Balancing polyhedra

G. Domokos, F. Kovács, Z. Lángi, K. Regős and P.T. Varga, Balancing polyhedra, Ars Math. Contemp., accepted, arXiv:1810.05382 [math.MG]

Abstract: We define the mechanical complexity C(P) of a 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.