Chirality

A shape is said to be chiral if it can't be transformed into its mirror image by rotations alone. All chiral shapes have a left-handed and right-handed version, which are also called enantiomorphs. If a shape has identical enantiomorphs, then it is said to be achiral. One well-known example of a chiral shape is the snub cube.

More specifically, a chiral shape cannot be transformed into its mirror image using only rotations and translations alone. In fact, for an n-dimensional shape, at least n+1 dimensions are required in order to do so.

A shape is chiral if its maximal symmetry belongs to a chiral symmetry group. All shapes contain a chiral subsymmetry; for example, the cube, which has BC3 symmetry, contains the chiral subsymmetry BC3+, the symmetry of the snub cube. This is because it contains the identity group I, which has no symmetry and is therefore found in all shapes.

Symmetry groups
A symmetry group can also be defined by chirality. In two dimensions, a symmetry group is chiral if it does not contain the mirror symmetry A1×I. This includes polygons with only central inversion symmetry, such as the parallelogram. However, there do exist unbounded shapes that are achiral yet have no mirror symmetry, because it contains a glide reflection, a property closely related to rotoreflection.

In three dimensions, however, chiral symmetry groups do not include those containing ±(I×I×I) (central inversion symmetry). This is because it contains a two-fold rotoreflection, which makes symmetry groups such as (G2+×A1)/2 achiral. Consequently, chiral three-dimensional shapes, in addition to having different mirror images, have no central inversion symmetry

Symmetry groups defined by swirl products, which in turn form swirl symmetries, are always chiral. While most finite chiral symmetry groups contain a finite achiral supersymmetry, this is not true in general for swirl symmetries.

It is conjectured that in four dimensions, and every even dimension in general, symmetry groups can be both chiral and contain central inversion symmetry at the same time, which is clearly seen in swirlchora such as the small swirlprism, while symmetry groups in odd dimensions behave like the three-dimensional ones.