Rigidity of circle polyhedra in the 2-sphere and of hyperideal polyhedra in hyperbolic 3-space

John C. Bowers, Philip L. Bowers, Kevin Pratt

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space E³ to the context of circle polyhedra in the 2-sphere S². We prove that any two convex and proper non-unitary c-polyhedra with Möbius-congruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the Riemann sphere as well as that of certain hyperideal hyperbolic polyhedra in H³.