The kissing number of a sphere is 12. This led Fejes Tóth to conjecture that in any unit sphere packing, the volume of any Voronoi cell around any sphere is at least as large as a regular dodecahedron of inradius 1. This statement is now known as the dodecahedral conjecture. It implies a bound of η<=0.754697... on the packing density for sphere packing, and thus provides a bound on the densest possible sphere packing. It is not, however, sufficient to establish the Kepler conjecture (which implies η = 0.74048). This long-outstanding conjecture was proved by Hales and McLaughlin using techniques of interval arithmetic and linear programming.

Kelvin's conjecture | Kepler conjecture | kissing number | sphere packing