Sphere Packings

Abstract

Sphere Packings is one of the most attractive and challenging subjects in mathematics. Almost 4 centuries ago, Kepler studied the densities of sphere packings and made his famous conjecture. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with othe subjects found. Thus, though some of its original problems are still open, sphere packings has been developed into an important discipline. This book tries to give a full account of this fascinating subject, especially its local aspects, discrete aspects and its proof methods. Cover -- Preface -- Basic Notation -- Table of Contents -- 1. The Gregory-Newton Problem and Kepler's Conjecture -- 2. Positive Definite Quadratic Forms and Lattice Sphere Packings -- 3. Lower Bounds for the Packing Densities of Spheres -- 4. Lower Bounds for the Blocking Numbers and the Kissing Numbers of Spheres -- 5. Sphere Packings Constructed from Codes -- 6. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres I -- 7. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres II -- 8. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres III -- 9. The Kissing Numbers of Spheres in Eight and Twenty-Four Dimensions -- 10. Multiple Sphere Packings -- 11. Holes in Sphere Packings -- 12. Problems of Blocking Light Rays -- 13. Finite Sphere Packings -- Bibliography.