The Farthest Point Map on the Regular Dodecahedron

Abstract

Let $X$ be the regular dodecahedron, equipped with its intrinsic path metric. Given $p \in X$ let $G(p)=-q$ where $q$ is the point on $X$ which maximizes the distance to $p$. (Generically, $G$ is single-valued.) We give a complete description of the map $G$ and as a consequence show that the $\omega$-limit set of $G$ is the $1$-skeleton of a subdivision of $X$ into $180$ convex quadrilaterals. $G$ is a piecewise bi-quadratic map, and each algebraic piece is defined by a straight line construction involving a rhombus. The rhombi involved have the same shapes as the ones in the Penrose tiling. Our proof is computer-assisted but rigorous.