On Delaunay’s Theorem Classifying Coincidences of Parallelohedra at Faces of Codimension 3

In 1929 B. N. Delaunay obtained the complete classification of all possible combinatorial coincidence types of parallelohedra at their faces of codimension 3. It appeared that every such coincidence is dual to one of the following five three-dimensional polytopes: a tetrahedron, a quadrangular pyramid, an octahedron, a triangular prism, or a parallelepiped. The present paper contains a new combinatorial proof of this result based on Euler formula. Using the classification, we have obtained several further properties of faces of codimension 3 in parallelohedral tilings. For instance, we showed that the Dimension Conjecture holds for faces of codimension 3, i.e. if we take the affine hull of centers of all parallelohedra containing a particular face of codimension 3, this affine hull is three-dimensional. Finally, we proved that the set of centers of all parallelohedra sharing a face of codimension 3 spans a three-dimensional sublattice of index one.