) is a property of asymmetry important in several branches of science. The word chirality is derived from the Greek χείρ (kheir), "hand", a familiar chiral object.

An object or a system is chiral if it is distinguishable from its mirror image; that is, it cannot be superposed (not to be confused with superimposed) onto it. Conversely, a mirror image of an achiral object, such as a sphere, cannot be distinguished from the object. A chiral object and its mirror image are called enantiomorphs (Greek, "opposite forms") or, when referring to molecules, enantiomers. A non-chiral object is called achiral (sometimes also amphichiral) and can be superposed on its mirror image.

The term was first used by Lord Kelvin in 1893 in the second Robert Boyle Lecture at the Oxford University Junior Scientific Club which was published in 1894:

I call any geometrical figure, or group of points, 'chiral', and say that it has chirality if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself.[1]

Human hands are perhaps the most recognized example of chirality. The left hand is a non-superposable mirror image of the right hand; no matter how the two hands are oriented, it is impossible for all the major features of both hands to coincide across all axes.[2] This difference in symmetry becomes obvious if someone attempts to shake the right hand of a person using their left hand, or if a left-handed glove is placed on a right hand. In mathematics, chirality is the property of a figure that is not identical to its mirror image.

Mathematics[edit]

An achiral 3D object without central symmetry or a plane of symmetry

A table of all prime knots with seven crossings or fewer (not including mirror images).

Main article: Chirality (mathematics)

In mathematics, a figure is chiral (and said to have chirality) if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from anticlockwise. See[3] for a full mathematical definition.

A chiral object and its mirror image are said to be enantiomorphs. The word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'. A non-chiral figure is called achiral or amphichiral.

The helix (and by extension a spun string, a screw, a propeller, etc.) and Möbius strip are chiral two-dimensional objects in three-dimensional ambient space. The J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space.

Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves, glasses (sometimes), and shoes. A similar notion of chirality is considered in knot theory, as explained below.

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

Geometry[edit]

In geometry, a figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. In two dimensions, every figure that possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. In three dimensions, every figure that possesses a plane of symmetry or a center of symmetry is achiral. There are, however, achiral figures lacking both plane and center of symmetry. In terms of point groups, all chiral figures lack an improper axis of rotation (Sn). This means that they cannot contain a center of inversion (i) or a mirror plane (σ). Only figures with a point group designation of C1, Cn, Dn, T, O, or I can be chiral.

Knot theory[edit]

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

Physics[edit]

Animation of right-handed (clockwise) circularly polarized light, as defined from the point of view of a receiver in agreement with optics conventions.

Main article: Chirality (physics)

In physics, chirality may be found in the spin of a particle, where the handedness of the object is determined by the direction in which the particle spins.[4] Not to be confused with helicity, which is the projection of the spin along the linear momentum of a subatomic particle, chirality is an intrinsic quantum mechanical property, like spin. Although both chirality and helicity can have left-handed or right-handed properties, only in the massless case are they identical.[5] In particular for a massless particle the helicity is the same as the chirality while for an antiparticle they have opposite sign.

The handedness in both chirality and helicity relate to the rotation of a particle while it proceeds in linear motion with reference to the human hands. The thumb of the hand points towards the direction of linear motion whilst the fingers curl into the palm, representing the direction of rotation of the particle (i.e. clockwise and counterclockwise). Depending on the linear and rotational motion, the particle can either be defined by left-handedness or right-handedness.[5] A symmetry transformation between the two is called parity. Invariance under parity by a Dirac fermion is called chiral symmetry.

Electromagnetism[edit]

Main article: Chirality (electromagnetism)

Electromagnetic waves can have handedness associated with their polarization. Polarization of an electromagnetic wave is the property that describes the orientation, i.e., the time-varying direction and amplitude, of the electric field vector. For example, the electric field vectors of left-handed or right-handed circularly polarized waves form helices of opposite handedness in space.

Circularly polarized waves of opposite handedness propagate through chiral media at different speeds (circular birefringence) and with different losses (circular dichroism). Both phenomena are jointly known as optical activity. Circular birefringence causes rotation of the polarization state of electromagnetic waves in chiral media and can cause a negative index of refraction for waves of one handedness when the effect is sufficiently large.[6][7]