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

A subset X in the k -dimensional Euclidean space 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 R 3 up to isomorphism.