Chirality (mathematics)

In mathematics, chirality (from the Greek kheir, "hand") describes a property of a figure or object that is not superimposable on its mirror image. Such an object and its mirror image are said to be enantiomorphs, and chirality is also called handedness.

More formally, an object is chiral if it cannot be mapped onto its mirror image by any combination of rotations and translations. This distinguishes chirality from symmetry under reflection (also called mirror symmetry), which requires the object to be identical to its reflection.

The concept of chirality is fundamental in geometry, topology, and knot theory. For instance, certain knots cannot be continuously deformed into their mirror images, making them chiral. Similarly, some polyhedra exist in chiral forms.

It is crucial to note that chirality is not the same as asymmetry. An object is asymmetric if it lacks any symmetry. A chiral object, however, may possess some symmetry operations, but never mirror symmetry. Therefore, chirality is a special type of asymmetry. A symmetrical object may possess a plane of symmetry, a center of symmetry, or an n-fold axis of rotational symmetry, and thus is achiral.

Whether an object is chiral or achiral is invariant under rotation, translation, and scaling. Thus, chirality is a geometric property.