Balancing polyhedra - Domokos, Gábor (Ars mathematica contemporanea, 2020)
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 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 ?$(1; 1)^E$?.
Keywords: f-vector; f-vektor; monostatic polyhedron; monostatični polieder; polieder; polyhedron; static equilibrium; statično ravnovesje