Zong C. Sphere Packings

New York: Springer-Verlag. – 1999. – 243 p. (Universitext) Sphere packings is one of the most fascinating and challenging subjects in mathematics. Almost four centuries ago, Kepler studied the densities of sphere packings and made his famous conjecture. Several decades later, Gregory and Newton discussed the kissing numbers of spheres and proposed the Gregory-Newton problem. Since then, these problems and related ones have attracted the attention of many prominent mathematicians. As work on the classical sphere packing problems has progressed, many exciting results have been obtained, ingenious methods have been created, related challenging problems have been proposed and investigated, and surprising connections with other subjects have been found. Thus, though some of its original problems are still open, sphere packings has developed into an important discipline. This tract gives full account of this subject. In addition to the classical sphere packing problems, it also deals with the contemporary ones; such as, blocking light rays, the holes in sphere packings, and finite sphere packings. Not only are the main results of the subject presented, but also its creative methods from areas such as geometry, number theory, and linear programming are described. The book also contains short biographies of several masters of this discipline and many open problems.Basic Notation.
The Gregory-Newton Problem and Kepler’s Conjecture.
Positive Definite Quadratic Forms and Lattice Sphere Packings.
Lower Bounds for the Packing Densities of Spheres.
Lower Bounds for the Blocking Numbers and the Kissing Numbers of Spheres.
Sphere Packings Constructed from Codes.
Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres I.
Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres II.
Claude Ambrose Rogers.
Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres III.
The Kissing Numbers of Spheres in Eight and Twenty—Four Dimensions.
Multiple Sphere Packings.
Holes in Sphere Packings.
Problems of Blocking Light Rays.
Finite Sphere Packings.