a listing of the changes in symmetry inherent in the atomic structure of crystals (crystal lattice). All 230 space groups were derived in 1890 and 1891 by the Russian crystallographer E. S. Fedorov and independently by the German mathematician A. Schoenflies. Geometrical changes of various objects (figures, bodies, functions), after which the object coincides with itself, are called symmetry operations. Since the crystal lattice has a three-dimensional periodicity, the operation of superposing the lattice onto itself by means of parallel displacements in three directions (translations) to distances (vectors) a, b, and c, which define the dimensions of the unit cell, is characteristic of the space symmetry of crystals. Other possible symmetry operations of the crystal structure involve rotations of 180°, 120°, 90°, and 60° about the axes of symmetry, reflections in planes of symmetry, inversions at symmetry centers, and symmetry operations with transitions (screw rotations, sliding reflections). Space symmetry operations may be combined according to certain rules established by mathematical group theory, and the operations themselves constitute a group.
A space group does not define a specific arrangement of atoms in the crystal lattice, but it does give one of the possible laws governing the symmetry of the relative positions of the atoms. Thus, space groups are of particular importance to the study of the atomic structure of crystals—each of the many thousands of structures so far investigated belongs to one of the 230 space groups. The determination of the space group is made by X-ray diffraction. A point group (class) of crystal symmetry—the set of symmetry transformations upon which one point of the crystal remains fixed (in the absence of translation)—should not be confused with a space group. A point group characterizes the symmetry of the external shape of crystals and the anisotropy of crystal properties. All 230 space groups have been tabulated in special handbooks.
B. K. VAINSHTEIN and M. P. SHASKOLSKAIA