Curso de Introducción a la Geometría Fractal Clase 1: Introducción a la Geometría Fractal. Historia Fractales Conocidos Definiciones Características Autosimilitud Clase 2: Distintos Tipos de Fractales Algo de Números Complejos y presentación de la ecuación de Mandelbro: Z=Z*Z + C Dimensión Fractal - Cálculos Clase 3: Teoría del Caos y Fractales Dualidad Orden-Caos Leyes del Caos Ejemplos de Sistemas no-lineales o caóticos Definición de Atractores Clase 4: Más consideraciones sobre Autosimilitud y Dimensión Fractal Ecuación no-lineal de distribución de poblaciones Curiosidades matemáticas referentes a los fractales Código en C de la Curva de Von-Koch y el Triángulo de Sierpinski Clase 5: Estudio de Fractales Tecnica de creación de imágenes fractales con el Fractint y el UltraFractal Clase de Repaso: Repaso de todos los temas vistos a pedido de los participantes del curso.
Clase 1 Palabras previas Bueno, finalmente llegó el momento de compartir este curso interactivo sobre “Introducción a la Geometría Fractal”. Como su nombre lo indica el mismo intentará dar un amplio panorama acerca de esta emergente rama de la matemática, mostrar
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sus aplicaciones en distintas y variadas disciplinas tales como la Ciencia, Medicina, Arte o Humanística. Hagamos primero un breve repaso por el título de este curso, y justamente esa es la primera palabra con la que nos topamos: “curso”. He intentado buscar otras palabras más adecuadas para denominar este trabajo, pero por más vueltas que di creo que terminó siendo la mejor opción. Quizás Q uizás lo que hubiese reflejado con más exactitud mi intención al crear este proyecto y lo que realmente deseo transmitir, podría haber sido algo así como: “intercambio interactivo de conocimientos multidisciplinarios”, pero bueno, está a la vista de todos que esta frase no suena muy bien como un título. Lo siguiente con lo que nos cruzamos es e s la palabra “ introducción”. Esto puede llegar a dar la imagen de algo básico o superficial, pero de ninguna manera será así. Afortunadamente se formó un grupo de personas que provienen de disciplinas totalmente distintas y unidas por la misma motivación: descubrir más sobre la Geometría Fractal. Al mismo tiempo esa diversidad puede significar que determinadas personas no estén interesadas en el 100% de los temas a tratar, por ejemplo cuando profundice en la parte puramente matemática y proponga ejercicios muy específicos. Entonces, si bien el propósito del curso será de carácter general, paralelamente haré hincapié y profundizaré en la parte de creación de algoritmos y como acabo de mencionar la parte puramente teórica de la matemática que involucra a los Fractales. Por último, y para concluir con estas palabras previas, me gustaría hacer una breve mención sobre la metodología que seguirá este curso. Al realizarlo de manera digital y a distancia puede significar o representar algo despersonalizado o frío y de una participación pasiva. He intentado crear todas las herramientas posibles dentro del sitio www.fractaltec.org , para tener la mayor interacción posible, y esto último depende de todos nosotros. La idea principal no es que solamente bajen estos textos a su PC y los lean, sino que además de eso, puedan participar activamente, haciendo consultas, contándonos sus puntos de vista, opiniones, compartir sus trabajos e ideas, resolver ejercicios, y participar de la manera que consideren de mayor utilidad posible para ustedes mismos y todos los integrantes de este grupo. Sinceramente espero que les resulte de interés, utilidad y que lo disfruten. Espero también que este sea el punto de inicio de varios proyectos que puedan surgir con el tiempo.
Comenzando Para introducirnos de lleno en el mundo de los Fractales, lo primero que quiero contarles son las primeras palabras del libro “Fractals Everywhere ” (“Fractales en todos Lados”) de Michael F. Barnsley, B arnsley, uno de los pioneros y más importantes divulgadores del tema que existen: “" La geometría Fractal cambiará a fondo su visión de las cosas. Seguir leyendo es peligroso. Se arriesga arrie sga a perder definitivamente la imagen inofensiva que tiene de nubes, bosques, galaxias, hojas,
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párrafo solo refleja la realidad y la visión de miles de científicos y artistas de todas las gamas sin importar de que rama cultural provengan o en que especialidad desarrollen sus actividades. Esta visión se incrementará más aún en todos aquellos que decidan dec idan utilizar programas de computadoras (los cuales veremos en clases futuras) para comenzar a navegar, crear y descubrir infinitas y hermosas estructuras matemáticas a las cuales llamamos Fractales. Una importante advertencia que siempre me gusta mencionar al comienzo de toda charla es la de tener un obsesivo cuidado de no ver estructuras fractales donde no las hay. Uno de los objetivos de esta primera clase es la de reconocer tales sistemas, lo cual profundizaremos cuando veamos más en detalle la Teoría del Caos. Como decía Barnsley en su impactante párrafo, la imagen de los objetos naturales que nos rodean cambiará de manera contundente, así que la advertencia queda hecha.
Un poco de historia Los Fractales son los objetos matemáticos que constituyen la Geometría de la Teoría del Caos. Los sistemas caóticos y dinámicos fueron conocidos y descubiertos mucho antes que los Fractales. De hecho se pueden encontrar y reconocer figuras con características fractales como la del triángulo de Sierpinski (figura 1) en grabados de tela de hace varias décadas atrás, hasta en los años de 1400 se hallaron grabados japoneses con estas estructuras.
Figura 1 – El triángulo de Sierpinski El mismo lo estudiaremos con detalle desde el punto de vista puramente puramente matemático y crearemos crearemos los algoritmos para generarlo en clases posteriores.
Antes de que Newton, Leibniz y colaboradores crearan en el siglo XVII lo que hoy conocemos como Calculus y estudiamos en la facultad fac ultad como Cálculo, Análisis Matemático o Cálculo Infinitesimal, se conocían funciones con abrutas irregularidades y discontinuidades, pero los científicos de aquella época supusieron que esas mismas funciones discontinuas eran muy escasas y que raramente surgirían en sistemas naturales, por lo que las consideraban excepciones a la matemática tradicional y simplemente las dejaban de lado. Un grupo de matemáticos comenzó a darse cuenta c uenta que en la naturaleza se daba muy seguido el fenómeno de irregularidades y que no eran excepciones como se suponía. Los primeros que comenzaron a demostrar teóricamente esta problemática fueron Cantor (con su famoso y casi cas i místico conjunto de Cantor – Figura 2) y Peano. Hasta llegar a los años de 1880 con Poincaré, al que se lo conoce como el padre de la Teoría del Caos.
