1. 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

2. 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

3. Typical Experimental Data Time0 500 x 5 -5

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

5. 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

6. Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) Solution of model equations Time0200 x 20 -20

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

8. Types of Attractors Fixed Point Limit Cycle TorusStrange Attractor SpiralRadial

9. 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

10. Stretching and Folding

11. 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

12. Natural Fractals

13. Human Evaluations

14. Aesthetic Evaluation

15. 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)

16. “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 10 11 different codes n ~0.01% are visually interesting n Would take 1 year to see interesting ones at a rate of 1 per second

17. Symmetric Icons 2 to 9 segments Original Image

18. 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

19. Artificial Neural Networks `Neurons’

20. 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)

21. 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

22. More Gorilla Art

23. Summary n Nature is beautiful n So is chaos

24. References n http://sprott.physics.wisc.edu/ lectures/monkey/ (This talk) http://sprott.physics.wisc.edu/ lectures/monkey/ n http://sprott.physics.wisc.edu/ fractals.htm (Fractal gallery) http://sprott.physics.wisc.edu/ 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 sprott@juno.physics.wisc.edu sprott@juno.physics.wisc.edu