WHAT ARE ALL THE BEST SPHERE PACKINGS IN LOW DIMENSIONS

We describe what may be all the best packings of nonoverlapping equal spheres in dimensions n less than or equal to 10, where ''best'' means both having the highest density and not permitting any local improvem ent. For example, the best five-dimensional sphere packings are parame trized by the 4-colorings of the one-dimensional integer lattice. We a lso find what we believe to be the exact numbers of ''uniform'' packin gs among these, that is, those in which the automorphism group acts tr ansitively. These assertions depend on certain plausible but as yet un proved postulates. Our work may be regarded as a continuation of Laszl o Fejes Toth's work on solid packings.