Average kissing numbers for non-congruent sphere packings (Q1901660)

The authors consider finite packings \(P\) of \(n\) unequal balls in \(\mathbb{R}^3\). If \(m\) denotes the number of contact points of these \(n\) balls then \(k(P)= 2m/n\) is called the average kissing number or contact number of the packing \(P\). Further let \[ k= \sup\{ k(P)\mid P \text{ is a finite packing of balls in } \mathbb{R}^3\}. \] Then the authors show that \(12.566\leq k< 14.928\). In particular the lower bound is surprising, because in \(\mathbb{R}^3\) the maximal kissing number of a single ball surrounded by congruent balls is 12. The construction for this lower bound uses the properties of one of the 6 convex 4-dimensional regular polytopes, which leads to an extremely dense sphere packing on \(S^3\), as was already shown by H. S. M. Coxeter and L. Féjes Toth in 1961. Here the authors use a refinement of the good kissing number properties.