sector elevation angle lies within the range (−π/2, π/2). Since the osculating normal is perpendicular to both g2 and g4 , we have g × g2 q=± 4 . (9.26) kg4 × g2 k

614


There is an ambiguity in sign, but that can be resolved based on the values of the sector angles: we take the (+) sign if α4 + α1 < π and the (−) sign if α4 + α1 > π. And if α4 + α1 = π? Then the vertex is straight-major, and Equation (9.26) is undefined (and the minor folds are either coaligned or unfolded). Given q, the sector elevation angles {σi } can be computed from rotations of the sector planes relative to the osculating vector about the major folds: σ1 σ2 σ3 σ4

= +∠(a1, q, g2 ), = −∠(a2, q, g2 ), = +∠(a3, q, g4 ), = −∠(a4, q, g4 ).

(9.27)

With all these parameters, it is now possible to construct 3D representations of individual vertices, of collections of vertices, and—most importantly—of the 2D crease patterns that give rise to them. ? ? ? 9.2.3. Degree-4 Vertex from Sector Elevation Angles Instead of starting with all four fold directions and computing the resulting four sector elevation angles, we may start with two folds and two sector elevation angles, and we would need to compute the remaining parameters of the vertex. This situation arises, for example, in the construction of a discrete space curve (which we will address in the next section), where the two major folds (which form the backbone of the space curve) are given, along with the two sector elevation angles on one side of the vertex. Specifically, assume we are given the two fold directions g2, g4 and the two sector elevation angles σ3, σ4 , as well as the fold angle expansion ρ. From given sector elevation angles σ3 and σ4 , the other two sector elevation angles are given by σ1 = 2 tan−1 ρ tan 12 σ3 , (9.28) σ2 = 2 tan−1 ρ tan 12 σ4 , and thus, the two major fold angles are γ2 = σ1 + σ2,

γ4 = σ3 + σ4 .

(9.29)


........

615

g2

q q

g3

a2 a1

Figure 9.8. Two possible configurations for a vertex for the two choices of osculating vector q with identical g2, g4, σ3, σ4 . Note that the choice of osculating vector effectively chooses the orientation of the white side of the paper. Top: q points up. Bottom: q points down.

aa33

g1

g3 a4

a3

g1

g3 g4

s3

a2

a2 g1 g2

a4

aa1 1

g2

s4

g4

g4 g2 a1

a2

g1

g3 a4

a3 g4

a4 g1 s4

g4

g2 g4

s3 g1

g2

g3 aa22 a3

q

a1

g3

In fact, though, there is one more factor that must be specified: the direction of the osculating vector q (which effectively determines the direction of the white side of the paper). There are two choices for q: g × g2 q=± 4 , kg4 × g2 k but since we don’t yet know the sector angles, we can’t choose the sign as we did in the previous subsection. Figure 9.8 illustrates the two possibilities for identical g2, g4 , σ3, σ4 . In the previous subsection, all four fold directions are given and there is no ambiguity in direction of the osculating vector (because the fold directions are specified CCW with respect to the osculating vector). But if we are only given two fold directions and two sector elevation angles, then the osculating vector is not yet fully specified. We would require a vector q0 that specifies the half-space into which q should point; if q · q0 > 0, we would take the positive sign and otherwise take the negative sign in our algorithm. Once we have chosen the direction of osculating vector q, we can proceed with the construction of the sector vectors and then

616


the remaining fold direction vectors. We have a1 a2 a3 a4

= R(−σ1, g2 ) · q, = R(+σ2, g2 ) · q, = R(−σ3, g4 ) · q, = R(+σ4, g4 ) · q.

(9.30)

The two remaining fold direction vectors may now be computed: a4 × a1 , ka4 × a1 k a3 × a2 g3 = ± , ka3 × a2 k g1 = ±

(9.31)

with the sign chosen so that g1 and g3 point into the same halfspace as q. The sector angles and any other desired vertex parameters may now be calculated from the formulas in the previous subsection. ???

9.3. Discrete Space Curve We can now put the formulas from the previous section into practice and compute the crease pattern for a folded form that follows an arbitrary discrete space curve when folded along a specified fold angle. Such a curve is piecewise-straight—that is, composed of discrete line segments—so this approach does not give us a formula for folding a smooth space curve. In practical computation, however, any continuous curve must be discretized to some level, and so the algorithm that follows is, in fact, quite a useful and practical one. We assume that we are given a set of points {p(i) }, i = 0, . . . , n, that trace out a space curve in 3D. In this analysis, we will use the same notation for directions and angles at each vertex for consistency, but we will use the superscript (i) to indicate which vertex we are considering, as shown in Figure 9.9. We further assume that we are given a vector q0 that defines the direction of the white side of the paper, initial sector elevation angles σL, σR that define the angles of the left and right sector planes relative to the first osculating vector, and desired widths w L, w R of the paper on either side of the space curve. Thus, the paper will initially be a flat strip of desired width w L + w R

........


617

p(n) r(n)

r(3) p(2) g3(1) a2(1)

Figure 9.9.

Labeling of points p(i) , curve directions r(i) , and crease directions g(i) j for a discrete space curve.

q0 sL p(0) sR

a3(1) r(1)

p(1) g4(1)

r(2)

g2(1)

a1(1) g1(1)

a4(1)

that traces out some path in the plane; when it is folded along the curve with the fold angle σL + σR , it will form the desired three-dimensional space curve. To begin, we define direction vectors along the curve and the lengths of each segment: p(i) − p(i−1)

, r(i) =

p(i) − p(i−1)

(i) (i) (i−1) l = p − p

.

(9.32)

We will number the fold directions, sector normals, etc., using the same counterclockwise numbering scheme that we have used throughout this chapter, but with a small modification: we will always start with g1, γ1 being the minor crease on the right side of the paper strip. With this definition, then, for each vertex (except the first and last), the two major fold directions are given by (i) g(i) 4 = −r , (i+1) g(i) . 2 = +r

(9.33)

We will need to solve for the two minor creases at each vertex in order to compute the entire pattern. Now we compute the boundary conditions. We will assume no edge torsion in the first segment, so that the osculating vectors

618


at the first two vertices are equal: q(0) = q(1) .

(9.34)

For the initial osculating vector q(0) , we know that it will be (1) equal to q(1) , which will be one of ±(g(1) 2 × g4 ). We choose the sign so that q(1) points into the same half-space as the specified q0 , and with that, we can now construct the remaining parameters of the first vertex. The first vertex p(0) and last vertex p(n) are “half-vertices,” since we only have sector planes on two sides of them. For simplicity, we will assume that there is zero bend at the vertex and the sector angles are all equal to π/2 at both ends of the strip, so that αi(0) = αi(n) = π/2. (9.35) The sector angles at the first vertex are specified, so that σ1(0) = σR,

(9.36)

σ2(0) = σL, and so the initial fold angle is simply γ2(0) = σ1(0) + σ2(0) = σR + σL .

(9.37)

So, for the first vertex, the sector normals are given by rotating the osculating vector about the major creases through the sector elevation angles, i.e., (0) a1 = R(+σR, g(0) 2 )·q ,

(9.38)

(0) a2 = R(−σL, g(0) 2 )·q .

The two minor fold directions can also be constructed by rotating the osculating vector: (1) (0) π g(0) 1 = R(+( 2 − σR ), r ) · q , (1) (0) π g(0) 3 = R(−( 2 − σL ), r ) · q .

(9.39)

Now, we march along the chain of vertices, using the formulas from Section 9.2.3 to compute the vertex parameters for each vertex, as follows. We start by calculating the osculating vector for the vertex, g(i) × g(i) 4 q(i) = ± 2(i) , (9.40) kg2 × g(i) k 4

........


619

taking the sign so that q(i) · q(i−1) > 0, i.e., the two consecutive osculating vectors point into the same half-space. We then find the edge torsion angle for the segment as the angle by which the osculating vector rotates from one segment to the next: τ (i) = ∠(q(i−1), q(i), r(i) ).

(9.41)

The incoming sector elevation angles for the vertex are given by the outgoing sector elevation angles of the previous vertex, modified by the edge torsion of the intervening segment: σ3(i) = σ2(i−1) − τ (i), σ4(i) = σ1(i−1) + τ (i) .

(9.42)

And now that we have the major fold directions, osculating vector, and sector elevation angles for the vertex, we can use the formulas from Section 9.2.3 to compute the remaining vertex parameters and, by this method, construct vertex parameters for all the remaining interior vertices. The final vertex, like the first vertex, is a special case since it, too, is a “half-vertex.” We assume no edge torsion in the last segment, so that q(n) = q(n−1) . (9.43) Then the final vertex parameters are σL0 ≡ σ3(n) = σ2(n−1), σR0 ≡ σ4(n) = σ1(n−1), γ4(n) = σL0 + σR0 , 0 (n) (n) π g(n) 1 = R(+( 2 − σR ), r ) · q , 0 (n) (n) π g(n) 3 = R(−( 2 − σL ), r ) · q ,

(9.44)

(n) g(n) 4 = −r , 0 (n) (n) a(n) 3 = R(−σL, g4 ) · q , 0 (n) (n) a(n) 4 = R(+σR, g4 ) · q ,

α3(n) = α4(n) = π/2. Having completed the construction of the vertex parameters in 3D, we can now turn our attention to constructing the crease

620


pattern, which will depend solely upon the sector angles that we computed at each vertex. If we assume that the first vertex is located at the origin (in 2D), the second vertex lies on the x-axis, then the path of the curve in 2D—the path taken by the spine of the folded sheet when unfolded—follows straightforwardly. As we have done before, we denote points and vectors of the crease pattern by the same symbol as in the 3D folded form, but primed. Let p0(i) be the coordinate of the ith vertex in 2D, and let r0(i) be the unit direction vector from the (i − 1)th vertex to the ith vertex in 2D. Then we have for the vertices of the crease pattern of the space curve, r0(1) ≡ (1, 0), r0(i) = R(α4(i−1) + α1(i−1) − π) · r0(i−1),

(9.45)

p0(i) = p0(i−1) + l (i) r0(i), where, as before, R(φ) is the 2D rotation matrix through an angle φ. The crease direction vectors emanating from each vertex follow immediately from the sector angles as well: (i) 0(i) g0(i) 1 = R(α4 ) · (r ), (i) 0(i) g0(i) 2 = R(α1 ) · (g1 ), (i) 0(i) g0(i) 3 = R(α2 ) · (g2 ),

(9.46)

(i) 0(i) g0(i) 4 = R(α3 ) · (g3 ).

This gives the 2D positions of the vertices {p0(i) } of the space curve itself, but to construct the crease pattern for the folded edge, we need to construct the creases on either side. For this, we assume that we have two widths w L and w R specified, which are, respectively, the desired widths of the strip on the left and right side of the folded edge. However, we cannot achieve a constant width in the general case; we can only approximate this width. Figure 9.10 illustrates why. For a single vertex, let us denote by wi(i) the width of the strip incident to sector αi(i) . Then widths w2(i) and w3(i) must meet at a point along crease γ3(i) , and similarly, widths w1(i) and w4(i) must meet at a point along crease γ1(i) . Let t L(i) be the distance of the left point from the vertex and t R(i) be the distance of the right point from the vertex. Then, from the

........


621

(i)

w1 (i)

w2

g3/(i)

Figure 9.10.

(i) a2(i) a1 (i) a3(i) a4

tL(i)

Schematic of a vertex of the folded edge and the strip of paper on either side.

w3(i)

g1/(i)

tR(i)

w4(i)

geometry illustrated, we must have t L(i) t R(i)

= =

w2(i) sin α2(i) w1(i) sin α1(i)

= =

w3(i) sin α3(i) w4(i) sin α4(i)

, (9.47) .

Clearly, there is no way in general for w2(i) and w3(i) to both be equal to some desired width w (i) L , since their ratio must be w2(i) w3(i)

=

sin α2(i) sin α3(i)

,

(9.48)

and similarly for w1(i) and w4(i) . A reasonable compromise is to set the geometric mean of the two variables equal to the desired width, so that v v u u t t (i) sin α3(i) sin α (i) (i) 2 w2 = w L, w3 = w L, sin α3(i) sin α2(i) (9.49) v v u u t t (i) (i) sin α1 sin α4 (i) w1(i) = w , w = w R, R 4 sin α4(i) sin α1(i) and

622

wL t L(i) = q , (i) (i) sin α2 sin α3


wR t R(i) = q . (i) (i) sin α1 sin α4

(9.50)

(i)

a2(i) a1 (i) a3(i) a4

(i) wL,max

a2(i-1)

l(i)

a1(i-1) (i-1) a3 a4(i-1)

Figure 9.11. Geometry of the maximum width on the left side of the strip.

The vertices on either side of vertex p0(i) are given, respectively, by (i) 0(i) (i) 0(i) p0(1) p0(1) (9.51) L = t L g3 , R = t L g1 . The crease assignments in the crease pattern are, of course, determined by the signs of the fold angles around each vertex in the folded form. There is one other factor to consider, however. On the inside of a curve, consecutive minor creases angle toward one another and eventually intersect. This sets an upper limit on the width of the strip on that side. For each segment of the polyhedral space curve, the sector angles at each end define this upper limit for that segment. If we choose a constant desired width for the entire strip, then the one worst-case segment will limit the width for the entire curve. Figure 9.11 illustrates the geometry of this maximum-width situation for a polyhedral curve that curves to the left. The maximum width for a single segment is clearly given by w (i) L,max = (i) w R,max

=

l (i) cot α2(i−1) + cot α(i) l (i) cot α1(i−1) + cot α4(i)

, (9.52) ,

with separate limits for both left and right. If the two creases angle (i) away from each other , then the value of w (i) L,max and/or w R,max may be negative, in which case they can be ignored.

........


623

To relate these values to the widths w L and w R chosen as the target values for the entire strip, we must recognize that w (i) L,max (i) and w R,max aren’t the maximum values of w L and w R , but rather

are the maximum values of w (i) j at the two adjacent vertices. This gives rise to four inequalities that must apply along each segment of the curve: v u t wL ≤ v u t wR ≤

sin α2(i) (i) w L,max, sin α3(i)

sin α1(i) (i) w R,max, sin α4(i)

v u t wL ≤

sin α2(i−1) v u t

wR ≤

sin α3(i−1)

w (i) L,max

sin α4(i−1) (i) w R,max sin α1(i−1)

if w (i) L,max > 0, (9.53) (i) if w R,max > 0.

Consequently, after analyzing the complete curve, one must check every segment of the curve to ensure that the chosen widths do not exceed these limits. To illustrate this algorithm, Figures 9.12 and 9.13 show two crease patterns that, when folded, curl up into 3D to form knots: specifically, a trefoil and a (1, 4) torus knot (which is topologically an unknot but nevertheless has an interesting space curve). A trefoil knot has a parametric representation f(t) = (sin t + 2 sin 2t, cos t − 2 cos 2t, − sin 3t) for t ∈ [0, 2π].

(9.54)

Discretizing this into a series of points gives a polygonal path, from which we can construct the crease pattern according to this prescription. Note that the crease pattern (almost) forms a closed loop. Of course, there needs to be a break in the crease pattern so that the folded form can take on the knotted state. Another knot, the (1, 4) torus knot, has a parametric representation f(t) = (cos t(cos 4t + 2), sin t(cos 4t + 2), sin 4t) for t ∈ [0, 2π].

(9.55)

The same treatment also gives a crease pattern for its polygonal discretization, shown in Figure 9.13. Many other interesting and beautiful curves are undoubtedly possible, and the same technique can be used to create discretized versions of 3D curves for incorporation into geometric designs.

624


Figure 9.12.

Left: crease pattern for a folded trefoil knot. Right: folded form.

Figure 9.13.

Left: crease pattern for a folded (1, 4) torus knot. Right: folded form.


........

625

When we discretize a space curve in this way, we have the choice of discretization: how closely spaced do we make our vertices? As we take ever more points for a given curve, the minor folds get more and more numerous, but shallower and shallower. In the limit as the quantization distance—the distance between consecutive points—goes to zero, the chain of major folds approaches a smooth curve, and the minor folds become ruling lines of curved-fold surfaces. The behavior of discretized curved folds can therefore give some insight into the behavior of folds and their geometric properties of smoothly curved folds—which otherwise require the tools of differential geometry [47, 23, 28, 21, 20] to describe. ???

9.4. Plate Model Most of the time, when we are trying to solve for an origami design, we seek to find a crease pattern and a folded form that satisfies some set of conditions defined by the application, aesthetics, or a combination of the two. We may have been given one of those, the other, or a mixed bag of requirements. That is, there are three potential classes of problem we might encounter: • Given a crease pattern and its fold angles, what is the 3D folded form? • Given a 3D folded form, what is its unfolded crease pattern? • Given some set of desired properties of the 3D folded form, find both folded form and crease pattern so that the former unfolds to the latter. The first two of these problems, and some instances of the third, can be addressed by using what is called a plate model of both the crease pattern and the folded form. We treat the facets of both crease pattern and folded form as a collection of fixedsize plates connected by the folds between the facets. This is particularly useful if one of either the crease pattern or folded form is fully known, because given one, there is a straightforward computational algorithm to realize the other.

626


? ? ? 9.4.1. Folding a Crease Pattern Suppose that the crease pattern is known, meaning we have values for all of the coordinates {pi } of the crease pattern. Then the 3D folded form is fully defined and can be constructed if we know (a) the position of one of the facets in 3-space and (b) all of the fold angles in the pattern. Let us call that first facet whose position and orientation we know the root facet. Its vertices in 3D can be written as a rigidbody motion of the same facet in the crease pattern: a combination of translation and rotation from its original position. That is, for each of the vertices of the root facet, if the crease pattern vertex is pi = (pi,x, pi,y ), the folded form vertex can be written as p0i = t + R(φ, w) · (pi,x, pi,y, 0)

(9.56)

for some rotation angle φ, axis w, and translation t. It is common and often simple to take φ = 0, t = (0, 0, 0), in which case we call the root facet the stationary facet—stationary because its position is the same in the crease pattern and folded form. To construct the 3D folded form of the other facets, think about how you might get from the stationary facet to any other facet if you were an ant crawling on the paper. Starting from the stationary facet, you would crawl across the face, and then each time you crossed an edge, there would be a rotation at that edge through the fold angle. You could build up the folded form by rotating each facet about the fold angle that connects it to one of its neighbors. To ensure that no facet is left out, we construct the spanning tree on the facets—connecting the facets in a network so that there are no loops in the chains of connection. This is very much like constructing the interior dual graph, except we delete any edges from the dual graph that would create a loop and refrain from deleting any edges that would disconnect parts of the pattern. Figure 9.14 shows the spanning tree on a crease pattern; the lower left facet is the stationary facet. The stationary facet is the root of the tree; the facets farther away from the root along each branch are the leaves of the tree. Once we have the spanning tree, we can construct the folded form by starting at each of the leaf facets of the tree and sequentially working our way back to the root, the stationary facet, at each

........


