Why random dice rolls draw a perfect fractal: the chaos game and the Sierpiński triangle
30seconds
Pick three dots, then keep jumping halfway towards a randomly chosen one. It sounds like it should leave a grey smudge, but after a few thousand jumps a razor-sharp Sierpiński triangle appears, hole within hole at every scale.
Play the chaos game yourself: drag the corners, change the jump fraction and watch the pattern bend, fill in or break into dust. Then see why every roll squeezes the picture into three half-size copies, and why a shape with no area can still have a dimension of about 1.585.
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