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...

Abstract. Two-dimensional colloidal suspensions subjected to laser interference patterns with decagonal symmetry can form an Archimedean-like tiling phase where rows of squares and triangles order aperiodically along one direction (J. Mikhael et al., Nature 454, 501 (2008)). In experiments as well as in Monte Carlo and Brownian dynamics simulations, we identify a similar phase when the laser field possesses tetradecagonal symmetry. We characterize the structure of both Archimedean-like tilings in detail and point out how the tilings differ from each other. Furthermore, we also estimate specific particle densities where the Archimedean-like tiling phases occur. Finally, using Brownian dynamics simulations we demonstrate how phasonic distortions of the decagonal laser field influence the Archimedean-like tiling. In particular, the domain size of the tiling can be enlarged by phasonic drifts and constant gradients in the phasonic displacement. We demonstrate that the latter occurs when the interfering laser beams are not ideally adjusted.