The two-distance sets in dimension four

TL;DR

This paper classifies all two-distance sets in four-dimensional Euclidean space using computer-aided methods, providing a complete understanding of their structure in this dimension.

Contribution

It offers the first complete classification of 2-distance sets in -dimensional space, combining theoretical insights with computational techniques.

Findings

Complete classification of 2-distance sets in D

Identification of all isometry classes in D

Methodology combining theoretical and computational approaches

Abstract

A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a $2$-distance set, if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $2$. In this note we classify the $2$-distance sets in $\mathbb{R}^4$ up to isometry with computer-aided methods.