## Sphere Packings, Lattices and Groups by J. H. Conway, N. J. A. Sloane (auth.)

By J. H. Conway, N. J. A. Sloane (auth.)

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Represent schools, the Voronoi cells are the school districts! Other names are nearest neighbor regions. Dirichlet regions. Brillouin zones and Wigner-Seitz cells (the last two are physicists' terms). The Voronoi cells of the hexagonal lattice, for example, are the regular hexagons shown in Fig. 3c. The Voronoi cells of many other lattices are described in Chaps. 4 and 21. The interiors of the Voronoi cells are disjoint, although they have faces in common. Each face lies in the hyperplane midway between two neighboring points Pi' The Voronoi cells are convex polytopes whose union is the whole of Rn.

2b) , one of the Archimedean polyhedra [Cunl, p. 981, [Fej91, [Holll, [Loel, p. 1291, [WeI4, p. 731, [Wenl, p. 211. 5456 ... 7043 .... Thus although the fcc lattice is the better packing, the bcc lattice is indeed a better covering. There is another difference between these two lattices. In the bcc lattice, as in the planar hexagonal lattice, there is only one kind of hole (all holes are deep), but in the fcc lattice there are two kinds (shallow and deep holes). See Fig. 2 and also Chap. 7. This phenomenon is particularly striking in the Leech lattice, where there are 23 kinds of deep hole and 284 kinds of shallow hole (Chaps .

Urn), v = (VI . vrn), their inner u I VI + ... } The determinant of A is then the determinant det A = det A. for the lattice. The (Given two vectors or scalar product either by u . 2. The plane divided into fundamental parallelotopes of a 2dimensional lattice. If M is a square matrix this reads det A = (det M)2. (5) Consider for example the familiar planar hexagonal lattice shown in Fig. 3a. An obvious generator matrix is M = [1~2 1/2~) ' (6) for which the Gram matrix is A = MMtr = and det A = det A generator matrix = 3/4.