Sierpinski Triangle — Chaos Game
Pick a random corner, jump halfway toward it, plot a dot, repeat. Watch the fractal
emerge dot by dot — the pace starts slow so you can see each jump, then accelerates.
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1
How it works: this is the "chaos game." Start anywhere inside the
triangle. On each step, pick one of the three corner vertices uniformly at random,
then move the current point exactly halfway between where it is and that corner —
and plot a dot at the new position. Repeat tens of thousands of times. Intuitively
you'd expect pure randomness to fill the triangle evenly, but it doesn't: because each
jump is a 50% contraction toward a fixed point, the process can never land inside the
triangle's three central sub-triangles — only in their corners' corners, recursively,
forever. The gaps that are never visited are exactly the holes of the Sierpinski
triangle, so the fractal emerges as a byproduct of what random halfway-jumps
structurally cannot reach.