Chiral polytope

In the study of abstract polytopes, a chiral polytope 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.

Definition
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 (the automorphisms of an abstract polytope). Chirality is meaningful for either type of symmetry but the two definitions classify different polytopes as being chiral or nonchiral.

Geometrically chiral polytopes
Geometrically chiral polytopes are relatively exotic compared to the more ordinary regular polytopes. It is not possible for a geometrically chiral polytope to be convex, and many geometrically chiral polytopes of note are skew.

In three dimensions
In three dimensions, it is not possible for a geometrically chiral polytope to have finitely many finite faces. For instance, the snub cube is vertex-transitive, but its flags have more than two orbits, and it is neither edge-transitive nor face-transitive, so it is not symmetric enough to meet the formal definition of chirality. The quasiregular polyhedra and their duals, such as the cuboctahedron and the rhombic dodecahedron, provide another interesting type of near-miss: they have two orbits of flags, but are mirror-symmetric, and not every adjacent pair of flags belongs to different orbits. However, despite the nonexistence of finite chiral three-dimensional polyhedra, there exist infinite three-dimensional chiral skew polyhedra of types {4,6}, {6,4}, and {6,6}.

In four dimensions
In four dimensions, there are a geometrically chiral finite polytopes. One example is Roli's cube, a skew polytope on the skeleton of the 4-cube.