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FRACTAL GEOMETRY IN MESOAMERICA
by Gerardo Burkle Elizondo Universidad Autonoma de Zacatecas Zacatecas, Mexico
© Symmetry Foundation. Digitized 2004 by permission of publisher.
Elizondo, G. B. (2001). Fractal geometry in Mesoamerica [Special issue of Symmetry: Culture and Science]. Symmetry in Ethnomathematics, 12(1-2), 201-214. Budapest, Hungary: International Symmetry Foundation.
This product was funded by the National Science Foundation (NSF) as a component of the National Science, Technology, Engineering, and Mathematics Education Digital Library (NSDL), award number DUE0121749. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of NSF.
Symmetry: Culture and Science Vol. 12, Nos. 1-2, 201-214, 2001
FRACTAL GEOMETRY IN MESOAMERICA Gerardo Burkle Elizondo
Doctorado En Historia Del Centro Interinstitucional De Investigaciones En Artes Y Humanidades De La Universidad Autonoma De Zacatecas. Address: Colina de San Antonio #105, Las Colinas, CP 98064, Zacatecas, Zac. México. E-mail:
[email protected].
Abstract: Fractal Dimension in different groups of mesoamerican artistic and architectural works was quantified in 106 structures. To make the analysis we used the “Benoit” program in order to calculate “Box Dimension”, “Information Dimension”, “Mass Dimension” and “Y” value about “Fractal Dimension”. We found a general average of 1.92 for fractal dimension with values by groups in a range between 1.883 and 2.038. These results are in a very good agreement with the fact that the values of complex fractals are usually situated between 1.5 and 2.0. Fractal dimension shows the efficiency with which an object fills the space that contains it, and is quantificated like the morphology of their complexity. Scaling properties exponent were proved, and specifically the fractal dimension obtained looks like a possible pattern. We think that part of the simple rules that could explain these complex dynamics findings is the fact that the artists and architects in Mesoamerica used mathematics in their work. The finding of “proportional systems” and “golden units” to measure in modular forms, like scaling rectangles and other techniques to get harmonic and constant units in the case of Mesoamerica, had been proved by some authors.
INTRODUCTION The study of geometric analysis and mathematics that were used in art composition and construction in Mesoamerica is not yet clearly established.
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Some approaches have been developed, and in general agree with the fact that there was a basic form that fundamentally consisted in the use of “golden unit measurements”. In her book, Margarita Martínez del Sobral (2000) speaks, as did before Beatríz de la Fuente about the Olmec sculpture (1984), about the existence of proportional systems consisting of rectangular standardized modules of standardized measurements. These rectangles were in fact golden and harmonic units. She found some constants in this way, basically different golden rectangle measures, and a developing in a spiral harmonic method related to growing rectangles. Therefore the figures appear like proportions in a module that can be geometrically analyzed. The rectangles can also move and grow in a spiral way to get, with the rotations, new and bigger rectangles but in proportion with its golden basic initial unit. Also she shows in her book the “Fibonacci serie” and finds many turn angles that were used in Mesoamerica in art and in architecture and urbanism. Both, De la Fuente and Martínez had proven that in the prehispanic world a system like this was used constantly by many cultures over the times, as a well developed geometric scheme by scribes (Tlacuilos), artists, sculptors and architects making of this a standardized technique and a universal tool in composition, and that these rules were transmitted like a tradition from one generation to another. The fact that the use of mathematics was a conscious planning device is not discussed now, but we need to try to understand better about the techniques that were used to calculate, design and create aesthetic proportions, in order to know what the evolution of art and history really was.
HYPOTHESES Mesoamerican astronomers developed, as is well known, a deep knowledge about the movement of stars and planets. They in fact made a connection between the cosmic geometric scheme that learned from the sky, and their own lives. To them all were related to sacredness and to the gods: their art, architecture and even the places and form of construction and the orientation of the cities. The part of archaeology that studies this matter is named “archaeoastronomy”.
