Chiral polytope

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 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}.