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

This paper reports on the full classification of Dirichlet–Voronoi polyhedra and Delaunay subdivisions of five‐dimensional translational lattices. A complete list is obtained of 110 244 affine types (L‐types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet–Voronoi polyhedra. Using a refinement of corresponding secondary cones, 181 394 contraction types are obtained. The paper gives details of the computer‐assisted enumeration, which was verified by three independent implementations and a topological mass formula check. The five‐dimensional Dirichlet–Voronoi polyhedra of lattices and their contraction types are classified. Computational enumeration yields 110 244 affine types and 181 394 contraction types.