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xn+1 = f (xn, xn-1, xn-2, ...) where f is some model equation with adjustable parameters ... http://sprott.physics.wisc.edu/ lectures/monkey/ (This talk) ...

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Can a Monkey with a Computer Create Art

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Can a monkey with a computer create art l.jpg

Can a Monkey with a Computer Create Art?

J. C. Sprott

Department of Physics

University of Wisconsin - Madison

Presented to the

Society for Chaos Theory in Psychology & Life Sciences

in Madison, Wisconsin

on August 4, 2001


Outline l.jpg

Outline

  • How this project came about

  • Properties of strange attractors

  • Search techniques

  • Aesthetic evaluation

  • The computer art critic

  • Samples


Typical experimental data l.jpg

Typical Experimental Data

5

x

-5

500

0

Time


Determinism l.jpg

Determinism

xn+1 = f (xn, xn-1, xn-2, …)

where f is some model equation with adjustable parameters


Example 2 d quadratic iterated map l.jpg

Example (2-D Quadratic Iterated Map)

xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2

yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2


Solutions are seldom chaotic l.jpg

Solutions Are Seldom Chaotic

20

Chaotic Data (Lorenz equations)

Chaotic Data

(Lorenz equations)

x

Solution of model equations

Solution of model equations

-20

0

Time

200


Probability of chaotic solutions l.jpg

Probability of chaotic solutions

100%

Iterated maps

10%

Continuous flows (ODEs)

1%

0.1%

Dimension

1

10


Types of attractors l.jpg

Types of Attractors

Limit Cycle

Fixed Point

Spiral

Radial

Torus

Strange Attractor


Strange attractors l.jpg

Strange Attractors

  • Limit set as t 

  • Set of measure zero

  • Basin of attraction

  • Fractal structure

    • non-integer dimension

    • self-similarity

    • infinite detail

  • Chaotic dynamics

    • sensitivity to initial conditions

    • topological transitivity

    • dense periodic orbits

  • Aesthetic appeal


Stretching and folding l.jpg

Stretching and Folding


Fractals l.jpg

Fractals

  • Geometrical objects generally with non-integer dimension

  • Self-similarity (contains infinite copies of itself)

  • Structure on all scales (detail persists when zoomed arbitrarily)


Natural fractals l.jpg

Natural Fractals


Human evaluations l.jpg

Human Evaluations


Aesthetic evaluation l.jpg

Aesthetic Evaluation


A simple 4 d example l.jpg

A Simple 4-D Example

xn+1 = a1xn + a2xn2 + a3yn + a4yn2 +a5zn + a6zn2 + a7cn + a8cn2 (horizontal)

yn+1 = xn (vertical)

zn+1 = yn(depth)

cn+1 = zn (color)


Infinite variety l.jpg

“Infinite” Variety

  • 8 adjustable coefficients

  • Like settings on combination lock

  • 26 values of each coefficient

  • 8-character name: KKGEOLMM

  • Compact coding! DOS filename

  • 268 = 2 x 1011 different codes

  • ~0.01% are visually interesting

  • Would take 1 year to see interesting ones at a rate of 1 per second


Symmetric icons l.jpg

Symmetric Icons

Original

Image

2 to 9

segments


Selection criteria l.jpg

Selection Criteria

  • Must be bounded (|x| < 100)

  • Must be chaotic (positive LE)

  • 1.2 < fractal dimension < 1.9

  • More than 10% of pixels on

  • Less than 50% of pixels on


Artificial neural networks l.jpg

Artificial Neural Networks

`Neurons’


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Computer Art Critique

  • Network trained on 100 “good” images and 100 “bad” images

  • Inputs are first 8000 bytes of gif file

  • Network has 16 neurons

  • A single output (can be + or -)

  • Gives ~85% accuracy on training set (200 cases)

  • Gives ~64% accuracy on out-of-sample data (different 200 cases)


Gorilla art l.jpg

Gorilla Art

http://www.koko.org/world/art.html

“It is part of ape nature to paint. Apes like

to use crayons, pencils and finger paints.

Of course, they also like to eat them.”

-- Roger Fouts


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More Gorilla Art


Summary l.jpg

Summary

  • Nature is beautiful

  • So is chaos


References l.jpg

http://sprott.physics.wisc.edu/ lectures/monkey/ (This talk)

http://sprott.physics.wisc.edu/ fractals.htm (Fractal gallery)

Strange Attractors: Creating Patterns in Chaos(M&T Books, 1993)

Chaos Demonstrations software

sprott@juno.physics.wisc.edu

References


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