627

-60° -90° 60°

60° -90°

90° 90°

-90° -60°

-60°

90° 60°

Figure 9.14. Setting up a spanning tree to calculate the folded form of a crease pattern. Left: crease pattern with fold angles on the interior folds. Right: spanning tree on the facets. The larger dot marks the stationary facet.

step folding the steadily lengthening branch of facets, as shown in Figure 9.15. In the intermediate stages, folding some creases but not others has the effect of breaking up the pattern, but once we have folded all of the creases along their proper fold angles (and assuming we have chosen the fold angles to be self-consistent), all of the facet edges will realign with each other along their proper folds. Thus, if we are given a crease pattern and a set of fold angles, we can easily construct the folded form. Almost the same procedure can be used to extract the crease pattern from an existing folded form. Given the folded form, we know the fold angles (or can easily compute them from the facet normals on either side of each crease). By following the same procedure, but applying the negative of each fold angle to its corresponding crease, we effectively unfold the folded form to a flat state (albeit still 3D); embedding the result in the x y-plane then transforms it into a crease pattern. ? ? ? 9.4.2. Fold Angle Consistency If we don’t know the fold angles to apply to a crease pattern, we might wish to choose them, for example, to see the 3D folded form for a given crease pattern. But we cannot choose them arbitrarily. We saw in the trigononometric treatment of vertices of Chapter 8 that we had conditions on sector angles: developability, which

628


Figure 9.15. Sequentially folding a crease pattern into a folded form. Red lines show the next crease to be folded at each step from crease pattern (upper left) to folded form (lower right).

ensures that a folded form can be unfolded to a flat sheet of paper, and, possibly, flat-foldability, which ensures that a crease pattern can fold into a flat shape. But there are other conditions, too, that link the sector and fold angles. As we saw, the fold angles {γi } and sector angles {αi } cannot be chosen entirely independently for a vertex. For a degree-4 vertex, given the sector angles, one can choose only a single fold angle (plus its major/minor status); from there, the other three fold angles are determined. Similar constraints apply to higher-order vertices. For a developable vertex the trace on the Gaussian sphere must have zero net area. Even if we don’t require developability, though, the trace must still be closed; if we travel around the vertex, we better end up where we started, whether traveling in real space or on the Gaussian sphere. This is a consistency requirement. Figure 9.15 hints at one implication of the consistency requirements. When we’re in an intermediate state, with some fold angles folded and others not, the faces of the pattern that are supposed to meet along folds don’t necessarily do so. If we fold a crease pat-

........


629

tern using only the fold angles encountered along the spanning tree of the facets and we haven’t chosen the fold angles consistently, the facet edges won’t line up along their proper folds when all rotations have been applied. There is a consistency requirement that must apply at every vertex of the pattern. In 2D, that consistency requirement was the Kawasaki-Justin Condition, which we encountered in Chapter 1. It has several incarnations, but they all boil down to the same concept: if you follow the layers of paper around a vertex in the folded form, using the fold angles and sector angles of the crease pattern as your guide, you must eventually get back to where you started. For flat folds, where we trace sector angles forward and backward around a circle, the mathematical expression of that concept is Theorem 1, the Kawasaki-Justin Theorem. We can express the same concept using the Gaussian sphere model of Chapter 8. If we construct the trace of a vertex on the Gaussian sphere, drawing an arc for every fold angle and making a turn for every sector angle, then as we go around the vertex, the trace must close on itself; it must form a closed spherical polygon. That’s the same notion. At the Second Internation Meeting of Origami Science and Scientific Origami, Toshikazu Kawasaki noted that this concept, a loop consistency condition, could be generalized to 3D (and even to higher dimensions) [62]. Broadly speaking, the process of “traveling around the vertex” could be expressed as a sequence of rotations about fold angles and sector angles. Then the general form of that consistency condition would be that the product of that sequence of rotations must give the identity transformation, i.e., you should end up where you started. Kawasaki expressed this concept as R(Γ) = I,

(9.57)

where R(Γ) is the product of rotations and I is the identity matrix. Using the rotation matrices from Section 9.1, we can construct R(Γ) for an arbitrary vertex and create an explicit formula for the Kawasaki-Justin Condition that applies to a partially folded vertex of any degree. We will assume a general degree-n vertex with fold angles {γi }, i = 1, . . . , n, and sector angles {αi }, i = 1, . . . , n. (For simplicity, we will illustrate with a degree-4 vertex.)

630


Figure 9.16.

ˆ coordinates system, with î = red, ˆj = green, kˆ = blue. The (î, ˆj, k) Left: local coordinate system with the î-axis along γ1 and the local îˆj-plane aligned with sector α1 . ˆ Middle: rotation about the new local k-axis through angle α1 . Right: rotation about the local î-axis through angle γ1 . Now the local îˆj-plane is aligned with sector α2 .

To begin, we will define a local coordinate system for our vertex consisting of three orthonormal vectors, î ≡ (i x, i y, iz ), ˆj ≡ ( j x, j y, jz ),

(9.58)

kˆ ≡ (k x, k y, k z ), where, for simplicity and definiteness, we will orient î along crease γ1 and put the îˆj-plane in the plane of sector α1 , as shown in Figure 9.16. We will then rotate that coordinate system around the vertex according to the sector and fold angles of the folded form and require that we end up where we started. ˆ Since we are rotating about local î- and k-axes, we call upon the two special cases of the general rotation matrix R(φ, w) previously defined: cos φ − sin φ 0 © ª Rz (φ) ≡ R(φ, zˆ ) = sin φ cos φ 0 ® , 0 1 ¬ « 0 which rotates a point about the global z-axis through angle φ, and 1 0 0 © Rx (φ) ≡ R(φ, xˆ ) = 0 cos φ − sin φ « 0 sin φ cos φ

ª ®, ¬

which rotates a point about the global x-axis through angle φ. If we left-multiply any vector by one of these rotation matrices, it will rotate it about the global zˆ - or xˆ -axis. We don’t want that;

........


631

we want to rotate our coordinate system about its own local î- or ˆj-axis. Specifically, we want to rotate the coordinate system by ˆ sector angle α1 about its local k-axis (blue in Figure 9.16), then rotate by fold angle γ2 about the new local î-axis (red), and so forth, all the way around the vertex. As it turns out, we can accomplish this very simply. We ˆ into a matrix M whose columns are arrange the vectors {î, ˆj, k} the local coordinate vectors: i © x M ≡ iy « iz

jx k x ª jy k y ® . jz k z ¬

(9.59)

If we right-multiply M by the relevant rotation matrix, the result is a new set of local coordinates that have undergone the desired rotation about the local axes. So, in Figure 9.16, in the left subfigure, the three local axes are the columns of M. In the middle subfigure, we have rotated through fold angle γ1 , and the new local axes are the columns of M · Rx (γ1 ). In the right subfigure, we have further rotated about the local z-axis, and the resulting local axes are the columns of M · Rx (γ1 ) · Rz (α1 ). And so we can continue, all the way around the vertex. The loop consistency condition, then, is that the product of all of the rotation matrices brings us back to where we started, i.e., that M · Rx (γ1 ) · Rz (α1 ) · . . . · Rx (γn ) · Rz (αn ) = M,

(9.60)

or, by right-multiplying by M−1 , C≡

n Ö

Rx (γi ) · Rz (αi ) = I.

(9.61)

i=1

The product on the left, C, is, of course, precisely Kawasaki’s R(Γ). We define it as the consistency matrix of the vertex. Setting this matrix equal to the identity matrix enforces consistency of the sector and fold angles of the vertex or, equivalently, closure of the trace on the Gaussian sphere. Much of the spherical trigonometry of the Gaussian sphere in the previous chapter is essentially wrapped up in this one equation. In the case of the 2D conditions—developability and flatfoldability—we saw that the matrix conditions could be expressed

632


in terms of angles (and trigonometric functions thereof) or, equivalently, as algebraic functions of the coordinates of the vertices surrounding the vertex under scrutiny. In 2D, the developability matrix condition was automatically satisfied when expressed in terms of the vertex coordinates of a plane embedding. In the same fashion, in 3D, the consistency matrix condition is automatically satisfied when expressed in terms of the vertex coordinates of a 3D embedding (which we have not yet created, but that is coming). However, when we express the 3D consistency matrix in terms of sector and fold angles, it is not automatically satisfied; instead, it captures the linkages between the N sector angles and N fold angles of a degree-N vertex and the relationship that they must satisfy. ? ? ? 9.4.3. Solving for Fold Angles If we are given a crease pattern, we can choose some subset of fold angles and must solve for the others by satisfying the consistency conditions at all of the interior vertices. The number of fold angles we can solve for are precisely the number of degrees of freedom from Chapter 7, which were given by Equation (7.85): Õ DOF = B − 3H + S − 3 − Pk (k − 3), k=4

where • B is the number of edges on the border, • H is the number of holes in the pattern, • S is the number of redundant constraints (typically geometry-specific), • Pk is the number of k-gon facets. An equivalent way of looking at this situation is to say that we start with DOF equal to the number of interior folds, then take away three degrees of freedom at each interior vertex due to the need to satisfy Equation (9.61) componentwise at each vertex. Now, Equation (9.61) says C = I, where both the consistency matrix C and the identity matrix I are 3 × 3 matrices. So, it would appear, there are nine, not three, equations: one for each component of C.

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633

However, C is a product of rotation matrices, which are individually (a) unitary and (b) skew-symmetric, meaning that their product is also a rotation matrix and so C is unitary and skewsymmetric. That means that C has the form C C1,2 C1,3 © 1,1 ª C = −C1,2 C2,2 C2,3 ® , «−C1,3 −C2,3 C3,3 ¬ its determinant is 1, and no element of C has magnitude greater than 1. If we force C1,2 = 0, C1,3 = 0, and C2,3 = 0, that will ensure that all of the off-diagonal terms are zero. Furthermore, since det C = 1, if the off-diagonal terms are zero, it must be that C1,1C2,2C3,3 = 1, and since none of the three can have magnitude greater than 1, they all must have magnitude equal to 1. That, in turn, means that exactly zero or two of them are −1 and one or three are +1. Of course, we want them all to be +1. If we solve for a vertex where C1,2 = C1,3 = C2,3 = 0 but two of the diagonal elements are −1, that is a spurious solution, corresponding to a non-physical configuration of the paper. A very simple example of this situation is shown in Figure 9.17 for a degree-2 vertex with two collinear creases. The crease pattern on the left with equal fold angles γ, which is clearly physically realizable, has a consistency matrix that is the identity matrix. The one on the right, which is (obviously) not

g

Figure 9.17.

g

g

g-p

A degree-2 vertex with two sector angles of 180◦ and two folds to be determined. Left: assigning the two folds the same fold angle γ gives the identity consistency matrix. Right: assigning one fold γ and the other γ − π gives a spurious consistency matrix.

634


physically realizable (at least not with flat facets), has a consistency matrix 1 0 0 © ª C = 0 −1 0 ® , «0 0 −1¬ which is one of the spurious solutions. So, if we incorporate the three off-diagonal consistency equations as equality constraints in a multi-dimensional root-finding framework to solve for fold angles, we need to take steps to ensure that spurious solutions will not be found. If the framework accepts inequality constraints, then we can add the constraint C1,1 + C2,2 + C3,3 ≥ 2,

(9.62)

which will do the trick nicely. You might wonder: since we want each of the diagonal terms to be equal to 1, why not just choose equations C1,1 = 1, C2,2 = 1, C3,3 = 1 as our three equations? There are two problems, both stemming from the fact that |Ci,i | ≤ 1 always. Since the desired value of each Ci,i is at an extremal point of the phase space, as we vary the fold angles in the neighborhood of the solution, |Ci,i | will vary quadratically, so the gradient of the system of equations will vanish in the neighborhood of a solution, leading to potential instabilities in root-finding. Worse: due to numerical round-off errors, it may actually be impossible to perfectly reach the value of Ci,i = 1; it may well be that when you plug in numerical values for a true solution, you might get Ci,i = 0.99999 . . .. Thus, using the off-diagonal elements of the consistency matrix for each interior vertex gives a more robust solution when solving for sets of compatible fold angles. There is another potential source of non-physical solutions: those that self-intersect. Even in a single-vertex crease pattern, it is possible to find solutions that self-intersect but that satisfy all of the vertex consistency conditions. In multi-vertex patterns, the problem of self-intersection arises regularly and can involve non-local intersections: two facets intersect in the folded form, even though they are far removed from one another in the crease pattern. Solving for feasibility while avoiding self-intersection is notoriously difficult. Even for flat-foldable crease patterns, the problem of determining a crease pattern that satisfies the Justin NonCHAPTER 9. 3D ANALYSIS

........

635

Crossing Conditions is known to be np-complete, meaning that there are certain crease assignment problems that are as hard as any problems in a broad class of computational challenges that include, for example, the famous Traveling Salesman Problem [11]. With the additional complication of 3D folding, where stacked layers present numerical challenges in distinguishing ordered layers, it is little wonder that most models focus just on the geometric relationships and rely on the desired configurations of interest being well removed from regions of self-intersection. Should you wish to set up your own simulation of rigid origami using the plate model, a thorough description of a computational model was presented by Tachi [113], who has also written and made available a software tool, Rigid Origami Simulator [119], based on this model. While creating a computational description of an origami pattern as input can be time-consuming, once the model is inside the simulator, manipulations of the pattern can be far simpler, and far more precise, than working with physical models. ???

9.5. Truss Model The plate model of origami is very well suited to modeling a folded form when the crease pattern is known, so that the edge lengths and facet corner angles can be regarded as fixed and the only variables needed are the fold angles. However, it is less useful when we are trying to solve for crease patterns and folded forms, where the vertices in both crease pattern and folded form can move around, thereby changing potentially all of the edge lengths and angles involved. Another approach is the truss model, in which we take as variables the vertex coordinates of the folded form, then impose constraints that ensure that the folded form is developable, i.e., that it unfolds to a flat sheet of paper. This approach was introduced to describe paper-folding by Resch and Christiansen [103]; a complete algorithm was given by Schenk and Guest [107]. The truss model has one very nice feature: it naturally incorporates considerations of stress, strain, and related deformations, allowing one to model non-idealities of the real world, such as facet bending and/or slight dimensional changes in crease lengths and vertex positions (which, in physical folds, arise from slight shifts of crease and vertex positions within the paper). Schenk and

636


Guest studied deformational modes of periodic structures such as the Miura-ori and showed, for example, that while the rigidly foldable planar expansion we have already seen is the single ideal motion under the assumption of flat facets, allowing facets to bend under applied stress gives rise to an anticlastic (saddle-shaped) bending mode as a low-energy mode of nonplanar deformation. Both plate model and truss model descriptions can be used to construct realizable folded forms from a specified crease pattern. In the plate model, there is one variable (fold angle) for each interior fold, so the number of variables is E − B, where E is the total number of edges in the pattern and B is the number of edges on the border. In the truss model, there are three variables for each vertex, so the number of variables is 3V. Generally, the number of variables for the plate model is smaller; for example, for an N × N grid of quadrilaterals, the number of interior edges is 2N(N − 1), while the number of vertex coordinates is 3(N + 1)2 . Still, these are of the same order of magnitude and are close enough that other considerations—the type of problem being solved, the need to include stress/strain—might well tip the balance in evaluating which best fits the problem at hand. Both plate models and truss models can be formulated the same way, as a constrained optimization. In each case, we have a bucket of variables (fold angles for plate model, coordinates for truss model), a bunch of equations that must be satisfied to describe a valid structure, and, optionally, further constraints or figures of merit that we would like satisfied and/or maximized. For the plate model, the equations were the consistency matrix conditions. For the truss model, we have a different set of equations: isometry on the folds. ? ? ? 9.5.1. 3D Isometry and Planarity Isometry captures the requirement that geodesic (shortest-path) distances measured on the crease pattern and the folded form have the same length. Foremost among those is the requirement that all creases have the same length in the crease pattern and folded form. Suppose, to begin with, that the crease pattern is given. That is, let {pi } be a set of 2D vertices of the crease pattern; we further have a list of (i, j) pairs that define all of the edges of the crease pattern, both border edges and interior (possibly folded) edges, where an edge (i, j) represents a fold or border line between vertices pi and CHAPTER 9. 3D ANALYSIS

........

637

p j . Then we know the length of the (i, j) edge in the crease pattern is some quantity li, j ≡ kpi − p j k. (9.63) The crease in the folded form must have the same length. So if the (variable) vertices of the folded form are {p0i }, they must satisfy a set of isometry equations kp0i − p0j k − li, j = 0.

(9.64)

If the crease pattern consists entirely of triangles, the set of equations like Equation (9.64) are all that is needed to enforce isometry between the crease pattern and the folded form (assuming that all of the triangular facets are planar). However, if there are facets that are quadrilateral or larger, enforcing isometry on the edges alone is not sufficient. You can think of an isometry constraint as modeling a facet by rigid, fixed-length sticks along the edges while allowing flexibility at the corners of the facet. If the edges of a triangle are fixed-length, the triangle is rigid. But quadrilaterals, and higher, would remain flexible; a square could be squashed into a rhombus, or even into a skew quadrilateral (a non-planar quadrilateral). Isometry can be preserved by adding (k − 3) edges to each degree-k facet, thereby breaking it into (k − 2) non-crossing triangles, as illustrated in Figure 9.18. We then add an isometry equation for each of the added edges. That is, for each of the dotted lines in Figure 9.18, we add an equality constraint of p¢4

p¢3

p¢4

p¢3

p¢5 p¢3

p¢1

p¢2

p¢1

p¢2

p¢1

p¢2

Figure 9.18. Triangulated facets. Left: a triangle is rigid and needs no additional isometry equations. Middle: a quadrilateral should be subdivided along a diagonal into two triangles. Right: a pentagon is subdivided into three triangles by two additional edges.

638


p¢4

p¢3

p¢4

p¢5 p¢3

p¢1

p¢2

p¢1

p¢2

Figure 9.19. Adding another isometry equation that crosses all of the others is sufficient to enforce planarity of a facet with more than three sides—at least, in principle.

the form

kp0i − p0j k − kpi − p j k = 0.

(9.65)

Effectively, this process breaks each facet into rigid triangles joined to each other by additional fictitious folds. In the absence of any other constraints, those folds are “floppy,” though; there is nothing that prevents a facet from bending along one of the newly added folds. So we need to add still more constraints: ones that prevent a facet from bending along a fictitious fold, i.e., that enforce planarity. There is a very simple method of doing this, which is simply to add crossing isometry constraints, as illustrated in Figure 9.19. This is, in principle, sufficient, but it is numerically brittle. The problem is that at the solution, the gradient of the constraint vanishes; worse, numerical round-off errors can result in there being no solution. Consider, by way of example, the simple case of a unit square whose edges are all of unit length where three of the corners are fixed by other means and we wish to find the position of the fourth vertex, ostensibly at position (1, 1, 0), as in Figure 9.20. z p¢4

y p¢3 (1,1,0)

p¢1 p¢2

x

Figure 9.20. A unit square with two crossing isometries to ensure planarity of the facet.

........


639

p¢4 n¢B

Figure 9.21.

n¢A

B

p¢1

p¢3

A

Normal vectors for the two triangles in a quadrilateral.

p¢2

In this figure, the edge length isometries set all edges to be unity, and √ the two crossing isometries set the diagonals to be of length 2. Clearly, diagonal p02 p04 satisfies its isometry, and if that isometry holds, point p03 can only travel along the circular arc √ shown. Its isometry equation, that kp01 − p03 k = 2, is satisfied only when p03 = (1, 1, 0), which is indeed the planar state. So what’s the problem? The problem is that in any computational implementation, the edge lengths are not equal to integer 1; √they’re equal to 1.000 . . .. And the diagonals are not equal to 2, they’re equal to 1.414214 . . .. If there is numerical round-off error, such that the length of kp01 − p03 k gets rounded down, but 1.414214 . . . gets rounded up, there will be no solution. There is another approach for enforcing planarity, though, that is more numerically robust, at the cost of somewhat more complex evaluations. Figure 9.21 shows how we will address planarity for a quadrilateral. Label the two triangles of the original triangulation A and B. Each triangle has its own unique facet normal, n0A and n0B , respectively. Prevention of bending means ensuring that the two facet normals point in the same direction. We can write the two facet normals in terms of the vertex coordinates (which, remember, are our variables): n0A

=

(p02 − p01 ) × (p03 − p01 ) k(p02 − p01 ) × (p03 − p01 )k

,

n0B

=

(p03 − p01 ) × (p04 − p01 ) k(p03 − p01 ) × (p04 − p01 )k

.

