Cube Orbit

repeat a sequence of turns and the cube comes back to solved. the dial is one ring per cycle of that sequence, and the cube is home when every ring is.

key insight: a ring's rotation is the permutation rather than a picture of one. a cycle's stickers sit on their ring in the order the sequence sends them, so one notch puts every one of them exactly where the sequence would have, and the dial and the cube cannot drift apart.

one number folds the net into the cube. --fold drives the five hinges, the angle it is seen from and the light on each face, and the faces hang off each other the way the paper does, so a hinge is one rotation about an edge two faces already share.