6 - Regular Polytopes on Space-Forms

Summary

In the traditional theory, the topological type of a polytope or tessellation is always determined by the geometry and topology of the ambient spherical, euclidean or hyperbolic space. In this chapter, we study the quotients of the regular tessellations in these spaces in the context of spherical, euclidean or hyperbolic space-forms.

After a short introduction to space-forms in Section 6A, we prove in Section 6B that the locally spherical abstract regular polytopes are precisely the (combinatorially) regular tessellations on space-forms. Then, in Section 6C, we briefly discuss the projective regular polytopes, which are the only regular tessellations on spherical space-forms which are not spheres.

In Sections 6D–6F, this is followed by a detailed investigation of the regular toroids of rank n + 1 (≥ 4), which are the regular tessellations on topological n-tori; the regular toroids of rank 3 have already been discussed in Section 1D. In Section 6G, the tori are proved to be the only compact euclidean space-forms which admit regular tessellations. We saw in Section 1D, that there are toroidal polyhedra which are chiral. In contrast, in Section 6H, we shall show that there exist no chiral toroidal polytopes of rank at least 4.

Finally, Section 6J presents some results on regular tessellations on hyperbolic space-forms, which try to explain why so little is known about them.