In the previous chapter, we extensively investigated the locally toroidal regular polytopes of types {4, 4, 3} and {4, 4, 4}, and obtained a nearly complete enumeration of those universal regular polytopes which are finite. In this chapter, we move on to the remaining types of locally toroidal 4-polytopes, namely, {6, 3, p} with p = 3, 4, 5, 6, and {3, 6, 3}, and carry out a similar classification. For the types {6, 3, p} our enumeration is complete, and for {3, 6, 3} substantial partial results are known.

It is striking that complex hermitian forms now enter the scene. Indeed, the structure of the universal polytopes is frequently governed by a complex hermitian form. We shall explain our basic enumeration technique in Section 11A, and then, when it applies, enumerate many finite universal polytopes in Sections 11B to 11E by appealing to the results of Chapter 9. In particular, we shall see that such a polytope is finite if and only if the corresponding hermitian form is positive definite. The automorphism group of a finite polytope is then a semi-direct product of a finite unitary reflexion group by a small finite group.

This link between polytopes and hermitian forms generalizes the classical situation, where the structure of a regular tessellation is determined by a real quadratic form which defines the geometry of the ambient space.