Balancing polyhedra

Domokos, G.; Kovács, Flórián; Lángi, Zsolt; Regős, Krisztina; Varga, Péter T.

doi:10.48550/arxiv.1810.05382

Cited by 1 publication

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“…We remark that the variant of Theorem 3 with respect to the center of mass of the polyhedron, which we state as Problem 1, proves that every combinatorial class has a representative whose every face, vertex and edge contains a static equilibrium point [8]. An affirmative answer to the problem, with many applications in mechanics [8,10,9], would be a discrete version of Theorem 1 in [10], stating that for every 3-colored quadrangulation Q of S 2 there is a convex body K whose Morse-Smale graph, with respect to its center of mass, is isomorphic to Q. These papers also describe possible applications of our problem in various fields of science, from physics to chemistry to manufacturing.…”

Section: Introductionmentioning

confidence: 99%

Centering Koebe polyhedra via Möbius transformations

Lángi¹

2018

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A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the problem of centering spherical configurations under Möbius transformations. In particular, Springborn proved that for any discrete point set on the sphere there is a Möbius transformation that maps it into a set whose barycenter is the origin, which implies that the combinatorial class of any convex polyhedron contains an element midsribed to a sphere with the additional property that the barycenter of the points of tangency is the center of the sphere. This result was strengthened by Baden, Krane and Kazhdan who showed that the same idea works for any reasonably nice measure defined on the sphere. The aim of the paper is to show that Springborn's statement remains true if we replace the barycenter of the tangency points by many other polyhedron centers. The proof is based on the investigation of the topological properties of the integral curves of certain vector fields defined in hyperbolic space. We also show that most centers of Koebe polyhedra cannot be obtained as the center of a suitable measure defined on the sphere.

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