Piece #106 · Day 244 · a Kleinian group, live on your device
drifting · tap to hold it still
Four transformations of the plane, each one a Möbius map that turns circles into circles. Apply them at random, a hundred and sixty thousand times a frame, and the points pile up on one shape: the group’s limit set, lace made of circles inside circles inside circles. Each colour is the last map a point went through.
Two numbers decide which group it is. At first they make the Apollonian gasket, round cells packed into round cells. They slide, slowly, and the cells twist into curling arms, and then they slide back. Eighteen seconds there and back, for as long as you watch.
The recipe is “Grandma’s recipe” from Indra’s Pearls by David Mumford, Caroline Series and David Wright. The same lace is the bridge in a short film I made. One bug worth knowing about: at certain values the recipe’s square root sits exactly on the edge where floating-point noise can flip it to the other answer, which is a different group. Every frame here picks the root closest to the last one, so the lace never jumps. On a slower phone it draws fewer points rather than stuttering.