Geodesic
Balancing polyhedra
Ars Mathematica Contemporanea, Tome 19 (2020) no. 1, pp. 95-124.

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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.
DOI : 10.26493/1855-3974.2120.085
Keywords: Polyhedron, static equilibrium, monostatic polyhedron, f-vector 
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Gábor Domokos; Flórián Kovács; Zsolt Lángi; Krisztina Regős; Péter T. Varga. Balancing polyhedra. Ars Mathematica Contemporanea, Tome 19 (2020) no. 1, pp. 95-124. doi : 10.26493/1855-3974.2120.085. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2120.085/

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