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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?

J. C. Sprott

Department of Physics

Presented to the

Society for Chaos Theory in Psychology & Life Sciences

on August 4, 2001

### Outline

• How this project came about

• Properties of strange attractors

• Search techniques

• Aesthetic evaluation

• The computer art critic

• Samples

5

x

-5

500

0

Time

### Determinism

xn+1 = f (xn, xn-1, xn-2, …)

where f is some model equation with adjustable parameters

### 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

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

100%

Iterated maps

10%

Continuous flows (ODEs)

1%

0.1%

Dimension

1

10

### Types of Attractors

Limit Cycle

Fixed Point

Spiral

Torus

Strange Attractor

### 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

### Fractals

• Geometrical objects generally with non-integer dimension

• Self-similarity (contains infinite copies of itself)

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

### 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

• 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

Original

Image

2 to 9

segments

### 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

`Neurons’

### 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

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

### Summary

• Nature is beautiful

• So is chaos

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