The symmetry of quasiperiodic crystals

Experimentally observed crystals range from periodic crystals, through incommensurately modulated crystals and composite crystals, to quasicrystals and even modulated quasicrystals. How does one characterize in a unified manner the symmetry of all these types of crystals? How does one classify all crystals according to their symmetry? These questions are answered through a review of the Fourier-space approach to crystal symmetry of Rokhsar, Wright, and Mermin. The notion of indistinguishability, which is central to the approach, is introduced and used as the basis for a generalization of the traditional space-group classification scheme, applicable to all types of crystals known to date.