As mentioned in the Preface, the lattice-point problems central to the geometry of numbers have extensive connections to modern mathematics and its applications. They arise in the theories of finite groups, of quadratic forms, of combinatorics, and of numerical methods for evaluating n-dimensional integrals. They appear in chemistry and physics, in crystallography in particular, and in the design of codes for transmitting, storing, and receiving data. In this appendix, we give a brief introduction to sphere packing, which, in turn, is critical to the development of error-detecting and error-correcting codes. In fact, finding dense packings of spheres into a given space is a problem equivalent to that of finding efficient error-correcting codes. Though our discussion here is brief, no introduction to the geometry of numbers could be complete without giving the reader at least a glimpse into this important and timely application.

Lattice-Point Packing

Let K be a convex set, or body, symmetrically placed about the origin, O. Suppose that we have an admissible lattice for K; that is, one that has no lattice point inside K other than the origin. If K is shrunk to half its linear dimensions, to ½K, and if this body is then translated to have its center at each lattice point, the resulting bodies will not overlap. Conversely, if the lattice had a point other than O inside K, the resulting bodies would overlap. Thus, an admissible lattice for K means precisely a lattice that provides a non-overlapping packing for the convex body ½K.