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Chaos Game

Pick a point at random inside a regular $n$-gon. Then draw the next point a fraction $r$ of the distance between it and a 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 case $(n,r)=(4,1/2)$ gives the interior of a Square with all points visited with equal probability.

\begin{figure}\begin{center}\BoxedEPSF{ChaosGame3_1-2.epsf scaled 500}\BoxedEPSF{ChaosGame5_1-3.epsf scaled 500}\end{center}\end{figure}

$(3, 1/2)$ $(5, 1/3)$

\begin{figure}\begin{center}\BoxedEPSF{ChaosGame5_3-8.epsf scaled 400}\BoxedEPSF{ChoasGame6_1-3.epsf scaled 400}\end{center}\end{figure}

$(5, 3/8)$ $(6, 1/3)$

See also Barnsley's Fern


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

Dickau, R. M. ``The Chaos Game.''

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 149-163, 1991.

mathematica.gif Weisstein, E. W. ``Fractals.'' Mathematica notebook Fractal.m.

© 1996-9 Eric W. Weisstein