Balancing polyhedra

Abstract

We define the mechanical complexity C(P) of a 3-dimensional convex polyhedron P, interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria; and the mechanical complexity C(S, U) of primary equilibrium classes (S, U)^E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U)^E with S, U > 1 is the minimum of 2(f + v − S − U) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral pair if there is a convex polyhedron with f faces and v vertices. In particular, we prove that the mechanical complexity of a class (S, U)^E is zero if and only if there exists a convex polyhedron with S faces and U vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes (1, U)^E and (S, 1)^E, and offer a complexity-dependent prize for the complexity of the Gömböc-class (1, 1)^E.

Dedicated to the memory of John Horton Conway.