Random vertex jumps produce deterministic fractal structures
Sierpinski triangle: jump ½ way to a random vertex each step
Points: 0
Chaos Game: Introduced by Michael Barnsley (1988), the chaos game works by repeatedly jumping a fraction r of the distance toward a randomly chosen vertex. Despite the randomness, the set of visited points converges to a fractal attractor — the IFS (Iterated Function System) attractor. For the triangle with r=½, this is the Sierpinski triangle (Hausdorff dimension log 3/log 2 ≈ 1.585). The square with a "forbidden opposite vertex" rule gives the Sierpinski carpet analog. The pentagon with r=1/φ (golden ratio inverse ≈ 0.618) produces the pentaflake (dim = log 5/log(1+φ) ≈ 1.672).