An important problem in classical geometry is the complete description of all regular polytopes and tessellations in spherical, euclidean or hyperbolic space. When asked within the theory of abstract regular polytopes, such an enumeration problem must necessarily take a different form, because an abstract polytope is not a priori embedded into the geometry of an ambient space. One appropriate substitute now calls for the classification of abstract regular polytopes by their “local” or “global” topological type.

The traditional theory of regular polytopes and tessellations is concerned with, and solves, the case where the topology is spherical. This is well known. In Chapter 6, we already moved on to other topological types and enumerated the regular toroids, which are the regular polytopes whose global topology is toroidal. Now, in this and the next two chapters, we investigate the locally toroidal regular polytopes. As we explain in Section 10A, such polytopes can only exist in ranks 4, 5 or 6.

In this and the next chapter, we treat the polytopes of rank 4, and obtain a nearly complete classification of the universal regular polytopes which are locally toroidal and finite. Then, in Chapter 12, we enumerate all such polytopes of rank 5, and produce a list of such polytopes of rank 6 which we strongly conjecture to be complete.