Cubic Tessellations of the Helicosms

TL;DR

This paper completes the classification of cubic tessellations on six orientable Euclidean 3-manifolds generated by fixed-point free crystallographic groups, extending previous classifications to the remaining four manifolds.

Contribution

It provides a complete classification of cubic tessellations on all six orientable Euclidean 3-manifolds generated by screw motions, filling gaps in prior work.

Findings

Classified cubic tessellations on four remaining orientable manifolds.

Extended previous classifications from the 3-torus and didicosm.

Provided a comprehensive understanding of tessellations in these manifolds.

Abstract

Up to isomorphism there are six fixed-point free crystallographic groups in Euclidean Space generated by twists (screw motions). In each case, an orientable 3-manifold is obtained as the quotient of E3 by such a group. The cubic tessellation of E3 induces tessellations on each such manifold. These tessellations of the 3-torus and the didicosm were classified as `equivelar toroids' and `cubic tessellations of the didicosm' in previous works. This paper concludes the classification of cubic tessellations on the remaining four orientable manifolds.