Higher-Dimensional Cases

Just as the exceptional regular polytopes of full rank are only of dimension at most four, so the exceptions of nearly full rank have dimension at most eight. The remaining regular polytopes and apeirotopes of nearly full rank are treated in this chapter, which completes their classification. The ‘gateway’ dimension five is crucial to the investigation, since there is a severe restriction on the possible symmetry groups, and hence on the corresponding (finite) regular polytopes. This dimension is first looked at only in general terms, since the polytopes not previously described fall naturally into families that are considered in later sections. However, one case is dealt with in full detail: there is a sole regular polytope in five dimensions (and none in higher dimensions) whose symmetry group consists only of rotations. The new families of regular polytopes of nearly full rank are closely related to the Gosset–Elte polytopes, so these are briefly described here. There are three families, which are dealt with in turn; however, a fourth putative family is shown to degenerate.