Classification of isosceles 7-point 3-distance sets in 3-dimensional Euclidean space
A subset \(X\) of the \(k\)-dimensional Euclidean space \(\mathbb{R}^k\) that contains \(n\) points 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. Evidently there are a lot of questions for existence of these special point-structures depending on \(k\), \(n\) and \(s\). Beforehand the author gives a short summary about known facts. The main result of this paper is the classification of all isosceles 7-point 3-distance sets in \(\mathbb{R}^3\): There exist exactly fifteen isomorphic-classes. Besides a multitude of properties of distance sets are discussed.
isosceles \(n\)-point \(s\)-distance set
scientific article; zbMATH DE number 5137575