ABSTRACT
In his paper On a class o f Euclidean polyhedra Venkov1 studied a tiling of n-dimensional Euclidean space with translates of a convex polyhedron which are joined together at their faces so that a face of one translate is wholly covered by a face of another. And he derived the necessary and sufficient conditions for the space to be tiled in this fashion without mutual overlappings of translates, i.e., for the polyhedron to be a parallelohedron. In this paper we generalize the Venkov theorem and give even a simpler result, which is actually a corollary of a well-known theorem in topology, namely, a simply connected space (or polyhedron) has no other covering than the space itself. 2 Here we give all the necessary definitions in order to make our paper comprehensible without the need for citing Venkov in the sequel.
