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

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

Apr 17, 2023 14 likes •119 views

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

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21/1 - 09 Emergent Systems, Jonny Pettersson, UmU

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

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21/1 - 09 Emergent Systems, Jonny Pettersson, UmU

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

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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!

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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

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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 nd

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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, Li (p) must also be a part

  • f the idealized MRCM fractal.

If Li (p) does not have an inverse, the

transformation must squeeze the input image into a line or a point

If Li (p) has an inverse, then Li

  • 1(p) must also be

part of the idealized MRCM fractal

If a point p is not part of an idealized MRCM

fractal, then Li (p) will be closer to the idealized MRCM fractal than p is.

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Iterated Functional Systems

Michael Barnsley

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Nonlinear Fractals

Iterative dynamical systems Complex numbers

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The Mandelbrot Set

xt + 1 = xt2 + c, x0 = 0 + i0 = 0 Questions:

With c = constant complex number, what

happens to xt when t goes to infinity?

What values of c makes xt diverges? (If a2 + b2 > 4, then xt diverges)

21/1 - 09 Emergent Systems, Jonny Pettersson, UmU

Mandelbrot - Algorithm

  • For each number, c, in a subset of the complex plane
  • Set x0 = 0
  • For t = 1 to tmax
  • Compute xt = xt

2 + c

  • If |xt| > 2, then break out of loop
  • If t < tmax, then color point c white
  • If t = tmax, then color point c black
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Mandelbrot - Infinity

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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, x0 , in a subset of the complex plane
  • For t = 1 to tmax
  • Compute xt = xt

2 + c

  • If |xt| > 2, then break out of loop
  • If t < tmax, then color point c white
  • If t = tmax, then color point c black
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A Mystery of the M-set

David Boll, 1991 Wanted to confirm

that the “neck” of the M-set at c = -3/4 + 0i is 0 in thickness

Tested: c = -3/4 + εi What is π doing there?

31415928 0.0000001 3141593 0.000001 314160 0.00001 31417 0.0001 3143 0.001 315 0.01 33 0.1 Iterations ε

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

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Next lecture

Chaos Producer-consumer dynamics