Fractals for Kids Clint Sprott Department of Physics University of Wisconsin - Madison Presented at the Chaos and Complex Systems Seminar in Madison, Wisconsin on January 28, 2014

November 6, 2013 Dear Professor Sprott We are book publishers based in the UK. Among our many titles is a series of colouring books; we have produced these in a range of subjects, ranging from flowers and impossible figures to stained glass, fairies and mandalas. My colleagues in Sales have suggested that it might be interesting to produce a fractals colouring book. For most of our colouring books we commission professional illustrators to provide mono pictures or patterns with outlines so that the purchaser has well defined areas to colour. Obviously, fractals are very different from the usual types of illustration we use for our colouring books. If the idea is to work, we are in need of expert input. Would you be interested in helping us to produce fractal images that could be used for our colouring book? I look forward to hearing from you. Best wishes Tessa Rose

Fractals According to Benoit B. Mandelbrot ( ), in “Fractals and Chaos”, Springer, 2004: Roughly speaking, fractals are shapes that look the same from close by and far away A self-explanatory term for “looking the same from close by and far away” is “self- similar.” I named them, tamed them into primary models of roughness, and helped them multiply As for the word fractal, I coined it on some precisely datable evening in the winter of 1975, from a very concrete Latin adjective, fractus, which denoted a stone’s shape after it was hit very hard. Lacking time to evolve, fractal rarely strayed far from the notion of roughness.

Iterated Function Systems

Lindenmayer Systems Turtle Graphics F: Move forward one step while drawing a line f: Move forward one step without drawing a line +: Turn left through a specified angle -: Turn right through a specified angle Typical axiom: F+F+F+F (draw a square) Typical rule: F  F+F-F-FF+F+F-F  Apply rule recursively with decreasing step size Result is a very long string of commands  This rule makes a Koch Island (snowflake)

Cellular Automata

References n lectures/coloring.ppt (this talk) lectures/coloring.ppt n sa/ (my chaos textbook) sa/ n (contact me)