Point reflection

In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance).

[1] In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation.

Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their physical properties.

dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where

In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1.

In dimension n, point reflections are orientation-preserving if n is even, and orientation-reversing if n is odd.

Given a vector a in the Euclidean space Rn, the formula for the reflection of a across the point p is In the case where p is the origin, point reflection is simply the negation of the vector a.

This mapping is an isometric involutive affine transformation which has exactly one fixed point, which is P. When the inversion point P coincides with the origin, point reflection is equivalent to a special case of uniform scaling: uniform scaling with scale factor equal to −1.

When P does not coincide with the origin, point reflection is equivalent to a special case of homothetic transformation: homothety with homothetic center coinciding with P, and scale factor −1.

The set consisting of all point reflections and translations is Lie subgroup of the Euclidean group.

It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise.

(see the paragraph below) In even-dimensional Euclidean space, say 2N-dimensional space, the inversion in a point P is equivalent to N rotations over angles π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P. These rotations are mutually commutative.

[2] Some molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry.

In many cases they can be considered as polyhedra, categorized by their coordination number and bond angles.

For example, four-coordinate polyhedra are classified as tetrahedra, while five-coordinate environments can be square pyramidal or trigonal bipyramidal depending on the bonding angles.

Tetrahedra, on the other hand, are non-centrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron.

Polyhedra with an odd (versus even) coordination number are not centrosymmtric.

[5] The later is named after C. V. Raman who was awarded the 1930 Nobel Prize in Physics for his discovery.

[6] In addition, in crystallography, the presence of inversion centers for periodic structures distinguishes between centrosymmetric and non-centrosymmetric compounds.

In other cases such as for metals and alloys the structures are better considered as arrangements of close-packed atoms.

[7] Real polyhedra in crystals often lack the uniformity anticipated in their bonding geometry.

Common irregularities found in crystallography include distortions and disorder.

Distortion involves the warping of polyhedra due to nonuniform bonding lengths, often due to differing electrostatic interactions between heteroatoms or electronic effects such as Jahn–Teller distortions.

Disorder can influence the centrosymmetry of certain polyhedra as well, depending on whether or not the occupancy is split over an already-present inversion center.

Crystals are classified into thirty-two crystallographic point groups which describe how the different polyhedra arrange themselves in space in the bulk structure.

The presence of noncentrosymmetric polyhedra does not guarantee that the point group will be the same—two non-centrosymmetric shapes can be oriented in space in a manner which contains an inversion center between the two.

Two tetrahedra facing each other can have an inversion center in the middle, because the orientation allows for each atom to have a reflected pair.

The inverse is also true, as multiple centrosymmetric polyhedra can be arranged to form a noncentrosymmetric point group.

Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by

In SO(2r), reflection through the origin is the farthest point from the identity element with respect to the usual metric.