Beware of 46-fold symmetry: The classification of two-dimensional quasicrystallographic lattices

Multiple-scale structures: from Faraday waves to soft-matter quasicrystals

For many years, quasicrystals were observed only as solid-state metallic alloys, yet current research is now actively exploring their formation in a variety of soft materials, including systems of macromolecules, nanoparticles and colloids. Much effort is being invested in understanding the thermodynamic properties of these soft-matter quasicrystals in order to predict and possibly control the structures that form, and hopefully to shed light on the broader yet unresolved general questions of quasicrystal formation and stability. Moreover, the ability to control the self-assembly of soft quasicrystals may contribute to the development of novel photonics or other applications based on self-assembled metamaterials. Here a path is followed, leading to quantitative stability predictions, that starts with a model developed two decades ago to treat the formation of multiple-scale quasiperiodic Faraday waves (standing wave patterns in vibrating fluid surfaces) and which was later mapped onto systems of soft particles, interacting via multiple-scale pair potentials. The article reviews, and substantially expands, the quantitative predictions of these models, while correcting a few discrepancies in earlier calculations, and presents new analytical methods for treating the models. In so doing, a number of new stable quasicrystalline structures are found with octagonal, octadecagonal and higher-order symmetries, some of which may, it is hoped, be observed in future experiments.

Properties of quasiperiodic functions

Today, quasiperiodic tilings are well known and have been studied in great detail since they are very useful to describe the properties of metallic and soft matter quasicrystals. A closely related topic are quasiperiodic functions which have also gained large interest recently. Different types of such functions and there interrelation will be presented here. The main topic will be quasiperiodic potentials generated by laser beams and their variability. The distribution of extremal points and local isomorphisms of quasiperiodic functions will also be addressed.

Mathematical diffraction of aperiodic structures

Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details.

Bravais colourings of planar modules with N-fold symmetry

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N-fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields.

Crystallographic surfaces

Nodal surfaces, defined by wave vectors in a reciprocal space, equipotential surfaces, defined by point charges in real space, and periodic minimal surfaces, ‘soap-film’ elements lying smoothly across the asymmetric domains in given crystallographic symmetry groups, are exact mathematical objects which may be of physical significance and which can help with the visualization of significant structure at a level above that of single atoms. They are two-dimensional manifolds, with local metrics different from the euclidean metric of the planar sheets of orthodox crystallography, and are of use in the development of the flexi-crystallography of the ‘soft matter’ proposed by de Gennes.