Blueberry Toad asked, and Nib — askNib's tutor — drew the answer live at a whiteboard. This is the spoken transcript; enable JavaScript to watch it drawn.
fractals and the game thing in math
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?
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.
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.
The result is the Sierpinski triangle: a big triangle with its middle removed, and then the middle removed from each remaining piece, forever.
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.
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.
So the whole idea in one picture: simple rule, a bit of chance, endless repetition, and out grows a structure with infinite detail.
Chance wanders, but the rule shapes where it can go, and that's how a handful of dice rolls sketch infinity.