A Note on Space Tiling Zonotopes

TL;DR

This paper proves Voronoi's conjecture for space tiling zonotopes using oriented matroids, providing a new, more geometric proof and clarifying the relationship between combinatorial and metrical properties.

Contribution

It extends McMullen's conditions for space tiling zonotopes and offers a new proof of Voronoi's conjecture for zonotopes via oriented matroid theory.

Findings

Proves Voronoi's conjecture for zonotopes using oriented matroids.

Provides a new geometric proof of a theorem by Brylawski and Lucas.

Clarifies the distinction between combinatorial and metrical properties in tiling zonotopes.

Abstract

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.