Figura 2 – Conjunto de Cantor
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posee.
Otra estructura matemática ya conocida en esa época y que más tarde pasó a formar parte de uno de los fractales más reconocidos es el de Koch “Snowflake” Curve, o la curva de “Copo de nieve” de Helge von Koch (figura 3). Figura 3 – El Copo de Nieve de von Koch ó Snowflake 3-A
En sus dos versiones más conocidas. Al igual que las anteriores imágenes las estudiaremos con más detalle matemático y crearemos los algoritmos para generarlas. En la próxima clase usaremos este modelo para describir como se genera un fractal. En general la primera de estas dos figuras (3-A) se utiliza para generar los algoritmos ya que es mucho más sencilla e intuitiva, la segunda se desprende directamente de la primera.
3-B Entonces, hasta el momento vimos de manera general como los sistemas dinámicos y caóticos se fueron introduciendo en el pensamiento de la comunidad científica y como fueron surgiendo nuevas necesidades para reformular teorías y crear nuevas herramientas que pudieran describir sistemas que hasta ese entonces eran considerados excepciones y a los cuales se les realizaban diferentes aproximaciones y se les aplicaban diversos “trucos” matemáticos para ajustarlos a las herramientas disponibles en esos tiempos. Más adelante veremos los problemas que surgen al aplicar estos tipos de técnicas de aproximación y omitir variables y detalles que con el trascurso del tiempo pueden derivar en sistemas caóticos y totalmente impredecibles, pero no nos adelantemos. Hemos visto también que objetos con estructuras que hoy reconocemos como fractales han existido durante muchos años, como ser todas las figuras anteriormente mencionadas, y es por ello que diversos libros y autores difieren al contarnos la historia de cómo surgió esta rama de la matemática. A pesar de toda esta introducción aún no les he contado como nacieron los fractales propiamente dichos ni como fueron formulados teóricamente.
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este tema debido a mis pobres conocimientos en el área de electrónica, al a l menos por el momento!). Lo que sí es cierto es que esas fluctuaciones no podían ser descriptas por la matemática estadística que existía. Mientras seguía adelante con sus tareas empezó a imaginar en que otros sistemas podría encontrar patrones similares que no puedan ser descriptos con exactitud por la matemática existente y que se comportaran de igual manera. Su visión lo llevó a hacerse una pregunta que para la mayoría de nosotros puede resultar obvia y hasta para muchos otros ser tonta o en el mejor de los casos sin sentido. Su famosa pregunta fue: Cuánto mide realmente la costa de Inglaterra? Ok, cualquier que tome un libro de geografía o un mapa va a poder contestar esto sin ningún tipo de problema. Imaginemos que el dato que encontramos es de 2.000 kilómetros (no conozco el valor real). r eal). Ahora bien, esos 2.000 KM., de donde provienen? Cómo se midieron? Para contestar esto voy a proponer 3 situaciones diferentes, con distintos puntos de vista: 1) 2)
3)
Si medim edimos os las las cos costa tass de de In Inglat glater erra ra desd desdee un un sat satél élit ite, e, vamos amos a ver ver que sus bordes son suaves, armónicos, con líneas casi rectas y ángulos prácticamente redondeados. Prob Probem emo os ah ahora medir edir la mism mismaa dis dista tanc ncia ia,, per pero o des desde de un aavvión ión qu que vuela mucho más bajo que el satélite. Qué pasa en este caso? Ahora que vemos las cosas con más detalle por estar más próximos, nos damos cuenta que los bordes no eran en e n realidad tan suaves como se había observado anteriormente, sino que notamos muchas más rugosidades. Imag Imagin inem emos os por por últ últim imo o un un ter terce cerr pun punto to de part partid ida, a, algo algo extr extrem emiista, sta, pero vale. Esta vez no estamos ni en un satélite, s atélite, ni en el avión; esta vez nos encontramos parados sobre la misma costa de Inglaterra con una regla como la que usábamos en la escuela, y nos ponemos a medir roca por roca, rugosidad por rugosidad, detalle por detalle.
Cuál creen que será el resultado de las distintas mediciones? Siempre habrá arrojado el mismo resultado? Si fue variando, cuál habrá sido el de mayor extensión? Sería bueno que antes de seguir leyendo sacaran sus propias conclusiones y las justificaciones de por qué pasa eso. La realidad y la geometría tradicional nos muestra que una figura con bordes rectos o redondeados tiene una longitud menor que otra equivalente pero llena de rugosidades. Comparemos por ejemplo un círculo perfecto per fecto con la figura que vimos anteriormente de von Koch, (figura 3-B). La segunda tiene una longitud mucho más grande en sus bordes que la circunferencia; c ircunferencia; y al que no he podido convencer a través de las imágenes espero poder hacerlo hacer lo la clase que viene cuando analicemos un fractal por dentro. Teniendo en cuenta esto volvamos al caso de la costa de Inglaterra y la pregunta de Mandelbrot. Si acabamos de decir que una longitud sin rigurosidades es menos extensa que una totalmente irregular, entonces podemos asegurar que los resultados de las 3 mediciones serán en todos los casos diferentes, y el de mayor extensión será el tercer caso, cas o, ya que es en el cual nos topamos con más detalles, a los cuales hubo que medir uno por uno. En realidad el resultado de este último caso se acercaría a infinito en el marco teórico. De qué dependerán nuestras mediciones, entonces? Justamente de la escala
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Qué son los Fractales? Cada vez que uno toma un libro sobre Geometría G eometría Fractal y busca una definición clara surgen por algún motivo diferentes enunciados. Lo que personalmente sospecho que no se trate de un problema de exactitud, sino que se requiere de tal abstracción para comprender el concepto de Fractal que los puntos de vista varían drásticamente por más que se esté hablando de lo mismo. Otra consecuencia puede llegar a ser que la Geometría Fractal se ha trasformado en una herramienta multidisciplinaria utilizada por científicos, artistas, psicólogos, sociólogos, etc ... entonces, un matemático no va a dar una definición de la misma forma que la dará un programador de computadoras o un artista plástico. Por lo tanto, para poder contarles que es un fractal, y al ser este que nos reúne un grupo multidisciplinario, decidí enunciar un conjunto de definiciones y pensamientos y al final intentar unirlos todos para que junto con mi propia visión, poder formular una definición lo más clara clar a y contundente posible. Empecemos. Por el momento entonces solamente voy a enumerar una serie de frases sueltas: -
1) Los Fractales son los objetos matemáticos que conforman la Geometría de la Teoría del Caos.
