Sphere Packings, Lattices and Groups

1. Finding the Closest Lattice Point

   • J. H. Conway, N. J. A. Sloane

   Pages 443-448

2. Voronoi Cells of Lattices and Quantization Errors

   • J. H. Conway, N. J. A. Sloane

   Pages 449-475

3. A Bound for the Covering Radius of the Leech Lattice

   • S. P. Norton

   Pages 476-477

4. The Covering Radius of the Leech Lattice

   • J. H. Conway, R. A. Parker, N. J. A. Sloane

   Pages 478-505

5. Twenty-Three Constructions for the Leech Lattice

   • J. H. Conway, N. J. A. Sloane

   Pages 506-512

6. The Cellular Structure of the Leech Lattice

   • R. E. Borcherds, J. H. Conway, L. Queen

   Pages 513-521

7. Lorentzian Forms for the Leech Lattice

   • J. H. Conway, N. J. A. Sloane

   Pages 522-526

8. The Automorphism Group of the 26-Dimensional Lorentzian Lattice

   • J. H. Conway

   Pages 527-531

9. Leech Roots and Vinberg Groups

   • J. H. Conway, N. J. A. Sloane

   Pages 532-553

10. The Monster Group and its 196884-Dimensional Space

    • J. H. Conway

    Pages 554-567

11. A Monster Lie Algebra?

    • R. E. Borcherds, J. H. Conway, L. Queen, N. J. A. Sloane

    Pages 568-571

12. Back Matter

    Pages 572-665

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The main themes. This book is mainly concerned with the problem of packing spheres in Euclidean space of dimensions 1,2,3,4,5, . . . . Given a large number of equal spheres, what is the most efficient (or densest) way to pack them together? We also study several closely related problems: the kissing number problem, which asks how many spheres can be arranged so that they all touch one central sphere of the same size; the covering problem, which asks for the least dense way to cover n-dimensional space with equal overlapping spheres; and the quantizing problem, important for applications to analog-to-digital conversion (or data compression), which asks how to place points in space so that the average second moment of their Voronoi cells is as small as possible. Attacks on these problems usually arrange the spheres so their centers form a lattice. Lattices are described by quadratic forms, and we study the classification of quadratic forms. Most of the book is devoted to these five problems. The miraculous enters: the E 8 and Leech lattices. When we investigate those problems, some fantastic things happen! There are two sphere packings, one in eight dimensions, the E 8 lattice, and one in twenty-four dimensions, the Leech lattice A , which are unexpectedly good and very 24 symmetrical packings, and have a number of remarkable and mysterious properties, not all of which are completely understood even today.

• Lattice
• Lie algebra
• algebra
• applied mathematics
• automorphism
• coding theory
• mathematics
• quadratic form
• quantization
• combinatorics

• Mathematics Department, Princeton University, Princeton, USA

  J. H. Conway

• Mathematical Science Department, AT&T Bell Laboratories, Murray Hill, USA

  N. J. A. Sloane

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