2004/02/04 by Frank Vallentin, Vallentin, Frank
Computer Science · Engineering · Materials Science · Mathematics · #Advanced Materials and Mechanics #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #math.CO #math.MG
paper · pdf · doi:10.48550/arxiv.math/0402053
7 pages, 1 figure
arxiv created 2004/02/04 · openalex publication_date 2004/02/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope to be a space tiling zonotope, i.e. a zonotope which admits a face-to-face tiling of space by translations. Implicitly, he related space tiling zonotopes to a special class of oriented matroids (regular matroids). We will extend his result to give a new proof of Voronoi's conjecture for zonotopes using oriented matroids. This enables us to distinguish between combinatorial and metrical properties and to apply the fact that oriented matroids considered here have an essentially unique realization. Originally, this is a theorem due to Brylawski and Lucas. By using oriented matroid duality we interpret a part of McMullen's arguments as an elegant geometric proof of this theorem in the special case of real numbers.