  An earlier paper describes a program to prove the Kepler conjecture on sphere
packings. This paper carries out the second step of that program. A sphere
packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are
divided into two classes. The first class of polyhedra, called quasi-regular
tetrahedra, have density at most that of a regular tetrahedron. The polyhedra
in the remaining class have density at most that of a regular octahedron (about
0.7209).

Metric Geometry