The Graphs Behind Reuleaux Polyhedra
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The Graphs Behind Reuleaux Polyhedra Luis Montejano1 · Eric Pauli2 · Miguel Raggi2 · Edgardo Roldán-Pensado3 Received: 29 April 2019 / Revised: 4 April 2020 / Accepted: 24 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, 3-connected and strongly self-dual. We study the question of when these conditions are sufficient. If G is any such graph, each vertex has an opposite face with isomorphism τ : G → G ∗ (where G ∗ is the unique dual graph), a metric mapping is a map η : V (G) → R3 such that the diameter of η(G) is 1 and for every pair of vertices (u, v) such that u ∈ τ (v) we have dist (η(u), η(v)) = 1. If η is injective, it is called a metric embedding. Our contributions are twofold: Firstly, we prove that any planar, 3-connected, strongly self-dual graph has a metric mapping to the vertices of a tetrahedron. Secondly, we develop algorithms that allow us to obtain every such graph with up to 14 vertices and we construct (numerically) metric embeddings for it. From these two facts we conjecture that every such graph is realizable as a Reuleaux polyhedron in R3 . In previous work the first and last authors described a method to construct a constant-width body from a Reuleaux polyhedron. So in essence, we also construct (numerically, but with very high precision) hundreds of new examples of constant-width bodies. Keywords Constant width bodies · Meissner solids · Reuleaux polyhedron · Self-dual planar graphs Mathematics Subject Classification 52A15 · 52B10 · 05C10
1 Introduction A Reuleaux polygon K ⊂ R2 is the intersection of finitely many circles of radius r in such a way that the vertices of K (i.e., the non-smooth points on the boundary of K )
Editor in Charge: János Pach Dedicated to the memory of Ricky Pollack This research was supported by PAPIIT Projects IA102118, IN112614 and IN116919, and CONACyT Project 166306. Extended author information available on the last page of the article
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Discrete & Computational Geometry
are precisely the centers of the circles defining K . It is a simple exercise to see that for any Reuleaux polygon, n must be odd and, conversely, that for every odd n ≥ 3 there exist Reuleaux polygons with n vertices. A ball polyhedron is the intersection of finitely many (at least three) 3dimensional balls of radius 1 in R3 . Ball polyhedra are natural objects used for studying several important problems in discrete geometry [3,9]. The geometry of the boundary of a ball polyhedron is of particular interest. Indeed, one can represent the boundary of a ball polyhedron ∂ as the union of vertices, edges, and faces defined in a natural way, giving rise to a graph G embedded in ∂ (see Sect. 2). The faces are closures of the connected components of ∂ \ G , where each face is a closed subset of a sphere of radius 1. Furthermore, each edge of G is an arc of a circle with radius smaller than 1. This structure is not necessarily a lattice, unless the
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