The Graphs Behind Reuleaux Polyhedra

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 $\tau :G \rightarrow G^*$ (where $G^*$ is the unique dual graph), a metric mapping is a map $\eta :V(G) \rightarrow \mathbb R^3$ such that the diameter of $\eta (G)$ is 1 and for every pair of vertices (u, v) such that $u\in \tau (v)$ we have ${{\,\mathrm{dist}\,}}{(\eta (u),\eta (v))}= 1$. If $\eta$ 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 $\mathbb R^3$. 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.