Elegance in Crystal Symmetry

Abstract

In N. David Mermin’s column in the March 2000 issue of Physics Today (page 11), I tasted the disappointment that the community of crystallographers has not en masse embraced his proposed description of periodic and aperiodic crystals, although it is so elegant. The explanation is simple. Mermin’s approach, using only reciprocal (or Fourier) space, was not as new as he claimed. Early in the study of modulated structures, the symmetry was described using irreducible representations of space groups. This is, in fact, a formulation in reciprocal space. The phase factors appearing there are exactly the gauge transformations Mermin discusses.In contrast, the description that has become standard uses either reciprocal space and three dimensions or direct space and more dimensions. These two formulations are equivalent, but sometimes one is more natural than the other. For example, the positions of atoms or the properties of tilings are more easily discussed in direct space than in reciprocal space. Mermin’s approach is very close to the reciprocal space formulation of the standard approach. To apply his approach to the Penrose tiling, one must first calculate the Fourier transform—an unnecessary detour. Therefore, in my opinion, Mermin’s approach is certainly elegant, but the standard approach proposed earlier is even more so.The reason that a serious commission struggles a long time with the nomenclature is simple. The higher-dimensional space groups studied are not used only for electron wavefunctions of aperiodic crystals. Structures, atomic positions, and especially deviations like phason strains are more easily visualized in direct higher-dimensional space, and are described there by space groups. Furthermore, higher-dimensional space groups are relevant not only for physics and crystallography. Quasicrystals have inspired a strong and interesting development in mathematics, for instance, in which problems concerning model sets, diffracting sets, tilings, and other objects are studied using symmetry arguments. Other mathematical topics such as the characterization of Lie groups or spaces of constant curvature use higher-dimensional space groups. Therefore, it is useful to have a nomenclature that satisfies the needs of the different users—mathematicians, physicists, and crystallographers—and that is understood by these groups. Developing such a nomenclature is time-consuming and requires a broad perspective.I agree with Mermin’s statement about the role of elegance in science, but I think that at least one of his examples is not very well chosen.© 2001 American Institute of Physics.