Space group

A space group is a set of isometries that maps a periodic arrangement onto itself, combining lattice translations with rotations, reflections, glide reflections, or screw rotations.

Bravais lattice: A lattice is an infinite periodic array of points; the 14 three-dimensional Bravais lattices classify possible lattice geometries. Its translations form the periodic framework that space-group operations must preserve.
Screw axis: A symmetry operation combining rotation about an axis with translation along that axis. Screw rotations are space-group operations absent from ordinary point groups.
International Tables for Crystallography: A reference work presenting crystallographic data, conventions, and mathematical tables for crystal symmetry. Its first volume tabulates the 230 three-dimensional space groups.
Wallpaper group: One of the 17 symmetry groups describing repeating patterns in two-dimensional Euclidean space. Wallpaper groups classify periodic plane patterns, while space groups extend the classification to three dimensions.
Point group: A point group is a set of symmetry operations that leaves at least one point fixed. Removing translations from a space group leaves its point-group symmetries.
Glide plane: A symmetry operation combining reflection across a plane with translation parallel to that plane. Glide reflections add symmetry that cannot be represented by a point operation alone.
X-ray crystallography: A method for determining atomic structures by analyzing how crystals diffract X-rays. Diffraction patterns reveal symmetry restrictions imposed by a crystal's space group.
Frieze group: One of seven symmetry groups for patterns repeating in one direction in the plane. Frieze groups have one repeating direction rather than a three-dimensional lattice.
Crystallographic restriction theorem: A theorem limiting rotational symmetries compatible with a periodic lattice to orders one, two, three, four, and six. It explains why periodic crystals exclude fivefold rotational symmetry.
Symmorphic space group: A space group whose operations can be generated by lattice translations and a point group sharing a common fixed point. It contrasts with groups whose rotations or reflections require fractional translations.

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