The extension theorem (Q1574575)

Given a finite polyhedral complex consisting of congruent convex polytopes in spherical, euclidean or hyperbolic space, one can ask whether or not it admits an extension to a global isohedral (tile-transitive) tiling of the ambient space. The Extension Theorem describes necessary and sufficient conditions for a finite polyhedral complex to admit a unique extension to an isohedral tiling. The conditions are expressed in terms of coronas of tiles and of their symmetries. The author also discusses several applications of the Extension Theorem. In one application, the well-known characterization of parallelohedra as centrally symmetric polytopes with centrally symmetric facets and with parallelogram-shaped or centrally symmetric hexagonal belts is derived from the Extension Theorem. In a second application, the author obtains the enumeration of the convex spherical or euclidean polytopes all of whose dihedral angles are of the form \(\pi/m_{ij}\) (Coxeter polyhedra). A third application describes conditions under which a finite set of points can be extended to an orbit of a point under some crystallographic group.