Chirality (mathematics)

Chirality and symmetry group

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 ${\displaystyle v\mapsto Av+b}$ with an orthogonal matrix ${\displaystyle A}$ and a vector ${\displaystyle b}$. The determinant 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. It does not refer to any orientation concept: an isometry 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.

Chirality in two dimensions

In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. (An axis 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, a triangle is achiral if it is equilateral or isosceles, and is chiral if it is scalene.

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 a frieze group generated by a single glide reflection.

Chirality in three dimensions

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) S_n axis of symmetry is achiral. (A plane 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. A center 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 a center axis.

Knot theory

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.