© 2003

Perfect Lattices in Euclidean Spaces

  • Long-awaited authoritative reference on this beautiful subject at the interface of geometry, number theory, coding theory and group theory

  • Complement to J.H. Conway and N.J.A. Sloane "Sphere Packings, Lattices and Groups" (Grundlehren der mathematischen Wissenschaften, Vol. 290)


Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 327)

Table of contents

  1. Front Matter
    Pages I-XXI
  2. Jacques Martinet
    Pages 1-35
  3. Jacques Martinet
    Pages 37-65
  4. Jacques Martinet
    Pages 67-108
  5. Jacques Martinet
    Pages 109-145
  6. Jacques Martinet
    Pages 147-188
  7. Jacques Martinet
    Pages 189-225
  8. Jacques Martinet
    Pages 227-262
  9. Jacques Martinet
    Pages 263-319
  10. Jacques Martinet
    Pages 321-362
  11. Jacques Martinet
    Pages 363-381
  12. Jacques Martinet
    Pages 383-426
  13. Jacques Martinet
    Pages 427-441
  14. Jacques Martinet
    Pages 443-465
  15. Jacques Martinet
    Pages 467-478
  16. Jacques Martinet
    Pages 479-488
  17. Jacques Martinet
    Pages 489-495
  18. Back Matter
    Pages 497-526

About this book


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.


Euclidean lattices Symbol coding theory eutactic lattices number theory perfect lattices sphere packings

Authors and affiliations

  1. 1.Institut de MathématiquesUniversité Bordeaux 1Talence cedexFrance

Bibliographic information

  • Book Title Perfect Lattices in Euclidean Spaces
  • Authors Jacques Martinet
  • Series Title Grundlehren der mathematischen Wissenschaften
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 2003
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-540-44236-3
  • Softcover ISBN 978-3-642-07921-4
  • eBook ISBN 978-3-662-05167-2
  • Series ISSN 0072-7830
  • Edition Number 1
  • Number of Pages XXI, 526
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Geometry
    Number Theory
  • Buy this book on publisher's site


From the reviews:

"It is worth saying at the outset that Perfect lattices in Euclidean spaces is a state-of-the-art research monograph (with exercises) by one of the leading experts in this rapidly developing field … . Martinet’s book appears in the same Springer series as Conway and Sloane’s epochal Sphere packings, lattices and groups and it will be similarly appreciated by researchers in this area as a carefully written, historically aware and authoritative companion volume focusing on local methods in lattice theory." (Nick Lord, The Mathematical Gazette, Vol. 88 (512), 2004)