### 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**

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MANUSCRIPTISSN: 1855-3974

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