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.
SEE ALSO: Barnsley's Fern,
Sierpiński
Sieve
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.
Referenced on Wolfram|Alpha:
Chaos Game
CITE THIS AS:
Weisstein, Eric W. "Chaos Game." From
MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChaosGame.html
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