What shape do random half-way jumps toward three corners draw? Within about 100 ticks the marks outline a triangle with an empty triangle in the middle and smaller empty triangles inside each corner copy.
Try this Choose Square — no consecutive corner (4 corners, switch-corner) and run it: touched-cells stops near 760 and the gaps repeat at smaller scales. Then switch corner-rule to unrestricted (it takes effect live; press Reset for a clean drawing): the points fill the whole diamond, about 1,760 cells.
About this model Where it ends up
The chaos game is Michael Barnsley's name, in Fractals Everywhere (1988), for drawing the attractor of an iterated function system by random iteration: pick a corner at random, move a fixed fraction of the way toward it, plot the point, repeat. With three corners and half-way jumps the points fall on the Sierpiński triangle.
Setup places one gold anchor per corner (corners, default 3) on an ellipse 68 wide and 50 tall, the first at the top, and walker-count hidden walkers near the center. Each tick every walker picks an anchor at random (under switch-corner, any anchor except the one it used last), moves jump-percent of the way toward it, and adds 1 to the hits of the patch it lands on. Patch color follows hits. The run stops when the tick count reaches duration.
The drawing is binned into the 81 × 61 patch grid, so only three or four levels of holes resolve. Walkers start at the center, which is not on the attractor, and their first few landings are plotted too, leaving faint marks at the centers of the holes. The anchors sit on an ellipse, so each shape is stretched; its attractor is an affine image of the regular one. corners and walker-count apply at setup; jump-percent, corner-rule and duration are read every tick.
