Space group
In mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a space group is a symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

, usually for three dimensions, that divides space into discrete repeatable domains.

In three dimensions, there are 219 unique types, or counted as 230 if chiral
Chirality (mathematics)
In geometry, a figure is chiral if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...

 copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach
Ludwig Bieberbach
Ludwig Georg Elias Moses Bieberbach was a German mathematician.-Biography:Born in Goddelau, near Darmstadt, he studied at Heidelberg and under Felix Klein at Göttingen, receiving his doctorate in 1910. His dissertation was titled On the theory of automorphic functions...

, and are discrete cocompact group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

s of isometries of an oriented Euclidean space.

In crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

, they are also called the crystallographic or Fedorov
Yevgraf Fyodorov
Yevgraf Stepanovich Fyodorov, sometimes spelled Evgraf Stepanovich Fedorov , was a Russian mathematician, crystallographer, and mineralogist....

, and represent a description of the symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 of the crystal.