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Balancing polyhedra. (arXiv:1810.05382v1 [math.MG])
来源于:arXiv
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 discuss the mechanical complexity of the
monostatic classes $(1,U)^E$ and $(S,1)^E$, and offer a complexity-dependent
prize for the compl 查看全文>>