Click Start to begin
Current Step

The chaos game starts by picking a random point inside the polygon. Each iteration, we jump halfway toward a randomly chosen vertex.

?
0.50 ?
50,000
Points
0
Jump Ratio
1/2
Vertices
3
Target
50K

Detailed Statistics

Points per Second: 0
Vertex Distribution: -
Fractal Dimension (est): ~1.585
Coverage Area: 0%
Last Vertex: -

The Paradox of Random Order

Pick any starting point. Repeatedly jump halfway toward a randomly chosen vertex. After thousands of jumps, a perfect Sierpinski Triangle emerges! Pure randomness creates exact structure.

The Algorithm

Start at point P. Each step:

P_new = P + r * (V - P)

Where V is a random vertex and r is the ratio (typically 0.5). For triangles, r=0.5 gives the Sierpinski gasket.

Iterated Function Systems

Michael Barnsley's book "Fractals Everywhere" (1988) proved this works because fractals are attractors of IFS. Any starting point converges to the same fractal pattern!

Different Shapes

With a square, ratio 0.5 fills everything. But with restrictions (can't jump to previous vertex), beautiful patterns emerge. Pentagons and hexagons create stunning variations with the right ratios!

Why Does This Work?

The key insight: points that would land outside the attractor become increasingly rare. The "forbidden regions" (like the center of the Sierpinski triangle) are self-similar - they appear at every scale, creating the fractal structure.