Sphere Packings, Lattices and Groups
We now apply the algorithm above to find the 121 orbits of norm -2 vectors from the (known) nann 0 vectors, and then apply it again to find the 665 orbits of nann -4 vectors from the vectors of nann 0 and -2. The neighbors of a strictly 24 dimensional odd
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Editors
S.S. Chern B. Eckmann P. de la Harpe H. Hironaka F. Hirzebruch N. Hitchin L. Hormander M.-A. Knus A. Kupiainen J. Lannes G. Lebeau M. Ratner D. Serre Ya.G. Sinai N. J. A. Sloane J. Tits M. Waldschmidt S. Watanabe Managing Editors
M. Berger J. Coates
S.R.S. Varadhan
Springer Science+Business Media, LLC
J.H. Conway N .J.A. Sloane
Sphere Packings, Lattices and Groups Third Edition With Additional Contributions by E. Bannai, R.E. Borcherds, J. Leech, S.P. Norton, A.M. Odlyzko, R.A. Parker, L. Queen and B. B. Venkov
With 112 Illustrations
'Springer
J.H. Conway Mathematics Department Princeton University Princeton, NJ 08540 USA conway@ math. princeton.edu
N.J.A. Sloane Information Sciences Research AT&T Labs - Research 180 Park Avenue Florham Park, NJ 07932 USA [email protected]
Mathematics Subject Classification (1991): 05B40, 11H06, 20E32, 11T71, 11E12 Library of Congress Cataloging-in-Publication Data Conway, John Horton Sphere packings, lattices and groups.- 3rd ed./ J.H. Conway, N.J.A. Sloane. p. em. - (Grundlehren der mathematischen Wissenschaften ; 290) Includes bibliographical references and index. ISBN 978-1-4419-3134-4 ISBN 978-1-4757-6568-7 (eBook) DOI 10.1007/978-1-4757-6568-7 1. Combinatorial packing and covering. 2. Sphere. 3. Lattice theory. 4. Finite groups. I. Sloane, N.J.A. (Neil James . II. Title. III. Series. Alexander). 1939QA166.7.C66 1998 511'.6-dc21 98-26950
Printed on acid-free paper. springeronline.com © 1999, 1998, 1993 Springer Science+B u siness Media New York Originally published by Springer-Verlag New York, Inc. in 1999 Softcover reprint of the hardcover 3rd edition 1999 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Allan Abrams; manufacturing supervised by Thomas King. Text prepared by the authors using TROFF.
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SPIN 10972366
Preface to First Edition
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 COYering problem, which asks for the least dense way to cove
