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
symmetry group, 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.
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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
. It is not possible for a geometrically chiral polytope to be convex,
and many geometrically chiral polytopes of note are
skew polytope.
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 geometrically chiral finite polytopes. One example is Roli's cube, a
skew polytope on the skeleton of the 4-cube.
Further reading
