🔺 Sierpinski Triangle Generator

The Sierpinski triangle is a self-similar fractal: take a solid triangle, remove the middle sub-triangle formed by joining the midpoints of its sides, and repeat the same step on each of the three remaining triangles forever. The result has a fractal dimension of log 3 / log 2 ≈ 1.585 — more than a line but less than a filled area. It was introduced by the Polish mathematician Wacław Sierpiński in 1915.

This tool offers two constructions. Recursive subdivision is deterministic: it repeatedly cuts each triangle into three smaller copies, so the shape is drawn exactly. The chaos game is random: fix the three corners of a triangle, start from any point, then repeatedly pick a corner at random and jump halfway toward it, plotting each landing spot. Both approaches produce the very same figure.

The three halfway-to-a-corner rules together form an iterated function system (IFS), and the unique attractor of that system is exactly the Sierpinski triangle. Because each rule shrinks distances toward a corner, repeated random iteration is pulled onto the attractor and can never land in the removed central holes. That is why scattering thousands of random points still traces out a perfectly ordered fractal.