(9.66)

Since these are unit vectors, a simple way of ensuring that they point in the same direction is to require that their dot product is 1: n0A · n0B − 1 = 0.

(9.67)

However, that is still too simple, and for the same reason that crossing isometries was a bad idea; because the maximum that the dot product can ever be is 1, the gradient of this constraint

640


will vanish at the solution, leading to numerical instability, and numerical round-off can lead to infeasibility. What we would like as a constraint equation is something that varies smoothly through the desired solution point, positive on one side of the solution, negative on the other. If we define θ AB as the angle of rotation from n0A to n0B , Equation (9.67) is equivalent to cos θ AB = 1 to ensure θ AB = 0. What we want is something akin to sin θ AB = 0. Well, we can construct exactly that. Knowing that both n0A and n0B will be perpendicular to edges p01 p03 , we have sin θ AB =

(n0A × n0B ) · (p03 − p01 ) kp03 − p01 k

.

(9.68)

This gives rise to a bending equality constraint, (n0A × n0B ) · (p03 − p01 )] kp03 − p01 k

= 0.

(9.69)

This formulation of a bending constraint gives a numerically robust equation that forces planarity of the facet. Perfect! Except, though, that it allows a spurious solution: θ A,B = π. If our optimization framework supports inequality constraints, though, then the additional inequality constraint n0A · n0B − 1 + ≥ 0

(9.70)

for some positive will eliminate the spurious solution. This approach is easily generalizable to higher-degree polygons: we simply add one additional planarity equality constraint (and, if desired, an inequality constraint to eliminate spurious solutions) for each added fictitious diagonal. That leaves open a question: how do we decide which way to triangulate a polygon? If we are going to force planarity, it shouldn’t matter. Well, it will matter a little, in how quickly a solution converges, but the feasible solution sets shouldn’t differ among the multiple choices of triangulation if all we want is planarity. If, however, we want to allow a little bit of bending, then triangulation starts to matter—as we now will see. ? ? ? 9.5.2. Explicit Stress/Strain One of the strengths of the truss model is that it naturally allows inclusion of stress/strain within the model by constructing the CHAPTER 9. 3D ANALYSIS

........

641

model as a constrained optimization, where equality and inequality constraints enforce isometry and screen out spurious solutions, and a figure of merit to be minimized is composed of a sum of strain energies contributed from things that are allowed to bend. Indeed, this was the rationale for Schenk and Guest’s original work: to include such deformations explicitly. We can allow facet bending, at the cost of creating a strain energy, by removing the equality constraint, Equation (9.69), and instead creating a contribution to the figure of merit that gives a strain energy that is at a smooth minimum at flatness and increases away from flatness. If all we are interested in is a qualitative model, then we can choose a simple model of strain energy where the energy per unit length is (1 − cos θ AB ), which takes on a minimum value of 0 at flatness and rises quadratically away from that minimum. With this simple model, the total strain energy for edge p01 p03 in Figure 9.21 would be E AB − kp03 − p01 k(1 − n0A · n0B ).

(9.71)

Summing over all such fictitious edges gives the total strain energy to be minimized. As I said, this is a very simple model, and yet it can capture quite a bit of useful behavior, such as the anticlastic bending of a Miura-ori and related patterns. It is possible to be considerably more accurate by using a more realistic model of the energy associated with bending and folding. Filipov et al. [27] have built such a model with analytic models of bend and fold energies (as well as stretching energies along bars) that allows the computation of both deformation modes and their associated stiffnesses for quite complex folded structures. There is a series of assumptions baked into this treatment, though: by triangulating all high-degree facets in a particular way, we are assuming that that particular triangulation is the preferred deformation mode for that facet. Considering just a quadrilateral, though, we know that a general quadrilateral could bend along either diagonal. For quadrilaterals, there is a mathematical way to allow either direction of bending, which relies on the fact that while bending allows one or the other diagonal to break isometry, at least one of the two must be isometric.

642


We can write the isometry equation for diagonal p01 p03 as kp03 − p01 k

−1=0

(9.72)

kp02 − p04 k − 1 = 0. kp2 − p4 k

(9.73)

kp3 − p1 k and for the diagonal p02 p04 as

An equation that captures the concept “at least one of these must be zero” is their product: 0 0 kp3 − p01 k kp2 − p04 k −1 − 1 = 0. (9.74) kp3 − p1 k kp2 − p4 k So using this expression as the isometry constraint will allow a quadrilateral to bend in either direction. (If we are minimizing strain energies, then the total strain energy would just be the sum of the contributions from the two diagonals.) This is a nice constraint to use when finding a 3D deformation of a crease pattern that contains only triangles and quadrilaterals, what we call a TQ mesh, where we will allow the quadrilateral to bend. It still has some weaknesses, though: • The constraint is numerically ill-behaved near total flatness; that is, numerical analysis is more robust if all of the quadrilateral facets are bent at least a little bit in one direction or the other. • Because the constraint has two distinct feasible branches, corresponding to the two directions of bending the quadrilateral, when starting a solution search from a flat state (or a slight perturbation of a flat state), the choice of which branch the solver goes down is not readily predictable, and depending on the root-finding algorithm, one might end up on a bentquadrilateral branch that takes more bend energy than some other solution. One is not guaranteed of finding the lowest-energy deformed state. And of course, this trick does not readily generalize for higherdegree polygons, where there are multiple (and more complex) choices of which diagonals the polygon could flex along. CHAPTER 9. 3D ANALYSIS

........

643

Nevertheless, TQ meshes are quite common in origami design, and using Equation (9.74) can often find valid 3D states for crease patterns where facets must bend in order to satisfy all of the constraints placed on the folding pattern. By way of example, most of the rotationally stretched pleated form images in Chapter 2 were generated using this type of mathematical model. ? ? ? 9.5.3. 3D Developability In the truss model as described above, the 2D crease pattern is assumed to be known, and we are solving for a 3D folded form subject to forces and constraints. But in many origami design problems, the crease pattern is unknown, or only partially known, and both crease pattern and folded form are to be solved for. In such cases, there are commonly more constraints than just those that enforce isometry: for example: “a portion of the folded form should follow a specified line,” as in the discrete space curve analysis of Section 9.3; or “a projection of the folded form should be isomorphic to a tree graph,” as in my own TreeMaker program [67, 69]; or “a portion of the folded form should follow a specified polyhedral surface,” as in Tomohiro Tachi’s Origamizer program [110, 119]. Many of these problems can be numerically addressed using some form of a truss model by the simple modification of adding 2D variables to represent the vertices of the crease pattern and solving for the expanded set of variables subject to assorted constraints on the desired form, the isometry constraints just discussed, and, if applicable, a figure of merit. We still have isometry equations, but they now contain both 2D and 3D variables, and we now have a total of 5V variables in the numerical system—more complicated, but still reasonably tractable. However, we don’t necessarily have to have a crease pattern in order to construct a folded form of some crease pattern. All that we need of the folded form is developability: that the sum of the sector angles around each vertex is precisely equal to 2π. For a crease pattern embedded in the plane in 2D, developability is automatically assured. For a 3D embedding, though, developability is not assured. If, though, we can ensure that the 3D folded form is developable at every vertex, then once we have solved for the 3D model, it can be unfolded to a flat sheet, by, for example, constructing a plate model and then unfolding it using the algorithm described in Section 9.4.1.

644


a2 p¢3

p¢2 p¢6

p¢4

a1 p¢1

Figure 9.22. Schematic of a folded form vertex p00 and its surrounding vertices {pi0 }.

p¢5 p¢0

To ensure developability, we add further constraints to our constrained optimization: one for each interior vertex. One such vertex is shown in Figure 9.22, with surrounding vertices indexed consecutively, for convenience. For each angular segment around the vertex, we can compute the sector angle αi from the vertex coordinates using Equation (9.19): αi = tan−1 ((p0i − p00 ) · (p0i+1 − p00 ), k(p0i − p00 ) × (p0i+1 − p00 )k). (9.75) Í Then our developability constraint is simply i αi − 2π = 0. The inverse trigonometry function can, in some circumstances, cause numerical problems due to its discontinuities. There is an alternative formulation that avoids the inverse trigonometry functions, at the cost of some additional algebraic complexity. Requiring that angles sum to 2π is the same as requiring that the product of 2D rotation matrices describing the angles between successive creases around a vertex be equal to a rotation by 2π. The developability condition is, as before, a 2D condition that D≡

n Ö

R(αi ) = I,

(9.76)

i=1

where the {R(αi )} are the 2 × 2 rotation matrices for the sector angles {αi } around the vertex under scrutiny. But now, the vertex (and its surrounding vertices) are 3-vectors and the angles around them must be measured in R3 . The 2 × 2 rotation matrices are, from Equation (1.23), cos αi − sin αi R(αi ) = . sin αi cos αi We can use Equations (9.17) and (9.18) to get sin αi and cos αi in terms of the coordinate variables {p0i }, and from there, we can construct the developability condition D = I.

........


645

As with the 2D developability condition and the 3D consistency condition, we only need to construct equalities for the offdiagonal terms of the resulting matrix equation. For analytic solutions, taking D1,2 = 0 will include all valid solutions. However, this expression may include spurious solutions as well, of two types. First, we can have D1,1 = D2,2 = −1, which is effectively Í equivalent to the condition i αi = (2k + 1)π. These can be suppressed by adding an inequality constraint of the form D1,1 > 0. The second spurious case is a bit more interesting: We can have D = I with multiply covered vertices—sector angles summing to larger multiples of 2π—and this is possible with high-degree vertices. For root-finding strategies that look for the closest feasible point in solution space, such spurious solutions will usually be avoided.

?

9.6. Time Efficiency As we have seen in earlier chapters, flat, 2D origami designs can often be designed from mathematical principles using nothing more than high school mathematics. For 3D origami design problems involving multiple vertices, though, using a vector description of the folded form is almost inescapable. Some design problems are pure “forward calculations,” like the discrete space curve; we start with a description of the desired result and, step by step, construct a mathematical description that culminates in a description of the 3D folded form and the underlying 2D crease pattern. For these, the existence of the result is known; we just have to “plug and crunch” our way to it. Other problems, though, require the solution of complex systems of equations that simultaneously satisfy or optimize criteria associated with the desired form and conditions on the mathematical description that enforce conditions specific to origami: consistency, developability, and/or facet planarity. These require the solution and/or optimization of systems of equations arising from the vector description. Solving such systems efficiently commonly requires close attention to optimizing the calculation: linearizing the systems of constraint equations, ensuring that computations are done efficiently by throwing away unneeded terms, and the like. I’ve not

646


gone into the details of how one might do that, but several references in this chapter give more detailed descriptions for both plate and truss models. With modern computer algebra and numerical analysis systems, it is often not necessary to spend a lot of effort on the most efficient computational implementation. It is often most efficient in terms of human time to simply dump the whole set of equations into a general-purpose constrained optimizer and go get a cup of coffee while the solver crunches out a solution. For pure forward calculations, even an inefficient implementation can often give a solution effectively instantaneously, allowing one to rapidly evaluate a range of forms on aesthetic criteria. In the next (and last) chapter, we will look at several classes of 3D forms that initially arose in the context of tessellations but have taken on a life of their own, and are readily designable using a vector description. ?

9.7. Terms Consistency matrix The matrix product of rotations for each sector angle and fold angle around an interior vertex of the folded form, which should be the identity matrix. Crease vectors At a vertex in a crease pattern, the 2D vectors pointing outward along each of the incident creases. Fold vectors At a vertex in the folded form, the 3D vectors pointing outward along each of the incident folds. Matching minor fold In a degree-4 vertex, the minor fold that has the same crease assignment as the two major folds. Mismatched minor fold In a degree-4 vertex, the minor fold that has the opposite crease assignment from the two major folds. Planarity The condition that a facet of degree 4 or higher remains planar. Plate model A representation of a folded form in which fold angles are used to describe the kinematic state of the form and the facets are treated as fixed plates. Rigid-body motion Some combination of translation and rotation. CHAPTER 9. 3D ANALYSIS

........

647

Sector vectors The vectors perpendicular to each of the sector planes in a 3D partially folded vertex. Spanning tree (of a folded form) A spanning tree of the interior dual graph of the crease pattern of the folded form. Spanning tree (of a graph) For a graph, a subgraph that includes every vertex, is connected, and contains no loops. Triangle-Quadrilateral (TQ) mesh A crease pattern composed only of triangles and quadrilaterals. Truss model A representation of a folded form in which vertex coordinates are used to describe the kinematic state of the form and the edges are treated as fixed (or flexible) truss elements.

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10

Rotational Solids ? ??

10.1. Three-Dimensional Twists

We have seen that the simple flat twist is a most versatile object. While it has a beauty and mathematical richness all its own, multiple twists can be combined into the intricate folded patterns called tessellations, and, as we have seen, an extraordinary variety of forms is possible just with those simple building blocks. But there is far more that can be done with a single twist fold, and in this chapter, we will explore several interesting variations of the SFT. We will discover a variety of beautiful structures both flat and 3D. A nice feature of many of these designs is that the crease pattern and folded form have the same border as an ordinary twist, so that more complicated twists can be “dropped into” otherwise ordinary tessellation patterns. ? 10.1.1. Puffy Twists The first question one might ask is: if we already have a flat twist, can we do something interesting with it? The answer is yes, and it is fairly easily implemented with any twist; by pushing in the sides of the twist, we can make it 3D. A simple example of this process is illustrated in Figure 10.1 for a hexagonal twist. You will find that to get this started, you may need to unfold the twist and insert your finger up inside the model to get the mountain fold creases started going the proper direction. Wouldn’t it be nice if there were a hole for one’s finger? Yes—you can make this figure from an open-back cyclic twist that has a hole for your finger. Try starting with a twist angle of, say, 45◦ , as shown in Figure 10.2, and follow the same procedure. 649

1. Begin with a 3×3 hexagon. Fold a closed-back twist along the grid lines shown.

4. Push in on the sides and form mountain folds on the existing creases so that the center Òpuffs up.Ó

2. Fold the corner to the center and unfold.

5. Finished hexagonal puffy star.

Figure 10.1. Folding sequence for a hexagonal puffy star.

Figure 10.2. Crease pattern for a hexagonal puffy star using a 45◦ twist angle.

650

........CHAPTER 10. ROTATIONAL SOLIDS

3. Repeat on the other five corners.

Figure 10.3.

Left: hexagonal puffy cylinder. Right: hexagonal puffy pyramid.

Once you’ve gotten the inside puffed up, you will find that there is a lot you can do simply by playing with the 3D portion of the twist. While the paper will naturally form curves in several places (which can be quite aesthetically appealing), it is also possible to turn this form into a polyhedral form (having flat facets with sharp creases), as shown on the left in Figure 10.3. It is also possible to vary the pinching of the “flanges” to create a pyramid in the middle of the twist, as shown on the right in Figure 10.3. In fact, as you play around with this shape, you should be able to create a variety of 3D forms from the middle of the twist, each with the same 6-fold rotational symmetry as the twist has. (You can also create forms that are non-symmetric, but we’ll stick to the rotationally symmetric ones for now.) And of course, there is nothing special about 6-fold rotational symmetry; it is possible to create similar shapes with any other rotation (3, 4, 5, and 8 give particularly interesting forms). All of these forms have the same general description: there is a (hollow) 3D polyhedron, whose edges are attached to flat “flanges,” which effectively soak up the excess paper not used in the construction of the polyhedron. The question then naturally arises: what types of polyhedra are possible using this general approach? And how do we construct them? The way I came to explore this type of structure was not from pushing in the sides of twists: it came about when I saw a folded model displayed by Chris K. Palmer at an origami convention in 2001, similar to that shown in Figure 10.4. It was a 3D twist,

........

CHAPTER 10. ROTATIONAL SOLIDS

651

Figure 10.4. My reconstruction of Palmer’s spherical “Polypouch” (ca. 2001).

clearly folded from a closed-back twist, but with the sides pushed in and shaped so that the central region took on a spherical form. Palmer had already realized many generalizations of these three-dimensional twists in a series of 3D boxes and other shapes that he called “Polypouches,” which he published in CD form [98], and had incorporated such 3D twists into his otherwise flat tessellations in the mid-1990s [93]. Examples of Palmer’s “Polypouches” are shown in Figure 10.5.

Figure 10.5. “Polypouch” designs by Chris Palmer from his 2001 CD “Polypouches” (folded by the author).

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When I saw the spherical form of Figure 10.4, I realized that it could be generalized: you weren’t restricted to a sphere, you could, in theory, choose any cross section you wanted for the shape, by shifting paper between the central shape and the flanges, as you might have done playing with the puffy twists above. In fact, as I eventually learned, Palmer had recognized the generalization as well, gaining his inspiration from a series of three-dimensional Japanese tato designs by the Japanese master Uchiyama. It is possible to create a shape such as that in Figures 10.4 and 10.5 just by playing with twists and pinching the flanges that connect the twist pleats to the central region. However, I wanted to be able to create not just a sphere, but an arbitrary 3D shape in the central region, and that led to a more wide-ranging exploration of three-dimensional twists and a wide range of subjects. It all started, though, with a sphere. ?? 10.1.2. Folding a Sphere Let’s suppose that we wished to construct a 3D sphere, or a reasonable approximation, from a flat twist. It’s not possible to create an actual sphere, but we could create an approximation of one, as shown, for example, in Figure 10.6. Why is it not possible to fold a perfect sphere from a flat sheet of paper? It is because paper is a ruled surface, a property that can be defined in several ways, but the simplest and most intuitive is this: at any smooth point on the surface (and by “smooth” I mean not a fold, not a corner), there must be some direction along which the paper forms a straight line in 3D. A way of visualizing this

Figure 10.6.

Left: a perfect sphere. Middle: a sphere with axial discretization. Right: a sphere with both axial and azimuthal discretization.

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Figure 10.7. The axially discretized sphere with two tangent lines. The green line touches at a point; the red line touches along the full width of the gore.

concept is to think of touching the middle of a pencil to the paper at the point, and rotating the pencil about the point of contact. There must be some angle at which the pencil contacts the paper all along its length—at least, until you reach a fold, a corner, or you run off the paper. As an example, take the middle figure in Figure 10.6. Here, the sphere is approximated by eight curved segments, called gores. In Figure 10.7, I have redrawn this figure and added two straight lines in 3D touching a point on the middle of one of the gores on the right side. The horizontal line (red) touches all along the width of the gore between the two sharp edges. The vertical line (green) only touches at a point on the equator of the gore. And if I rotated the green line segment about the point of contact, it would still only touch at a single point, until it became perfectly horizontal—at which point, it would touch along the full width of the gore. So, since the red line touches the gore across its full width, there is a line that is part of the gore that is straight. In fact, for this figure, every horizontal line within a gore is a straight line. A line within a surface that is straight in 3D is called a ruling line1 of the surface. In this figure, all of the ruling lines are horizontal. That brings us back to origami. Any surface that can be folded from a flat sheet of paper must be a ruled surface: there must be a ruling line for every smooth point of the surface. (In fact, the requirement is even stricter: it must be a developable surface, 1

You may recall that the angle between the minor folds of a degree-4 vertex in 3D was the ruling angle. The use of “ruling” is no coincidence; if we discretize a curved fold, the ruling lines of the curved surface become the minor folds of the discretized vertices.

