Mathematics · Study 045
Chaos Game
Pick a random corner, jump part of the way there, plot a dot, repeat. The chaos game builds Sierpinski's triangle and its cousins from noise.
What you're looking at
Start anywhere. Choose one corner of the polygon at random and move a fixed fraction of the way toward it, then mark the spot. For a triangle and a fraction of one half, the dots never land in the central hole or any of the smaller holes, and the Sierpinski triangle appears. Other polygons need different fractions, and the square needs a rule against picking the same corner twice.
The idea behind it
The surprise is that a random process has a fixed, intricate destination. Each jump is a contraction, and the only set that maps onto itself under all the contractions together is the fractal. Randomness merely decides the order in which its points are visited.
Try this
Click or tap to cycle through three, four, five and six corners.
