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.