Chaos Game

CindyJS geometry fractals chaos

The chaos game turns pure randomness into perfect order. Mark the corners of a polygon, start from any point, and repeat one simple step over and over:

1. Pick one corner at random.
2. Move a fixed fraction of the way from your current spot toward that corner.
3. Mark the new spot.

In formula form, each new point lands a fraction $r$ of the way from the old point $p_k$ toward the chosen corner $v_i$:

\[p_{k+1} = (1-r)\,p_k + r\,v_i\]

Here is the surprise: even though the corner is chosen at random every time, the dots never scatter into a shapeless cloud. They always settle into the same intricate, repeatable pattern—a fractal. With a triangle and $r = \tfrac{1}{2}$, that pattern is the famous Sierpiński triangle.

Use the N slider to change the number of corners (3 to 36) and the r slider to set how far each step moves. It snaps to special ratios (such as $N/(N+3)$) that produce especially clean shapes. The speed slider controls how fast new dots appear. Toggle COLOR to tint each dot by the corner it jumped toward, and GUIDE to show or hide the corners and the current step.