The coincidence problem for shifted lattices and crystallographic point packings

Abstract

A coincidence site lattice is a sublattice formed by the intersection of a lattice Γ in {\bb R}^d with the image of Γ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of Γ. The more general case allowing any affine isometry is considered here. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, the shifted square lattice and the diamond packing are discussed in detail.