Voronoi and Delaunay Tilings for Lattices

Summary form only given. In this paper the author considers two outstanding problems in the theory of Delaunay and Voronoi tilings for lattices. There is a new classification problem that has arisen from a new structure theorem. Sergei Ryshkov proved that: The Minkowski sum of two Voronoi polytopes is a Voronoi polytope, if and only if the corresponding Delaunay tilings are commensurate. This result raises the question of classifying all Minkowski irreduciable Voronoi poloytopes-the dual description is to determine all edge forms in Voronoi's theory of lattice types. The author describes the first steps that have been taken in this direction.