The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices

In this short note we report on the full classification of Dirichlet-Voronoi polyhedra and Delone subdivisions of five-dimensional translational lattices. We obtain a complete list of $110244$ affine types (L-types) of Delone subdivisions and it turns out that they are all combinatorially inequivalent. Therefore we find the same number of combinatorial types of Dirichlet-Voronoi polyhedra of translational lattices.