This is a very interesting question. Yes, I think this is a proof that any convex polyhedron rolling on its mirror image has trivial holonomy. (Meaning: If they both start by sharing a common face in 3-space, a reflection in whose plane identifies each polyhedron with its mirror image.) Sketch of proof: Then rotating one of them by angle theta about an edge of the latest shared face is equivalent to rotating both of them in opposite directions by an angle theta/2 about the same edge. As for rolling one sphere on an identical one (with neither slipping nor twirling about the line connecting their centers): Indeed, essentially the same proof sketch as above shows that once each point of a sphere is identified (once and for all) with its mirror image, then (after some rolling) reflection about the tangent plane separating the spheres will always carry a point of one sphere to its (original) mirror image on the other sphere. * * * Here's a related problem I worked on for a while but only got as far as a good approximation to the answer: Given a sphere rolling on a horizontal plane (without slipping or twirling): Suppose it begins tangent to the origin (0,0) of the plane, with its point of tangency equal to the south pole S of the sphere. Suppose it's now rolled along a simple closed curve C of the plane so that when it returns to (0,0), its final point of tangency is the north pole N. Then, among the simple closed curves C for which this occurs: 1) Which C is the shortest? 2) Which C surrounds the smallest area? —Dan
On Friday/25December/2020, at 2:31 PM, James Propp <jamespropp@gmail.com> wrote:
Does a cube rolling around the outside of another cube have trivial monodromy? Is this true for every convex polyhedron rolling around its mirror-image? I *think* so, and I even have an idea for how to prove it (use a moving coordinate system in which the two polyhedra are mirror-images of each other in a fixed plane at all times). But then it would be true for spheres as well, and this smells wrong to me.