Sierpiński Gasket: The Chaos Game (100 Million Samples)

The so-called chaos game [1, 2] refers to a type of iterated map that can be used to approximate fractal patterns. A popular example is given by the following procedure: Suppose one has an equilateral triangle {A, B, C} along with a single, randomly chosen point inside of its volume. By iteratively choosing one of the triangle's vertices and placing a new sample in between the vertex and the previous sample, one finds that the resulting set of points lies dense within the Sierpiński gasket.

The plot on the right is generated based on 100`000`000 samples. Note that the first few samples of the iteration have been dropped to exclude transient behavior.

[1] R. L. Devaney, Chaos Rules!, Math Horizons 12(2), 11 (2004).