Point Group

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O(d). Point groups can be realized as sets of orthogonal matrices M that transform point x into point y:

y = Mx

where the origin is the fixed point. Point-group elements can either be rotations (determinant of M = 1) or else reflections, improper rotations, rotation-reflections, or rotoreflections (determinant of M = −1). All point groups of rotations with dimension d are subgroups of the special orthogonal group SO(d).

Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number. These are the crystallographic point groups.