On the Cover of the Rolling Stone

We construct a convex polytope of unit diameter that when placed on a horizontal surface on one of its faces, it repeatedly rolls over from one face to another until it comes to rest on some face, far away from its start position: that is, the horizontal distance between the footprints of the start and final faces can be larger than any given threshold. According to the laws of physics, the vertical distance between the center of mass of the polytope and the horizontal surface continuously decreases throughout the entire motion. The speed of the motion is irrelevant. Specifically, if the polytope is manually stopped after each tumble, the motion resumes when released (unless it stands on the final stable face). Moreover, such a polytope can be realized so that (i) it has a unique stable face, and (ii) it is an arbitrary close approximation of a unit ball. As such, this construction gives a positive answer to a question raised by Conway (1969). The arbitrarily large rolling distance property investigated here for the first time raises intriguing questions and opens new avenues for future research.