ar X iv : h ep - t h / 91 10 04 3 v 1 1 5 O ct 1 99 1 Copernican Crystallography

Redundancies are pointed out in the widely used extension of the crystallographic concept of Bravais class to quasiperiodic materials. Such pitfalls can be avoided by abandoning the obsolete paradigm that bases ordinary crystallography on microscopic periodicity. The broadening of ordinary crystallography to include quasiperiodic materials is accomplished by defining the point group in terms of indistinguishable (as opposed to identical) densities. A periodic material (crystal) is characterized by a lattice of vectors [1] specifying the translations that leave its density unchanged. A century and a half ago Frankenheim classified such lattices by their symmetry, counting 15 types. A few years later Bravais pointed out that two of the Frankenheim classes contained identical lattices, and today the edifice of crystallography rests on a foundation of 14 Bravais classes [2]. Within the past two decades the discovery of many quasiperiodic materials (dis-placively modulated crystals, substitutionally modulated crystals, incommensurate inter-growth compounds, quasicrystals) has stimulated efforts to extend the crystallographic classification scheme to include these novel structures. Roughly speaking quasiperiodic materials have the property — reminiscent of but weaker than periodicity — that sub-regions of arbitrary size can be found reproduced elsewhere in the material at distances of the order of that size. This notion is made precise by the Fourier space definition of quasiperiodicity, which gives a simple and natural expression of the close connection between periodic and quasiperiodic materials. Densities of either type of material are su-perpositions of plane waves whose wave vectors can be expressed as a lattice of integral 1 linear combinations of 3+d primitive wave vectors that span a three dimensional space and are linearly independent over the integers. A material is periodic if d = 0 and quasiperi-odic if d > 0. To emphasize that the vectors in such lattices are wave vectors rather than translations, one may refer to them as reciprocal lattices. Although originally viewed as containing lattices of translations, the Bravais classes of periodic materials can equally well be regarded as classes of reciprocal lattices. Since quasiperiodic materials have no 3-dimensional translational symmetry but continue to be described by a lattice of wave vectors, it is only in Fourier space that the concept of Bravais class can directly be applied to them. The first attempt at such a classification, using the less direct superspace formalism described below, was made over a decade ago by Janner, Janssen, and de Wolff (JJdW) [3,4] for the simplest quasiperiodic …