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2) La Geometría Fractal es también conocida como la “Geometría de la Naturaleza.
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3) La palabra Fractal, enunciada por Mandelbrot proviene del latín y significa roto, quebrado. (esto se asocia con las discontinuidades de funciones matemáticas que mencionaba en párrafos anteriores).
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4) La Geometría Fractal es un nuevo lenguaje; ya que los puntos, rectas, esferas, elipses y demás objetos de la geometría tradicional son reemplazados por algoritmos iterativos computacionales que permiten describir sistemas naturales, caóticos y dinámicos.
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5) Los Fractales son objetos cuya dimensión es no entera o fraccionaria.
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6) Un objeto fractal es aquél que su dimensión fractal de Hausdorff -Besicovich supera a su dimensión topológica.
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7) Un objeto fractal es aquél que posee las siguientes dos características: a) Autosimilitud, b) Dimensión Fractal
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8) Un fractal es un objeto en el cual sus partes tienen “alguna” relación con el todo. (esto está íntimamente ligado a la Autosimilitud)
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especial se da en la Física. La teoría del Caos no es la excepción a la regla, y se sustenta, entre otras cosas, sobre la Geometría Fractal. La segunda nos dice, que a la Geometría Fractal se la conoce como la “Geometría de la Naturaleza”, y este caso no lo voy a explicar con palabras, sino con imágenes:
Sigamos adelante. El tema de dimensión fractal y de Autosimilitud es el que viene en la próxima clase. Por lo tanto ahora solo daré algún ejemplo para par a al menos dejarlos con la idea de que se trata.
a) Autosimilitud:
Cada porción de un objeto tiene las mismas características
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La primera de estas cuatro imágenes es el conjunto de Mandelbrot en su estado original, o sea, sin ninguna iteración, o para par a que se entienda mejor, sin haber hecho ningún ZOOM dentro de la imagen. Las siguientes figuras se generan ampliando un sector del fractal y viendo que se encuentra dentro. Por ejemplo, a la segunda se le hizo un ZOOM, a la tercera cinco ampliaciones consecutivas y por último a la cuarta se le aplicaron 10 ZOOM. Nótese que estoy hablando de un ZOOM inicial en un área ár ea determinada, si hubiese elegido otro lugar de fractal donde comenzar a interactuar,
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- Dimensión 0 - Dimensión 1 - Dimensión 2 - Dimensión 3
---- ---- ---- ----
Un punto Una línea recta Un plano El espacio
Existe una quinta que es la de un conjunto vacío, se dice que el mismo posee una dimensión de –1. Existen también otros conceptos distintos de dimensión como los que mencionaba en una de las definiciones anteriores de fractales, y una de ellas es la dimensión de Hausdorff –Besicovich y se basa en la autosemejanza de objetos. Me toca en la próxima clase clas e definirla al igual que darle un sustento matemático a lo que voy a decir a continuación y que por el momento deberán creerme. Basados en la dimensión de Hausdorff –Besicovich se puede decir que la ya conocida por ustedes Curva de von Koch tiene una dimensión de 1.2619 y supera su dimensión topológica de 1 por generarse a partir de una recta. No se asusten, en unos días retomo estos temas. Solo para darles una idea de lo que es una dimensión fractal, fraccionaria o no entera, segunda característica que poseen TODOS los fractales. Hasta aquí hemos recorrido un camino relativamente largo e intenso para llegar tan solo a poder encontrar una definición lo más justa posible de la Geometría Fractal y que englobe varios temas abstractos y tal vez nada intuitivos. Así que acá va el primer ensayo para una posible definición: “La “La Geometría Fractal, llamada también "Geometría de la Naturaleza", es un conjunto de estructuras irregulares y complejas descriptas a través de algoritmos matemáticos y computacionales; los cuales reemplazan a los puntos, rectas, circunferencias y demás figuras provenientes de la matemática tradicional. Estos objetos tienen como características fundamental las propiedades de Autosimilitud y la de convivir en extraños paisajes formados por dimensiones fraccionarias.”
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FIN de la Primera Clase Ejercicios
Por lo visto en esta primera clase, no son muchos los ejercicios que puedo proponer, así que solamente voy a enunciar tres que me parecen importantes para tener las primeras nociones básicas y poder seguir adelante de una manera mucho mas clara. 1) Releer la explicación de escalas de medición en el ejemplo dado para calcular la longitud de las costas de Inglaterra. Entender por qué la tercera opción arroja un resultado de una extensión mucho mayor que las anteriores, e intentar deducir por qué en ese mismo caso, la dimensión puede acercarse a cercarse a infinito. 2) Volver a ver las imágenes en la sección de Autosimilitud, y encontrar e ncontrar todos los detalles que luzcan parecidos en las 4 figuras dadas. 3) Encontrar ejemplos de sistemas en la naturaleza, o fuera de ella, que sean fractales, justificar por qué lo son, y encontrar sistemas que posean una estructura fractal, obviamente también justificarlo.
