Point group

In mathematics, a point group is a group of geometric symmetries (isometries) leaving a point fixed.

Overview
Point groups can exist in a Euclidean space of any dimension. A discrete point group in 2D is sometimes called a rosette group, and is used to describe the symmetries of an ornament. The 3D point groups are heavily used in chemistry, especially to describe the symmetries of a molecule and of orbitals forming covalent bonds, and in this context they are also called molecular point groups.

There are infinitely many discrete point groups in each number of dimensions. However, the crystallographic restriction theorem demonstrates that only a finite number are compatible with translational symmetry. In 1D there are 2, in 2D 10, and in 3D 32 such groups, called crystallographic point groups.

In two dimensions
Point groups in 2D fall into two distinct families, according to whether they consist of rotations only, or include reflections. The cyclic groups, Cn (abstract group type Zn), consist of rotations by 360°/n, and all integer multiples. For example, a swastika has symmetry group C4, consisting of rotations by 0°, 90°, 180°, and 270°. The symmetry group of a square belongs to the family of dihedral groups, Dn (abstract group type Dihn), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S1 is distinct from Dih(S1) because the latter explicitly includes the reflections.

An infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/√2, which does not include rotation by 180°. Depending on its application, homogeneity up to an arbitrarily fine level of detail in a transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored.

Cn and Dn for n = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups.

In three dimensions
More complex symmetries arise in 3D, see point groups in three dimensions.

Generalization
In any dimension d, the continuous group of all possible fixed point isometries is the orthogonal group, denoted by O(d); and its continuous subgroup of all possible rotations is the special orthogonal group, denoted by SO(d). This is not Schönflies notation, but the conventional names from Lie group theory.