Quasicrystalline Patterns Research Articles

Quasicrystals and strong interactions between square modes

We analyze the possibility of finding almost hexagonal and quasicrystalline patterns, in finite square containers, as the result of the nonlinear interaction of four-dimensional and eight-dimensional representations of ${D}_{4}$. We report the possibility of sustained oscillations between simple and nonsimple squares as the result of these interactions.

Pattern formation in weakly damped parametric surface waves

We present a theoretical study of nonlinear pattern formation in parametric surface waves for fluids of low viscosity, and in the limit of large aspect ratio. The analysis is based on a quasi-potential approximation to the equations governing fluid motion, followed by a multiscale asymptotic expansion in the distance away from threshold. Close to onset, the asymptotic expansion yields an amplitude equation which is of gradient form, and allows the explicit calculation of the functional form of the cubic nonlinearities. In particular, we find that three-wave resonant interactions contribute significantly to the nonlinear terms, and therefore are important for pattern selection. Minimization of the associated Lyapunov functional predicts a primary bifurcation to a standing wave pattern of square symmetry for capillary-dominated surface waves, in agreement with experiments. In addition, we find that patterns of hexagonal and quasi-crystalline symmetry can be stabilized in certain mixed capillary–gravity waves, even in this case of single-frequency forcing. Quasi-crystalline patterns are predicted in a region of parameters readily accessible experimentally.

The Formation of a Paracrystalline Structure from Muscle F-Actin and Oligolysine

At very low ionic strength (gamma less than 0.05) oligohomopolymers of lysine cause lateral association of muscle F-actin filaments into ordered structures which appear at low magnification in electron-micrographs as rigid needles. At higher magnification these aggregates display regular quasicrystalline patterns. The structures dissolve reversibly when the ionic strength is raised suggesting that F-actin filaments are crosslinked by oligolysine due to electrostatic forces.

Amorphous NiB seen as a disordered quasicrystal

The icosahedral quasicrystalline structure has recently been proposed as a possible prototype for the structure of metallic glasses. We have examined this suggestion in some detail, by comparing experimental partial radial distribution functions for the metallic glass Ni81B19 with those for a computer-built quasicrystalline structure of composition Ni84B16. The results cast serious doubts on the proposed existence of quasicrystalline structural patterns in amorphous structures.

Commensurate states on incommensurate lattices.

Can commensurate states exist on incommensurate lattices? We introduce and study by simulated annealing a simple 1D model related to flux quantization on superconducting networks or charged particles on a substrate to answer this question. For periodic and quasicrystalline patterns we find a set of low-energy states related to both decimation symmetry and periodicity.