Bueno amigos, hemos llegado al final de la primera clase de las siete que están estipuladas para el total del curso. Espero que les haya resultado de interés y ahora le toca a cada uno de ustedes realizar todos los comentarios, preguntas, consultas, manifestar sus dudas y emitir sus puntos de vista acerca del tema, todas las opiniones son válidas por igual para debatirlas. Para todo ello ustedes cuentan con foros de discusión en: http://www.fractales.fractaltec.org/foros.htm Tienen también una poderosa herramienta para estar en contacto permanente entre todos los participantes que es la lista de distribución de mails, y para ello pueden suscribirse desde cualquier página del sitio, el casillero para dicha suscripción se encuentra al tope de cada página justo a la derecha del logo de FractalTec. Otra alternativa es que pueden enviarme sus s us comentarios por privado, al mail
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Curso de Introducción a la Geometría Fractal Clase 2 Más sobre los Fractales En la clase anterior hemos hecho una pequeña introducción sobre la historia de los Fractales, como han surgido sur gido y por qué. Todo en matemática tiene un motivo de ser y una razón de existir, y por supuesto una necesidad que cubrir. Leyes, teoremas, postulados, axiomas y teorías, no están ahí solamente para satisfacer una inquietud intelectual de los matemáticos o construir entes abstractos en la mente de sus creadores. La matemática es una herramienta fundamental en nuestra vida cotidiana. Todas las disciplinas exactas, naturales y técnicas han progresado en un alto porcentaje gracias a ella, y los fractales no son la excepción, han nacido para cubrir una necesidad específica: la de comprender determinados sistemas naturales que por sus complejas características internas, no pueden ser descriptos y estudiados de una manera detallada y convincente por la geometría o matemática tradicional. Anteriormente hemos transitado también un largo camino de conceptos y propiedades de la Geometría Fractal el cual nos guió hacia el final a una definición general que intenta englobar sus características más importantes, a continuación la enunciaré nuevamente:
“La Geometría Fractal, llamada también "Geometría de la Naturaleza", es un conjunto de estructuras irregulares y complejas descriptas a través de algoritmos matemáticos matemáticos y computacion computacionales; ales; los cuales reemplazan a los puntos, rectas, circunferencias y demás figuras provenientes de la matemática tradicional. Estos objetos tienen como características fundamental las propiedades de Autosimilitud y la de convivir en extraños paisajes formados por dimensiones dimensiones fraccionarias.” Bien, ahora nos toca estudiar de una manera más detallada cada uno de estos aspectos, de no hacerlo incurriríamos entonces en un mero texto de divulgación, lo cual no es mi intención para nada.
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•
1) Existen dos tipos bien definidos de fractales. Los LINEALES y
los NO LINEALES . Estoy seguro que intuitivamente y con las imágenes que hayan visto con anterioridad en diferentes galerías los podrían reconocer sin problema. De hecho uno de los participantes del curso, Geova, planteó que hace un tiempo había creado los algoritmos y programas de manera autodidacta para generar fractales como el Triángulo Tr iángulo de Sierpinski y la Curva de von Koch, y que otros como el Conjunto de Mandelbrot no le habían salido fácilmente. Bueno, los primeros fractales son justamente los lineales y se generan a través de algoritmos conocidos por la matemática euclídea, el Conjunto de Mandelbrot se genera a través de números complejos, y tiene una dificultad mucho mayor. Igualmente ahora les voy a contar sus diferencias difer encias y a dar ejemplos, muchos de ellos ya conocidos por ustedes. Los fractales lineales son aquellos que se construyen con un simple cambio en la variación de sus escalas. Esto implica algo muy importante, los fractales lineales son exactamente idénticos en todas sus escalas hasta el infinito. En un rato cuando estemos construyendo fractales espero que quede más claro. Por el momento aquí van los ejemplos.
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Figura 2-A
Figura 2-B
Figura 2-C
• 2) De esta última propiedad (1) que hemos visto, podemos sacar una conclusión sumamente importante, y a la que hay que prestarle MUCHA atención.
Los Fractales pueden ser generados a partir de elementos de la matemática tradicional (fractales lineales), o a través de números complejos. Estos últimos seguramente los han estudiado en la secundaria. (estoy preparando un apunte aparte sobre números complejos)
De hecho, el conjunto de Mandelbrot se genera a través de iterar una cierta cantidad de veces la ecuación compleja:
Z
Z^2 + C
Donde Z^2 significa Z al cuadrado. El símbolo ^ se utiliza para representar una potencia.
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Por potencies sucesivas de los números complejos sabemos que i^2 = -1, por lo tanto nuestra expresión nos quedaría: X^2 + 2XYi – Y^2 Ya tenemos la primera parte, Z^2, ahora nos toca sumarle el complejo C: X^2 + 2XYi – Y^2 + a + bi Para sumar lo que debemos hacer es agrupar la parte real y luego la parte compleja, entonces tendríamos:
X^2 – Y^2 + a + 2XYi + bi He puesto en color rojo la parte real (X) y en azul la parte imaginaria (Y) Ya tenemos la expresión final de la ecuación que genera el Conjunto de Mandelbrot, lo que debemos hacer ahora iterarla, para ello, hay que elegir justamente, el número de iteraciones. Los fractales matemáticos perfectos y teóricos tienen un número infinito de iteraciones y detalles. Ahora retomo este tema!
3) Qué son las iteraciones? Es repetir y volver sobre sí mismo una cierta cantidad de veces. En el caso de los fractales lo que iteramos son fórmulas o ecuaciones como la que recién hemos visto para generar el Conjunto de Mandelbrot. Cómo iteramos esas fórmulas sería la siguiente pregunta ... bueno, la respuesta es sencilla, hacerlo realidad es más complicado. Lo hacemos mediante algoritmos •
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tanto no sería un fractal perfecto, per fecto, que como ya hemos deducido hace un momento solo existen en el campo teórico.
4) Dijimos muy superficialmente en la clase anterior que todos los próximo tema), pero no todos los fractales DEBEN tener una dimensión fractal ( próximo fractales son autosimilares, estos últimos son los menos, pero es necesario mencionarlos. Un ejemplo de ellos son los fractales plasmáticos, y tienen una forma muy indefinida. Veamos una imagen. •
Imagen de un fractal plasmático. Generado con el Fractint
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- Dimensión 3 ---- El espacio Como los fractales están compuestos por elementos cada vez más pequeños de sí mismo (Autosimilitud), el concepto de longitud pasa a ser algo muy complejo y hasta casi sin sentido, de esta manera el concepto de dimensión comienza a jugar un papel fundamental. De ahora en adelante no deberíamos preguntarnos pr eguntarnos cuanto mide un fractal, sino cual es su dimensión. Calcular la dimensión fractal de un objeto puede ser una tarea por demás complicada, de hecho hay muchos fractales a los que aún no se les ha podido calcular su dimensión con precisión. Es todo un tema de debate la dimensión fractal de la corteza del cerebro humano por ejemplo, aunque se dice que ronda los 2.8... Para poder comprender estos nuevos conceptos voy a trabajar ahora con fractales res ultar mucho más sencillo que con los no lineales, ya que como lineales, lo cual va a resultar acabo de mencionar, estos últimos pueden resultar muy tediosos y se necesitan otros conocimientos que escapan a este curso. (Para quienes lo deseen podemos debatir este tema por otros medios). Empecemos entonces. Para ello, y por le momento, tengamos presente que partiendo de una dimensión entera, debemos llegar a una no entera.
