Just as in the previous two chapters we almost completed the classification of the locally toroidal regular polytopes of rank 4, we shall now give an almost complete description of those regular polytopes of higher rank whose vertex-figures and facets are either spherical or toroidal, with at least one of each. We shall briefly refer to these as higher toroidal polytopes.

As a necessary preliminary, in Section 12A we look at certain of the regular hyperbolic honeycombs, and the relationships among them. We then consider the higher toroidal polytopes of rank 5 in Section 12B, and those of rank 6 in Sections 12C–12E.

The techniques which we bring to bear on these classification problems are various. On occasions, a more or less direct geometric construction will suffice to settle a problem. More frequently, though, we shall rely heavily on twisting methods; we shall thus make many references to Chapter 8. A stock in trade is then to pass from a twisted group to a suitable quotient group, which will be that in which we are ultimately interested.

Hyperbolic Honeycombs in ℍ4 and ℍ5

In Sections 6D and 6E we have already described the regular toroids which are the potential candidates for facets or vertex-figures of higher toroidal polytopes.