THE EXTENSION THEOREMNikolai

To Ludwig Danzer on the occasion of his 70th birthday Abstract Given a compact convex polyhedron, can it tile space in a transitive (or in a regular) way? We discuss in the paper the so-called extension theorem giving conditions under which there is unique extension of a nite polyhedral complex, which consists of replicas of the given polyhedron, to a global isohedral tiling. The extension theorem gives a way to get all possible regular tilings with the given polyhedron. The well-known results on fundamental domains in the case of a translational group or of a Coxeter group generated by mirrors follow from the extension theorem too. The extension theorem gives a method of describing which nite point sets can admit extension to a regular point orbits with respect to crystallographic groups.