Es una línea recta un fractal? Espero que todos hayan contestado que NO! Que debe pasar entonces para que no lo sea? Espero que todos hayan contestado que su dimensión topológica coincide con la de Hausdorff – Besicovich o a lo sumo es menor que ella. Si alguno de ustedes tuvo algún tipo de inconvenientes para responder estas dos preguntas, sugiero que antes de seguir leyendo repase las definiciones de dimensión que vimos anteriormente.
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Dos Segmentos S = 2 Longitud 2L L = 2 (dos veces nuestra escala de medición) Esto significa que si tenemos la recta original de 1 metro, y la medimos con una regla de 50 cm. obtendremos 2 segmentos de 50 cm cada uno, o sea, dos veces nuestra escala de medición. Llegamos a la parte más importante de lo que va del curso. Que es definir la Dimensión Fractal, pero esta vez expresada matemáticamente. Para ello utilizaremos las mediciones de las rectas que realizamos hace un momento. momento. Pero antes pensemos un poco ... Resulta muy fácil para nosotros decir que la dimensión de una recta es 1, la de un plano es 2 y el espacio espac io donde vivimos es 3, pero de que hablamos realmente re almente cuando hablamos de dimensión?, como la definimos? Realmente no me ha resultado nada fácil encontrar una definición clara para contarles, por lo tanto haré mi mejor esfuerzo. La dimensión está directamente ligada con los grados de libertad. Cuando la
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Log S = D* Log L
(* denota multiplicación)
Por último divido ambos miembros por Log L y obtengo: D = Log S / Log L (/ denota división) división) Ya tengo la expresión final para calcular cualquier dimensión. Nótese que en ningún momento hasta ahora usé propiedades fuera de la matemática tradicional. Volvamos a nuestro ejemplo de las rectas r ectas que medimos anteriormente. En la primera experiencia la longitud del segmento (L) era de un metro, o sea 1 ... y la cantidad de segmentos (S) también era 1, por lo tanto nuestra ecuación quedaría: D = Log 1 / Log 1 ----- D = 1
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1) Imagen generadora Imagen 2-a
2) Aplicación del algoritmo
Imagen 2-b 3) Se comienza a generar el fractal. (Iteración)
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para ver si se lo puede considerar como fractal o no .. así que les muestro esa propiedad y luego lo discutimos. Traigamos frente a nuestros ojos nuevamente la figura 1-A
Tenemos otra vez como imagen generadora una línea recta (se genera de arriba hacia abajo). Estudien con detalle lo que ven y notarán que ahora S = 2 y L = 3. Por el contrario contrario de lo que sucede con con el Conjunto de Cantor, Cantor, el algoritmo nos divide ahora la recta en 3 segmentos iguales, pero elimina a uno de ellos y no agrega nada como sucedía antes. Por lo tanto tenemos:
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Teoría del Caos
Clase 2 Palabras Previas
Hemos llegado a la tercera clase de este curso de “Introducción a la Geometría Fractal”. Con ella finalizamos la primera de las tres etapas estipuladas. Haciendo un pequeño balance, podemos decir que las dos clases anteriores sumadas a esta que hoy nos reúne, han sido de carácter general, teniendo como único propósito el de “generar nuevos conceptos”. Aproximadamente el 20% de los participantes ya tenían conocimientos previos y avanzados sobre el tema, muchos de ellos ya habían creado sus propios códigos y publicado trabajos. El 80% restante había oído hablar de los Fractales, habían visto figuras como el Conjunto de Mandelbrot o el Triángulo de Sierpinski pero desde un
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Lo primero que quiero comentarles es una observación de uno de los participantes de este curso: krukmat. Cuando expliqué en la clase anterior como calcular la dimensión fractal de una línea, usamos la siguiente fórmula:
S = LD Donde S es la longitud
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que vimos la clase anterior, y algo más curioso aún ocurre con las curvas de Peano y Hilbert como muy bien lo destacó Bond007 en los foros de debate. Veamos las imágenes:
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El título solo nos da un panorama de lo que veremos a continuación: CAOS. Este tema tiene además un fuerte trasfondo filosófico. Preguntas tales como: Juega Dios a los dados? haciendo alusión al azar con que se presentan los hechos en la Naturaleza; el eterno enfrentamiento entre el Orden y el Caos, o el Reduccionismo y Determinismo y figuras tales como la Entropía de la mecánica clásica y
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Qué sistemas describen realmente estas ecuaciones? En primer lugar, obviamente se aplican a sistemas naturales y mecánicos reales, si no, la Física no tendría ningún sentido. En segundo lugar, y el que nos interesa particularmente a nosotros, estas fórmulas se aplican a sistemas IDEALES. A qué me refiero con ideal? Si bien ecuaciones como las de los ejemplos anteriores tienen en cuenta detalles como el rozamiento, la resistencia del aire y otra serie de variables, JAMAS van a poder contemplar la totalidad de variables que se presentan en las condiciones iniciales de un sistema. Vamos a dar dos ejemplos que se encuentran en los extremos y ello nos va a
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aire y como se potencia el rumor y cuales son los resultados.) -
Hace aproximadamente 6 años la Tierra tardó en girar alrededor del Sol un segundo más de lo estipulado. Se cree cr ee que se debió en principio a las mareas. Ese segundo de más causará ca usará que en cientos de millones de años a ños la Tierra pierda su órbita. (para nosotros mucho tiempo, para la Naturaleza un tiempo razonable)
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programa para generar fractales. Pero todo es cuestión de tiempo y perseverancia. En los foros de debate (http://www.foros.fractaltec.org http://www.foros.fractaltec.org)) en la Clase 5 abrí un foro especial llamado Software Fractal, para que puedan realizar todas sus consultas, pero háganme caso, antes de realizar preguntas intenten por ustedes mismos y agoten todas las opciones posibles, no porque no quiera contestarles como se hace algo, ya que me gusta hacerlo y lo disfruto, al igual que muchos otros participantes, participa ntes, lo digo porque sin lugar a dudas es la mejor manera de aprender, por ustedes mismos; y tomen la clase de hoy como una simple guía para poder comenzar. Los espero entonces en los foros, y si s i quieren pueden enviarme las imágenes que hayan hecho y las subiré al sitio así los demás participantes pueden verlas. Si les parece una buena idea, hasta podemos organizar un concurso de imágenes y dentro de un tiempo otro de códigos de programación en diferentes lenguajes. ¿Qué les parece? Recuerden que el próximo viernes hay clase de repaso de Dimensión Fractal y Autosimilitud. Mientras vayan creando sus imágenes fractales, no las vean solo como arte, también véanlas como piezas matemáticas y comiencen a detectar la Autosimilitud, así la clase que viene todo está más claro. Saludos para todos
Juan Pablo
Curso de Introducción a la Geometría Fractal Clase de Repaso
Palabras Previas Como les prometí en la clase anterior, hoy voy a hacer un repaso de los temas de Autosimilitud y Dimensión Fractal, ya que son los de mayor relevancia en este curso y necesarios para seguir adelante.
