Chaos game

The chaos game is a stochastic method for constructing the attractor of an iterated function system. Rather than applying all transformations simultaneously, one selects a transformation at random at each step and applies it to the current point. Counterintuitively, the resulting sequence of points, plotted over many iterations, converges to the same deterministic attractor as the full IFS. This reveals that randomness at the local level can produce structure at the global level. The chaos game demonstrates that IFS attractors are not merely geometric objects but dynamical equilibria of stochastic processes. The method has been generalized to non-contractive systems and used to visualize strange attractors in chaotic dynamics. It also connects to the study of chaos control through the analysis of invariant measures under random iteration.