 Hans Wondratschek

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]