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The Maya mathematical system was a vigesimal one. Eric Thompson (1950, p. 143) found the “uinic” like a standard measure whose value was 20 units in a cord-length used to measure the land. Before him the linguist Daniel Brinton (1890, pp. 433-451) reported that he found “words for measurings” in nahuatl and maya-yucateco languages that correspond to some of the dimensions of the human body, for example the arm and the foot and others. Some authorities like V. Garth Norman had found a very probable mesoamerican cultural marker in a “codo” equivalent, whose measurement is 49.5 centimeters or 19.49 inch, and that this and their multiples and divisions were used, for example, in Theotihuacan and in many parts of the Maya area. The engineer Hugh Harleston (1974) who made measurements of the Theotihuacan pyramids finally also found the smallest fraction represented by the number 19 and other standards very close to a meter. In a previous paper we reported the finding of Fractal Dimension in many mesoamerican art and architectural works (Burkle 2001). Now we collected a greater number of structures, especially a big and representative number of pyramids to increase the value. From the foregoing evidence, we think that the use in this field of “fractal geometry” would make sense in two ways: first we hypostatize that mesoamerican art, sculpture, aesthetics and architecture, with particular attention to the mayas, are “fractals” or at least have fractal dimension in their basic form. This could mean that the designs, the projects and their conceptions are easier to understand in a base of complex patterns with up or down scale properties, that are characterized by a fractal exponents with particular magnitudes. We found these values between 1.883 and 2.038 in the averages of ten groups that we submit to analysis. The second point is that we think that probably the mayas knew and used mathematics that today science calls “non-linear”, “scalant”, “boundaries”, “Fibonacci series”, “extrange attractors”, “self-affinity”, “self-similarity”, “fractals”, concepts that belong to “chaos theory”.
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Confirmation about the discovery of a standard unit requires different ways of study and precise and consistent use checking, and we think that the “fractal and chaos theory” application could help to increase the understanding of the real dimension and grade of mesoamerican cultural development.
METHODS Today fractal analysis is a universal tool that had been applied to many sciences. It seems that no previous study had been done about its application to understand ancient mathematics, especially those that were used in México during the prehispanic period. We know that they were very good mathematicians, and in fact our findings in this study confirm this, especially because in our fractal analysis we found a very close geometric pattern has is shown in the results. The study was divided in 10 groups: 1. Walls sculptures of Palenque temples 2. Maya stelas of different places 3. Maya hieroglyphs from Palenque 4. Pyramids and temples from various places of Mesoamerica 5. Different calendars in stones and codex (Tonalamatl) 6. Various codex pages, specially from the Dresde one 7. Murals of different parts of Mesoamerica 8. Great stone monuments like the Aztec calendar, Coatlicue, a Tula giant, stela D from Copán and a colossal head from La Venta 9. Astronomic stones and 10. Figures in ceramic from the Maya culture. The ten groups give a total of 106 figures that were analysed and that are described in Table 1. Basically the images correspond first to Maya culture, developed at Chiapas and Yucatán in México, Tikal at Guatemala and Copán Honduras, and to the Aztec or Mexica cultures developed at Mexican Central High-plains, and some to the Toltec and Olmec cultures from México.
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God Coyoxauqui of the Aztec culture at México
Temple Pyramid of Tikal of the Maya culture at Guatemala
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Disc metal plate of the Cenote Sagrado of Chichen Itzá, of the Maya culture at Yucatán, México
Codex Fejérváry-Mayer page 1, Mixteca culture from México
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Temple of the Sun Panel at Palenque Chiapas, México
To make the analysis these figures were first scanned and saved as bitmap files. Thereafter the images were analysed with the “Benoit Program version 1.3” for Fractal Analysis System. Then we calculated three main characteristics about the fractal dimension of each one: its Box Dimension, Information Dimension, and the Mass Dimension with the “Y” value in each case with their respective standard errors, and intercepts on log-log plots graphs. The selection of the structures and images were non-random.
