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J. H. Conway

Mathematics Department, Princeton University, Princeton, USA

N. J. A. Sloane

Mathematical Science Department, AT&T Bell Laboratories, Murray Hill, USA
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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 ndimensional space with equal overlapping spheres; and the quantizing problem, important for applications to analogtodigital 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 twentyfour 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.
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Open access
19 October 2023
Table of contents (30 chapters)

Front Matter
Pages ixxvii

 J. H. Conway, N. J. A. Sloane
Pages 130

 J. H. Conway, N. J. A. Sloane
Pages 3162

 J. H. Conway, N. J. A. Sloane
Pages 6393

 J. H. Conway, N. J. A. Sloane
Pages 94135

 John Leech, N. J. A. Sloane
Pages 136156

 J. H. Conway, N. J. A. Sloane
Pages 157180


 J. H. Conway, N. J. A. Sloane
Pages 206244





 A. M. Odlyzko, N. J. A. Sloane
Pages 337339

 E. Bannai, N. J. A. Sloane
Pages 340351

 J. H. Conway, N. J. A. Sloane
Pages 352405

 J. H. Conway, N. J. A. Sloane
Pages 406420



 J. H. Conway, A. M. Odlyzko, N. J. A. Sloane
Pages 439442
Authors and Affiliations

Mathematics Department, Princeton University, Princeton, USA
J. H. Conway

Mathematical Science Department, AT&T Bell Laboratories, Murray Hill, USA
N. J. A. Sloane