1 Fractals Coined by Benoit Mandelbrot To differentiate from pure - PDF document

Last time � Concepts � Emergence, emergent systems, … � Life � Real life � Artificial life � Topics � NetLogo � Assignment 1 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Outline for this lecture � Fractals - general � Lindenmayer Systems � The Multiple Reduction Copy Machine � Iterated Functional Systems � Non linear fractals � The Mandelbrot Set 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 1

Fractals � Coined by Benoit Mandelbrot � To differentiate from pure geometric figures � Two interesting qualities � Self-similar on multiple scales � Fractional dimension 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Examples of Fractals � The Cantor Set � The Koch Curve � The Peano Curve � Fractional dimension 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Random Fractals � Random processes in nature are often self-similar on varying temporal and spatial scale 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 2

White Noice and Brownian Motion � White Noise � Describe things believed to be formed by random walk-like processes � Brownian Motion � Particles in liquids 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Diffusion Limited Aggregation � Particles with Brownian motion stop moving when they touch stationary objects � 2-dimensional � 3-dimensional 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Fractal growth � Fractals are effective at compressing info � Natural fractals � Must grow! 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 3

Lindenmayer Systems � Aristid Lindenmayer, 1968 � Mathematical description of plant growth � Very compact � Axiom: seed cell � Production rules: describe growth � Strings can be interpreted 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Turtle Graphics � Seymour Papert � A simple computer language that children could use to draw graphical pictures � Can be used to interpret L-system strings � NetLogo is an extension of this 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU L-systems 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 4

21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Linear Algebra � Translation � Scaling � Reflection � Rotation � Composing 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU The Multiple Reduction Copy Machine Algorithm � Uses 2 or more linear transformations � Problem: � n = # transform � d = depth � � n d 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 5

The Multiple Reduction Copy Machine Algorithm - Problem 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Iterated Functional Systems � An ”idealized MRCM fractal” � Composed entirely of a set of points � If some point p is part of an idealized MRCM fractal, then, for all i , L i ( p ) must also be a part of the idealized MRCM fractal. � If L i ( p ) does not have an inverse, the transformation must squeeze the input image into a line or a point � If L i ( p ) has an inverse, then L i -1 ( p ) must also be part of the idealized MRCM fractal � If a point p is not part of an idealized MRCM fractal, then L i ( p ) will be closer to the idealized MRCM fractal than p is. 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Iterated Functional Systems � Michael Barnsley 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 6

Nonlinear Fractals � Iterative dynamical systems � Complex numbers 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU The Mandelbrot Set � x t + 1 = x t2 + c, x 0 = 0 + i0 = 0 � Questions: � With c = constant complex number, what happens to x t when t goes to infinity? � What values of c makes x t diverges? � (If a 2 + b 2 > 4, then x t diverges) 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Mandelbrot - Algorithm For each number, c , in a subset of the complex plane • Set x 0 = 0 • For t = 1 to tmax • 2 + c • Compute x t = x t • If | x t | > 2, then break out of loop If t < tmax , then color point c white • If t = tmax , then color point c black • 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 7

Mandelbrot - Infinity 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Mandelbrot – Self-similar 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU The Master Julia Set Set c to some constant complex value • For each number, x 0 , in a subset of the complex plane • For t = 1 to tmax • 2 + c • Compute x t = x t • If | x t | > 2, then break out of loop If t < tmax , then color point c white • If t = tmax , then color point c black • 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 8

21/1 - 09 Emergent Systems, Jonny Pettersson, UmU A Mystery of the M-set � David Boll, 1991 ε Iterations � Wanted to confirm 0.1 33 that the “neck” of the 0.01 315 M-set at c = -3/4 + 0 i 0.001 3143 is 0 in thickness 0.0001 31417 � Tested: c = -3/4 + ε i 0.00001 314160 � What is π doing there? 0.000001 3141593 0.0000001 31415928 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU Summary � Fractals - general � Lindenmayer Systems � The Multiple Reduction Copy Machine � Iterated Functional Systems � Non linear fractals � The Mandelbrot Set 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 9

Next lecture � Chaos � Producer-consumer dynamics 21/1 - 09 Emergent Systems, Jonny Pettersson, UmU 10