The Farthest Point Map on the Regular Dodecahedron

TL;DR

This paper thoroughly describes the farthest point map on a regular dodecahedron, revealing its complex structure and limit set, with connections to Penrose tiling, using a rigorous computer-assisted proof.

Contribution

It provides a complete characterization of the farthest point map on the dodecahedron and links its algebraic structure to Penrose tiling rhombi.

Findings

The map G is piecewise bi-quadratic with algebraic pieces defined by rhombus constructions.

The omega-limit set of G is the 1-skeleton of a subdivision into 180 convex quadrilaterals.

The proof combines geometric analysis with computer-assisted rigor.

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.