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Author : Thomas Callister Hales
Publisher : Cambridge University Press
Page : 286 pages
File Size : 50,9 Mb
Release : 2012-09-06
Category : Mathematics
ISBN : 9780521617703
The definitive account of the recent computer solution of the oldest problem in discrete geometry.
Author : J.H. Conway,N.J.A. Sloane
Publisher : Springer Science & Business Media
Page : 724 pages
File Size : 42,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475722499
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 : Chuanming Zong
Publisher : Springer Science & Business Media
Page : 242 pages
File Size : 45,7 Mb
Release : 2008-01-20
Category : Mathematics
ISBN : 9780387227801
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 : Thomas Callister Hales
Publisher : Unknown
Page : 287 pages
File Size : 41,6 Mb
Release : 2014-05-14
Category : MATHEMATICS
ISBN : 1139568701
The definitive account of the recent computer solution of the oldest problem in discrete geometry.
Author : Jeffrey C. Lagarias
Publisher : Springer Science & Business Media
Page : 456 pages
File Size : 45,9 Mb
Release : 2011-11-09
Category : Mathematics
ISBN : 9781461411291
The Kepler conjecture, one of geometry's oldest unsolved problems, was formulated in 1611 by Johannes Kepler and mentioned by Hilbert in his famous 1900 problem list. The Kepler conjecture states that the densest packing of three-dimensional Euclidean space by equal spheres is attained by the “cannonball" packing. In a landmark result, this was proved by Thomas C. Hales and Samuel P. Ferguson, using an analytic argument completed with extensive use of computers. This book centers around six papers, presenting the detailed proof of the Kepler conjecture given by Hales and Ferguson, published in 2006 in a special issue of Discrete & Computational Geometry. Further supporting material is also presented: a follow-up paper of Hales et al (2010) revising the proof, and describing progress towards a formal proof of the Kepler conjecture. For historical reasons, this book also includes two early papers of Hales that indicate his original approach to the conjecture. The editor's two introductory chapters situate the conjecture in a broader historical and mathematical context. These chapters provide a valuable perspective and are a key feature of this work.
Author : Denis Weaire,Tomaso Aste
Publisher : CRC Press
Page : 147 pages
File Size : 53,8 Mb
Release : 2000-01-01
Category : Mathematics
ISBN : 9781420033311
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
Author : Jacques Martinet
Publisher : Springer Science & Business Media
Page : 535 pages
File Size : 51,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662051672
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 : Wu Yi Hsiang
Publisher : World Scientific
Page : 425 pages
File Size : 46,7 Mb
Release : 2001
Category : Mathematics
ISBN : 9789810246709
The dense packing of microscopic spheres (i.e. atoms) is the basic geometric arrangement in crystals of mono-atomic elements with weak covalent bonds, which achieves the optimal ?known density? of B/û18. In 1611, Johannes Kepler had already ?conjectured? that B/û18 should be the optimal ?density? of sphere packings. Thus, the central problems in the study of sphere packings are the proof of Kepler's conjecture that B/û18 is the optimal density, and the establishing of the least action principle that the hexagonal dense packings in crystals are the geometric consequence of optimization of density. This important book provides a self-contained proof of both, using vector algebra and spherical geometry as the main techniques and in the tradition of classical geometry.
Author : Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo
Publisher : World Scientific
Page : 5396 pages
File Size : 47,7 Mb
Release : 2019-02-27
Category : Mathematics
ISBN : 9789813272897
The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.
Author : Brian E. Clancy
Publisher : Unknown
Page : 28 pages
File Size : 53,7 Mb
Release : 1966
Category : Electronic
ISBN : UOM:39015086587501
Author : John H. Conway,Neil J.A. Sloane
Publisher : Springer
Page : 665 pages
File Size : 44,8 Mb
Release : 2013-02-14
Category : Mathematics
ISBN : 1475720173
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 : John Conway,Neil J. A. Sloane
Publisher : Springer Science & Business Media
Page : 778 pages
File Size : 45,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475765687
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 : Thomas M. Thompson
Publisher : Unknown
Page : 252 pages
File Size : 48,6 Mb
Release : 1983
Category : Error-correcting codes (Information theory)
ISBN : 0883850001
Author : Kenneth Stephenson
Publisher : Cambridge University Press
Page : 380 pages
File Size : 41,9 Mb
Release : 2005-04-18
Category : Mathematics
ISBN : 0521823560
Publisher Description
Author : Chuanming Zong
Publisher : Springer Science & Business Media
Page : 245 pages
File Size : 51,8 Mb
Release : 1999-08-19
Category : Mathematics
ISBN : 9780387987941
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.