Table 1: Art and Architecture Analysed Groups
I. PALENQUE AND OTHER TABLETS (Milbrath 1999). II. MAYA STELAS AND OTHERS (Aveni 1997). III. MAYA HIEROGLYPHS (Milbrath 1999). IV. PYRAMIDS AND TEMPLES OF MESOAMERICA (Marquina 1964).
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V. CALENDARS (TONALAMATL) AND ASTRONOMIC STONES (Thompson 1950). VI. CODEX PAGES (Harleston 1974). VII. MURALS OF MESOAMERICA CITYES (Aveni 1997). VIII. GREAT STONE MONUMENTS (Tyler 1995). IX. ASTRONOMIC STONES (Broda Prucha 1991). X. CERAMIC VASEES AND OTHERS (Schele and Grube 1997).
EXPLANATION ABOUT THE ANALYSIS (BENOIT FRACTAL ANALYSIS SYSTEM) To get our data we used the program (TruSoft Int’l Inc. Benoit, version 1.3: Fractal Analysis System. 20437th Ave. No. 133, St. Petersburg, FL 33704, USA) we had mentioned before. This analysis system permits to get different kinds of fractal calculations (fractal dimensions). We chose three characteristics to be studied because it seems to be specifically useful to the understanding of the dynamics we were studying. These three methods are: a)
Box Dimension;
b) Information Dimension and c)
Mass Dimension. In each case we get too the “Y” value.
The three methods belong to self-similar pattern “x” analysis, and can be better understood if we pay attention to some of the relevant aspects: first, if we begin using some important aspects of chaos theory, actually in this case the use of concepts like “order” can help us to look at the fundamental tasks of this work and, second: fractals are in fact “the geometry of chaos”, but a special kind of chaos in which order emerges with laws governing the description of the system. Starting from the properties of a “fractal” as the principles that make reference to its own characteristics like: 1. The value of its dimension, that is between 1 and 2 depending on the occupied space and the dynamic (Hausdorff dimension).
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2. Self similarity is another characteristic that means that the form does not change when the scale does, but that becomes more and more complex (complexity) in the nonlinear fractals, especially the 2 nearest values. 3. The self-similarity in fact mathematically means a potential relation between the size and the measurement or scale. 4. Its dimension is fractional or fractal and non-Euclidean. 5. They have an infinite longitude. The data set has scale invariance characteristics and exhibits a fractional motion with statistical properties in a curve with a scaling factor between 1 and 2 (Hurst exponent). This scale invariance can be tested studying the set in many length scales, which can be done with the “Benoit Fractal Analysis System” that we used to get the information about the figures to verify first if they were or were not possible fractals, their fractal dimensions and get the exponent and try to see if this is a pattern that can be may be recognized like a characteristic marker in the results. About “Dimensions” - Box Dimension Estimation Method Interface shows “the number of boxes of linear size x necessary to cover a data set of points distributed in a twodimensional plane”. In the “Benoit” program each box and grid rotates 90 degrees size through dividing to get the minimum value of the x dimension. Information Dimension Estimation Method Interface “assigns weights to the boxes”. These boxes that contain more points or those that contains less, are analyzed from the point of view of their masses contained in “each box” (information entropy). Mass Dimension Estimation Method Interface implicates the number of points inside a circle around the data set in a two-dimensional plane to define the mass; this can be measured in “circles of increasing radius”. These three methods will reveal first if the set have fractal dimension, and its complexity degree. About “Y” - Fractal dimension is expressed like the slope in the graph. The factor represents the lagunarity or empty spaces distribution related to those non-empty, and the exponent represents the fractal dimension. The graphs a, b and c correspond to Fractal Analysis of the Glyph 14 from Palenque Palace tablet, respectively Box Dimension (Db), Information Dimension (Di) and Mass Dimention (Dm).
