Tool

Chaos game

Random vertex picks + a fixed contractive rule reveal a deterministic fractal. Switch presets to see Sierpinski triangle, Sierpinski pentagon, and "forbidden-move" variants.

Vertices N: 3

Step factor r: 0.500

Restriction rule

Points: 80,000

Chaos game rule

vertices = 3 regular polygon
step:    P_{n+1} = P_n + 0.500 · (V_random − P_n)
similarity dim (rule = none): 1.5850

Pure randomness + a fixed contractive rule = deterministic fractal. Barnsley's theorem guarantees convergence. Try N=4 with factor=0.5 and rule=none (fills a square) vs no-same (Sierpinski-carpet-like cousin).

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