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Noviembre.
Repaso Los Fractales nacen de la siguiente idea, Mandelbrot, en su libro “Geometría Fractal de la Naturaleza” dice algo así como: “Las “Las copas de los árboles no son triángulos, las montañas no son conos, las nubes no son elipses... ” Lo que quiso expresar con esto es muy claro, las herramientas y figuras de las que consta la Geometría Tradicional o Euclidiana no describen con exactitud o de una manera convincente diversos sistemas que se encuentran en la Naturaleza. (Recuerden de la Clase 4, lo importante de describir con sumo detalle todos los sistemas dinámicos y caóticos) Como fue explicado en la Clase 1, Mandelbrot estaba estudiando el “ruido” que existía en determinados fluidos eléctricos. En los mismos detectó un patrón de semejanza, y el mismo mostraba que a diferentes escalas estas perturbaciones se repetían de una manera similar o auto-similar. No hace falta decir que tales observaciones le provocaron una gran inquietud y lo llevó a pensar en que otros sistemas podría encontrar tales patrones. Lo primero que formuló fue su famosa pregunta: ”¿Cuánto mide la costa de Inglaterra?” (ver clase 1) Entonces, de acuerdo a estas observaciones, resultados obtenidos y a su nueva “visión de la Naturaleza” formula el concepto de Fractal, al cual le asigna dos características fundamentales: -
Autosimilitud Dimensión Fractal
Antes de comenzar con estas propiedades, recordemos que existen dos tipos de fractales bien definidos: -
Lineales Complejos
Los primeros se generaban justamente a partir de conceptos, figuras y algoritmos lineales, como ser rectas o triángulos. (ejemplos de ellos son el Triángulo de Sierpinski, la Curva de Von Koch o el Polvo de Cantor) Los segundos se generan partiendo de iteraciones complejas, como ser:
Z
Z^2 + C
Con Con est estaa ite itera raci ción ón se gene genera ra el Conj Conjun unto to de Ma Mand ndel elbr brot ot.. fractal complejo es el Conjunto de Julia.
Otro Otro ejem ejempl plo o de de
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Había definido esta propiedad de la siguiente manera:
“cada porción de un fractal tiene la misma forma o se asemeja al fractal completo” Había dividido también la categoría de Autosimilitud en dos subconjuntos más: -
Estrictamente Autosimilares Estadísticamente Estadísticamen te Autosimilares
Estas dos subdivisiones causaron varios inconvenientes en los foros de debate, voy a ver si puedo explicarlos de una mejor manera:
A) Estr Estricta ictament mente e Autosimila Autosimilares res En primer lugar quiero recordar que esta categoría se asocia a los Fractales LINEALES. Si comenzamos a sumergirnos dentro del Triángulo de Sierpinski por ejemplo, siempre encontraremos un nuevo triángulo exactamente igual al original (fractal inicial.) Veamos las figuras:
Uno puede decir que ambas figuras son las mismas, pero no! La primera es la figura original del Triángulo de Sierpinski Sier pinski y la segunda es un trozo de la primera ampliado o iterado. Por supuesto que ambos se verán iguales, y eso se debe justamente a que son s on ESTRICTAMENTE AUTOSIMILARES . Sucede lo mismo si uno toma la Curva de Von Koch por ejemplo, y elige una zona adecuada para ampliar de dicho fractal. Encontré estas animaciones que estoy seguro que pueden ayudar mucho. Como los archivos DOC y PDF tienen problemas para agregar animaciones, les hice los siguientes enlaces: http://www.curso.fractaltec.org/SierTamiZoom.gif http://www.curso.fractaltec.org/SierTriZomm.gif Les aconsejo que los vean ya que desde mi punto de vista les aclarará definitivamente el tema de “Estrictamente Autosimilar.” Una de esas imágenes es el ya conocido por ustedes Triángulo de Sierpinski, la otra es una variante de la misma, también creada por Sierpinski, y es el Tamiz que lleva su nombre.