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which is both ruled and has zero Gaussian curvature—which is a concept we have already encountered in Chapter 8.) The important thing to take away from these two figures is that we cannot fold a perfectly smooth sphere from a flat sheet, at least, not with a finite number of folds. We must be content with folding some sort of approximation. It could be one with smoothly curved gores, as in Figure 10.7, or it could even be one with flat polygonal facets, as shown in the right image in Figure 10.6. Although the middle and right figures of Figure 10.6 look pretty different—one is smooth, the other is faceted—mathematically, they are not all that different. If we fold a faceted version, we can vary the number of facets, subdividing the paper more and more finely, and come as close as we want to a perfectly smooth sphere. In fact, the middle figure, which looks like it is smoothly curved, is actually also polygonally faceted, but there are 100 subdivisions from pole to pole, which is fine enough that it looks smooth. If we were making this shape with multiple sheets of paper, we could make each gore from a single sheet, with each gore either curving smoothly (as in Figure 10.7) or being folded polyhedrally (as on the right in Figure 10.6), and then tape individual gores to each other at their edges. If we are folding the shape from a single sheet of paper, though, then we need to figure out how to get all of the gores from the single sheet. One way to do this would be to cut the gores almost entirely apart, leaving them joined at the South Pole of the sphere, and then uncurl the gores so that they all lie flat, as shown in Figure 10.8. Once the gores are unwrapped, we can embed them into a single sheet of paper and then add folds to hide the excess paper between them as we re-wrap the gores around the sphere. That would be the white regions in Figure 10.8. Since each white

Figure 10.8. The sphere with gores unwrapped to a flat surface.

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Figure 10.9. The folded sphere for 8 axial divisions. Left: crease pattern. Middle: folded form. Right: folded form, turned over.

region is mirror-symmetric, an obvious way of hiding the paper suggests itself: fold each distinct white region in half, then tuck it out of the way, for example, off to one side. And this method works! If you do this with the gores from a sphere, you will get the crease pattern and folded form shown in Figure 10.9. We call this conceptual approach an azimuthal unfolding or radial unfolding of the sphere. In this pattern, the curved mountain fold follows one edge of the gore; there is no fold along the other edge. The gore surfaces conform nicely to the surface of our desired sphere, but the flanges are also noticeable, especially near the North Pole, where there is the most paper to hide. We can reduce the appearance of the flanges, though, by increasing the number of axial divisions—for example, from 8 to 16 and then 32, as illustrated in Figures 10.10 and 10.11. The higher the axial order—the number of divisions around the equator—the smaller the individual flanges are, and the closer the shape comes to the desired sphere. There is a tradeoff, though: the folds get closer and closer together, particularly in the very center of the crease pattern. While a theoretically computed pattern will fold into a near-perfect sphere, with real paper and real folding, the barely separated folds will mush out where they all come together, giving an imperfect folded form. In practice, I have found that axial divisions between about 8 and 16 strike a good balances between capturing the shape and not having too many creases coming together in the middle of the polygon.

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So, that’s the concept; but what is the precise shape of the creases? Consider a point somewhere along the edge of the gore as shown in Figure 10.12. We denote its position in 3D by p0 (red dot) and its position on the unwrapped gore by p (green dot). For the unwrapped gore, it would suffice to know the two distances r—the radial distance outward on the flattened gore—and w—the distance from the center line of the flattened gore—for each point on the edge of the gore.

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Figure 10.12. Geometry of a single gore, both wrapped around the sphere and unwrapped.

Let’s first look at a vertical slice through the sphere, as illustrated in Figure 10.13. We’ll parameterize each point by its angle θ, as measured from the center of the sphere and starting at the South Pole. In that case, since our sphere is a unit sphere (radius = 1), the distance r from the South Pole is simply r = θ,

(10.1)

and the distance x from the axis of rotation is x = sin θ.

(10.2)

For the distance w, we’ll look at a horizontal slice through the sphere, as illustrated in Figure 10.14. Now, because we have divided the surface into m gores (with m = 8 for this example), we have a series of 2m right triangles arrayed about the center, and we can work out the distance w, which is the second distance needed for the gore. We define, for convenience, the angle φ ≡ π/m. Then from Figure 10.14, we have w = x tan φ = sin θ tan mπ .

(10.3) z

x

Figure 10.13.

Left: a vertical slice through the sphere. Right: the cross section.

658


p¢

q r x

y

f

p¢ w x

x

Figure 10.14.

Left: a horizontal slice through the sphere. Right: the cross section.

Armed with these formulas, we can construct the gores for a sphere of any rotational order m. And once we have the gores, we can place then into an m-gon, add diagonal folds to the corners, and complete the full crease pattern. But, even more useful, we can generalize this concept for arbitrary cross sections, making it very powerful indeed. ???

10.2. Thin-Flange Algorithm We’ll now set up the machinery for constructing a m-fold rotationally symmetric form from a regular m-gon with an arbitrary cross section. For compactness, we’ll use vector notation. Figure 10.15 shows the basic concept for a very simple cross section consisting of three straight line segments. We define the cross section, shown on the left, and specify the rotational order m. From that we seek to compute the crease pattern, shown in the middle, and the 3D representation of the folded form, shown on the right. Rather than parameterizing on an angle θi as we did for the sphere, we will define the cross section as a series of discrete point pairs (xi, zi ), i = 1 . . . N, where xi is the distance from the axis of rotation and zi is the height above the base of the 3D “pot.” Now we will solve for the parameters of the gore—and also of the other folds in the pattern. Because of the rotational symmetry of the crease pattern and folded form, we only need to solve for a single unit of the pattern that is then duplicated and rotated m times to make up the full crease pattern. A logical choice for what to choose as the unit

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659

z 2.5 2.0 1.5 1.0 0.5 0.5

-1.5 -1.0 -0.5

1.0

1.5

x

Figure 10.15.

Left: example cross section. Middle: crease pattern. Right: 3D rendering of the folded form.

would be to choose the gore, plus whatever other folds are needed on one side or the other. It turns out, though, that there is some benefit to splitting each gore down the middle and then taking the boundaries of our unit to be the midlines of two consecutive gores, plus whatever creases fall between them, as illustrated in Figure 10.16. We will call this region a wedge of the crease pattern. We will label the vertices of the crease pattern, with a set {ai, bi, di, ei, fi } being vertices in the crease pattern and {a0i, b0i, d0i, e0i, f0i } being their corresponding vertices in the folded form for the horizontal slice that corresponds to the cross section point (xi, zi ). (Yes, we’ve skipped ci ; there’s a reason, which we’ll y

z

y fi

f i¢

ei

ei¢ di

di¢ f i¢

bi¢ ai¢ f

bi wi ri

ai

x

y

Figure 10.16.

x

Left: a single wedge of the crease pattern. The two half-gores are shaded. Middle: the folded form of the wedge. Right: a top view of a horizontal slice of the folded form.

660


di¢

ei¢ bi¢ wi xi

ai¢

x

come to.) Not all of these vertices lie on actual folded creases: in particular, the lines connecting the set {bi } are unfolded creases. For completeness, though, we will include and calculate them. We also note that the points b0i and e0i are (theoretically) coincident in the folded form; they are shown here slightly spread, for clarity (and they must be slightly spread when the structure is folded from real paper, of course). With these definitions in place, we can now calculate the values of all of the vector coordinates. It all begins from the distance as measured along the surface of the middle of the gore, which we denote by ri and which should be computed recursively. We have r1 = 0, p ri = ri−1 + (xi − xi−1 )2 + (zi − zi−1 )2 .

(10.4)

And the gore half-width, wi , is wi = xi tan mπ .

(10.5)

For notational simplicity, we introduce several unit vectors. In 2D, ˆ u(θ) ≡ (cos θ, sin θ), and in 3D,

uˆ 0(θ) ≡ (cos θ, sin θ, 0), zˆ 0 ≡ (0, 0, 1).

Then the crease pattern points are as follows (given in order of their dependencies): ˆ ai = ri u(0), ˆ π2 ), bi = ai + wi u( ˆ π2 ), di = ai + ri tan φ u( (10.6) ˆ fi = ri u(2φ), ˆ ei = fi + wi u(2φ − π2 ). The points in the folded form are given by a0i = xi uˆ 0(0) + zi zˆ 0, b0i = a0i + wi uˆ 0( π2 ), d0i = a0i + ri tan(φ) uˆ 0( π2 ), e0i = b0i, f0i = xi uˆ 0(2φ) + zi zˆ 0 .

(10.7)

Similarly, e0i depends on f0i . CHAPTER 10. ROTATIONAL SOLIDS

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Figure 10.17. An assortment of rotationally symmetric pots and other containers, designed and folded in the early 2000s from various weights of watercolor paper.

With these definitions, we can construct a single gore of both the crease pattern and the folded form and then, by duplicating and rotating both m times, create representations of the complete rotational form. Using this technique, I designed and folded quite a few rotationally symmetric forms throughout the decade of the 2000s. Palmer’s “Polypouches” were, like the original tatos, envisioned as containers that closed around their contents, but the notion of choosing the cross section resonated with me as reminiscent of the way a potter shapes a pot on the wheel. Accordingly, most of my designs using this form were (and continue to be) inspired by pottery. And indeed, I tend to call this family of my designs “generalized pots.” Several of my early examples are shown in Figure 10.17. One of the challenges of these designs is that the curved folds are incredibly sensitive to errors in the smoothness of the curve; even a slight deviation in curvature results in a noticeable buckle in the 3D surface. This bothered me, a lot. In the mid-2000s, though, I began playing with the concept of laser scoring (thanks to a generous offer of facility usage from Saul Griffith, at Squid

662


Figure 10.18. More rotationally symmetric pots, folded from Canson and Elephant Hide papers.

Labs). Eventually, I obtained my own laser cutter, and using that for scoring, I was able to make my folds more precise, which allowed me to expand the range of forms I was exploring. Several more recent designs are shown in Figure 10.18. As time went on, the resonance with pottery, and with American Southwestern pottery in particular, grew stronger, and I sought ways of strengthening the connection between these designs and their original inspiration (while enjoying the contrast between the handcraft of the original work and the mix of mathematics and technology that enabled my own). One of the connections came through the paper: I found a series of handmade papers created from yucca fiber. The yucca plant and its fibers had been used by the first people of the Americas for thousands of years, and I found that paper made from this fiber was amenable to rotationally symmetric forms. I created and folded two pots from this paper, shown in Figure 10.19, and named them, respectively, for a New Mexican pueblo still in existence and known for its pottery, and for a humble Mexican potter who rejuvenated an entire village by reconstructing their ancestral pottery craft. There is much further potential in this particular family of rotationally symmetric structures, and I (and others) have explored them further. But there is also more potential in the concept, as we will see.

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663

Figure 10.19.

Left: “AcomanPot12” (2011), by the author. Right: “QuezadaPot13” (2009), by the author. Both folded from yucca fiber paper.

? ???

10.3. Thick-Flange Structures

? 10.3.1. Mosely’s “Bud” There are two dimensions of design variation with these rotationally symmetric shapes: we can vary the rotational order m discretely, and we can vary the cross section continuously. Actually, there’s a third way to vary: we can vary the direction that the flanges point, either clockwise or counterclockwise, in the same way that a flat twist can be either clockwise or counterclockwise. But those two directions are the only possibilities: for any given flange, if you crease it along the gore edges, you can snap it from one direction to the other, but for most structures, there is no possibility for an intermediate state. Or is there? A few years after I saw Chris Palmer’s 3D spherical twist, Boston geometer Jeannine Mosely began showing and describing a remarkable curved form, which she named “Bud” [90, 91, 92]. My folding of her original design is shown in Figure 10.20, along with its crease pattern. The design was elegant, it used curved folds (which was relatively rare at that time), and the arcs were ideal circles or sufficiently close that one could easily construct the crease pattern with a compass and ruler. But what made it especially interesting to me was that the shape seemed to be in the same general family as the 3D twist: a rotationally symmetric shape, defined by gores, with the excess paper gathered together and forming flanges around the outside of the central core.

664


Figure 10.20.

Left: crease pattern for Jeannine Mosely’s “Bud” (2003). Right: folded version, folded by the author.

What made this shape different and interesting, though, was that the flanges were thick and 3D, rather than the thin, doublelayered flanges of the previous section. The flanges had volume. That made the resulting structure more interesting and striking: it made the resulting form entirely volumetric. And its design was no accident. In a single wedge of a thinflange form, the radial mountain folds in each wedge lie in a common vertical plane in the folded form, as shown in Figure 10.21. For any nontrivial cross section, there are only two solutions for the paper on either side of the plane: it can continue straight across unchanged, or it can double back (via the mountain fold), in which

Figure 10.21. A single wedge with a vertical plane (gray) running through the border between the gore and flange. Left: counterclockwise-pointing flange. Right: clockwise-pointing flange.

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Figure 10.22. Schematic top view of the flange; dashed line indicates the position of the vertical plane. Left: counterclockwise flange, both layers point left, left layer is folded. Middle: clockwise flange, both layers point right, right layer is folded. Right: both layers are folded; they must double back to meet in the middle.

the folded layers are (at least locally around the fold) the mirror image of the incident layers. In the left figure of Figure 10.21, the paper on the left side of the vertical lane doubles back while the paper on the right side extends straight across the plane. In the right figure, it’s the other way around. Let’s look at the top of the flange from directly above, as shown schematically in Figure 10.22. The first two images show the two configurations that were shown in 3D in Figure 10.21. In the thin-flange version, there are two possible configurations: one layer continues straight, the other is folded, and depending on which is which, the flange points left or right, as shown on the left and in the middle. But there is a third option, which Mosely realized: what if both layers are folded, as shown on the right in Figure 10.22? Obviously, the two gores must be connected to one another, and we have no choice in how much paper is used for the connection: the total amount of paper in the connection must be exactly the same amount that was used in the thin flange. Now, though, there are four creases in each wedge, rather than two, as before. From the geometry of Figure 10.22, we can work out the lengths that the new creases have to be, and when we do, an elegant result ensues, illustrated in Figure 10.23. The two radial mountain folds of the thick-flange design lie in exactly the same place as the mountain folds of the two directions of the thin-flange design. This is to be expected; these are just the edges of the gores. There are two unexpected (or at least, less-expected) features of the solution, though.

666


y

y

fi ei

di

ci si b si i

wi x

ri

ai

ri

wi x

Figure 10.23. Comparison of thin-flange and thick-flange designs. Left: creases and dimensions of the thin flange. Right: creases and dimensions of the thick flange.

The first is that the creases that cross the edges of the gores remain straight. (In fact, we saw this happening in the thin-flange solution.) This behavior follows from the property that in the folded form, the edges of the gores—the {bi }-lines and {ei }-lines—lie in a vertical plane (the gray plane shown in Figure 10.21). That means that all of the vertices along those folds share a common osculating plane and each vertex is a straight-minor vertex. Thus, the minor folds are collinear across the vertex at each vertex along both lines. The second (and, to my mind, more surprising) result is that the vertices of the radial valley folds—the {ci }-lines and {di }-lines— lie exactly halfway between the gore vertices and the points on the diagonal line (former valley fold of the thin flange) that are immediately above them. That is, the distances marked si in Figure 10.23 are equal and are given by si =

1 2

(ri tan φ − wi ) .

(10.8)

That permits a simple geometric construction of the thick-flange version from a thin-flange crease pattern: extend the axial creases and add vertices halfway between the gore vertices and the midline of the wedge. Once folded, the thick-flange structures are distinctly different from their thin-flange brethren. Figure 10.24 compares thin-flange and thick-flange versions for the cross section that we used for Figure 10.15. Of course, the pattern can also be computed; there are formulas analogous to those of the previous section, which I now provide.

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Figure 10.24. Comparison of thin-flange and thick-flange examples. Left: crease pattern and folded form for a clockwise thin-flange structure. Right: crease pattern and folded form for a thick-flange structure.

? ? ? 10.3.2. Thick-Flange Algorithm Figure 10.25 shows a single wedge of the crease pattern and its folded form with vertices labeled, similarly to Figure 10.16. We have more creases and so more vertices to compute: {ai, . . . , fi } for the crease pattern, {a0i, . . . , f0i } for the folded form. (And now we’ll allow ci to show its face.) We define ri and wi as before. Then for the crease pattern, the vertices are (given in order of their dependencies) ˆ ai = ri u(0), ˆ π2 ), bi = ai + wi u(

ˆ π2 ), ci = bi + 12 (ri tan φ − wi ) u( ˆ fi = ri u(2φ), ˆ ei = fi + wi u(2φ − π2 ),

(10.9)

ˆ di = ei + 12 (ri tan φ − wi ) u(2φ − π2 ), and for the folded form, they are a0i = xi uˆ 0(0) + zi zˆ 0, b0i = a0i + wi uˆ 0( π2 ),

c0i = b0i + 12 (ri tan φ − wi ) uˆ 0(2φ − π2 ), f0i = xi uˆ 0(2φ) + zi zˆ 0, e0i = f0i + wi uˆ 0(2φ − π2 ),

d0i = e0i + 12 (ri tan φ − wi ) uˆ 0( π2 ).

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(10.10)

y

z

y fi ei

f i¢ di

ai

f i¢

ai¢ ci si

ri

di¢ ei¢ b¢ i c i¢

wi

f

bi x

y

di¢

ei¢ bi¢ xi

wi

c i¢

ai¢

x

Figure 10.25.

Left: a single thick-flange wedge of the crease pattern. The two half-gores are shaded. Middle: the folded form of the wedge. Right: a top view of a horizontal slice of the folded form.

As with the thin-flange structures, one can compute both crease pattern and folded form by duplicating and rotating copies of the individual wedges. My first design using this concept was an explicit hommage to Mosely’s “Bud.” I called it “Buh” (because it was half of a “Bud”). The crease pattern and folded form are shown in Figure 10.26. (There are some extra folds near the corners of the square; I use these to lock the bottom of the model together.)

Figure 10.26. “Buh” (2004), by the author. Left: crease pattern. Right: folded version.

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x

Figure 10.27. “Lavalamp” (2004), by the author. Left: crease pattern. Right: folded version.

