Cubic Tessellations of the Helicosms

Up to isomorphism, there are six fixed-point free crystallographic groups in Euclidean 3-space E 3 generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of E 3 by such a group. The cubic tessellation of E 3 induces tessellations on each such manifold. The corresponding classification for the 3-torus and the didicosm were classified as ‘equivelar toroids’ and ‘cubic tessellation of the didicosm’ in previous works. This paper concludes the classification of cubic tessellations on the remaining four orientable manifolds.