An upper bound for a valence of a face in a parallelohedral tiling

TL;DR

This paper establishes an upper bound on the valence of faces in face-to-face parallelohedral tilings of Euclidean space, generalizing known bounds for Voronoi tilings without assuming affine equivalence.

Contribution

It proves a universal upper bound for face valence in parallelohedral tilings, extending known results beyond Voronoi tilings without requiring affine equivalence.

Findings

Valence of a face is at most 2^k in such tilings.

The bound applies generally, not just to Voronoi tilings.

Provides a new perspective on face adjacency in tilings.

Abstract

Consider a face-to-face parallelohedral tiling of $\mathbb R^d$ and a $(d-k)$-dimensional face $F$ of the tiling. We prove that the valence of $F$ (i.e. the number of tiles containing $F$ as a face) is not greater than $2^k$. If the tiling is affinely equivalent to a Voronoi tiling for some lattice (the so called Voronoi case), this gives a well-known upper bound for the number of vertices of a Delaunay $k$-cell. Yet we emphasize that such an affine equivalence is not assumed in the proof.