To exercise the notion of choosing an arbitrary cross section, I created a combination of sinusoids that went in and out, which resulted in a shape, shown in Figure 10.27, that I called “Lavalamp,” because it reminded me of the 1970s-vintage icon. Thin-flange structures are inherently handed; the flanges point either clockwise or counterclockwise. These thick-flange structures are mirror-symmetric, which I find more elegant, and so I have tended to favor using thick-flange patterns in my own designs. When I began to follow the Muse that said “pots,” I started using the thick-flange concept for my designs. Thin-flange patterns naturally give rise to crease patterns that are regular m-gons. Since each contour of constant height is a regular m-gon, if we cut the shape at any contour, an m-gon is what we get. With thick-flange structures, though, the contours of constant height are m-gons with the corners trimmed off, giving rise to a 2m-gon, which isn’t usually regular. It is possible, of course, to extend the corner of the paper in the middle of the thick flange to create a regular m-gon, which might be considered a more graceful and elegant crease pattern. This extension gives rise to an additional aesthetic bonus. If the top of the pattern is curved back inward to be horizontal, with the edge chosen to be a constant-height contour, a portion of the “inside” of the paper is visible along the edge. This could be considered a decorative finish, but if we extend the paper to create a regular

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Figure 10.28. An assortment of pots using the thick-flange algorithm with various cross sections, folded from Canson and Elephant Hide paper (2004–2013), by the author.

n-gon, then the valley folds of the flanges hit the edges at 90◦ angles, and the extended corners precisely cover up the exposed underside of the paper. Most of my thick-flange designs are from a regular m-gon, as in the examples in Figure 10.28. For large values of the rotational order m, the polygon begins to look a lot like a circle, which then raises the notion of simply using a circle as the polygon by extending and/or truncating the constant-height edge of the polygon. There are very few origami designs that come from a circle, in part, perhaps, because it is difficult to make use of the curved edge of a circle in a natural way. Creating rotationally symmetric forms is a very natural way to use a circle, though, and creating rotationally symmetric patterns within a circular sheet of paper provides yet another variant on this concept. ? ? ? 10.3.3. Specified Gores Mosely’s crease pattern in Figure 10.20 has two aesthetic properties that, in my view, give it a special appeal: First, it’s from a square, which is always desirable when we’re in the world of

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671

origami. Second, the curved creases can be drawn as circular arcs, so it’s possible to construct and score the crease pattern with nothing more complicated than a pencil, ruler, and compass. The curved flange creases are not precisely circular arcs, even if the gores are; but they are close enough that for practical purposes, circular arcs may be used. It also raises an interesting question. There is an inverse problem here. We started from the assumption that we were given the cross section points {xi, zi }, and from that we computed the shapes of the gores (and everything else). But what if we were given a set of gore shapes and wanted to know what the cross section of the completed surface would be? And just to make things more complicated: suppose that the gore shapes are, as in Mosely’s “Bud,” truly smooth curves, rather than discrete points? Let’s address both questions at once. We assume the gore to be described by a continuous vector-valued function (r, w(r)), as shown on the left in Figure 10.29; this should give rise to a cross section that is also described by a continuous vector-valued function (x(r), z(r)), as shown on the right. The distance r is the horizontal position along the gore, as shown on the left, but it is also the distance as measured along the cross section, and that is the key to finding the relationship that gives the two unknown functions x(r) and z(r). We first note that from Equation (10.5), w(r) and x(r) are simply related: π x(r) = w(r) cot , (10.11) m where m is the rotational order. If we take the cross section r(r) ≡ (x(r), z(r)) to be a parametric curve, parameterized on the arc length r, then r(r) is a z

Figure 10.29.

Left: schematic of a gore. Right: cross section of the rotational solid with continuous parameterization.

672

(x(r), z(r))

x(r)

y (r, w(r)) z(r)

r

w(r) r


x

x

unit-speed curve, which must satisfy s 2 2 dx dz 0 kr (r)k = + = 1. dr dr This lets us solve for dz/dr: s 2 dz dx = 1− , dr dr

(10.12)

(10.13)

and thus, z(r) =

∫

r

s 1−

cot2

0

2 π dw(s) ds. m ds

(10.14)

For Mosely’s “Bud,” we have m = 4, so x(r) = w(r) because tan(π/4) = 1. If we assume the length of the gore is 1 (i.e., r ∈ [0, 1]), then we find that the parameterization of the cross section is 1 p x(r) = w(r) = 1 + 4r − 4r 2 − 1 . (10.15) 2 It turns out that z(r) has a considerably more complex expression involving elliptic integrals (which I will not give here, but is readily obtained with a little computer algebra). Despite the complexity of z(r), the shape is simple and approximately elliptical; it is shown in Figure 10.30. While one usually goes the other direction—pick a cross section, then solve for the gore—this approach can be used if you wanted to force the crease pattern to consist of particular shapes, like circles. Of course, perhaps the easiest way to find out what the cross section might be would be to simply fold one up. z 0.8

0.6

0.4

0.2

-0.2-0.1

Figure 10.30. 0.1 0.2

x

Computed cross section of Mosely’s “Bud.”

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673

r=+1

r ~1/2

r=0

r~-1/2

r =-1

Figure 10.31.

Top view of a single flange as the flange parameter ρ varies from +1 to −1.

? ? ? 10.3.4. Generalized Flanges In the thin-flange algorithm, all of the excess paper outside of the gores is gathered into either a clockwise- or counterclockwisepointing flange. In the thick-flange algorithm, the excess paper is divided symmetrically between two flanges that point in opposite directions. Is there a configuration that represents an intermediate state: a thick-flange design where the flanges are asymmetric? There is indeed, and now that we have set up models for thin and thick flanges, we can generalize the flange design to make both cases part of a single continuum of design. We can define a parameter ρ, for which ρ = −1 creates a clockwise-pointing flange, ρ = +1 creates a counter-clockwise-pointing flange, and ρ = 0 splits the paper evenly, giving the thick-flange structure. Then for every value of ρ in the range [−1, 1], there should be a structure with (usually) asymmetric (usually) thick flanges, which collectively make up a continuum of design, as illustrated schematically in Figure 10.31. There is one more generalization we can make. In both the thin-flange and thick-flange designs, the paper touches itself at the vertical plane of the wedge, that is, the gray plane shown on the left in Figure 10.32. But there is no absolute requirement that the paper comes together; we could leave a gap between the two edges, so that adjacent gores no longer touch each other, as illustrated on the right in Figure 10.32. This choice could be made for artistic reasons (to shift more paper into the flanges) or for structural reasons (to allow a drainage path for a functional container). We define δ as the half-angle between the two planes that define the sides of the gap. Then we can set up points in the crease pattern and folded form for the generalized flange, as illustrated in Figure 10.33.

674


Figure 10.32. A single wedge. Left: an asymmetric flange (ρ = 0.5) with a vertical plane (gray) where the flange meets the gores. Right: a symmetric flange but with a gap between the two sides of the flange.

We previously defined the quantity si as the length of the short angled portion of the thick flange. Now that we have asymmetric flanges, the two lengths are different, and so we introduce ti as the second length, as illustrated in Figure 10.33. We can now set up the coordinates of the crease pattern and folded form, this time accounting for the gap half-angle δ and the varying distances si and ti . To accommodate the gap, we have a reduced gore half-width: wi ≡ xi tan(φ − δ).

(10.16)

y fi

y ei

ti f i′

di

ei′

ci si bi ai ri

wi

di′

bi′ s i wi ai′

φ x

ti

δ c i′ x

xi

Figure 10.33.

Left: a single generalized-flange wedge of the crease pattern. The two half-gores are shaded. Right: a top view of a horizontal slice of the folded form.

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675

Then for the crease pattern (again, given in order of their dependencies), we have ˆ ai = ri u(0), ˆ π2 ), bi = ai + wi u( ˆ π2 ), ci = bi + si u( ˆ fi = ri u(2φ),

(10.17)

ˆ ei = fi + wi u(2φ − π2 ),

ˆ di = ei + ti u(2φ − π2 ), and for the folded form, a0i = xi uˆ 0(0) + zi zˆ 0, b0i = a0i + wi uˆ 0( π2 ),

c0i = b0i + si uˆ 0(2(φ − δ) − π2 ), f0i = xi uˆ 0(2φ) + zi zˆ 0,

(10.18)

e0i = f0i + wi uˆ 0(2φ − π2 ),

d0i = e0i + ti uˆ 0( π2 + 2δ).

The question is, what are the values of si and ti ? Let us consider first the case with no gap, δ = 0. We know three solutions: thin-flange clockwise,

ρ = −1 : si = (ri tan φ − wi ), ti = 0;

thick-flange, thin-flange counterclockwise,

ρ = 0 : si = ti = 12 (ri tan φ − wi ); ρ = +1 : si = 0, ti = (ri tan φ − wi ).

(10.19)

In order to describe the continuum of possibilities and relate it to the parameter ρ already defined, we specify both si and ti in terms of a new parameter ui : si ≡ (ri tan φ − wi )ui 1−ρ , 2 (10.20) 1+ρ ti ≡ (ri tan φ − wi )ui 2 . Equation (10.20) gives the appropriate split between si and ti as a function of ρ, but it pushes the unknown into the variable ui . For the three special cases we already know, ui = 1, which suggests that this might be the case for all intermediate values of ρ. But is it?

676


And for that matter, there is another question: could we actually vary ρ from one level i to the next level i + 1? That is, could the flange be clockwise-pointing at the bottom of the shape, symmetric in the middle, and then counterclockwise at the top? We do not have total freedom in making this choice; there are two conditions that must be satisfied. The first is isometry: fold lengths in the crease pattern must be equal to their corresponding lengths in the folded in form. In particular, considering points ci and di in the crease pattern and their corresponding points c0i and d0i in the folded form, we require that kdi − ci k = kd0i − c0i k.

(10.21)

The second condition is subtler. We have assumed in our construction that all of the facets in the folded form are planar, and thus far, for the shapes we’ve constructed, they have been so. Many of the facets are quadrilaterals, though, and there is the potential for those facets to be skew quadrilateral (i.e., bent along one or the other diagonal). We need to check this possibility. Consider two successive levels of the pattern, as illustrated in Figure 10.34. We are looking at a projection of a single wedge, levels i and i + 1, both projected into the xy-plane. Now, in 3D, there are quadrilateral facets between each corresponding pair of edges: for example, between edge a0i b0i and a0i+1 b0i+1 , and so on. For those facets to all be planar, each ith edge must be parallel to its corresponding (i + 1)th edge. Any facet between two non-parallel edges will be, at best, skewed (folded along its diagonal) and may, in fact, violate isometry along one or both of its diagonals. y

f i+1 ′ f i′

di+1 ′ di′

ti+1

ti

ei+1 ′ bi+1 ′

ei′ bi′ wi

φ xi

ai′

si

si+1 c i+1 ′

c i′ ai+1 ′

x

Figure 10.34. Top view, in 3D, showing two successive levels of a single wedge.

........


677

y

f i+1 ′ di+1 ′ f i′

ti+1

di′

vi

Top view, in 3D, showing two successive levels of a single wedge with a nonzero gap.

ei′

g i+1 ′

g i′

Figure 10.35.

δ φ

φ−δ xi

vi+1

ei+1 ′

ti

bi+1 ′ bi′ wi ai′

si+1

si

c i+1 ′

c i′ ai+1 ′

x

In order to ensure planarity of the facets, the individual edges at each level must be parallel to their corresponding edges at the next level, and that, in turn, requires that triangles 4(c0i, d0i, e0i ) and 4(c0i+1, d0i+1, e0i+1 ) be geometrically similar. For that to be the case, the ratio ti : si must be the same for each level. That means that, as implied by Equation (10.20), the splitting ratio ρ must indeed be a constant for the entire wedge. And now that we know that ρ must be constant for each level, we can plug expressions for the points into Equation (10.20), and we will find that for all values of ρ, the solution that satisfies isometry is ui = 1. Next, we consider the case of nonzero gap: δ > 0. Once again, we must have similar triangles from one level to the next, as illustrated in Figure 10.35, but the legs of the triangles are no longer si and ti . In order for the edges at level i to be parallel to their corresponding edges at level i + 1, the triangles that must be geometrically similar are triangles 4(c0i, d0i, g0i ) and 4(c0i+1, d0i+1, g0i+1 ), where points g0i are the intersections of the extensions of edges b0i c0i and d0i e0i . That means that for the general case, δ , 0, we must choose a different parameterization for si and ti , one that enforces geometric similarity. We take si ≡ (ri tan φ − wi )ui 1−ρ − vi, 2 (10.22) 1+ρ ti ≡ (ri tan φ − wi )ui 2 − vi, where vi is the distance from g0i to b0i (or e0i ) and is given by vi = xi sin δ sec(φ − δ) sec(φ − 2δ).

678


(10.23)

We can now plug this parameterization into the isometry equation, Equation (10.21), and solve for ui . There is an analytic solution (and it is not ui = 1); in fact, ui will, in general, differ for each i. The analytic expression is quite complex (and I will not present it here). But it is solvable, and, given that solution for ui , it is then straightforward to construct both the crease pattern and folded form for any cross section and combination of flange parameters ρ and gap half-angle δ. Well, almost any combination. It is clear from the form of Equation (10.22) that for nonzero δ, there are values of ρ that give negative values for si and/or ti . In fact, it is clear that for ρ = ±1 there is no possible solution that makes both si and ti positive. Variables si and ti are crease lengths; we can’t have a negative crease length in either the crease pattern or folded form. And so, if we introduce gaps between the gores, the only possible solutions are thick-flange structures. A further consideration arises from the observation that with thick-flange structures, adjacent flanges extend toward each other, and since the flange width generally grows as we move away from the origin, at some point, adjacent flanges will intersect one another. In practice, this can be dealt with in several ways; sometimes, the flanges can be arranged to overlap each other (thereby making the model deviate slightly from theory; it is usually necessary to bend some of the facets slightly). Alternatively, overlapping corners can be reverse-folded so as not to interfere with one another. An advantage of the thin-flange structure is that since all of the flanges point in the same direction going around the structure, they will always naturally overlap, never interfere. Figure 10.36 shows an example I designed using this algorithm; Figure 10.37 shows its folded realization. ???

10.4. Axial Unfoldings Back in Section 10.1.2, we “unwrapped” our sphere by peeling the gores from the North Pole down to the South Pole and flattening them out into a flat sheet—an azimuthal unfolding. There is another way of unwrapping a sphere to a flat sheet, though, and that is to unwrap the gores at both poles and map them onto a cylinder wrapped around the equator of the sphere as illustrated in Figure 10.38—called an axial unfolding or cylindrical unfolding. If we then uncurl the cylinder to a flat sheet, we will get an

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Figure 10.36. Design for “RingTower11.” Left: cross section. Middle: crease pattern. Right: computed folded form.

Figure 10.37. “RingTower11” (2017), by the author.

Figure 10.38.

Left: a polyhedral sphere. Middle: the sphere plus unwrapped gores. Right: the unwrapped gores on a flat surface.

680


y

z

y fi ei

bi

h

ri

f i¢

wi

f i¢

ai¢

ti d i si ci

f

wi ai

di¢ ei¢ bi¢ c i¢

x

di¢

ei¢ bi¢ xi

wi

c i¢

ai¢

x

y

x1

Figure 10.39.

Left: a single panel of the crease pattern. Middle: a single wedge of the folded form. Right: top view of a single level of the folded form.

arrangement of gores that has translational symmetry, rather than radial symmetry, but we can still approach creating the threedimensional shape in the same way: hide the excess paper between the gores. We can approach this design in the same way as the radial structure, but now, each wedge in 3D will be folded from a rectangle, rather than a wedge, in 2D. Once again, we split each gore along the mid-line; then we will plan to assemble the full crease pattern from an array of individual panels, each of which contains two half-gores and the flange material in between, as illustrated in Figure 10.39. As with the radial arrangement, we will place the “cut edge” of a panel on the x-axis with its lower left corner at the origin. We can set up the crease pattern panel and folded form wedge with the same generality as we did for the radially symmetric structure, allowing for a flange asymmetry parameter ρ and a gap δ between the gore edges. And there is one more bit of freedom: with the radial pattern, all of the gores had to come together at a point, so the first point of the cross section in 3D had to be the point (0, 0, 0)—that is, we required that x1 = 0. We can now relax that requirement, as illustrated in the middle subfigure. As with the radial pattern, the axial creases of the folded form (which are now precisely vertical in the crease pattern) are

........


681

d x

continuous across the bi and ei radial creases, which are now roughly horizontal in the crease pattern. Once again, the half-width of the gore at each level is wi ≡ xi tan(φ − δ).

(10.24)

We have freedom in choosing the half-height h of the panel; it must be large enough to accommodate the widest part of each gore, i.e., h ≥ max{wi }. (10.25) i

Then for the crease pattern, our vertex coordinates are given by (in order of dependency) ai bi ci fi ei di

= (ri, 0), = ai + (0, wi ), = bi + (0, si ), = (ri, 2h), = fi − (0, wi ), = ei − (0, ti ),

(10.26)

and for the folded form, a0i = xi uˆ 0(0) + zi zˆ 0, b0i = a0i + wi uˆ 0( π2 ), c0i = b0i + si uˆ 0(2(φ − δ) − π2 ), f0i = xi uˆ 0(2φ) + zi zˆ 0, e0i = f0i + wi uˆ 0(2φ − π2 ), d0i = e0i + ti uˆ 0( π2 + 2δ). Once again, we parameterize si and ti : si ≡ (h − wi )ui 1−ρ − vi, 2 − vi, ti ≡ (h − wi )ui 1+ρ 2

(10.27)

(10.28)

where, as before, vi = xi sin δ sec(φ − δ) sec(φ − 2δ).

(10.29)

Variables si and ti must satisfy the isometry condition, which is, as before, kdi − ci k = kd0i − c0i k. (10.30)

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Figure 10.40.

Left: crease pattern for Mosely’s “Pearl.” Right: “Pearl” (2017), by Jeannine Mosely.

The solutions are different, though, owing to the different parameterizations of the crease pattern points, folded form points, and variables si and ti . For no gap, δ = 0, there are, as before, analytic solutions for ui : ρ = ±1 : ui = 1, φ ρ = 0 : ui = sec2 , (10.31) q 2 2 2 2 (1 + ρ ) + (1 − ρ ) cos 2φ − 2 otherwise : ui = . (1 − ρ2 ) sin2 φ For δ > 0, there is also an analytic solution, but again, it is extremely complex and so I will omit it here. It is also possible to take sections of an azimuthal unfolding and reassemble them to get something that is similar to an axial unfolding but is composed of alternating azimuthal wedges. Jeannine Mosely, in her explorations of the rotational solid concept, developed a lovely demonstration of the concept shown in Figure 10.40. ???

10.5. Variations on the Theme

? ? ? 10.5.1. Twist Lateral Shifts After working out the mathematics for thin-flange, thick-flange, and generalized-flange patterns, with both azimuthal and axial unfoldings, the field seemed like it had been thoroughly plowed. But then, a few years later, artist and mathematician Rebecca

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Gieseking burst onto the scene with some of the most mindblowing designs I had ever seen, four of which are shown in Figure 10.41. She describes her work thus: Origami is a synthesis of my two interests: the scientific and logical in harmony with the artistic and creative. My design process reflects the interdependence of these seemingly disparate modes of thought. I apply logic and math to engineer new ways of transforming flat paper into three-dimensional forms, continually pushing the boundaries of what is possible. At the same time, I am mindful of the aesthetic qualities of the forms I create and gravitate toward designs where the complex series of folds produces simple, elegant figures. Following the traditions of origami, each piece is folded from an uncut rectangle of paper. I start by painting the paper, carefully measuring many reference points so the painted pattern will be aligned correctly on the folded shape. I then construct the form with a series of precise straight and curved creases, using water to mold the paper into shape. Since it is impossible to completely remove folds, any mistakes or changes to the design will be visible in the finished piece. By employing traditional folding techniques to create forms inspired by ceramics, wood, and glass, I transform the flat surface of paper into a sculptural object. Origami is folding with no cutting, but the combinations of the sharp folded and color transitions, offsets, and, especially, diagonal shifts have every fiber of your perception crying out that these must have been cut and reassembled. But no, they haven’t. In fact, the magic arises from a structural concept we have already seen: the triangulated cylinder of Chapter 2. Recall that in Section 2.4.6, we analyzed the bistability of each twisted section of a tube consisting of multiple twisted sections. In Figure 2.93, we assumed that the top and bottom of each tube lay in a plane. But we can relax that assumption, and that will give new degrees of freedom to use in design.

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Figure 10.41. Four designs by Rebecca Gieseking. Top left: “Double Diagonal Shift Vase 3”Ó (2014). Top right: “Double Diagonal Shift Vase 2” (2013). Bottom left: “Double Diagonal Shift Variant Vase” (2014). Bottom right: “Doubly Divided Vase” (2014).


