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.

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.^[1]

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.^[2]

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,^[3] and many geometrically chiral polytopes of note are skew.

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}.^[2]

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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.^[4][5]

• Monson, Barry; Pisanski, Tomaž; Schulte, Egon; Weiss, Asia Ivić (2007), "Semisymmetric graphs from polytopes", Journal of Combinatorial Theory, Series A, 114 (3): 421–435, arXiv:math/0606469, doi:10.1016/j.jcta.2006.06.007, MR 2310743, S2CID 10203794.
• Hubard, Isabel; Weiss, Asia Ivić (2005), "Self-duality of chiral polytopes", Journal of Combinatorial Theory, Series A, 111 (1): 128–136, doi:10.1016/j.jcta.2004.11.012, MR 2144859.
• Conder, Marston; Hubard, Isabel; Pisanski, Tomaž (2008), "Constructions for chiral polytopes", Journal of the London Mathematical Society, Second Series, 77 (1): 115–129, doi:10.1112/jlms/jdm093, MR 2389920, S2CID 14658184.
• Monson, Barry; Ivić Weiss, Asia (2008), "Cayley graphs and symmetric 4-polytopes", Ars Mathematica Contemporanea, 1 (2): 185–205, doi:10.26493/1855-3974.79.919, MR 2466196.