Pick a point at random inside a regular -gon. Then draw the next point a fraction 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 gives the interior of a Square with all points visited with equal probability.
See also Barnsley's Fern
References
Barnsley, M. F. and Rising, H. Fractals Everywhere, 2nd ed. Boston, MA: Academic Press, 1993.
Dickau, R. M. ``The Chaos Game.''
http://forum.swarthmore.edu/advanced/robertd/chaos_game.html.
Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 149-163, 1991.
Weisstein, E. W. ``Fractals.'' Mathematica notebook Fractal.m.