Chirality (mathematics)

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. In 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane. 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. In 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane. 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'. A non-chiral figure is called achiral or amphichiral. 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. 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 figure is achiral if and only if its symmetry group contains at least one orientation-reversing isometry. (In Euclidean geometry any isometry can be written as v ↦ A v + b {displaystyle vmapsto Av+b} with an orthogonal matrix A {displaystyle A} and a vector b {displaystyle b} . The determinant of A {displaystyle A} is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.) See for a full mathematical definition of chirality. In three dimensions, every figure that possesses a mirror plane of symmetry S1, an inversion center of symmetry S2, or a higher improper rotation (rotoreflection) Sn axis of symmetry is achiral. (A plane of symmetry of a figure F {displaystyle F} is a plane P {displaystyle P} , such that F {displaystyle F} is invariant under the mapping ( x , y , z ) ↦ ( x , y , − z ) {displaystyle (x,y,z)mapsto (x,y,-z)} , when P {displaystyle P} is chosen to be the x {displaystyle x} - y {displaystyle y} -plane of the coordinate system. A center of symmetry of a figure F {displaystyle F} is a point C {displaystyle C} , such that F {displaystyle F} is invariant under the mapping ( x , y , z ) ↦ ( − x , − y , − z ) {displaystyle (x,y,z)mapsto (-x,-y,-z)} , when C {displaystyle C} is chosen to be the origin of the coordinate system.) Note, however, that there are achiral figures lacking both plane and center of symmetry. An example is the figure which is invariant under the orientation reversing isometry ( x , y , z ) ↦ ( − y , x , − z ) {displaystyle (x,y,z)mapsto (-y,x,-z)} and thus achiral, but it has neither plane nor center of symmetry. The figure