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Lo que quiero decir con ello, es que en este caso la Autosimilitud no es “perfecta” como en los que acabamos de ver, sino que es muy aproximada. Veamos un ejemplo a ver si de esta manera queda más claro:
Bien, en la primera figura tenemos el Conjunto de Mandelbrot original, sin haberlo iterado aún. En la segunda, tenemos una sección secc ión del mismo conjunto, pero esta vez luego de haberle aplicado 5 ZOOMs o iteraciones con el programa UltraFractal. (ver clase anterior) Lo que nos muestran estas dos imágenes, es que la segunda tiene características muy similares a la primera, pero no es exactamente igual a como sucedía con el caso anterior del Triangulo de Sierpinski donde sus ZOOMs nos revelaban siempre la misma imagen, (exactamente igual a la original.) Cuando sucede esto decimos que el fractal es “Estadísticamente Estadísticamente Autosimilar”. Y son los que se presentan en la Naturaleza, en el mundo real. Como ya dijimos varias veces, los primeros son solamente teóricos, matemáticos. El siguiente cuadro resume estos puntos:
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2) Dimensión Fractal
El otro tema que hay que repasar es el concepto de Dimensión Fractal. Para ello, nos enfrentamos a dos problemas diferentes que surgieron en los foros de debate: Problemas para entender el concepto. Diferentes puntos de vista y falta de acuerdo en determinados aspectos. -
Para salvar el primer inconveniente voy a intentar explicar más claramente estos conceptos y detallarlos con ejemplos. Con respecto a los diferentes punto de vista, no hay mucho que pueda hacer, voy a contarles lo que dice la mayoría de los libros, expertos y lo que he escuchado en diferentes debates, y luego seguir intercambiando ideas y quizás nos pongamos todos de acuerdo. Primero vamos a situarnos en esta problemática. Tenemos tan grabado en nuestros cerebros el concepto de dimensión que seguramente s eguramente muy pocos se han puesto a pensar lo que realmente significa ese término. De hecho la respuesta no es nada fácil, ni siquiera antes de haberse descubierto los fractales. Quien definió con propiedad la dimensión fue Euclides, y la asoció a los grados de libertad que necesita un objeto para existir o el espacio e spacio requerido donde “vivir”. Así un punto tiene una dimensión de 0, una recta de 1, un plano de 2, y por último el espacio de 3. Desde un punto de vista un poco más matemático podría decir que la dimensión es la cantidad de vectores independientes que se necesitan para generar o describir un espacio (no me refiero al espacio 3D.) Hasta aquí nada nuevo, esto es más que conocido por todos nosotros. Ahora bien, cuando Mandelbrot comenzó a estudiar el tamaño de las costas de Inglaterra se dio cuenta que ésta arrojaba diferentes resultados dependiendo de la escala con que se la midiese. Recuerden el ejemplo de la primera clase cuando se repetía la misma experiencia y se medía desde un satélite, un avión y luego caminando sobre la misma costa. Entonces, obviamente el concepto c oncepto de tamaño de un fractal ya no puede asociarse con la longitud, volumen, área o perímetro al que estamos acostumbrados en la Geometría Euclidea. Nace así el concepto de Dimensión Fractal, el cual se lo utiliza para responder a la pregunta: ¿Cuánto mide un fractal?
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Resumen hasta el momento: La Dimensión Fractal nace para solucionar los siguientes problemas:
Determinar cuanto mide un objeto fractal Estudiar el tamaño de un fractal, independientemente de la escala con la que se lo esté midiendo. Importante : recuerden que los fractales teóricos tienen detalle infinito. Esto significa dos cosas:
a) Su longitud sería infinita si la viéramos desde el punto de vista matemático tradicional. (o cero para el Conjunto de Cantor... después vuelvo sobre esto)
b) Cada vez que tomemos una sección de un fractal y la ampliemos siempre ampliemos siempre vamos a encontrar una imagen estadísticamente o estrictamente similar al fractal original, y esto sin importar cuantas veces hemos ampliado o iterado el fractal.
Noten también que esto sucede solo en el campo teórico, en la Naturaleza y en el mundo real los fractales no tienen detalle infinito, llega un momento donde la Autosimilitud se pierde.
Sigamos adelante. Ya dijimos que las dimensiones Euclídeas (0-1-2-3) no nos sirven para describir el tamaño de un fractal, por lo tanto necesitamos otra manera de medirlo. Existe una ecuación que describe la dimensión de objetos autosimilares. Es la llamada dimensión de Hausdorff - Besicovitch, la cual no es ni más ni menos que una generalización de la dimensión Euclidea.
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Y por propiedades de los logaritmos podemos decir: Log S = D * Log L Solo nos falta pasar Log L dividiendo para obtener D:
Log S / Log L = D
Ya tenemos despejada nuestra incógnita y solo nos queda hacer cuentas. Para que quede más claro aún podemos decir que:
S es la longitud de los segmentos, o en su defecto, el número de piezas autosimilares. L la escala de medición o el factor de magnificación (aumento)
Antes de continuar hagamos un resumen, así nadie se pierde. Hasta aquí solo he presentado una nueva ecuación para calcular dimensiones, y he descripto cada una de sus componentes, y como despejar la incógnita que nos interesa, D, la dimensión. Por ahora nada más, solo “trucos” algebraicos. Así que sigamos avanzando. Vamos a comenzar a mostrar un ejemplo grafico de cómo aplicar esta fórmula, primero a figuras de la geometría tradicional para mostrar que también funciona con ellas, ya que como dije es una extensión de la dimensión descripta desc ripta por Euclides. Luego pasaré a aplicarla a fractales lineales conocidos. Aclaración Importante: Como este tema ya lo expliqué anteriormente con
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Ahora, si medimos esa misma recta, rec ta, pero con un instrumento del doble de precisión tendremos L = 2 y S = 2 (ver cuadro) Ahora, generalizando, si medimos la recta con c on un instrumento de n veces de precisión, la recta se dividirá en n segmentos, por lo tanto, L = n y S = n Veamos en el grafico:
Haciendo cuentas, tenemos:
D = Log 2 / Log 2 = 1 D = Log 3 / Log 3 = 1 . . . D = Log n / Log n = 1 Esto significa que sin importar que escala de medición o instrumento de precisión utilicemos para medir la recta, su dimensión será siempre 1. MUY IMPORTANTE!
b) Cu Cuad adra rado do Si ahora tenemos en lugar de una recta, rec ta, un cuadrado y lo medimos:
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Les dejo ahora dos cuadros para ver lo que pasa con las dimensiones topológicas y sus relaciones con la dimensión de Hausdorff - Besicovitch y luego paso a los fractales.
Figu Fi gura ra Eu Eucl clíd ídea ea
Dime Di mens nsió ión n No No.. de Se Segm gmen ento toss
Línea
1
2 = 21
Cuadrado
2
4 = 22
Cubo
3
8 = 23
Similitud al duplicar
d
n = 2d
Este cuadro muestra la cantidad de segmentos autosimilares que se forman al medir una recta, cuadrado o cubo con un instrumento de medición el doble de preciso al original. Veamos ahora un cuadro más grafico:
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Dimensión de Hausdorff - Besicovitch en los fractales fractales
Acabamos de ver como aplicamos esta fórmula para calcular dimensiones a figuras Euclídeas, traslademos esto ahora a nuestros fractales.
