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 n How this project came about n Properties of strange attractors n Search techniques n Aesthetic evaluation n The computer art critic n Samples

Typical Experimental Data Time0 500 x 5 -5

Determinism x n+ 1 = f (x n, x n- 1, x n- 2, …) where f is some model equation with adjustable parameters

Example (2-D Quadratic Iterated Map) x n+1 = a 1 + a 2 x n + a 3 x n 2 + a 4 x n y n + a 5 y n + a 6 y n 2 y n+1 = a 7 + a 8 x n + a 9 x n 2 + a 10 x n y n + a 11 y n + a 12 y n 2

Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) Solution of model equations Time0200 x

Probability of chaotic solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 110 Dimension

Types of Attractors Fixed Point Limit Cycle TorusStrange Attractor SpiralRadial

Strange Attractors n Limit set as t   n Set of measure zero n Basin of attraction n Fractal structure u non-integer dimension u self-similarity u infinite detail n Chaotic dynamics u sensitivity to initial conditions u topological transitivity u dense periodic orbits n Aesthetic appeal

Stretching and Folding

n Geometrical objects generally with non-integer dimension n Self-similarity (contains infinite copies of itself) n Structure on all scales (detail persists when zoomed arbitrarily) Fractals

Natural Fractals

Human Evaluations

Aesthetic Evaluation

A Simple 4-D Example x n+1 = a 1 x n + a 2 x n 2 + a 3 y n + a 4 y n 2 + a 5 z n + a 6 z n 2 + a 7 c n + a 8 c n 2 (horizontal) y n+1 = x n (vertical) z n+1 = y n (depth) c n+1 = z n (color)

“Infinite” Variety n 8 adjustable coefficients n Like settings on combination lock n 26 values of each coefficient n 8-character name: KKGEOLMM n Compact coding! DOS filename n 26 8 = 2 x different codes n ~0.01% are visually interesting n Would take 1 year to see interesting ones at a rate of 1 per second

Symmetric Icons 2 to 9 segments Original Image

Selection Criteria n Must be bounded (|x| < 100) n Must be chaotic (positive LE) n 1.2 < fractal dimension < 1.9 n More than 10% of pixels on n Less than 50% of pixels on

Artificial Neural Networks `Neurons’

Computer Art Critique n Network trained on 100 “good” images and 100 “bad” images n Inputs are first 8000 bytes of gif file n Network has 16 neurons n A single output (can be + or -) n Gives ~85% accuracy on training set (200 cases) n Gives ~64% accuracy on out-of- sample data (different 200 cases)

Gorilla Art “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

More Gorilla Art

Summary n Nature is beautiful n So is chaos

References n lectures/monkey/ (This talk) lectures/monkey/ n fractals.htm (Fractal gallery) fractals.htm n Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993) Strange Attractors: Creating Patterns in Chaos n Chaos Demonstrations software Chaos Demonstrations n