14 - Gosset–Elte Polytopes

Summary

As already observed, the Gosset–Elte polytopes play an important role in the theory of regular polytopes of nearly full rank; this chapter collects some more facts about them. In particular, their realization domains are of interest, since they provide good examples of how the general theory of realizations expounded in Chapters 3 and 4 works. In addition, some simple projections of the Gosset–Elte polytopes into the plane can reveal a lot about their structure. The purpose of these projections is not to display the large amount of their symmetry, but rather to illustrate suitable sections, to show how components of the polytopes fit together. After a brief discussion of the Gosset–Elte polytopes in general terms, with two exceptions they and their realization domains are described. The exceptions have too many vertices to be amenable to our treatment, but in any case they do not underlie regular polytopes of nearly full rank. Two of the cases that are treated also have many vertices; both pose considerable problems.