Chiral polytope

This article is about polytopes with full rotational symmetry. For polytopes lacking mirror symmetry, see Chirality.

The flags of Heawood map under its automorphism group form two orbits, colored here in black and yellow.

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

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:

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

Symmetry groups

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

In the chiral case, the generators satisfy:

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

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

References

1. 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 .
2. Pellicer, Daniel (2012). "Developments and open problems on chiral polytopes". Ars Mathematica Contemporanea . 5 (2): 333–354. doi: 10.26493/1855-3974.183.8a2 .
3. Bracho, Javier; Hubard, Isabel; Pellicer, Daniel (2014), "A Finite Chiral 4-polytope in ℝ⁴", Discrete & Computational Geometry
4. Coxeter, H. S. M.; Moser, W. O. J. (1980). Generators and Relations for Discrete Groups (4th ed.). Springer-Verlag. ISBN   0-387-09212-9.
5. 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.

Further reading

• Monson, Barry; Pisanski, Tomaž; Schulte, Egon; Weiss, Asia Ivić (2007), "Semisymmetric graphs from polytopes", Journal of Combinatorial Theory , 114, doi:10.1016/j.jcta.2006.06.007