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

A subset XX in the kk-dimensional Euclidean space RkRk that contains nn points (elements) is called an isosceles nn-point ss-distance set if there are exactly ss distances between two distinct points in XX 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 R3R3 up to isomorphism.