= 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.<Symmetry by L. S. Dent Glasser>[Sections 2–3: symmetry operations, crystal classes and periodic arrays]{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 as
$$
(x,y,z)\longmapsto(-x,-y,z+\tfrac12).
$$
Applying 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>.<Symmetry by L. S. Dent Glasser>[Section 2, Table 1, and section 3: crystal classes and translational symmetry]{ref}
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