CRYSTALLOGRAPHIC SCALE TRANSFORMATIONS

Quasi-periodic structures have strictly speaking no three-dimensional space group symmetry. Their symmetry group is in fact a space group in a higher dimensional space. Besides this symmetry there may also be other invariances, in particular scale symmetry. The occurrence of such symmetries in the most common families of quasicrystals is discussed, in relation to their n-dimensional space group symmetry. The scale transformations may be combined with translations and n-dimensional orthogonal transformations to form symmorphic and non-symmorphic scale-space groups. The application of these symmetries is discussed.