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

TL;DR

This paper provides a comprehensive classification of five-dimensional Dirichlet-Voronoi polyhedra and Delaunay subdivisions of translational lattices, revealing over 110,000 affine types and nearly 181,400 contraction types through computer-assisted enumeration.

Contribution

It presents the first complete classification of five-dimensional Dirichlet-Voronoi polyhedra and Delaunay subdivisions, including detailed enumeration and verification methods.

Findings

110244 affine types of Delaunay subdivisions

181394 contraction types identified

Enumeration verified by multiple independent methods

Abstract

In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. We obtain a complete list of $110244$ 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, we obtain $181394$ contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check.