polygon fractal

An algorithm originally described by Barnsley in 1988. 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 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 (n, r). The above plots show the chaos game for 10000 points in the regular 3-, 4-, 5-, and 6-gons with r = 1/2. The case (n, r) = (4, 1/2) gives the interior of a square with all points visited with equal probability.

Barnsley's fern | Sierpiński sieve