Sphere packings is one of the most fascinating and challenging subjects in mathematics. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. This book gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and its proof methods. The book includes both classical and contemporary results and provides a full treatment of the subject.
Author(s): Chuanming Zong, John Talbot (editor)
Series: Universitext
Edition: 1999
Publisher: Springer
Year: 1999
Language: English
Pages: 242
Cover
Title
Copyright
Preface
Basic Notation
Contents
1. The Gregory-Newton Problem and Kepler's Conjecture
1.1 Introduction
1.2 Packings of Circular Disks
1.3 The Gregory-Newton Problem
1.4 Kepler's Conjecture
1.4.1. L. Fejes Tóth's Program and Hsiang's Approach
1.4.2. Delone Stars and Hales' Approach
1.5. Some General Remarks
2. Positive Definite Quadratic Forms and Lattice Sphere Packings
2.1. Introduction
2.2. The Lagrange-Seeber-Minkowski Reduction and a Theorem of Gauss
2.3. Mordell's Inequality on Hermite's Constants and a Theorem of Korkin and Zolotarev
2.4. Perfect Forms, Voronoi's Method, and a Theorem of Korkin and Zolotarev
2.5. The Korkin-Zolotarev Reduction and Theorems of Blichfeldt, Barnes, and Vetčinkin
2.6. Perfect Forms, the Lattice Kissing Numbers of Spheres, and Watson's Theorem
2.7. Three Mathematical Geniuses: Zolotarev, Minkowski, and Voronoi
3. Lower Bounds for the Packing Densities of Spheres
3.1. The Minkowski-Hlawka Theorem
3.2. Siegel's Mean Value Formula
3.3. Sphere Coverings and the Coxeter-Few-Rogers Lower Bound for δ(Sn)
3.4. Edmund Hlawka
4. Lower Bounds for the Blocking Numbers and the Kissing Numbers of Spheres
4.1. The Blocking Numbers of S3 and S4
4.2. The Shannon-Wyner Lower Bound for Both b(Sn) and k(Sn)
4.3. A Theorem of Swinnerton-Dyer
4.4. A Lower Bound for the Translative Kissing Numbers of Superspheres
5. Sphere Packings Constructed from Codes
5.1. Codes
5.2. Construction A
5.3. Construction B
5.4. Construction C
5.5. Some General Remarks
6. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres I
6.1. Blichfeldt's Upper Bound for the Packing Densities of Spheres
6.2. Rankin's Upper Bound for the Kissing Numbers of Spheres
6.3. An Upper Bound for the Packing Densities of Superspheres
6.4. Hans Frederik Blichfeldt
7. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres II
7.1. Rogers' Upper Bound for the Packing Densities of Spheres
7.2. Schläfli's Function
7.3. The Coxeter-Böröczky Upper Bound for the Kissing Numbers of Spheres
7.4. Claude Ambrose Rogers
8. Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres III
8.1. Jacobi Polynomials
8.2. Delsarte's Lemma
8.3. The Kabatjanski-Levenstein Upper Bounds for the Packing Densities and the Kissing Numbers of Spheres
9. The Kissing Numbers of Spheres in Eight and Twenty–Four Dimensions
9.1. Some Special Lattices
9.2. Two Theorems of Levenštein, Odlyzko, and Sloane
9.3. Two Principles of Linear Programming
9.4. Two Theorems of Bannai and Sloane
10. Multiple Sphere Packings
10.1. Introduction
10.2. A Basic Theorem of Asymptotic Type
10.3. A Theorem of Few and Kanagasahapathy
10.4. Remarks on Multiple Circle Packings
11. Holes in Sphere Packings
11.1. Spherical Holes in Sphere Packings
11.2. Spherical Holes in Lattice Sphere Packings
11.3. Cylindrical Holes in Lattice Sphere Packings
12. Problems of Blocking Light Rays
12.1. Introduction
12.2. Hornich's Problem
12.3. L. Fejes Tóth's Problem
12.4. László Fejes Tóth
13. Finite Sphere Packings
13.1. Introduction
13.2. The Spherical Conjecture
13.3. The Sausage Conjecture
13.4. The Sausage Catastrophe
Bibliography
Index
