On Delaunay's classification theorem on faces of parallelohedra of codimension three

TL;DR

This paper provides a combinatorial proof of Delaunay's 1929 classification theorem on the five types of face coincidences in parallelohedra of codimension three, and explores additional properties of these faces.

Contribution

It offers a new combinatorial proof of Delaunay's theorem and extends understanding of three-codimensional faces in parallelohedral tilings.

Findings

Confirmed the five types of face coincidences in codimension three

Proved additional properties of three-codimensional faces

Extended the theoretical framework of parallelohedral tilings

Abstract

In 1929 B.~N.~Delaunay proved that there are exactly 5 types of coincidence of parallelohedra at faces of codimension 3. We give a combinatorial proof of this theorem and prove several additional statements on three-codimensional faces of parallelohedral tiling. -- The original paper appeared in 2013 in MAIS (see the bibref) and was in Russian. This is the English version.