Chaos Game: Three Dimensions

In the 3 dimensional version of the Chaos Game we start with a regular polyhedron and mark selected points which will typically be the vertices. These points will be called endpoints and will be marked with red squares. The game begins by randomly choosing a starting point and one of the endpoints. Mark a new point at a fixed distance ratio from the starting point to the endpoint (e.g., halfway to the endpoint). Select another endpoint at random and, with the most recently created point, repeat the process to generate the next point and continue. By applying the right distance ratio the resulting set of points can converge to a beautiful image known as a fractal. For each polyhedron the required distance ratio to yield a fractal will be provided, but try different settings to see what other patterns may arise!