 Crystallographic symmetry

Crystallographic symmetry consists of the spatial transformations that leave an ideal crystal structure unchanged, including the identities of its chemical atoms. A symmetry of the crystal lattice alone need not preserve the atomic basis of a crystal. The full set of crystal-preserving transformations forms a space group; its translational part distinguishes it from a crystallographic point symmetry description.^[ref]

In fractional unit cell coordinates a space group operation has the form $x\mapsto Wx+w$. The matrix $W$ describes its linear action and $w$ its translation. A screw axis combines a rotation with a translation parallel to the axis; a glide plane combines reflection with translation within the plane. For example, a twofold screw can act asApplying it twice gives $(x,y,z+1)$, a full unit cell translation. Replacing it by a pure twofold rotation would lose this relation between orientation and position.

For ordinary three-dimensional periodic crystals, the possibilities are organized into 32 crystallographic point groups and 230 space groups. These counts concern spatial symmetry without additional magnetic labels. A square outline alone does not establish fourfold symmetry: decorating alternate corners with different atoms can break the rotation. In structure determination, the symmetry therefore constrains both the lattice geometry and which sites can carry equivalent chemical atoms.^[ref]