Space group

In mathematics and physics, 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)).

History

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.