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RESULTS We could summarize our findings in the following order: 1. In all the 106 mesoamerican artistic and architectural images that were analysed we found fractal dimension, and all were “complex”, with a Df between 1.8 and 2.0. 2. We find that this is a very close range between the different dimensions of the ten groups of elements that were analysed. This could mean that this may be a fractal marker for Mesoamerica measurement (1.92) with the method we used. 3. We find a pattern present in the average of each group, specifically in the more complex groups. This probably reflects the use of golden units we had mentioned before that had been reported by other authors. 4. In fact this method of study, the fractals and chaos theory applied to art, history and ancient scientific culture seems to be a good and interesting tool of complexity theory for analysis to still studying. 5. We can see about the range we spoke about, that it is at the highest level of complexity. This could be understood as order emerging from the edge of chaos critically, which in fact define the concept of “complexity”. In Table 2 we can see the Box Dimension (Db), Information Dimension (Di) and the Mass Dimension (Dm) averages by groups of study. GroupNo.
Db ± SD
Di ± SD
Dm ± SD
I II III IV V VI VII VIII IX X
1.918±0.010 1.923±0.007 1.910±0.008 1.926±0.011 1.921±0.008 1.918±0.009 1.919±0.006 1.917±0.009 1.900±0.006 1.883±0.013
1.932±0.002 1.940±0.001 1.903±0.003 1.933±0.006 1.926±0.002 1.924±0.003 1.929±0.002 1.914±0.003 1.877±0.003 1.888±0.003
2.018±0.111 1.887±0.060 2.036±0.088 1.975±0.010 1.937±0.051 2.038±0.269 1.964±0.058 1.954±0.053 1.975±0.047 1.966±0.214
Table 2
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The images of the next page belongs to some of the structures that were studied such as a Jaguar sculpture from Teotihuacan México with Db = 1.917±0.003, Di = 1.915±0.001 and Dm = 1.934±0.017. The next image belongs to the Stela F from Quirigua Guatemala with Db = = 1.940±0.011, Di = 1.930±0.009 and Dm = 2.142±0.276. The analysis of the disc of Xochimilco from the Aztec culture at México shows a Db = = 1.9008±0.0075, Di = 1.874±0.003 and a Dm = 1.994±0.048. These are some examples about the results that we got in our study.
GENERAL CONCLUSIONS AND COMMENTS It is very interesting to discover that for all the 106 cases that were studied of mesoamerican culture images, all of them have Fractal Dimension. Some of these belong to those kinds of fractals or fractal geometry that can be discovered at first sight in its proportions and its dimensions, and are in general figures that imitate the nature’s aesthetics in some way. Others have greater complexity, maybe because its self-similarity or self-affinity is statistic and anisotropic just as real objects are. In both cases that we found out there could be some kind of metapatterns that need to be understood better. It is necessary to make more studies to get conclusions, but with the information we gathered, we think that there undoubtedly existed a mathematical system and a deep geometrical development in mesoamerican art and architecture, and that they used patterns and “golden units”. We also think that these wise men used complex mathematics to make their works, maybe non-linear and chaotic, trying to imitate the movements in the sky and the complexity of the nature’s phenomena to establish some kind of similitude from this side, and the world were the gods live, trying then to understand in a holy way their designs and random wishes.
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They needed to get the best possible information about the world, the universe, the implied order that exists between them and the sacred cosmos that governed their uncertain lives. Dr. Joana Broda (1991, p. XII) says that in the archaic past, the science was historically determinate in civilizations that, like now, assumed science “like a part of a social whole”. The use of fractals and chaos theory as a tool to study history for certain will open new doors of understanding between social sciences and things that we perhaps can not imagine now. In his book “Metapatterns”, Tyler Volk (1995, p. 206) asks the following metaphoric idea: “…Are boarding crossings in space the inevitable tool for portraying time’s breaks in diagrams and myths?”.
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