Introduction

The following sections are included:

• Sphere Packings and the Sphere Packing Problem

  • Density of an infinite packing

• Kepler's Conjecture on Sphere Packings

  • Another mathematical formulation of the sphere packing problem and Kepler's Conjecture

• Density of Finite Packings without a Container and the Least Action Principle of Crystal Formation

  • Hexagonal close packings

  • Least action principle and crystal formation

  • Local cell decomposition, local density and the Dodecahedron Conjecture

• Locally Averaged Density and the Intrinsic Density of the Second Kind

• Main Theorems on Sphere Packings

  • Relationship between the two kinds of relative densities and the proof of Kepler's Conjecture