This tag is for questions on mathematical crystallography.
Mathematical crystallography is the study of external forms of crystals and their internal spatial structure and is based on the conception that the particles forming the crystal lattice are arranged in an ordered, periodic, three-dimensional configuration.
Crystals grown under equilibrium conditions have the form of symmetric regular convex polyhedra. To describe crystals, one uses symmetry groups of which the most important are the space, or Fedorov, groups describing the atomic structure of crystals and the point groups describing the crystal's external form.
Operations characteristic for the space group include three non-coplanar translations that correspond to the three-dimensional periodicity of the atomic structure of crystals. Because it is possible to combine translations and point symmetry operations in the lattice, the space groups also contain operations and corresponding symmetry elements with a translation component—screw axes of various orders and sliding planes.
There are 230 space groups; any crystal belongs to one of those groups. The translation components of the microsymmetry elements do not appear macroscopically; therefore, each of the 230 space groups is macroscopically similar to one of the 32 point groups. The set of translations in a given space group is a translation subgroup of it, or a Bravais lattice; there are 14 such lattices.
