Monolayers on crystalline surfaces often form complex structures with physical and chemical properties that differ strongly from those of their bulk phases 1 . Such hetero-epitactic overlayers are currently used in nanotechnology and understanding their growth mechanism is important for the development of new materials and devices. In comparison with crystals, quasicrystalline surfaces exhibit much larger structural and chemical complexity leading, for example, to unusual frictional 2 , catalytical 3 or optical properties 4,5 . Deposition of thin films on such substrates can lead to structures that may have typical quasicrystalline properties. Recent experiments have indeed showed 5-fold symmetries in the diffraction pattern of metallic layers adsorbed on quasicrystals 6,7 . Here we report a real-space investigation of the phase behaviour of a colloidal monolayer interacting with a quasicrystalline decagonal substrate created by interfering five laser beams. We find a pseudomorphic phase that shows both crystalline and quasicrystalline structural properties. It can be described by an archimedean-like tiling 8,9 consisting of alternating rows of square and triangular tiles. The calculated diffraction pattern of this phase is in agreement with recent observations of copper adsorbed on icosahedral Al 70 Pd 21 Mn 9 surfaces 10 . In addition to establishing a link between archimedean tilings and quasicrystals, our experiments allow us to investigate in real space how single-element monolayers can form commensurate structures on quasicrystalline surfaces.Quasicrystals are unusual materials: they are aperiodic but retain true long-range order 11 . Although quasicrystalline structures have been theoretically also predicted in systems with a single type of particle 12,13 , experimentally their spontaneous formation has been observed only in binary, ternary or even more complex alloys 14 . Accordingly, their surfaces exhibit a high degree of structural and chemical complexity and show unexpected mechanical, electrical and optical properties 15 . To understand the origin of those characteristics it is useful to disentangle structural and chemical aspects; this can be achieved by growing single-element monolayers to quasicrystalline surfaces 16,17 . Apart from adding to our understanding of how quasicrystalline properties can be transferred to such monolayers 18 , this approach might permit the fabrication of materials with previously unobserved properties. Heteroepictatic growth experiments on decagonal and icosahedral surfaces did indeed show the formation of Bi and Sb monolayers with a high degree of quasicrystalline order as determined by low-energy electron diffraction and elastic heliumatom scattering experiments 6,18 . In comparison with reciprocal space studies, it was only recently that scanning tunnelling microscopy permitted an atomic resolution of the adsorbate morphology 7 . Even then, however, it was difficult to relate the structure of the adsorbate to that of the underlying substrate.Here we report an experi...

The coincidence problem for planar patterns with N -fold symmetry is considered. For the N -fold symmetric module with N < 46, all isometries of the plane are classified that result in coincidences of finite index. This is done by reformulating the problem in terms of algebraic number fields and using prime factorization. The more complicated case N ≥ 46 is briefly discussed and N = 46 is described explicitly.The results of the coincidence problem also solve the problem of colour lattices in two dimensions and its natural generalization to colour modules.

In recent computer simulations of a simple monatomic system interacting via the Dzugutov pair potential, freezing of the fluid into an equilibrium dodecagonal quasicrystal has been reported [M. Dzugutov, Phys. Rev. Lett. 70, 2924Lett. 70, (1993]. Here, using a combination of molecular dynamics simulation and thermodynamic perturbation theory, we conduct a detailed analysis of the relative stabilities of solid-phase structures of the Dzugutov-potential system. At low pressures, the most stable structure is found to be a bcc crystal, which gives way at higher pressures to an fcc crystal. Although a dodecagonal quasicrystal and a σ-phase crystal compete with the bcc crystal for stability, they remain always metastable.

Quasicrystals provide a fascinating class of materials with intriguing properties. Despite a strong potential for numerous technical applications, the conditions under which quasicrystals form are still poorly understood. Currently, it is not clear why most quasicrystals hold 5-or 10-fold symmetry but no single example with 7-or 9-fold symmetry has ever been observed. Here we report on geometrical constraints which impede the formation of quasicrystals with certain symmetries in a colloidal model system. Experimentally, colloidal quasicrystals are created by subjecting micron-sized particles to two-dimensional quasiperiodic potential landscapes created by n ¼ 5 or seven laser beams. Our results clearly demonstrate that quasicrystalline order is much easier established for n ¼ 5 compared to n ¼ 7. With increasing laser intensity we observe that the colloids first adopt quasiperiodic order at local areas which then laterally grow until an extended quasicrystalline layer forms. As nucleation sites where quasiperiodicity originates, we identify highly symmetric motifs in the laser pattern. We find that their density strongly varies with n and surprisingly is smallest exactly for those quasicrystalline symmetries which have never been observed in atomic systems. Since such high-symmetry motifs also exist in atomic quasicrystals where they act as preferential adsorption sites, this suggests that it is indeed the deficiency of such motifs which accounts for the absence of materials with e.g., 7-fold symmetry.7-fold symmetry | growth mechanism | light patterns T he presence or lack of order is of primary importance in a broad range of fundamental phenomena in science. Until the early 1980s, it was unanimously established that ordered matter is always periodic (1). Accordingly, the rotational symmetry in real space was thought to be limited to N ¼ 2, 3, 4 and 6. However some metal alloys (2), polymers (3), micelles (4), and even recently colloidal nanoparticles (5) and nonspherical particles (6), have defied these crystallographic rules and selforganized into so-called quasicrystals. These structures form a unique type of matter which-unlike periodic crystals or amorphous materials-exhibit long-range positional order but are not periodic. Quasicrystals show many interesting properties which are quite different compared to that of periodic crystals. Accordingly, they are considered as materials with high technological potential e.g., as surface coatings, thermal barriers, catalysts, or photonic materials (7).Since the properties of quasicrystals are strongly connected to their atomic structure, a better understanding of their growth mechanisms is of great importance (8-11). Perhaps one of the most interesting questions in this context is why all observed quasicrystals have only 5-, 8-, 10-, and 12-fold symmetry but no single quasicrystal with 7-, 9-,11-, and 13-fold symmetry was ever found (12). For a classification of different surface symmetries it is helpful to consider the rank D, i.e., the number of incommensurate wa...

Among the distinctive features of quasicrystals-structures with long-range order but without periodicity-are phasons. Phasons are hydrodynamic modes that, like phonons, do not cost free energy in the long-wavelength limit. For light-induced colloidal quasicrystals, we analyze the collective rearrangements of the colloids that occur when the phasonic displacement of the light field is changed. The colloidal model system is employed to study the link between the continuous description of phasonic modes in quasicrystals and collective phasonic flips of atoms. We introduce characteristic areas of reduced phononic and phasonic displacements and use them to predict individual colloidal trajectories. In principle, our method can be employed with all quasicrystalline systems in order to derive collective rearrangements of particles from the continuous description of phasons.