Integral Lattice

a set of points in the plane or in space whose coordinates in some rectilinear coordinate system are integers.

Integral lattices play an important role in various problems of crystallography, the theory of functions, and number theory. For example, the problem of the classification of crystal systems is connected with the study of the symmetry of integral lattices. In the theory of functions of a complex variable the set of periods of doubly periodic functions forms an integral lattice.

The systematic use of integral lattices in number theory was begun by K. Gauss and led to the creation by H. Minkowski of the geometry of numbers, in which many problems connected, for example, with quadratic forms and the approximation of irrational numbers with rational numbers, are solved on the basis of geometric considerations. The geometry of numbers was developed further by such Russian mathematicians as G. F. Voronoi and B. N. Delone. Delone also contributed papers on the application of integral lattices to crystallography.