Locally Optimal 2-Periodic Sphere Packings

Abstract

The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi presented an algorithm to enumerate all local optima in a finite computation, which has been implemented in up to d = 8 dimensions.We generalize Voronoi's algorithm to m > 1 and use this new algorithm to enumerate all locally optimal 2-periodic sphere packings in d = 3, 4, and 5.In particular, we show that no 2-periodic packing surpasses the density of the optimal lattice in these dimensions. A partial enumeration is performed in d = 6.Keywords: sphere packing, periodic point set, quadratic form, Ryshkov polyhedron 2010 MSC: 52C17, 11H55, 52B55, 90C26