Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. Usually the term is applied to groups that repeat in all three dimensions, but the line groups and the wallpaper groups also apply, or can apply, to three-dimensional arrangements. The space groups that repeat in all three dimensions are classified into 219 distinct types, or 230 types if chiral copies are considered distinct. Space groups are discrete cocompactgroups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3, they are sometimes called Bieberbach groups.

In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. Large collected sources of this information are available.