How long is the Chaos Game?

TL;DR

This paper analyzes the chaos game algorithm for fractal generation, providing an asymptotic formula for the expected time to produce a dense subset of self-similar fractals, connecting fractal theory with Markov chain covering times.

Contribution

It introduces a new asymptotic formula for the expected duration of the chaos game on self-similar fractals using Markov chain theory.

Findings

Derived an asymptotic formula for the chaos game's expected time

Connected fractal generation with Markov chain covering times

Provided insights into the efficiency of fractal sampling methods

Abstract

In the 1988 textbook "Fractals Everywhere" M. Barnsley introduced an algorithm for generating fractals through a random procedure which he called the "chaos game". Using ideas from the classical theory of covering times of Markov chains we prove an asymptotic formula for the expected time taken by this procedure to generate a $\delta$-dense subset of a given self-similar fractal satisfying the open set condition.