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

The technical definition of a chiral polytope is a polytope that has exactly two orbits of flags under its group of symmetries, such that adjacent flags always lie in different orbits.

If $\Phi$ represents the set of all flags of the polytope ${\mathcal {P}}$ , and $\Gamma ({\mathcal {P}})$ is the symmetry group, the chirality condition is expressed as: $|\Phi /\Gamma ({\mathcal {P}})|=2$

This implies that the polytope is vertex-transitive, edge-transitive, and face-transitive, as each element must be represented by flags in both orbits. However, it cannot be mirror-symmetric, as any reflection would necessarily map a flag to an adjacent flag, thereby collapsing the two orbits into one.^[1]

Geometrically chiral polytopes

Geometrically chiral polytopes are exotic structures that cannot be convex.^[2] Many geometrically chiral polytopes are skew, meaning their vertices do not all lie in a single hyperplane.

In three dimensions

In Euclidean 3-space, there are no finite chiral polyhedra. While the snub cube is vertex-transitive and lacks mirror symmetry, it is not a chiral polytope because its flags form more than two orbits. However, there exist three types of infinite chiral skew polyhedra:

$\{4,6\}$ (quadrilateral faces, six around each vertex)
$\{6,4\}$ (hexagonal faces, four around each vertex)
$\{6,6\}$ (hexagonal faces, six around each vertex)

In four dimensions

In four dimensions, finite geometrically chiral polytopes do exist. A prominent example is Roli's cube, a skew polytope constructed on the skeleton of the 4-cube.^[3]

Combinatorial chirality and maps

Combinatorial chirality refers to the properties of an abstract polytope regardless of its geometric realization. Many chiral abstract polytopes are realized as maps on surfaces.

For a map of type $\{p,q\}$ on a surface, the map is chiral if its automorphism group ${\text{Aut}}({\mathcal {M}})$ has two flag orbits. On a torus, chiral maps are often denoted using the notation $\{4,4\}_{b,c}$ or $\{3,6\}_{b,c}$ . These maps are chiral if and only if $bc(b-c)\neq 0$ .^[4]

Symmetry groups

The symmetry group $\Gamma ({\mathcal {P}})$ of a chiral polytope of rank $n$ is generated by elements $\sigma _{1},\sigma _{2},\dots ,\sigma _{n-1}$ . In a regular polytope, these would be defined by reflections $\rho _{i}$ such that $\sigma _{i}=\rho _{i-1}\rho _{i}$ .

In the chiral case, the generators satisfy:

$\sigma _{i}^{p_{i}}=1$
$(\sigma _{i}\sigma _{i+1}\dots \sigma _{j})^{2}=1$ for $1\leq i<j\leq n-1$

Crucially, there is no automorphism $\phi$ such that $\phi ^{2}=1$ that acts as a reflection on the flags, meaning the full Coxeter group is not realized.^[5]

References

^ 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.
^ 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
^ Coxeter, H. S. M.; Moser, W. O. J. (1980). Generators and Relations for Discrete Groups (4th ed.). Springer-Verlag. ISBN 0-387-09212-9.
^ 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.

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