Chaos Game

An algorithm originally described by Barnsley in 1988. Pick a point at random inside a regular -gon. Then draw the next point a fraction  of the distance between it and a polygon vertex picked at random. Continue the process (after throwing out the first few points). The result of this "chaos game" is sometimes, but not always, a fractal. The results of the chaos game are shown above for several values of .

The above plots show the chaos game for  points in the regular 3-, 4-, 5-, and 6-gons with . The case  gives the interior of a square with all points visited with equal probability.

The above plots show the chaos game for  points in the square with , 0.4, 0.5, 0.6, 0.75, and 0.9.

REFERENCES:

Borwein, J. and Bailey, D. "Pascal's Triangle." §2.1 in Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 47-48, 2003.

Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993.

Bogomolny, A. "Sierpinski Gasket Via Chaos Game." http://www.cut-the-knot.org/Curriculum/Geometry/SierpinskiChaosGame.shtml.

Dickau, R. M. "The Chaos Game." http://mathforum.org/advanced/robertd/chaos_game.html.

Jeffrey, H. J. "Chaos Game Representation of Genetic Sequences." Nucleic Acids Res. 18, 2163-2170, 1990.

Jeffrey, H. J. "Chaos Game Visualization of Sequences." Comput. & Graphics 16, 25-33, 1992. Reprinted in Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 5-13, 1998.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Fractals for the Classroom, Part 1: Introduction to Fractals and Chaos. New York: Springer-Verlag, pp. 41-43, 1992.

Pickover, C. A. (Ed.). Fractal Horizons: The Future Use of Fractals. New York: St. Martin's Press, pp. 27, 57-59, and 169-171, 1996.

Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, pp. 226-239, 1999.