Two-Distance Sets and the Golden Ratio

Abstract

A point set in d-dimensional Euclidean space E ^d with pairwise distances 1 and α, α > 1, is called a two-distance set with diameter α. The maximum diameter of all two-distance sets with n points in E ^d is denoted by σ(d, n) for n ≥ d + 2. It is known that two-distance sets do not exist for $n > \left( {\begin{array}{*{20}{c}} {d + 2} \\ 2 \\ \end{array} } \right)$ (see [1]). However, existence is proved for $n \leqslant \left( {\begin{array}{*{20}{c}} {d + 1} \\ 2 \\ \end{array} } \right)$ (see [4]). For n =4 and d = 2 Figure 1 shows all six two-distance sets (see [2]), and the last two of them determine $\sigma \left( {2,4} \right) = \sqrt {{2 + \sqrt {3} }} = \left( {1 + \sqrt {3} } \right)/\sqrt {2}$. The regular unit pentagon is the only example for n = 5 and d = 2. It follows that $\sigma \left( {2,5} \right) = \tau = \frac{{1 + \sqrt {5} }}{2}$. This golden ratio τ also occurs in E ³ since σ(3,6) = τ is realized in subsets of the unit icosahedron, for example, one vertex point together with all its five neighbors. Here we will ask for further maximum diameters σ(d, n) with value τ.