El ejemplo 1 sería la Curva de von Koch, pero como ya está explicada en la clase 2 no voy a volver a repetirla con tanto detalle.
Tan solo para recordar diré que se calculaba haciendo: Log 4 / Log 3 = 1,26185... y la deducción se puede observar en el siguiente grafico:
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Log 4 / Log 3 = 1,26185... Log 16 / Log 9 = 1,26185... Log 64 / Log 27 = 1,26185...
Hemos llegado aquí a lo más importante. Noten, que sin importar la escala de medición (1, 2, 3,..., n) la dimensión de la Curva de Von Koch sigue dando el mismo resultado: resultado: 1,26185.. 1,26185.... Justamente lo que planteaba planteaba Mandelbrot, buscar un método que permita medir un objeto fractal, independiente de la escala con el que se lo viera.
daba:
Otro ejemplo que también ya vimos es la Curva o Polvo de Cantor, el cual nos
D = Log 2 / Log 3 = 0.6309 y la imagen para interpretarlo es la siguiente:
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El Triángulo de Sierpinski se forma de la siguiente manera: -
-
Se inicia con un triángulo equilátero, conectado desde sus puntos medios de sus tres lados formando un nuevo triángulo equilátero invertido. Luego se remueve ese triangulo invertido o de color blanco en la figura. En cada iteración el algoritmo repetirá el mismo procedimiento cada vez que se tope con un con un triángulo nuevo (negro ya que el blanco es un espacio vacío.)
También les dejo a ustedes volver a hacer las cuentas como hice con la Curva de Von Koch y calcular los diferentes Logaritmos y obtener siempre el mismo resultado para los diferentes niveles de iteración. Como último ejemplo veremos la Curva de Peano la que también generó bastante polémica en los foros. La figura inicial para construirlo es una recta, luego el grafico de interpretación es el siguiente:
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Para finalizar esta parte de la clase (ya falta poco!) les dejo otro cuadro comparativo.
Escala S
Número de Segmentos
Dimensión D=logN /log /log s
L
Nuestra Fórmula S =L =LD
Conjunto de Cantor Curva de Koch
1/ 3
2
2=3D
D=log2/log3=0,631
1/ 3
4
4=3D
D=log4/log3=1,262
Curva de Peano
1/ 3
9
9=3D
D=log9/log3=2
Triángulo de Sierpinski
1/ 2
3
3=2D
D=log3/log2=1,585
Bueno, para terminar esta clase de repaso vamos a hablar del tema más controversial que ha salido en los foros de debate, y son los siguientes:
Las Excepciones Todo lo que vimos hasta ahora son definiciones y características muy estudiadas. Pero ahora entra en e n juego el punto de vista de cada uno en particular, lo cual siempre resulta algo confuso, ya que la matemática es una disciplina exacta, y las excepciones no están contempladas. Así que estudiemos es tudiemos con más detalle esta
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Otra particularidad importante, que no la he visto nombrar en muchos lados, es el Triángulo de Sierpinski. Su dimensión es de 1.585... Pero, su figura original es un triángulo, por lo tanto de dimensión topológica igual a 2, entonces, su dimensión fractal vuelve a ser MENOR que su dimensión topológica, y según la definición de Mandelbrot, Sierpinski no sería un fractal, frac tal, pero al mismo tiempo está reconocido por expertos como el fractal más famoso. Este es un caso similar al Cantor, si ustedes ven el grafico de Sierpinski notarán triángulos negros y otros blancos, bueno, los blancos son trozos del fractal que se han eliminado. Por lo tanto, el fractal en lugar de ir creciendo va disminuyendo como acabo de decir pasa con Cantor. Si pudiéramos poner una lupa sobre esta imagen, podríamos ver infinitos agujeros (triángulos blancos eliminados) El último detalle curioso, que algunos lo encuadran en la categoría de excepción y otros no, son las curvas de Peano y Hilbert. Es muy cierto que su dimensión fractal es MAYOR que su topológica, por lo tanto no tendría problemas con la definición de Mandelbrot, pero esta tiene la particularidad de ser una dimensión entera, 2. Todos estamos acostumbrados a ver dimensiones fraccionarias, por lo tanto esta es una curiosidad a tener en cuenta, lo cual es muy lógico, ya que esta curva nace de una recta y luego de iterarla, llega a cubrir el plano de dim. 2 completamente.
Observaciones Finales:
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es decir, que son directamente proporcionales Otra cosa que se dice es que: la definición formal de un fractal, es una figura cuya dimensión fractal es mayor que su dimensión topológica.! Y es de esperarse, porque por que si partimos de una línea que tiene una dimensión 1, y el resultado es a la vista más complejo, tiene que tener una dimensión mayor. Pero qué pasa con el Conjunto de Cantor. Algo anda mal!! Al menos el sentido común no!, porque el resultado es más complejo que su origen . A ver si llegamos a conluir algo... Más aún. Había visto a 007 utilizando una variante de la fórmula de dimensión, y ésta es un tercer método utilizado para fractales fra ctales realmente complejos!. Este método consiste en mapear el fractal mediante cuadraditos y contando los que ocupa; digamos que se divide el espacio que ocupa el fractal en N partes y que los que ocupa el mismo son M, entonces D se calcula como: D = log(M) / log (1/N) je, je. Eureka!, lo encontré! Aplicando ésta fórmula se puede calcular la complejidad (como prefiero llamarlo) de un cultivo de bacterias. Muyyyy interesante! Antes de que se haga demasiado largo, la rgo, les comparto una última curiosidad: los fractales y el cáclulo. El principio de hallarle la derivada a una curva o función se basa en que conforme los intervalos se hacen más pequeños, el segmento _tiende_ a ser una línea. Pero con los fractales sucede todo lo contrario; mientras más acercamientos le hacemos, obtenemos el mismo detalle. Conclusión, no son derivables!!! Bueno, Aquí el esperado enlace:
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Saludos para todos, y recuerden que la clase del lunes 11 pasa para el 18.
Juan Pablo.