Place some points. Start anywhere, pick one of your points at random, and jump part of the way toward it. Mark the spot, and do it again. Pure chance, and yet a perfect pattern appears.
Start with three points and a jump of one half: Sierpiński's triangle appears, a triangle with a triangular hole, each corner a smaller copy of the whole. Now try a square at one half. You get grey fog: four halfway jumps fill the square evenly. But add a rule, never the same point twice in a row, and the fog turns into a pattern. Five points need a jump of 0.618, the golden ratio, to make a snowflake of pentagons; press best for this many points to find the jump that makes the copies just touch for any number.
Click to add points (up to 24), drag them about, and right-click or long-press to remove one. Each dot is coloured by the point it jumped toward, so you can see which corner made which part.
Wacław Sierpiński described his triangle in 1915, by cutting the middle out of a triangle, then out of each of the three that are left, forever. Seventy years later Michael Barnsley showed that you can draw it, and far more complicated shapes, by playing this game of chance, and named it the chaos game in his 1988 book Fractals Everywhere. The same idea, with a few different jumps, draws ferns, trees and coastlines.