........

685

y

tiÐ1

x

ti

qi

q¢i+1 q¢iÐ1

Figure 10.42. Geometry of a generalized twist tube joint. Left: folded form. Right: unfolded crease pattern.

p¢iÐ1

ti qi+1

q¢i

p¢i+1

p¢i

pi

pi+1 bi

y

biÐ1

x

bi

Let us assume that we have a twisted segment between two polygonal tubes, as illustrated in Figure 10.42. We will assume that we are given the polygonal cross section of the top and bottom regions, but we make no assumptions about the shape of the polygon formed by the vertices of the top and bottom—what I will call the twist surfaces of the two tubes. Each twist surface could be planar, as in the original triangulated cylinder; it could be planar, but tilted; or it could be something entirely different. So, we will assume that the folded form coordinates are of the form p0i ≡ (p0i,x (0), p0i,y (0), gi ), 0 (0) 0 (0) q0i ≡ (qi,x , qi,y , hi ),

(10.32)

0 (0), q0 (0) } are given quantities that define where {p0i,x (0), p0i,y (0), qi,x i,y the bottom and top cross sections, and {gi, hi } are unknowns to be solved for. If we assume an m-gon for the top and bottom, then there are m values for each of the sets of vertices of the twist surfaces. If we assume that the polygonal tubes on the top and bottom run vertically, that simplifies some of the correspondences between the crease pattern and folded form and also imposes a consistency requirement between the top and bottom: the girth of the top tube

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must be equal to the girth of the bottom tube. This translates into a consistency requirement on our original choice of the top and bottom polygon cross sections. We construct the width of the ith bottom panel as r 2 2 (0) (0) (0) (0) 0 0 0 0 bi ≡ pi+1,x − pi,x + pi+1,y − pi,y , (10.33) and the width of the ith top panel as r 2 2 (0) (0) 0 0 (0) 0 0 (0) . ti ≡ qi+1,x − qi,x + qi+1,y − qi,y We must have that

Õ i

bi =

Õ

ti .

(10.34)

(10.35)

i

Our goal is to find the crease pattern that folds into this shape, so we have another set of variables, the vertices of the crease pattern {pi, qi } (no primes for the crease pattern variables), which we define as pi ≡ (pi,x, pi,y ), qi ≡ (qi,x, qi,y ).

(10.36)

The crease pattern variables must satisfy isometry with respect to the folded form variables, so that constrains their values. We can assume without loss of generality that the y-coordinate of one crease pattern vertex, say, pi , is the same as the z-coordinate of its corresponding folded form vertex. But if the polygonal tubes at the top and bottom have constant cross section in z, the relative heights of the bottom vertices must match between the crease pattern and folded form, which means that this correspondence holds for all of the bottom vertices. Thus, we must have that pi,y = gi

(10.37)

for all vertices. Similarly, the relative heights of the top vertices must match between the crease pattern and folded form, but they generally don’t match from top to bottom; in the folded form, the top vertices will be closer to the bottom vertices than in the crease pattern. We can incorporate this observation by introducing a new variable, δh: qi,y = hi + δh.

(10.38)

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687

For crease pattern x-coordinates, we can choose one bottom vertex x-coordinate arbitrarily; when we do so, each of the other coordinates is completely determined from the bottom panel widths, so that p1,x ≡ 0, (10.39) pi+1,x = pi,x + bi . For the top vertex x-coordinates, the relative spacing is determined from the top panel widths, but there is an unknown x-offset, which we accommodate by introducing a variable δx: q1,x ≡ δx, qi+1,x = qi,x + ti .

(10.40)

Next, we identify the isometry equations. There are two types: with respect to Figure 10.42, the lengths of the mountain folds must match between the crease pattern and folded form, kp0i − q0i k = kpi − qi k,

(10.41)

and similarly, the lengths of the valley folds must match: kp0i − q0i+1 k = kpi − qi+1 k.

(10.42)

Note that we’ll need to accommodate wrap-around effects in the indexing; if there are m vertices, we would define p0m+1 ≡ p01, q0m+1 ≡ q01,

(10.43)

but since the crease pattern does not wrap around, Õ pm+1 ≡ p1 + (bi, 0), qm+1 ≡ q1 +

i Õ

(ti, 0).

(10.44)

i

Now, let us take stock of variables and equations. We have 2m + 2 variables: {gi }, {hi }, δx, and δz. We have 2m isometry equations: those of Equations (10.41) and (10.42). That is nearly balanced, but it leaves two free parameters: things that we can choose. Looking back at Figure 10.42, while we’ve specified the cross section of the top and bottom twists, we haven’t specified the zcoordinates of either in any way. The z-coordinates in the folded

688


form are given by the variables {gi } and {hi }. We could choose the value of one of each of these sets, which would precisely balance the number of equations and variables, or we could do something equivalent. My preferred choice is to introduce a parameter δz(0) that specifies the average separation between the top and bottom sets of vertices, which then gives the last two equations: 1Õ 1 gi = − δz(0), m i 2 (10.45) Õ 1 1 hi = δz(0) . m i 2 Now, we have a well-defined system of equations, where the number of equations matches the number of variables. In general, there is no analytic solution (there is for certain special cases), but such a system is easily solved numerically. The simplest and most symmetric cases arise by assuming that the cross sections are regular polygons, with the upper polygon rotated by some angle relative to the bottom. Figure 10.43 shows crease patterns and folded forms for three different top/bottom separations δz.

Figure 10.43. Three twist tube joints with unit-radius 12-gon cross sections with the top rotated relative to the bottom by 60◦ . Top row: crease patterns. Bottom row: folded forms. Left: δz = 0.5. Middle: δz = 1.0. Right: δz = 1.5.

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689

Figure 10.44.

Three twist tube joints with unit-radius 12-gon cross sections with δz = 1 and the top rotated relative to the bottom by varying angles δφ. Top row: crease patterns. Bottom row: folded forms. Left: δφ = 50◦ . Middle: δφ = 95◦ . Right: δφ = 140◦ .

Figure 10.44 shows crease patterns and folded forms for three different rotation angles δφ from bottom to top. Now, this formalism works for any pair of polygonal cross sections, as long as the number of panels and total girth match from top to bottom. Things get interesting if we introduce a lateral offset between the top and bottom, as in Figure 10.45, which shows the result for three different offsets (with values of δz adjusted to avoid self-intersection of the surfaces). Lateral offsets introduce a behavior that is both unsurprising and surprising: that the top and bottom twist polygons become tilted and the tilt angle increases as the lateral offset increases. That the twist polygons are tilted at all is not in itself all that surprising. We can think about the region of triangulation as having to “steal” material from the top and bottom tubes in order to make the connection between them. With a lateral offset as in the figure, the twists need to steal more material in the x-direction, the direction of the offset, than in the y-direction, where there is no offset. Hence, a tilt.

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Figure 10.45.

Three twist tube joints with unit-radius 12-gon cross sections with δφ = 120◦ and varying offsets in the x-direction. Note that as the lateral offset increases, the diagonal tilt increases as well. Top row: crease patterns. Middle row: folded forms, side view. Bottom row: folded forms, top view. Left: offset = 0.1, δz = 0.1. Middle: offset = 0.4, δz = 0.2. Right: offset = 0.7, δz = 0.5.

But the surprising part (at least, it was surprising to me) was that the tilted surfaces are exactly planar, and they seem to be so for any values of offset and rotation angle δφ, as long as the tube cross sections are regular polygons. This behavior is the key to many of Gieseking’s mind-blowing designs: one is tempted to say,“wow, she achieved both a lateral offset and a diagonal surface.” But in truth, they’re not independent effects: they come together, as part of the package. A contributing factor to the effects seen in her designs are the use of relatively high rotation orders, which give smooth, nearCHAPTER 10. ROTATIONAL SOLIDS

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Figure 10.46. “When Rebecca Met Shuzo 1,” “. . . 2,” and “. . . 3” (2016), by the author.

round surfaces. I decided to look in some different directions: what happens at very low rotational orders, like m = 4? I developed a trio of designs based on square cross section tubes. The twist closures at the top and bottom are similar to what Shuzo Fujimoto used in his iconic cube, so I titled the series, “When Rebecca met Shuzo.” They are shown in Figure 10.46. Like most mathematically designed 3D figures, most of these twisted offset tube patterns have crease patterns that must be computed and then transferred to the paper. A few, though, have sufficient regularities that they can be folded with bootstrapped reference points, and the first in this series falls into that category. I give the crease pattern in Figure 10.47 with some construction lines that will help in finding the necessary reference points. The planarity of the two twist surfaces is not universal; it only happens when we use regular polygons. An ellipse, for example,

Figure 10.47. Crease pattern for “When Rebecca Met Shuzo 1.” Construction lines are in light blue. Horizontal lines divide the paper evenly into 13ths; vertical lines divide the paper evenly into 5ths.

692


Figure 10.48. An elliptic polygonal cross section tube with a twist inserted into the tube. Left: crease pattern. Right: folded form.

Figure 10.49. An elliptic polygonal cross section tube with the top rotated by 90◦ from the bottom. Left: crease pattern. Right: folded form.

gives saddle-shaped twist surfaces—even with no lateral shift at all, just a pure twist connection, as in Figure 10.48. We could insert a lateral shift, of course; but we can also change the shape of the tube, for example, using the same top but rotated 90◦ with respect to the bottom, as shown in Figure 10.49. As we did with tiles and tessellations, we can construct mating gadgets—crease patterns and folded forms that connect to one another—to build up more complex shapes. Figure 10.50 shows an example, constructed from these elliptical twist connections (and with another twist to close the ends). The general concept of “join two tubes with a triangulated section” can be generalized even more broadly. With relatively straightforward extensions, one can use the same concept to connect two tubes with different apparent diameter, as in Figure 10.51, in which we both reduce the diameter of the second tube by pleating it and introduce a lateral shift between the two tube centers— which, not surprisingly, tilts the twist surfaces. One could even change the direction of one of the tubes, as in Figure 10.52. Each of these connections can be considered to be a building block, from which larger crease patterns and folded forms can be assembled.

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Figure 10.50. “Golden Ellipse Twist Pillar” (2017), by the author. Left: crease pattern. Right: folded artwork.

Figure 10.51. Connecting two tubes with different diameters and a lateral shift. Left: crease pattern. Right: folded form.

Figure 10.52. Connecting two tubes that are tilted with respect to one another. Left: crease pattern. Right: folded form.

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? ? ? 10.5.2. Triangulated Gores One of the aethestically pleasing features of the azimuthal thinflange designs is that because the flanges get thicker as they climb toward the top of the shape, the folded edge of the flange climbs helically, creating a definite handedness to the pot and, for smoothly curved shapes, adding to the sense of roundedness. The amount of helical motion is determined, though, completely by the cross section of the pot and the chosen rotational symmetry. I wondered: is there any way to enhance (or reduce) the helical sense by incorporating concepts from the triangulated cylinder geometry into an azimuthal unfolding of the desired shape? A triangulated cylinder has a series of horizontal polygonal cross sections, each one rotated a bit from the previous one down. We can think of that as an axial unfolding of a shape with helically triangulated gores that just happen to be constant width. An azimuthal unfolding of a thin-flange form, by contrast, has smooth, uncreased gores whose width varies with height. What we’d like to do is to put the two concepts together: use helically triangulated gores, but with an azimuthal unfolding of the desired shape. Figures 10.53 and 10.54 show the idea. For a given cross section, we can construct an azimuthal thin-flange form as in Figure 10.53, using the algorithm of Section 10.2. If we triangulate the gores, though, as in Figure 10.54, we can add an additional twist to the 3D folded form. The design of such a structure for arbitrary cross section can be done by following an approach that, I hope, is becoming familiar:

Figure 10.53. An azimuthal thin-flange pot. Left: cross section. Middle: crease pattern. Right: folded form.

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Figure 10.54. An azimuthal thin-flange with twisted triangulated gores. Top left: crease pattern. Top right: folded form. Bottom left: side view of the folded form, showing three levels of horizontal folds. Bottom right: top view of the folded form, showing rotation of the polygons at each level.

parameterize the crease pattern and folded form in a way that captures the desired features of the 3D form; then force isometry to solve for the unknown variables in the description. We adopt notation and indexing as shown in Figure 10.55. Here I use unprimed variables for the crease pattern vertices and primed variables for the folded form. Each vertex gets two indices: the first gives its radial position; the second, its rotational position. As before, we assume a desired cross section specified by a set of points (xi, zi ), i = 0, . . . , N, with m-fold rotation symmetry. And, as before, for convenience, we define the angle φ ≡ π/m and the ith gore half-width, wi = xi tan φ.

(10.46)

In both the crease pattern and folded form, we have gores and flanges. Both can be described by three sets of vertices: {pi, j , qi, j , ri, j } for the crease pattern, and their primed counterparts for the flanges. Now, if we are folding a thin-flange design, the flange valley folds (running through the qi, j points) must lie on straight line for each value of j, as shown in the figure. In the conventional thin-flange form, that line emanates from the origin at an angle

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q3,0

r3,1

p3,2

r2,1

p2,2

q2,0 z

p1,2 p1,3

p0,i p1,4

f

p3,1

p2,1

p1,1

p¢1,3 p¢3,2

p¢2,1 p¢2,2 p¢ 1,2

p¢1,1

x3 x2

p¢0,i p1,5

Figure 10.55. Notation and indexing of a twisted azumuthal form. Left: crease pattern. Gores are shaded; thin flanges are unshaded. Right: folded form, top view. One gore is shaded. Flanges are not shown.

(2k + 1)φ, but for a twisted form, it can run at some other angle, which we can characterize by an angular offset ζ, which is a design parameter that we can choose. However, the flange valley fold cannot start from the first vertex p0,i , because that would simply give the standard thin-flange form with an overall rotational offset. Thus, we must start the flange at p1, j , with the jth flange valley fold running at angle (2 j − 1)φ + ζ with respect to the x-axis. This, in turn, means that the bottom of the shape must be flat; we must have z0 = z1 , and by convention, we can take x0 = z0 = z1 = 0. So, now we have the parameters that specify the shape: {xi, zi } and ζ. We can explicitly construct the first level of the crease pattern: p0, j = q0, j = r0, j = (0, 0), p1, j = q1, j = r1, j = R((2 j − 1)φ) · (x1, w1 ).

(10.47)

Crease pattern vertices at higher radial indices are still unknown. CHAPTER 10. ROTATIONAL SOLIDS

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697

Now, let us turn our attention to the folded form. At each level of the folded form, the cross section should be a regular m-gon at height zi with side length 2wi . In the standard thinflange algorithm, that gives an explicit expression for the points {p0i, j }, but in this twisted geometry, each level is rotated relative to its predecessor by some unknown angle. We introduce ξi as the rotation angle of the ith level relative to that of the untwisted form; then the folded form vertices of the gores may be written as p0i, j = Rz ((2 j − 1)φ + ξi ) · (xi, wi, zi ).

ri,1

(10.48)

So, at each level i, we have these unknowns: crease pattern vertices pi, j , qi, j , ri, j , and the unknown folded form rotation angle ξi . If we focus just on the gores, we can ignore the vertices qi, j . And we only need vertices for a single j-value, because all of the others can be obtained by multiplying by the appropriate rotation matrix. Furthermore, if we have pi, j , we can obtain ri, j , because it is the mirror image of pi, j , reflected through the flange valley fold. So, for example, ˆ π2 + φ + ζ) (pi,1 − p1,1 ) · u( ˆ π2 + φ + ζ) , = p1,1 + (pi,1 − p1,1 ) − 2u( (10.49) ˆ where, as earlier, u(θ) ≡ (cos θ, sin θ). So, for the gores, at least, at level i > 1, there are really only two unknowns: the x- and y-components of pi,1 , and the folded form rotation angle ξi . Now we can turn our attention to isometry. The gores are fully triangulated, so satisfying their isometry equations is sufficient to ensure isometry between the crease pattern and folded form. For the ith level, isometry is satisfied for these three equations for any one j-value: kp0i, j − r0i, j−1 k = kpi, j − pi, j−1 k, kp0i, j − p0i−1, j k = kpi, j − pi−1, j k, kr0i, j

−

p0i−1, j+1 k

(10.50)

= kpi, j − pi−1, j+1 k.

Note that we could have chosen to bisect each gore quadrilateral along the other diagonal, which would give a different family of solutions; in that case, the third isometry equation would be kp0i, j − r0i−1, j−1 k = kpi, j − pi−1, j−1 k.

698


(10.51)

Figure 10.56. Twisted pots by the author. Left: crease pattern for “TwistPotD_16.” Right: clockwise from top left, “TwistPotD_16” (2017); “TwistPotB_12” (2017); “TwistPotF_9” (2017); “TwistPotA_8” (2017).

We have (at each level) three equations and three unknowns. Solving them sequentially for each level gives a complete solution for the gore vertices in both the crease pattern and folded form. What about the flange vertices, {qi, j }? We can solve for these similarly; they have their own isometry equations, the determination of which I will leave as an exercise for the interested reader (they are hinted at by the faint lines in Figure 10.55). In practical designs, I have found that the flange vertices are not terribly important, because all of the folds are quite shallow. When scoring a pattern for folding, if we leave out the flange vertices and their connecting folds, then the flanges become gently curved, which provides a nice contrast to the triangulated gores. Several examples of pots constructed with this algorithm are shown in Figure 10.56. ?

10.6. Artists of Revolution In the world of mathematics, there are two worldviews: Kantians believe that mathematics are entirely the product of human creativity; Platonists believe that mathematics exists independently of humanity, and we are merely rediscovering preexisting truths.

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699

While I am no expert in the philosophical arguments (see, e.g., [79]), I find that I incline toward the Platonist position: mathematical concepts are “out there,” and we are rediscovering them. I find the strongest evidence for this position in the regular phenomenon of independent invention (or rediscovery?), which happens all of the time in mathematics and, not surprisingly, happens not infrequently in art, especially when there is a strong mathematical underpinning to this art. Over time, many origami artists have come to explore the concept of rotationally symmetric solids. We have come to them via different avenues (in my case, “Polypouches”), but quite a few have ended in the same general category. Still, though, mathematical origami is a mixture of both mathematics and art, and the artistic side of things offers wide latitude for personalization, exploration, and uniqueness. Several artists have developed their own concept and realization of rotationally symmetric 3D forms. Some I have already mentioned (Chris Palmer, Jeannine Mosely). There are many others, and I would like in this last section to highlight a few I have run across, and whose work is distinctive, beautiful, and creative. In the latter part of the first decade of the 21st century, Japanese mathematician Jun Mitani independently developed the mathematical prescriptions for thin-flange designs [80], thick-flange designs [81], and other rotationally symmetric generalizations. He described the mathematics and gave many examples in his book, Spherical Origami [83, 84], and in a software tool he wrote, ORIREVO [82], with which one can design many such patterns. A few of Mitani’s many artworks are shown in Figure 10.57. Artist and engineer Cheng Chit Leong also independently developed an algorithm for designing such patterns, drawing on his background in naval architecture. He describes his design process as being based on “curved couplets,” a combination of a straight mountain fold with a curved valley fold (or vice versa) that, in 3D, creates apparent curvature in two directions in a surface [76]. This curved-couplet motif can be seen in all of the thin-flange family of designs, including the examples of Leong’s designs shown in Figure 10.58. I noted at the beginning of this chapter that it’s possible to develop a 3D rotational form by simply manipulating the edges of a flat twist, shaping it into 3D. Doing so, one can develop an intuitive understanding of the relationship between the shape

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Figure 10.57. Rotationally symmetric designs by Jun Mitani. Top row: left to right, “Spheres” (2010), “Bowls” (2014), “Spring” (2012). Bottom: a collection of rotational designs (and a few based on other principles) by Mitani.

