| It is shown that the Fourier-space reformulation of the conventional symmetry classification of crystals, which abandons the traditional reliance on periodicity, not only unifies the treatment of periodic and quasiperiodic crystals but also provides for a unified treatment of the various types of quasiperiodic crystals--modulated crystals, composite crystals, and quasicrystals. The approach is more coherent than the conventional high-dimensional "superspace" approach (which treats the types of quasiperiodic crystals differently, producing a separate classification for each type) and has the added benefit of working in three dimensions. As a pedagogical example, a complete enumeration of all Bravais classes for the simplest incommensurate reducible lattices--all 3-dimensional rank-4 Bravais classes and all 3-dimensional rank-6 cubic and tetrahedral Bravais classes--is given. This is followed by a classification of all Bravais classes and space groups for hexagonal and trigonal crystals of arbitrary finite rank. In the process, general techniques for the enumeration of Bravais classes and space groups are illustrated. |
