Sphere Packings, Lattices and Groups

  title={Sphere Packings, Lattices and Groups},
  author={John H. Conway and N. J. A. Sloane},
  booktitle={Grundlehren der mathematischen Wissenschaften},
The second edition of this 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… 
Packing Lines, Planes, etc.: Packings in Grassmannian Spaces
A reformulation of the problem gives a way to describe n-dimensional subspaces of m-space as points on a sphere in dimension ½(m–l)(m+2), which provides a (usually) lowerdimensional representation than the Plucker embedding and leads to a proof that many of the new packings are optimal.
Covering of Discrete Quasiperiodic Sets: Concepts and Theory
The packing of congruent, convex, impenetrable bodies in 3-space has obvious practical applications. Mathematical analysis has extended the search for optimal packings to spaces of dimension n > 3.
Lattices, packings and tilings in the plane
Preliminary note. For the arrangement of Chaps. IX and X, we have adopted the following policy concerning the interplay between dimension 2 and dimensions 3 and higher. In the present chapter - in
Unit Sphere Packings
Unit sphere packings are the classical core of (discrete) geometry. We survey old as well new results giving an overview of the art of the matters. The emphases are on the following five topics: the
Counting lattice points in pyramids
  • P. Solé
  • Mathematics, Computer Science
    Discret. Math.
  • 1995
Kissing numbers, sphere packings, and some unexpected proofs
The ikissing number problemi asks for the maximal number of white spheres that can touch a black sphere of the same size in n-dimensional space. The answers in dimensions one, two and three are
Low-dimensional lattices. IV. The mass formula
  • J. Conway, N. Sloane
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1988
The mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them. We restate the formula so as to make it easier to


From error-correcting codes through sphere packings to simple groups
1. The origin of error-correcting codes an introduction to coding the work of Hamming the Hamming-Holbrook patent the Hamming codes are linear the work of Golay the priority controversy 2. From
Twenty-three constructions for the Leech lattice
  • J. Conway, N. Sloane
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1982
In a recent paper with R. A. Parker we classified the points at maximum distance from the Leech lattice (the ‘deepest holes’ in that lattice), and showed that there are 23 classes of such holes, the
Extremal self-dual lattices exist only in dimensions 1 to 8
It is known that if A is a self-dual lattice in R", then min {u-u | u e A, u # 0} < (M/8) + 1. If equality holds the lattice is called extremal. In this paper we find all the extremal lattices: there
Kepler's Spheres and Rubik's Cube
"How many spheres of radius r may simultaneously be tangent to a fixed sphere of the same radius?" This question goes back at least to the year 1694, when it was considered by Isaac Newton and his
A bound for the covering radius of the Leech lattice
  • S. Norton
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1982
In this paper a bound is obtained for the covering radius of the Leech lattice that is close to the subsequently obtained true value, by a method which may have more general use.
Winning Ways for Your Mathematical Plays
In the quarter of a century since three mathematicians and game theorists collaborated to create Winning Ways for Your Mathematical Plays, the book has become the definitive work on the subject of
A Monster Lie Algebra
We define a remarkable Lie algebra of infinite dimension, and conjecture that it may be related to the Fischer-Griess Monster group.