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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]

Hans Wondratschek developed mathematical treatments of crystallographic symmetry and wrote the teaching pamphlet Matrices, Mappings and Crystallographic Symmetry by Hans Wondratschek. It connects vectors, matrices and coordinate transformations with the operations of space groups.^[ref]

A symmetry operation can be written as an affine mappingwhere $W$ is its linear part and $w$ its translation. Combining two operations gives $(W_2{W}_1,W_2{w}_1+w_2)$, so the translation cannot simply be added without accounting for the intervening linear map. For example, inversion followed by translation differs from translation followed by inversion. The formula turns that geometric distinction into a concrete calculation.

His pamphlet also distinguishes changes of coordinate description from physical symmetry operations. Re-expressing the same crystal in another basis changes its matrix descriptions while preserving the underlying structure.^[ref]

Ian B. Smith studied oriented chemical transformations in manganese-containing solids with L. S. Dent Glasser. Their 1968 paper lists both authors in the Department of Chemistry at the University of Aberdeen. Its subject connects chemical reactions with crystal orientation: the arrangement of a product can retain a reproducible relationship to the starting material even when the material's composition changes.^[ref]

The distinction is useful experimentally. Identifying which substance formed answers a different question from identifying how its crystal lattice is oriented relative to the original crystal. An orientation relationship can constrain a proposed reaction mechanism beyond what chemical composition alone establishes.

Where their observations could be interpreted, Smith and Dent Glasser related the orientations to the oxygen framework and substantial migration of manganese. They also reported unexplained changes in crystallographic symmetry. Their work thus illustrates how structural observations can support a mechanism while leaving particular transformations unresolved.^[ref]