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Sphere Packings, Lattices and Groups

  • J. H. Conway
  • N. J. A. Sloane

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 290)

Table of contents

  1. Front Matter
    Pages i-xxvii
  2. J. H. Conway, N. J. A. Sloane
    Pages 1-30
  3. J. H. Conway, N. J. A. Sloane
    Pages 31-62
  4. J. H. Conway, N. J. A. Sloane
    Pages 63-93
  5. J. H. Conway, N. J. A. Sloane
    Pages 94-135
  6. John Leech, N. J. A. Sloane
    Pages 136-156
  7. J. H. Conway, N. J. A. Sloane
    Pages 157-180
  8. N. J. A. Sloane
    Pages 181-205
  9. J. H. Conway, N. J. A. Sloane
    Pages 206-244
  10. N. J. A. Sloane
    Pages 245-266
  11. J. H. Conway
    Pages 267-298
  12. J. H. Conway
    Pages 299-330
  13. J. H. Conway
    Pages 331-336
  14. A. M. Odlyzko, N. J. A. Sloane
    Pages 337-339
  15. E. Bannai, N. J. A. Sloane
    Pages 340-351
  16. J. H. Conway, N. J. A. Sloane
    Pages 352-405
  17. J. H. Conway, N. J. A. Sloane
    Pages 406-420
  18. R. E. Borcherds
    Pages 421-426
  19. B. B. Venkov
    Pages 427-438
  20. J. H. Conway, A. M. Odlyzko, N. J. A. Sloane
    Pages 439-442
  21. J. H. Conway, N. J. A. Sloane
    Pages 443-448
  22. J. H. Conway, N. J. A. Sloane
    Pages 449-475
  23. J. H. Conway, R. A. Parker, N. J. A. Sloane
    Pages 478-505
  24. J. H. Conway, N. J. A. Sloane
    Pages 506-512
  25. R. E. Borcherds, J. H. Conway, L. Queen
    Pages 513-521
  26. J. H. Conway, N. J. A. Sloane
    Pages 522-526
  27. J. H. Conway, N. J. A. Sloane
    Pages 532-553
  28. R. E. Borcherds, J. H. Conway, L. Queen, N. J. A. Sloane
    Pages 568-571
  29. Back Matter
    Pages 572-665

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • J. H. Conway
    • 1
  • N. J. A. Sloane
    • 2
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Mathematical Science DepartmentAT&T Bell LaboratoriesMurray HillUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-2016-7
  • Copyright Information Springer-Verlag New York 1988
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-2018-1
  • Online ISBN 978-1-4757-2016-7
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site