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The Complete Classification Of Five-Dimensional Dirichlet–Voronoi Polyhedra Of Translational Lattices, Mathieu Dutour Sikiric, Alexey Garber, Achill Schürmann, Clara Waldmann

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.