Chaos Game

The chaos game is a stochastic process for generating fractals, most famously the Sierpinski triangle, by repeatedly moving a point a fixed fraction of the distance toward a randomly chosen vertex. Despite its reliance on randomness, the process converges to a deterministic geometric attractor — a counterintuitive result that reveals deep connections between probability and structure. The chaos game generalizes to any iterated function system with probabilities, and its convergence properties are governed by the contraction mapping theorem. It is a pedagogical gateway to understanding that random processes can have non-random limits, and that the attractor of a dynamical system is often more structured than the process that generates it. The Barnsley fern — a naturalistic plant shape generated by a four-map IFS chaos game — demonstrates that the same mathematics produces both abstract fractals and organic forms.