What is the chaos game and how do fractals work?

1the bet

Here's a wild bet: pick three dots, play a game of pure dice rolls, and a perfect, intricate pattern appears. Randomness drawing order. How?

2the rules

Here are the rules. Pick any starting point. Roll a die to choose corner A, B, or C. Jump exactly halfway toward that corner, and drop a dot.

3the surprise

Repeat thousands of times. You'd expect a smudge of dots everywhere. Instead, the dots dodge a hole in the middle, and holes inside the holes.

4Sierpinski

The result is the Sierpinski triangle: a big triangle with its middle removed, and then the middle removed from each remaining piece, forever.

5why it works

Why does it work? Every jump halves the distance, so a dot is always pulled into one of three half-size copies. Zoom into any copy and the game repeats.

6fractal

That self-similarity is what a fractal is: zoom in and the part looks like the whole. Coastlines, ferns, lightning, and your lungs do this too.

7the whole idea

So the whole idea in one picture: simple rule, a bit of chance, endless repetition, and out grows a structure with infinite detail.