Chiral polytope

In mathematics, there are two competing definitions for a chiral polytope. One is that it is a polytope that is chiral (or "enantiomorphic"), meaning that it does not have mirror symmetry. By this definition, a polytope that lacks any symmetry at all would be an example of a chiral polytope. The other, competing definition of a chiral polytope is that it is a polytope that is as symmetric as possible without being mirror-symmetric, formalized in terms of the action of the symmetry group of the polytope on its flags. By this definition, even highly-symmetric and enantiomorphic polytopes such as the snub cube are not chiral. Much of the study of symmetric but chiral polytopes has been carried out in the framework of abstract polytopes, because of the paucity of geometric examples. Many polytopes lack mirror symmetry, and in that sense form chiral polytopes. The simplest example is a scalene triangle. It is possible for polytopes to have a high degree of symmetry, but yet to lack mirror symmetry; a simple example is the disphenoid when its faces are not congruent to an isosceles triangle; an other example is the snub cube, which is vertex-transitive and chiral in this sense. The more technical definition of a chiral polytope is a polytope that has two orbits of flags under its group of symmetries, with adjacent flags in different orbits. This implies that it must be vertex-transitive, edge-transitive, and face-transitive, as each vertex, edge, or face must be represented by flags in both orbits; however, it cannot be mirror-symmetric, as every mirror symmetry of the polytope would exchange some pair of adjacent flags. For the purposes of this definition, the symmetry group of a polytope may be defined in either of two different ways: it can refer to the symmetries of a polytope as a geometric object (in which case the polytope is called geometrically chiral) or it can refer to the symmetries of the polytope as a combinatorial structure (an abstract polytope).