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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]}

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]}

**^**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 ℝ^{4}",*Discrete & Computational Geometry***^**Monson, Barry (2021),*On Roli's Cube*, arXiv:2102.08796

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