Figure 10.58. Rotationally symmetric designs by Cheng Chit Leong. Left: “Greek Vases” (2009). Middle: “Wineglasses” (2009). Right: “Baseball Cap” (2010).


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701

Figure 10.59. Rotationally symmetric designs by Philip Chapman-Bell. Left: “Onion” (2008). Middle: “Triple Frangipani Box” (2008), unlocked form. Right: “Triple Frangipani Box,” locked form.

of the curved folds and the resulting folded form, and so one can design rotational forms without actually having to carry out vector mathematics (albeit, possibly with a bit more design iteration required). One artist who has developed several rotational forms using this practical approach is Philip Chapman-Bell [14], two of whose designs are shown in Figure 10.59. His “Triple Frangipani Box” makes use of a nice bit of physics: the springiness of paper combined with the pleats coming together in the center allows for two stable states, locked and unlocked. There is no rigidly foldable path between the two states, but as with many of the bistable mechanisms we have seen, practical folding can often toggle easily between the two forms. ?

10.7. Terms Axial unfolding A mapping of a rotationally symmetric solid onto a crease pattern that has translational symmetry. Azimuthal unfolding A mapping of a rotationally symmetric solid onto a crease pattern than has rotational symmetry. Developable (curved) surface A ruled surface that can be flattened to a plane; equivalently, a surface for which the direction of maximum curvature is perpendicular to the direction of the ruling line at every point.

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Gore A single segment of a rotationally symmetric surface that, if replicated and rotated, gives the full surface. Panel (of a rotational solid) A segment of a rotational solid consisting of two half-gores plus the paper between them in an axial unfolding. Ruled surface A surface with the property that at every point, there is some direction along which the surface follows a straight line in 3D. Ruling line A line on a ruled surface that is straight in 3D. Twist surface The 3D polygon composed of the vertices on one side or the other of a triangulated segment that connects two tubes of specified cross section. Unit-speed curve A parameterized curve for which the magnitude of its derivative (its speed) is everywhere equal to 1. Wedge (of a rotational solid) A segment of a rotational solid consisting of two half-gores plus the paper between them in an azimuthal unfolding.


........

703

Afterword For hundreds of years, origami artists created their designs by drawing on their intuitive understanding of the relationships between folds in paper and the resulting effects on the shape of the latter. Despite the development of formal mathematical techniques in the past few decades, most still do. But now, for many, mathematical methods add to the arsenal of design techniques, and for some classes of shape, mathematical methods are the only way to construct the crease patterns that will give rise to the desired form. The mathematical methods I’ve presented in this book are by no means comprehensive; the scope of geometric origami outgrew long ago what a single document could describe. The field of mathematical origami design is still growing by leaps and bounds, and there are topics I’ve not touched on—flashers, Ron Resch tessellations, and curved folds, to name a few—that could warrant an entire second book. (And perhaps, at some point, they will.) What I’ve tried to do is provide a framework for how to address the construction of geometric origami mathematically, and a few tools that will assist in that construction. I am often asked: what is the direction that origami is moving in the future? And my answer is, it is not a single direction: it is a radiation outward, with people exploring many different directions and choosing many different goals to shape their explorations. There are new applications—space, architecture, medicine—and new materials—wood, glass, polymers, composites. There has evolved a rich interplay between the purely artistic and the technologically applied, with concepts from art informing the engineering applications, and materials and processes from engineering enriching and enhancing the art. Part and parcel of the latter is the use of formal mathematical methods. 705

Mathematics in origami, as in so many other fields, has two aspects that drive its development. There is, first, the idea of mathematics as a tool. If we can describe our desired field of endeavor in mathematical language, then we can draw upon the centuries of mathematical development across many specializations to accomplish the goals we have set for ourselves. But there is also the exploration of mathematics for its own interest—as much aesthetics-driven as any art form. We recognize that mathematics describes the regularities of the world—its “unreasonable effectiveness,” in the words of the physicist Eugene Wigner—and so the study of the mathematics of origami can give us a view and appreciation of origami in ways that, perhaps, no other approach can. Whether your interest in origami is driven by the engineering goal of accomplishing a particular structure or mechanism; the aesthetic goal of a particular combination of line, form, and evocation; or—like mine—an untangleable mixture of the two, my hope is that the techniques described in this book, and those you might develop atop them, will lead you to a rich, satisfying endeavor. Happy folding!

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........AFTERWORD

Acknowledgements This book was conceived in early 2000 but has evolved enormously in the years since. It and I have benefitted from the input of and interaction with numerous people over the years. I’d like to thank the following: ? Roger Alperin, for many useful discussions about mathematics, corrections too numerous to count, and in particular, pointing me to Brocard points and polygons. ? The students at the Royal Academy of Art and Architecture in Copenhagen, Denmark, who tried out some of the tiling-based tessellations and provided much useful feedback. And to Ali Tabatabai, who invited me. ? The J-term students at Gustavus Adolphus College, who braved −30◦ temperatures in 2011 to serve as guinea pigs for much of the tiling-based tessellations, along with Barbara Knight Kaiser, who invited me and team-taught the course. ? The original senseis of origami geometrics, who I have been fortunate enough to meet: Shuzo Fujimoto, Yoshihide Momotani, and Koryo Miura, all of whom shared their pioneering work in origami tessellations and periodic structures through books, through displays, and in person. ? Ron Resch, who shared his work and passion for periodic folding patterns through his sculptures, videos, publications, and presentations at the Gathering for Gardner meetings. ? David Huffman, who, sadly, I never met, but whose work continues to influence and guide my own explorations; I also thank Elise Huffman and the Huffman family for allowing me to examine his work and notes. 707

? The modern masters of the origami tessellation and geometric arts, who graciously granted permission for inclusion of their work: Alex Bateman, Jeff Beynon, Philip Chapman-Bell, Thomas Crain, Andrey Ermakov, Tomoko Fuse, Nicolás Gajardo Henriquez, Ilan Garibi, Rebecca Gieseking, Eric Gjerde, Paul Jackson, Yves Klett, Goran Konjevod, Ralf Konrad, Cheng Chit Leong, Jun Maekawa, Jun Mitani, Jeannine Mosely, Uyen Nguyen, Yuko Nishimura, Chris K. Palmer, Ray Schamp, Yuri and Katrin Shumakov, Tomohiro Tachi, and Polly Verity. ? Those researchers who have taken origami into the application domain and permitted me to share their technological accomplishments: Stavros Georgakopoulos, Larry Howell, Yves Klett, Kaori Kuribayashi, Spencer Magleby, Koryo Miura, David Morgan, and Zhong You. ? Alex Bateman (again), who wrote the first origami tessellation software, Tess, and who collaborated with me on putting the shrink-rotate algorithm onto a firm analytic footing. ? Erik and Martin Demaine, who have been fast friends and mathematical collaborators on projects from the foundations of folding theory to algorithms for mathematical design. Also, their students and collaborators with whom I have had the pleasure of working or meeting through their efforts: Zachary Abel, Jason Ku, Anna Lubiw, and David Eppstein, to name a few. ? Thomas Hull and sarah-marie belcastro, who have provided mathematical advice and critique over many years; whatever mathematical rigor you may find in this book, their efforts brought it well above my usual slovenly ways. ? The late Klaus Peters, whose vision brought my first design book, Origami Design Secrets, to reality, and whose prodding moved this book from a concept onto the path to fruition. Klaus, I miss you and am sorry you could not see this one realized. ? My editors at CRC Press, Sunil Nair and Sarfraz Khan, who patiently endured the years of delay and evolution and retained enthusiasm and support for the project. Also, now that I think of it, Douglas Adams, for this: “I love deadlines. I love the whooshing noise they make as they go by.” ? The U.S. National Science Foundation and Air Force Office of Scientific Research, who funded research under the

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........ACKNOWLEDGEMENTS

Emerging Frontiers in Research—Origami Design for Integration of Self-assembling Systems for Engineering Innovation (EFRI-ODISSEI) program that supported the development of some of the underlying mathematics in this book under grants EFRI-ODISSEI-1240417, EFRI-ODISSEI1240441, EFRI-ODISSEI-1332249, and EFRI-ODISSEI1332271. Also to Glaucio Paulino of NSF (now at Georgia Tech) for his vision in championing the topic. ? My collaborators on EFRI-ODISSEI and NSF EAGER projects: Larry Howell, Spencer Magleby, David Morgan, Terri Bateman, and Denise Halverson at Brigham Young University, along with students Shannon Zirbel, Bryce Edmonson, Thomas Evans, and Todd Nelson; Chris Santangelo and Ryan Hayward at University of Massachusetts, Amherst with students Arthur Evans and Jun-hee Na; Itai Cohen at Cornell University; Thomas Hull at Western New England University; Mark Kuzyk at Washington State University, Pullman; Julie Kornfield, Azita Emami, Sergio Pellegrino, and Y. C. Tai at Caltech; Carol Livermore at Northeastern University; Roger Alperin at San Jose State University; and Martin Culpepper and Sangheeta Bhatia at the Massachusetts Institute of Technology. ? During the evolution of this book, a near-uncountably large number of typographical (and occasionally, mathematical) errors were caught by Roger Alperin, Alessandro Beber, Denise Halverson and her students, Violeta Vasilevska, Kathy Andrist, Thomas Hull, Diane Lang, and others. My hope is that the number of any remaining is at least finite and, needless to say, they are entirely my fault! ? In addition to vast quantities of error-catching, my editor, Charlotte Byrnes, provided a wealth of stylistic, grammatical, and typographical knowledge and an amazing amount of LATEX guruness. She also designed the lovely layout. ? Last, I offer a special thank-you to my wife, Diane, for over 30 years of encouragement, support, proofreading of this (and other) manuscripts, and generally making my life wonderful.

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ACKNOWLEDGEMENTS

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Index Page numbers in boldface indicate a listing in a Terms section. 2-uniform tiling, 371

fold direction, 522 Kawasaki, 554 Abel, Zachary, 516, 528 limits for twists, 238 adjacent fold angles, 558 osculating, 139, 191, 557 AKJC, see Algebraic Kawasaki-Justin Condition rotation, 408 Albers, Josef, xvi, 114, 177 rotation, of a twist, 215 Algebraic Kawasaki-Justin Condition, 68, 73 ruling, 139, 192, 554 algorithm sector, 16, 76, 461, 478, 521, 573, 629 discrete space curve, 617 sector elevation, 569, 604, 613 flagstone vertex pattern, 447 solid, 545, 550 rigidly foldable quadrilateral mesh, 524 tilt, 281, 343 shrink-rotate, 406, 473 twist, 209, 270, 408 woven tessellation, 459 winding, 397 aluminum, 511 antenna, microwave, 159 Ammann bars, 452 anticlastic, 172, 190, 637, 642 angle antipode, 585, 603 bend, 139, 190, 576 anto Brocard, 259, 268, 387 crease, 25 central, of a cyclic polygon, 322, 342 pleat, 228, 268 dihedral, 12, 74, 479 sector, 25, 30, 73, 225 directed, 608 aperiodicity, 80, 190 edge torsion, 578, 603 aramid fiber, 189 exterior, 248 Archimedean tiling, 347, 402 fold, 12, 75, 477, 521, 629 arctangent, four-quadrant, 608

723

area of a spherical triangle, 481 of a triangle, 609 aspect ratio parameter, 410 assignment, see crease assignment automated folding, 188 axial order, 656 axial unfolding, 679, 702 azimuthal unfolding, 656, 695, 702 Barreto, Paulo Taborda, xvi, 118, 498 bas-relief, 186 basis vectors, 91, 190 Bateman CP-FF Duality Theorem, 412 Bateman, Alex, 195, 266, 395, 412, 432, 455 Bauhaus, xvi, 114, 177 Beber, Alessandro, 378 bend angle, 139, 190, 576 bending equality constraint, 641 Besenyei, Ádám, 331 Bettens, Christiane, 392 Beynon, Jeff, 164 Big-Little-Big Angle Condition, 26, 73 Big-Little-Big Angle Theorem, 25, 34, 36, 46, 73, 212, 229, 230, 244, 247, 249, 274, 508 binding condition, 547, 567 binding crease, 548, 603 binding state, 548, 603 bird’s-foot pattern, general, 516, 533 bird’s-foot vertex, 37, 73, 97, 135, 139, 530 bistability, 161, 190 Blanco Muñoz, Miguel Angel, 392 BLBA, see Big-Little-Big Angle Theorem BLBA sector, 27, 73 Blockfaltung, 531, 533 bootstrapped sequence, 390, 402, 692 border, 15 border crease, 15, 73

724

........INDEX

border edge, 425 border vertex, 425 bow-tie trace, 543 brass, 505 Brocard angle, 259, 268, 387 Brocard point, 259, 268 Brocard polygon, 259, 268, 331 Cairo tiling, 385, 402, 453 cake mold, 184 Calladine, C. R., 154 carbon fiber, 188 carbon nanotubes, 1 Cauchy, Augustin-Louis, 517 center of mass, 407 centered twist tile, 280, 342, 356 folded form, 306 vertex coordinates, 299 central angle, 322, 342 central polygon, 208, 268 retrograde, 333 central tessellation, 393 centroid, 407, 471 Chapman-Bell, Philip, 702 Chen, Yan, 477 chicken wire pattern, 98, 102, 190 Christiansen, Hank, 636 circle great, 479, 533, 539, 596 Lexell, 585 small, 585, 604 circulant matrix, 323 circumcenter, 314, 342, 407 clockwise twist, 214, 268 closed-back twist, 214, 255, 268 cloth, 1, 265 collapse, 206, 268, 475 color-up, 14, 74

colored side, 11 composite, 503, 705 composite origami, xv computational complexity, 10 concyclicity, 320, 331, 342 conformal transformation, 217 conjugate-gradient root-finding technique, 71 Connelly, Robert, 517 consistency, 629 consistency matrix, 632, 637, 647 constrained optimization, 466, 637, 642 constraint, 466, 645 bending equality, 641 equality, 72, 466, 635, 638, 641, 642 inequality, 72, 466, 635, 641, 646 Cooper, Joel, 267 corrugation, 184, 190, 194 counterclockwise twist, 214, 268 CP, see crease pattern CP coloring, 9, 74 Crain, Thomas, 505 crane, xv crease, 5, 74 anto, 25 binding, 548, 603 border, 15, 73 iso, 25 mountain, 12 pleat, 209, 269 unfolded, 12, 76 valley, 12 crease assignment, 8, 14, 74 in shrink-rotate tessellations, 433 crease pattern, 5, 74 invalid, 11, 75 unassigned, 14, 76 valid, 9, 77 of a vertex, 18, 77

crease point, 5 crease type majority, 29, 76 minority, 29, 76 crease vector, 612, 647 creep, 2 crimp, 137, 190 crimping, see sector reduction critical design, 163 critical twist angle, 236, 268, 289 cross product, 606 crossing embedding, 100, 190, 333, 430 curved origami, 4, 577 cyclic overlap condition, 236, 268 cyclic permutation, 292 cyclic polygon, 320, 331, 342 cyclic twist, 212, 268, 272 cylindrical unfolding, see axial unfolding da Vinci, Leonardo, 102 degree, 20, 74 degree-2 vertex, 32 Degree-2 Vertex Half-Planes Theorem, 41 degree-4 vertex, 33, 543 construction of, 37 Degree-4 Vertex Half-Planes Theorem, 41 degree-6 vertex, 92 degrees of freedom, 103, 108, 426, 516, 633 Delaunay triangulation, 439, 471 Demaine, Erik D., 30 determinant of a matrix, 62 developability, 19, 63, 71, 74, 628, 632 3D, 644 condition on angles, 63 developability condition, 645 developable curved surface, 654, 702 digraph, see directed graph dihedral angle, 12, 13, 74, 479, see also fold angle

........

INDEX

725

Diophantine equation, 291, 342 directed angle, 608 directed graph, 493, 533 direction vector, 537, 603, 606 discrete space curve, 617 divot, shallow-angle, 136, 192 DOF, see degrees of freedom dot product, 606 doubly periodic, 91, 190 doubly periodic tiling, 348 dual graph, 417, 471 duality, 422, 542 duo paper, 29, 74 edge orientation, 403, 434 of a tiling, 358 edge torsion angle, 578, 603 edge, border, 425 edge-to-edge tiling, 383, 402 Elephant Hide, 663 embedding crossing, 100, 190, 333, 430 of a graph, 417, 471 orthogonal, 421, 472 straight-line, 417, 473 enantiomorphic, 352, 402 Equal and Opposite Angles Theorem, 488 Eretz Israel Museum, 33 Ermakov, Andrey, 188 errors, round-off, 72, 635, 639, 641 Escher, M. C., 345 Euler relation, 425, 427, 428 Euler totient function, 222 Euler, Leonhard, 584 Evans, Thomas, 510 even degree, 296 exterior angle, 248

726

........INDEX

fabric, 101 facet, 5, 74 flagstone, 443 point, 5 root, 627 stationary, 627 facial subgraph, 429, 472 fan, paper, 177 figurate origami, see representational origami figure of merit, 466, 642, 644 Filipov, Evgueni, 642 fixed point, 318, 342 flagstone facet, 443 flagstone tessellation, 202, 269, 443 flange, 651 generalized, 674 flat-foldability, 5, 71, 211, 222, 571, 629, 632 local, 225, 244 matrices, 68 matrix formulation, 67 of multiple vertices, 42 flat-foldable, 5, 74 globally, 42, 75 isometrically, 247, 269 locally, 42, 76 flat-foldable origami, 4 flat-foldable vertex inequality, 492 flat-unfoldability condition, 462 flexible polyhedra, 517 fold major, 109, 191, 487, 488, 613 minor, 109, 191, 488, 613 mountain, 7, 76 negative length, 99, 303, 679 valley, 7, 77 fold angle, 12, 75, 477, 521, 629, see also dihedral angle fold angle expansion, 571, 615

fold angle graph, 493, 533 fold angle magnitude, 488 fold angle multiplier, 495, 509, 520, 533 fold direction angle, 17, 75, 522 fold direction vector, 611 fold length, negative, 99, 303, 679 fold vector, 611, 647 folded form, 3, 75 folding sequence, xix forcing method for rigid foldability, 528, 533 forward problem, 545, 646 four-quadrant arctangent, 59, 608 Fujimoto, Shuzo, xvi, 170, 174, 177, 193, 264, 391, 692 fully folded, 14, 75 Fuse, Tomoko, 155, 157, 164 gadget, 502 Gardner, Martin, 98 Garibi, Ilan, 39, 180 Gauss map, 538, 594, 604 Gauss’s formulas, 484 Gaussian curvature, 540, 655 Gaussian sphere, 535, 538, 604, 611, 629 general bird’s-foot pattern, 516, 533 generalized Mars pattern, 121 generalized Miura-ori, 144 generating line, 120, 135, 190 generating strip, 129 generating vertex, 93, 190 generic, 330, 342, 517 genus, 282, 342, 346 Georgakopoulos, Stavros, 157, 159 ghost paper, 43, 75, 274, 567 Giegher, Mattia, 112 Gieseking, Rebecca, 684, 691 Girard’s Theorem, 29, 482, 489, 544, 553 girth, 686 Gjerde, Eric, 391, 443, 453, 455

glass, 705 globally flat-foldable, 42, 75 Goldberg tiling, 393, 402 Goldberg, Michael, 393 Google Doodle, 131 gore, 654, 703 gradient, 71 grafting, 532 graph directed, 493, 533 dual, 417, 471 fold angle, 493, 533 interior dual, 416, 472 local flat-foldability, 46 planar, 418, 472 plane, 416, 418, 472 primal, 417, 472 weighted fold angle, 496, 534 graph embedding, 417 great circle, 479, 533, 539, 596 grid tessellation, 206, 269, 389 Griffith, Saul, 662 Guest, Simon, 154, 636, 642 half-plane properties, 41 Harbin, Robert, 8 haversine, 483 helix, 583 Henriquez, Nicolás Gajardo, 400 herringbone weave, 469 hexagonal twist, 198 Hochfeld, Henry, 112 honeycomb cores, 188 Howell, Larry, 510 Huffman grid, 93, 152, 174, 190, 525, 591 Huffman’s Major-Angle Identity, 556, 565, 588 Huffman’s Minor-Angle Identity, 558, 560, 563, 575, 588 Huffman’s Sector Elevation Identity, 570, 588

........

