Chirality (mathematics)

The footprint here demonstrates chirality. Individual left and right footprints are chiral enantiomorphs in a plane because they are mirror images while containing no mirror symmetry individually.

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.

A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'.

Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule.

Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves and shoes. Right shoes differ from left shoes only by being mirror images of each other. In contrast thin gloves may not be considered chiral if you can wear them inside-out.[citation needed]

The J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Individually they contain no mirror symmetry in the plane.

A general definition of chirality based on group theory exists.[1] It does not refer to any orientation concept: an isometry is direct if and only if is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.[2]

also is achiral as the origin is a center of symmetry, but it lacks a plane of symmetry.

But if one prolongs the pattern in both directions to infinity, one receives an (unbounded) achiral figure which has no axis of symmetry. Its symmetry group is a frieze group generated by a single glide reflection.

A knot is called achiral if it can be continuously deformed into its mirror image, otherwise it is called a chiral knot. For example, the unknot and the figure-eight knot are achiral, whereas the trefoil knot is chiral.