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A crystal lattice records the translation symmetry of a periodic crystal. With a primitive basis of three linearly independent lattice vectors $a_1,a_2,a_3$, its translation vectors formThe crystal structure additionally specifies the arrangement of atoms associated with each lattice point. A lattice point is therefore a symmetry marker, not necessarily one atom.^[ref]

For example, a one-dimensional chain with alternating species $A,B$ at spacing $a$ has translation period $2a$, not $a$: shifting by $a$ exchanges distinct species. It can be described by lattice $2a\mathbb{Z}$ with a two-atom basis at offsets zero and $a$. In a diffraction measurement, the translation lattice selects possible reciprocal-space peak positions, while the atomic basis affects their intensities and systematic absences. This distinction lets diffraction reveal internal structure rather than just the visible shape of the specimen.

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]