Lattices And Sphere Packings In Euclidean Space
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Author | : J.H. Conway |
Publisher | : Springer Science & Business Media |
Total Pages | : 724 |
Release | : 2013-03-09 |
Genre | : Mathematics |
ISBN | : 1475722494 |
The second edition of this timely, definitive, and popular book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to examine related problems such as the kissing number problem, the covering problem, the quantizing problem, and the classification of lattices and quadratic forms. Like the first edition, the second edition describes the applications of these questions to other areas of mathematics and science such as number theory, coding theory, group theory, analog-to-digital conversion and data compression, n-dimensional crystallography, and dual theory and superstring theory in physics. Results as of 1992 have been added to the text, and the extensive bibliography - itself a contribution to the field - is supplemented with approximately 450 new entries.
Author | : John Conway |
Publisher | : Springer Science & Business Media |
Total Pages | : 778 |
Release | : 2013-06-29 |
Genre | : Mathematics |
ISBN | : 1475765681 |
The third edition of this definitive and popular book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also examine such related issues as the kissing number problem, the covering problem, the quantizing problem, and the classification of lattices and quadratic forms. There is also a description of the applications of these questions to other areas of mathematics and science such as number theory, coding theory, group theory, analogue-to-digital conversion and data compression, n-dimensional crystallography, dual theory and superstring theory in physics. New and of special interest is a report on some recent developments in the field, and an updated and enlarged supplementary bibliography with over 800 items.
Author | : Jacques Martinet |
Publisher | : Springer Science & Business Media |
Total Pages | : 535 |
Release | : 2013-03-09 |
Genre | : Mathematics |
ISBN | : 3662051672 |
Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sphere packing in a given dimension is a fascinating and difficult problem: the answer is known only up to dimension 3. This book thus discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory, centering on the study of extreme lattices, i.e. those on which the density attains a local maximum, and on the so-called perfection property. Written by a leader in the field, it is closely related to, though disjoint in content from, the classic book by J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, published in the same series as vol. 290. Every chapter except the first and the last contains numerous exercises. For simplicity those chapters involving heavy computational methods contain only few exercises. It includes appendices on Semi-Simple Algebras and Quaternions and Strongly Perfect Lattices.
Author | : John H. Conway |
Publisher | : Springer |
Total Pages | : 665 |
Release | : 2013-02-14 |
Genre | : Mathematics |
ISBN | : 9781475720174 |
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.
Author | : Stephanie L. Vance |
Publisher | : |
Total Pages | : 62 |
Release | : 2009 |
Genre | : Sphere packings |
ISBN | : |
Author | : J. H. Conway |
Publisher | : |
Total Pages | : 732 |
Release | : 2014-01-15 |
Genre | : |
ISBN | : 9781475722505 |
Author | : Thomas Callister Hales |
Publisher | : Cambridge University Press |
Total Pages | : 286 |
Release | : 2012-09-06 |
Genre | : Mathematics |
ISBN | : 0521617707 |
The definitive account of the recent computer solution of the oldest problem in discrete geometry.
Author | : John H. Conway |
Publisher | : Springer Science & Business Media |
Total Pages | : 690 |
Release | : 2013-04-17 |
Genre | : Mathematics |
ISBN | : 1475720165 |
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.
Author | : Thomas M. Thompson |
Publisher | : |
Total Pages | : 252 |
Release | : 1983 |
Genre | : Error-correcting codes (Information theory) |
ISBN | : 9780883850008 |
Author | : Denis Weaire |
Publisher | : CRC Press |
Total Pages | : 147 |
Release | : 2000-01-01 |
Genre | : Mathematics |
ISBN | : 142003331X |
In 1998 Thomas Hales dramatically announced the solution of a problem that has long teased eminent mathematicians: what is the densest possible arrangement of identical spheres? The Pursuit of Perfect Packing recounts the story of this problem and many others that have to do with packing things together. The examples are taken from mathematics, phy