Crystallographic point group

In crystallography, a crystallographic point group is a point group whose symmetry operations are compatible with the translational symmetry of crystallographic lattices. According to the crystallographic restriction theorem, point groups may only contain one-, two-, three-, four- and sixfold rotational symmetries. This reduces the number of crystallographic point groups to 10 in 2-dimensional space and 32 in 3-dimensional space.

In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations, that is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the (geometric) crystal class of the crystal.

The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro-optical features such as the Pockels effect.