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
[edit]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
[edit]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
[edit]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]
In four dimensions
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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]
References
[edit]- ^ Schulte, Egon; Weiss, Asia Ivić (1991), "Chiral polytopes", in Gritzmann, P.; Sturmfels, B. (eds.), Applied Geometry and Discrete Mathematics (The Victor Klee Festschrift), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, Providence, RI: American Mathematical Society, pp. 493–516, MR 1116373.
- ^ a b Schulte, Egon (2004), "Chiral polyhedra in ordinary space. I", Discrete and Computational Geometry, 32 (1): 55–99, doi:10.1007/s00454-004-0843-x, MR 2060817, S2CID 13098983.
- ^ Pellicer, Daniel (2012). "Developments and open problems on chiral polytopes". Ars Mathematica Contemporanea. 5 (2): 333–354. doi:10.26493/1855-3974.183.8a2.
- ^ Bracho, Javier; Hubard, Isabel; Pellicer, Daniel (2014), "A Finite Chiral 4-polytope in ℝ4", Discrete & Computational Geometry
- ^ Monson, Barry (2021), On Roli's Cube, arXiv:2102.08796
Further reading
[edit]- 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.