On the Density of Packings of Spheres in Spherical 3-Space*

Abstract

In an n-dimensional Euclidean, spherical or hyperbolic space consider a packing of at least four spheres with given radius r. The well-known simplicial density bound dnðrÞ gives an upper bound to the density of such a packing. In spherical 3-space there are exactly four packings for which the density d3ðrÞ is attained. These packings are formed by the inspheres of the regular tilings of type fa; 3; 3g, for a ¼ 2; 3; 4 and 5. In this paper it is proved that d3 is a strictly decreasing function of r. This implies the existence of the upper bound 0.77963 ... to the density of any packing of at least four congruent spheres in S 3 .