# Space group

In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions. In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space.

In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography (Hahn (2002)).

Space groups in 2 dimensions are the 17 wallpaper groups which have been known for several centuries, though the proof that the list was complete was only given in 1891, after the much harder case of space groups had been done.

In 1879 Leonhard Sohncke listed the 65 space groups (sometimes called Sohncke space groups or chiral space groups) whose elements preserve the orientation. More accurately, he listed 66 groups, but Fedorov and Schönflies both noticed that two of them were really the same. The space groups in 3 dimensions were first enumerated by Fedorov (1891) (whose list had 2 omissions (I43d and Fdd2) and one duplication (Fmm2)), and shortly afterwards were independently enumerated by Schönflies (1891) (whose list had 4 omissions (I43d, Pc, Cc, ?) and one duplication (P421m)). The correct list of 230 space groups was found by 1892 during correspondence between Fedorov and Schönflies. Barlow (1894) later enumerated the groups with a different method, but omitted four groups (Fdd2, I42d, P421d, and P421c) even though he already had the correct list of 230 groups from Fedorov and Schönflies; the common claim that Barlow was unaware of their work is a myth. Burckhardt (1967) describes the history of the discovery of the space groups in detail.

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices, each of the latter belonging to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 different space groups describing all possible crystal symmetries.

The elements of the space group fixing a point of space are rotations, reflections, the identity element, and improper rotations.