INDEX

727

Huffman, David, xvii, 93, 483, 536, 556, 558, 570, 585 Hull, Thomas, 26, 30, 43, 49, 223, 380 incenter, 407 infinitesimal trace, 586 injection, see injectivity injectivity, 2, 42, 75, 413, 527 interior dual graph, 416, 425, 472, 594, 627 interior vertex, 29, 75, 291, 425 intersection of two lines, 61 intuition, 705 invalid crease assignment, 11 invalid crease pattern, 75 inverse problem, 461, 545, 591, 672 iso crease, 25 pleat, 228, 269 sector, 25, 75, 225 iso-area, 198, 202, 216, 217, 269 symmetry, 198 twist, 218 isohedral tiling, 382, 402 isometric, see isometry isometric transformation, 217 isometrically flat-foldable, 247, 269 isometry, 2, 17, 42, 71, 75, 637, 677, 687, 696 isometry equations, 638, 688 Jackson, Paul, 33, 182, 186 JAXA, 108 jazz origami, xix JIT, see Justin Isometry Theorem join line, 277, 342 Joisel, Eric, xix Justin Isometry Theorem, 44 Justin Non-Crossing Conditions, 21, 46, 223, 249, 461, 636 Justin Non-Twist Theorem, 46

728

........INDEX

Justin path, 43, 75 Justin, Jacques, 20, 21, 26, 28, 43 k-uniform tiling, 348, 371, 403 Kapton, 1 Kasahara, Kunihiko, 188 Kawasaki angle, 554 Kawasaki excess, 337, 342, 548 Kawasaki, Toshikazu, 20, 26, 49, 216, 630 Kawasaki-Justin Condition, 21, 68, 75, 118, 223, 250, 325, 336, 413, 414, 421–424, 432, 450, 461, 487, 547, 630 Kawasaki-Justin Theorem, 17, 20, 27, 30, 45, 75, 97, 535, 548, 630 Kayaragusa, 6 kinetic sculpture, 518, 529 Klett, Yves, 188, 530 knot, 624 Konjevod, Goran, 186 Konrad, Ralf, 195, 267 Kosmulski, Michał, 341 Kresling pattern, see triangulated cylinder Kresling, Biruta, 154 Kuribayashi-Shigetomi, Kaori, 174 L’Hôpital’s rule, 561, 563 L’Huilier’s Theorem, 483, 550, 553 laser cutter, 206, 505 lattice, 350, 403 lattice patch, 350, 403 of Archimedean tilings, 356 lattice vector, 354, 403 Law of Cosines, 481, 574, 590 Law of Sines, 481, 574, 588, 590 leather, 1 leaves, 627 Leong, Cheng Chit, 700 Lexell circle, 585 Lexell’s Theorem, 585

Lexell, Anders Johan, 584 LFF graph, see local flat-foldability graph Li Tre Trattati, 112 line, 12, 55 generating, 120, 135, 190 intersection, 61, 299 join, 277, 342 tile, 86, 192 vector representation, 55, 605 linear chains, 88 Liu, Xueli, 159 local flat-foldability, 225, 244 local flat-foldability graph, 46 locally flat-foldable, 42, 76 logarithmic spiral, 396 loop in LFF graph, 48 in weighted fold angle graph, 494

consistency, 632, 637, 647 determinant, 62 developability, 63 diagonal terms, 71 flat-foldability, 68 identity, 607 pseudoinverse, 324 row-major, 62 singular, 323 skew-symmetric, 634 maximal twist, 227, 269 Maxwell Condition, 427, 428, 472 Maxwell, James Clerk, 423, 445 mechanical advantage, 514, 534 mechanical metamaterial, 149, 172, 191 mechanical yield, 2 mechanism, 518, 534 merit function, 466 Maekawa, Jun, 28, 188, 217 mesh Maekawa-Justin Condition, 28, 76, 450 quadrilateral, 518 Maekawa-Justin Theorem, 28, 30, 34, 41, 76, 92, TQ, 643 247, 482, 544 metal, 182, 503 “Magic Ball”, 172 metamaterial, mechanical, 149, 172, 191 Major Angle Equality Theorem, 556 midfold point, 586 major fold, 109, 191, 613 minimal twist, 225, 269 of vertex, 487, 488, 534 Minor Angle Equality Theorem, 558 majority crease type, 29, 76 minor fold, 109, 191, 613 Mars pattern, 119, 511, 598 matching, 613, 647 generalized, 121 mismatched, 613, 647 matching minor fold, 613, 647 of vertex, 488, 534 material thickness, 2 minority crease type, 29, 76 Mathematica, 467 mismatched minor fold, 613, 647 mating, 272, 278, 289, 342 Mitani, Jun, 700 pairwise, 276 Miura, Koryo, 99, 106, 107, 147 Principle of, 274 Miura-ori, 108, 152, 174, 191, 476, 488, 498, matrix 511, 518, 530, 533, 597, 637, 642 2D rotation, 58, 298, 645 3D rotation, 607 generalized, 518

........

INDEX

729

modular origami, xv Momotani’s “Wall”, 201–203 Momotani, Sumiko, 193 Momotani, Yoshihide, xvi, 193, 201, 264, 391 Mona Lisa, 102 monostability, 161, 191 Morgan, David, 184, 529 mosaic, xvi, 345 Mosely, Jeannine, 664, 683, 700 mountain crease, 12 mountain fold, 7, 76 mountain-like vertex, 29, 76 Mylar, 1 nano tessellation, 267 Napier’s analogies, 485 napkin-folding, xvi, 114 necklace problem, 222 negative area of a trace, 540 negative fold length, 99, 303, 679 negative Poisson’s ratio, 148 negative twist angle, 411 Nguyen, Uyen, 157 Nishimura, Yuko, 184 Nojima, Taketoshi, 155, 164 non-degenerate vertex, 613 non-local interactions, 11 non-rigidly foldability, 476 non-self-intersection, see injectivity non-stretchy paper, see isometry nonregular polygon, 382 norm of a vector, 57 normal, 537 np-complete, 636 null space of a matrix, 324 O’Rourke, Joseph, 30 oak, 505 offset of a translation, 57

730

........INDEX

offset twist tile, 287, 342, 366, 503 folded form, 311 vertex coordinates, 304 open-back twist, 214, 269 openness of a vertex, 547 ORI-REVO, 700 orientation, 358, 403, 434 origami sekkei, xix origami tessellation, xvi origami, definition of, xv, 1 Origamizer, 644 orthocenter, 407 orthogonal embedding, 421, 472 orthonormal unit vectors, 606 Oru, 265 osculating angle, 139, 191, 557 osculating normal, 546, 604 osculating normal vector, 611, 613 osculating plane, 537, 545, 604, 667 overlap, cyclic, 249 Palmer, Chris K., xvi, 25, 194, 195, 264, 380, 391, 405, 432, 498, 651, 664, 700 panel of rotational solid, 681, 703 paper Canson, 663 duo, 29, 74 Elephant Hide, 663 ghost, 43, 75, 274, 567 yucca fiber, 663 Partially Folded Vertex Half-Planes Theorem, 42 Penrose tiling, 452 pentagon tiling, 384 period, 80, 191 periodic, 80, 191 periodic tiling, 348 periodicity, 80 1D, 80 2D, 90

Perl, 406 petal fold, 3 phyllotactic spiral, 442 plain weave, 456, 472 planar graph, 418, 472 planarity, 639, 647 plane osculating, 537, 545, 604, 611, 613, 667 ruling, 549, 604, 613 sector, 569 tangent, 536 plane graph, 416, 418, 472 plastic (material), 1, 157 plastic deformation, 2 plate model, 626, 644, 647 pleat, 83, 191, 201, 208 all-anto assignment, 231 all-iso assignment, 230 anto, 228, 268 iso, 228, 269 pleat creases, 209, 269 pleat offset, 287 pleat vector, 444, 472 pleat width, 281 Poincaré map, 436 point, 12 Brocard, 259, 268 crease, 5 facet, 5 fixed, 318, 342 vector representation, 2D, 54 vector representation, 3D, 605 vertex, 5 point+perpendicular representation, 56 Poisson’s ratio, 148, 191 negative, 148 polygon Brocard, 259, 268, 331

central, 208, 268 cyclic, 331, 342 cyclic Brocard, 381 generic, 330 nonregular, 382 regular, 346 self-crossing, 450 spherical, 480, 534 polygonal tiling, 345, 403 polyhedral origami, 4 polyhedral vertex, 541, 604 polymer, 511, 705 Polypouches, 652, 662 pottery, 662 predistortion, 138, 142, 191 primal graph, 417, 472 primal-dual tessellation, 432, 472 Principle of Mating, 274 product cross, 606 dot, 606 triple, 606 pseudoinverse, 324, 342 puffy twist, 649 pyramid, 651 quadrilateral mesh, 518, 534 rigidly foldable, 519 radial unfolding, see azimuthal unfolding rainbow tessellation, 435 Randlett, Samuel L., 8 ray, 55 reciprocal figure, 424, 443, 472, 594 reflection, 60 regular polygons, 211 regular simple flat twist, 212, 269 representational origami, xv Resch, Ron, xvii, 636, 705

........

INDEX

731

resin-impregnated textiles, 188 retrograde central polygon, 333 rhombus tiling, 414, 418 right degree-4 vertex, 37 rigid foldability, 97, 108, 146, 191, 198, 219, 476 Rigid Origami Simulator, 636 rigid-body motion, 627, 647 rigidly foldable, 198 rigidly foldable quadrilateral mesh, 519 Robertson, S. A, 20 Rodrigues’s Formula, 607 root, 627 root facet, 627 root-finding, 71 roots, multiple, 71 rotation 2D, 58 3D, 607 about local axes, 608 resulting from periodicity, 84 rotation angle in 2D, 408 in 3D, 609 rotation matrix 2D, 58 3D, 607 rotational symmetry, 84, 192 round-off errors, 72, 635, 639, 641 row-major matrix, 62 ruled surface, 653, 703 ruling angle, 139, 192, 554 ruling line, 626, 654, 703 ruling plane, 549, 604 safe twist angle, 240, 269, 271, 287, 300 Sakoda, James Minoru, 118 Sallas, Joan, 114 sandwich panels, 188 scalar product, see dot product

732

........INDEX

scalar triple product, 606 scaling parameter, 408 scaling symmetry, 396, 403 Schamp, Ray, 185 Schenk, Mark, 636, 642 Second International Meeting of Origami Science and Scientific Origami, 118, 265 sector, 18, 76 anto, 25, 30, 73, 225 Big-Little-Big Angle, 27, 73 iso, 25, 75, 225 sector angle, 16, 76, 461, 478, 521, 573, 629 sector elevation angle, 569, 604, 613 sector plane, 569 sector reduction, 30 sector vector, 611, 648 self-contained tile, 302, 343 self-crossing polygon, 450 self-intersection, 283, 527, 567, 635 self-similar tilings, 395 self-similarity, 396, 403 semifoldable, 43, 76 semigeneralized Miura-ori, 128, 179, 192 semiregular tiling, 347, 403 SFT, see simple flat twist SGMO, see semigeneralized Miura-ori Shafer, Jeremy, 265 shallow-angle divot, 136, 192 shape-memory alloy, 174 shrink-rotate algorithm, 406, 443, 473 shrink-rotate tessellation, 301, 407, 473 Shumakov, Katrin, 172 Shumakov, Yuri, 170, 172 sign cutter, 206 signed length, 431 silicon, 511 simple flat twist, 194, 208, 269, 649 simple over-and-under weave, see plain weave

simple woven tessellation, 456, 473 Single Vertex Flat-Foldable Test, 31 singular multiplier, 512, 534 singular value decomposition, 324 singular vertex, 512, 534 skew quadrilateral, 638, 677 skew-symmetric, 634 small circle, 585, 604 solid angle, 481, 545, 550 spanning tree, 627, 648 species, 282, 343, 346 sphere, 653 spherical excess, 482, 570 spherical geometry, 480 spherical image, see Gauss map spherical mapping, see Gauss map spherical polygon, 480, 534 spiderweb, 431, 473 spiderweb condition, 445 spiral, logarithmic, 396 split twist, 334, 343 “Spring Into Action”, 164, 165 spurious solutions, 634, 646 square twist, 195–197 squash fold, 3 Squid Labs, 663 stacking order, 9, 21, 23, 219 standard position, 298, 355 stationary facet, 627 stent, 174 steradians, 481 straight-line embedding, 417, 473 straight-major, 556, 604, 615 straight-minor, 558, 604, 667 strain, 636 stress, 431, 636 stretched pleats, 177

structure, 518, 534 sub-tile, 319 supercritical twist, 239, 269 surface developable curved, 654, 702 ruled, 653 twist, 686, 703 symmetric bird’s-foot vertex, 37, 76, see also bird’s-foot vertex symmetry, 84, 192 helical, 97 rotational, 84, 192, 651 scaling, 396 translational, 84, 192, 348 synclastic, 172, 192 Tachi, Tomohiro, 144, 147, 152, 477, 491, 516, 518, 636, 644 Tachi-Miura polyhedron, 146 tangent plane, 536 tensile forces, 2 Tess, 406, 412, 432 tessellation, xvi, 193, 270 degree-4, 491 flagstone, 202, 269, 443 grid, 206, 269 nano, 267 primal-dual, 432, 472 rainbow, 435 shrink-rotate, 301, 473 simple woven, 456, 473 woven, 289, 454, 473 tessera, xvi tetrahedron, 610 textile weaving, 456 thick origami, 4 Three Facet Crease Assignment, 24 Three Facet Theorem, 24, 46, 76, 467

........

INDEX

733

tile, 85, 192, 271, 277, 600 centered twist, 280, 342, 356 crease pattern, 86 folded form, 86 nonregular, 381 offset twist, 287, 342, 366, 503 offset twist, general polygons, 330 offset twist, general quadrilateral, 326 pathological twists, 332 positioning of the twist, 278 self-contained, 302, 343 split-twist, 334 tile line, 86, 192 tiling, 345, 403 2-uniform, 371 Archimedean, 347, 402 Cairo, 385, 402, 453 doubly periodic, 348 edge-to-edge, 383, 402 from pentagons, 384 Goldberg, 393, 402 isohedral, 382, 402 k-uniform, 348, 371, 403 non-periodic, 391 of rhombuses, 418 P4 − 52, 384 Penrose, 452 periodic, 348 polygonal, 345, 403 of rhombuses, 414 self-similar, 395 semiregular, 347, 403 spiral, 395 two-colorable, 366, 403 two-colorable 2-uniform, 375 uniform, 347, 403 tiling lattice, 350

734

........INDEX

tilt angle, 281, 343 time efficiency, 646 torus knot, 624 TQ mesh, 643, 648 trace, 538, 604, 611, 629 bow-tie, 543 infinitesimal, 586 transitivity class, 371 translation, 57, 84 translational symmetry, 84, 192, 348 translucent paper, 413 TreeMaker, 644 trefoil knot, 624 triangle area, 609 triangle centers, 407 triangle tiles centered twist, 312 offset twist, 316 triangle twist, 508 triangle, spherical, 480 Triangle-Quadrilateral mesh, see TQ mesh triangulated cylinder, 154, 192, 528, 684 triangulation, 638 triple product, 606 Troublewit, 131, 175, 184 truss, 425 truss model, 636, 648 tsuru, xv twill weave, 469 twist, 195, 270 clockwise, 214, 268 closed-back, 214, 255, 268 counterclockwise, 214, 268 cyclic, 212, 234, 268, 272 cyclic irregular, 249 distinct, 220 hexagonal, 198

irregular polygon, 242 iso-area, 218 maximal, 227, 269 minimal, 225, 269 open-back, 214, 269 puffy, 649 simple flat, 194, 269 split, 334, 343 square, 195 supercritical, 239, 269 triangle, 243 twist angle, 209, 270, 408 critical, 236, 268, 289 negative, 411 safe, 240, 269, 271, 287, 300 twist surface, 686, 703 two-colorable tiling, 366, 403 two-coloring, 15, 76, 366, 395, 457 Tyvek™, 529 Uchiyama, K¯osh¯o, 653 unassigned crease pattern, 14, 76 unfolded crease, 12, 76 unfolding axial, 679 azimuthal, 656 cylindrical, see axial unfolding radial, see azimuthal unfolding uniform tiling, 347, 403 Unique Largest Angle Theorem, 35 unique-largest-sector vertex, 34, 77 unique-smallest-sector vertex, 34, 77 unit vector, 299, 354, 606 unit-speed curve, 673, 703 valid crease pattern, 9, 77 valley crease, 12 valley fold, 7, 77 valley-like vertex, 29, 77

vector, 82, 192 2D, 54 3D, 605 direction, 606 fold, 611 fold direction, 611 representation, 3D, 55 to describe periodicity, 82 unit, 606 vector product, see cross product vector sum, 444 vector-valued function, 672 veneer laminate, 182 Verity, Polly, 505, 532 vertex, 5, 16, 77 bird’s-foot, 139, 530 border, 425 degree-4, general, 543 degree-6, 92 even degree, 296 generating, 93, 190 interior, 29, 75, 291, 425 mountain-like, 29, 76 non-degenerate, 613 partially folded, 480 polyhedral, 541, 604 reflex, 333 singular, 512, 534 symmetric bird’s-foot, 37, 76 unique-largest-sector, 34, 77 unique-smallest-sector, 34, 77 valley-like, 29, 77 vertex crease pattern, 18, 77 vertex figure, 290, 343, 347 Vertex Folded Form Half-Planes Theorem, 41 vertex point, 5 vertex type, 29 Voronoi diagram, 439, 473

........

INDEX

735

“Wall”, Momotani’s, 203 warp, 456 Waterbomb tessellation, 164, 174, 192, 603 wedge, 209, 270 rotational solid, 660, 703 wedge vector, 444, 473 weft, 456 Weierstrass substitution, 485, 490 Weierstrass, Karl, 486 weighted fold angle graph, 496, 534 white side, 11 white-up, 14, 77 Wigner, Eugene, 706 Wilson, Andy, 266

736

........INDEX

winding angle, 397 wood, 182, 503, 705 wood composite, 529 wood veneer, 504 woven tessellation, 289, 454, 473 Yoshimura pattern, 98, 101, 152, 192, 476, 525, 533 Yoshimura, Yoshimaru, 101 Yoshimura-Miura hybrid patterns, 126 Yoshizawa, Akira, 8, 131 You, Zhong, 174 Zero-Torsion Planarity Theorem, 581

Robert J. Lang - Twists, Tilings and Tessellations

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