Ingeometry, a figure ischiral (and said to havechirality) if it is not identical to itsmirror image, or, more precisely, if it cannot be mapped to its mirror image byrotations andtranslations alone. An object that is not chiral is said to beachiral.

A chiral object and its mirror image are said to beenantiomorphs. The wordchirality is derived from the Greekχείρ (cheir), the hand, the most familiar chiral object; the wordenantiomorph stems from the Greekἐναντίος (enantios) 'opposite' +μορφή (morphe) 'form'.

Thetetrominos S and Z are enantiomorphs in 2-dimensions

S	Z

Some chiral three-dimensional objects, such as thehelix, can be assigned a right or lefthandedness, according to theright-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 theminside-out.^[1]

The J-, L-, S- and Z-shapedtetrominoes of the popular video gameTetris 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 itssymmetry group contains at least oneorientation-reversing isometry. In Euclidean geometry anyisometry can be written as${\displaystyle v\mapsto Av+b}$ with anorthogonal matrix${\displaystyle A}$ and a vector${\displaystyle b}$. Thedeterminant of${\displaystyle A}$ is either 1 or −1 then. If it is −1 the isometry is orientation-reversing, otherwise it is orientation-preserving.

A general definition of chirality based on group theory exists.^[2] It does not refer to any orientation concept: anisometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime.^[3][4]

In two dimensions, every figure which possesses anaxis of symmetry is achiral, and it can be shown that everybounded achiral figure must have an axis of symmetry. (Anaxis of symmetry of a figure${\displaystyle F}$ is a line${\displaystyle L}$, such that${\displaystyle F}$ is invariant under the mapping${\displaystyle (x,y)\mapsto (x,-y)}$, when${\displaystyle L}$ is chosen to be the${\displaystyle x}$-axis of the coordinate system.) For that reason, atriangle is achiral if it isequilateral orisosceles, and is chiral if it isscalene.

Consider the following pattern:

This figure is chiral, as it is not identical to its mirror image:

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 afrieze group generated by a singleglide reflection.

In three dimensions, every figure that possesses amirror plane of symmetryS₁, an inversioncenter of symmetryS₂, or a higherimproper rotation (rotoreflection)S_n axis of symmetry^[5] is achiral. (Aplane of symmetry of a figure${\displaystyle F}$ is a plane${\displaystyle P}$, such that${\displaystyle F}$ is invariant under the mapping${\displaystyle (x,y,z)\mapsto (x,y,-z)}$, when${\displaystyle P}$ is chosen to be the${\displaystyle x}$-${\displaystyle y}$-plane of the coordinate system. Acenter of symmetry of a figure${\displaystyle F}$ is a point${\displaystyle C}$, such that${\displaystyle F}$ is invariant under the mapping${\displaystyle (x,y,z)\mapsto (-x,-y,-z)}$, when${\displaystyle C}$ 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

${\displaystyle F_0=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,-2,-1)\right\}}$

which is invariant under the orientation reversing isometry${\displaystyle (x,y,z)\mapsto (-y,x,-z)}$ and thus achiral, but it has neither plane nor center of symmetry. The figure

${\displaystyle F_1=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}}$

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

Achiral figures can have acenter axis.

Aknot is calledachiral if it can be continuously deformed into its mirror image, otherwise it is called achiral knot. For example, theunknot and thefigure-eight knot are achiral, whereas thetrefoil knot is chiral.

• Chiral polytope
• Chirality (physics)
• Parity (physics)
• Chirality (chemistry)
• Asymmetry
• Skewness
• Vertex algebra

• Flapan, Erica (2000).When Topology Meets Chemistry. Outlook. Cambridge University Press and Mathematical Association of America.ISBN 0-521-66254-0.

• Symmetry, Chirality, Symmetry Measures and Chirality Measures: General Definitions
• Chiral Polyhedra byEric W. Weisstein,The Wolfram Demonstrations Project.
• Chiral manifold at the Manifold Atlas.

• Knot theory
• Polyhedra
• Chirality
• Topology

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