A proof of a Dodecahedron conjecture for distance sets

TL;DR

This paper proves that the largest 5-distance set in three-dimensional space has 20 points, which correspond to the vertices of a regular dodecahedron, confirming a conjecture about such configurations.

Contribution

It establishes that the maximum size of a 5-distance set in ^3 is 20, and characterizes all such sets as similar to a regular dodecahedron.

Findings

Maximum 5-distance set size in ^3 is 20.

All 20-point 5-distance sets are similar to a regular dodecahedron.

Confirmed the conjecture linking 5-distance sets to the dodecahedron.

Abstract

A finite subset of a Euclidean space is called an $s$-distance set if there exist exactly $s$ values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in $\mathbb{R}^3$ is 20, and every $5$-distance set in $\mathbb{R}^3$ with $20$ points is similar to the vertex set of a regular dodecahedron.