The Symmetry of Quasiperiodic Crystals with N.D. Mermin. Within the framework of a reformulation of the conventional symmetry classification of crystals, we have shown that the Fourier-space approach of Rokhsar, Wright, and Mermin, 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 (refs. 1-3). Aside from the obvious benefits of working in three dimensions, our 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, with a particularly awkward and redundant treatment of composite crystals (ref. 4) .
Symmetry Changes in Rank-Lowering Phase Transitions with N.D. Mermin. We developed a procedure to determine the possible low-rank space groups that may arise from a given high-rank space group through a continuous phase transition (ref. 5) . Specific examples are the incommensurate to commensurate phase transitions observed in modulated crystals. Our procedure exploits the fact that in Fourier-space both the high-rank and the low-rank structures are described in the same 3-dimensional space.
CURRENT RESEARCH
Tetrahedral and Icosahedral Quasicrystals with J. Draeger and N.D. Mermin. We are investigating the detailed relations between the symmetry of icosahedral quasicrystals and that of periodic and the recently discovered quasiperiodic tetrahedral crystals. These relations bear directly on the variety of structures appearing in the intricate phase diagrams of many quasicrystals. Our description of these structures in Fourier-space allows us to deal with them in a unified manner.
The Symmetry of Quasiperiodic Tensor Fields with N.D. Mermin. We are extending the Fourier-space approach to the symmetry classification of quasiperiodic multicomponent fields with applications to such systems as magnetic crystals and liquid crystals, and to the problem of color symmetry (ref. 6) . The generalization of point group on which Fourier-space crystallography rests is here extended to encompass the rich group theoretical structure of these systems.