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

Φ

{\displaystyle \Phi }

represents the set of all flags of the polytope

P

{\displaystyle {\mathcal {P}}}

, and

Γ

(

P

)

{\displaystyle \Gamma ({\mathcal {P}})}

is the symmetry group, the chirality condition is expressed as:

|

Φ

/

Γ

(

P

)

|

=

2

{\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.

Geometrically chiral polytopes

Geometrically chiral polytopes are exotic structures that cannot be convex. 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

}

{\displaystyle \{4,6\}}

(quadrilateral faces, six around each vertex)

{

6

,

4

}

{\displaystyle \{6,4\}}

(hexagonal faces, four around each vertex)

{

6

,

6

}

{\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.