The automorphism group is one of the most natural groups that acts on polytopes, since it captures its level of symmetry. The connection group is another group that acts naturally on the flags of a polytope; it can be interpreted as a recipe to recover the structure of the polytope from its flags. The chirality index is a measure of how far a chiral polytope is from being regular, and it is linked to a group called the chirality group. This chapter addresses these three groups and their interactions.

The first part of this chapter is devoted to studying three families of chiral polytopes grouped by common properties. The first family consists of chiral polyhedra, which are chiral polytopes of the lowest possible rank. The second family groups the tight chiral polytopes, which are the natural candidates to be the chiral polytopes with the smallest number of flags in each rank where they exist (3 and 4). The third family includes chiral polytopes obtained from geometric constructions, like Coxeter’s twisted honeycombs and Roli’s cube. The second part of the chapter deals with ideas to construct chiral polytopes of high ranks, a task that since the origin of the notion of abstract chiral polytope has proved to be difficult. One of these ideas is to build a polytope of rank n+1 from a set of building blocks that are isomorphic polytopes of rank n, while keeping the vertex-figures all isomorphic to some other prescribed polytope; such a polytope of rank n+1 is said to be an amalgamation of the other two. The second idea is similar, but without prescribing the isomorphism type of the vertex-figure. The polytopes of rank n+1 constructed in this way are said to be extensions of the polytope of rank n.

In this chapter the main topic of the book is put in context.

Abstract chiral polytopes are intrinsically combinatorial objects. In this chapter a geometric meaning is given to many of them. This follows the ideas of Grünbaum of skeletal polyhedra. As part of the discussion, chiral polyhedra in Euclidean three-dimensional space are described in a different way from the one in which they were originally found by Schulte. Chiral polytopes of full rank are those that attain a certain upper bound with respect to their dimensions; this is the same bound used to define regular polytopes of full rank. It is proven that chiral polytopes of full rank exist only in ranks 4 and 5. This is an unexpected contrast with regular polytopes of full rank, which exist in every rank.

The idea of this chapter is to show ways to construct polytopes from other polytopes, with emphasis on constructions yielding chiral polytopes. Some of these constructions can be performed through the connection group, and include the well-understood dual and Petrial operations. Other constructions use geometric or topological ideas, like embeddings, quotients and covers. The last construction, called the mix, uses two polytopes to obtain a third polytope that covers the other two.

The main definitions and structural results of abstract polytopes, regular polytopes and chiral polytopes are presented in this chapter. Here the reader can also find equivalent definitions of abstract polytopes and of chiral abstract polytopes, that prove useful in their study. In order to illustrate some of these topics, a summary of symmetries of toroidal polytopes is provided, with special emphasis on chiral toroidal polytopes.