Classification of isosceles 7-point 3-distance sets in 3-dimensional Euclidean space

Abstract

A subset $X$ in the $k$-dimensional Euclidean space ${\mathbb{R}}^k$ that contains $n$ points (elements) is called an isosceles $n$-point $s$-distance set if there are exactly $s$ distances between two distinct points in $X$ and if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that there exist exactly fifteen isosceles 7-point 3-distance sets in ${\mathbb{R}}^3$ up to isomorphism.