Point group

In chemistry, 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. The discrete point groups in two dimensions, also called rosette groups, are used to describe the symmetries of an ornament. The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular 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.

The Bauhinia blakeana flower on the Hong Kong flag has C₅ symmetry; the star on each petal has D₅ symmetry.

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, C_n (abstract group type Z_n), consist of rotations by 360°/n, and all integer multiples. For example, a four legged chair 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, D_n (abstract group type Dih_n), 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.

C_n and D_n 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.

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.

See also

• Crystallography
• Crystallographic point group
• Molecular symmetry
• Space group
• X-ray diffraction
• Bravais lattice

External links

• Molecular symmetry examples
• Web-based point group tutorial (needs Java and Flash)
• Subgroup enumeration (needs Java)