Regular sphere packings (Q1347778)

A collection of not necessarily congruent non-overlapping spheres is a packing, which is \(k\)-regular if every sphere is touched by exactly \(k\) spheres. The authors determine, for \(k\leq 6\), all possible numbers \(n\) of spheres which may form a connected \(k\)-regular packing (e.g., \(k= 4\), \(n= 6\) and \(n\geq 8\)) and, for \(7\leq k\leq 10\), give bounds on \(B(k)=\) the number of spheres which can form a \(k\)-regular packing (e.g., \(B(8)\leq 20\)). These results advance considerably the known results. The problem of finiteness of \(B(11)\) remains open.