What are all the best sphere packings in low dimensions?

Abstract

We describe what may be all the best packings of nonoverlapping equal spheres in dimensions n ≤10, where "best" means both having the highest density and not permitting any local improvement. For example, the best five-dimensional sphere packings are parametrized by the 4-colorings of the one-dimensional integer lattice. We also find what we believe to be the exact numbers of "uniform" packings among these, that is, those in which the automorphism group acts transitively. These assertions depend on certain plausible but as yet unproved postulates. Our work may be regarded as a continuation of László Fejes Tóth's work on solid packings. © 1995 Springer-Verlag New York Inc.