In crystallography, a crystallographic point group is a three-dimensional point group whose symmetry operations are compatible with the translational symmetry of three-dimensional crystallographic lattices. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions (Note that inversion centers and mirror planes are included as equivalent operations to one-fold and two-fold rotoinversions). This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 groups are the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms. In 1867 Axel Gadolin, who was unaware of the previous work of Hessel, found the crystallographic point groups independently using stereographic projection to represent the symmetry elements of the 32 groups.
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.
