The chaos game is a random walk that steps halfway between your current position and a set of predefined points. If the points are on an equilateral triangle, the resulting set is Sierpinski's triangle. I wanted to see what would happen if the points were chosen uniformly from a circle and the result is pretty interesting:
It looks like there are two peaks in the radial density. If you make a histogram of the radial density and multiply by $1/r$ (to account for the expected spreading in 2D) you get two peaks around 0.35 and 0.60.
Is it possible to derive the location of these peaks, and maybe even the full radial distribution function?