Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices

Abstract

When two-dimensional pattern-forming problems are posed on a periodic domain, classical techniques (Lyapunov-Schmidt, equivariant bifurcation theory) give considerable information about what periodic patterns are formed in the transition where the featureless state loses stability. When the problem is posed on the whole plane, these periodic patterns are still present. Recent work on the Swift-Hohenberg equation (an archetypal pattern-forming partial differential equation) has proved the existence of quasipatterns, which are not spatially periodic and yet still have long-range order. Quasipatterns may have 8-fold, 10-fold, 12-fold and higher rotational symmetry, which preclude periodicity. There are also quasipatterns with 6-fold rotational symmetry made up from the superposition of two equal-amplitude hexagonal patterns rotated by almost any angle α with respect to each other. Here, we revisit the Swift-Hohenberg equation (with quadratic as well as cubic nonlinearities) and prove existence of several new quasipatterns. The most surprising are hexa-rolls: periodic and quasiperiodic patterns made from the superposition of hexagons and rolls (stripes) oriented in almost any direction with respect to each other and with any relative translation; these bifurcate directly from the featureless solution. In addition, we find quasipatterns made from the superposition of hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). We consider the periodic case as well, and extend the class of known solutions, including the superposition of hexagons and rolls. While we have focused on the Swift-Hohenberg equation, our work contributes to the general question of what periodic or quasiperiodic patterns should be found generically in pattern-forming problems on the plane.

Rectangle-triangle soft-matter quasicrystals with hexagonal symmetry

“…More elaborate potentials, perhaps involving threebody interactions, may be required for other tilings or indeed to make the structures discussed here the global minima of F . The number of aperiodic tilings found so far is large [53], and includes structures that may be relevant to two-dimensional materials such as bilayer graphene [44,54] and three-dimensional quasicrystals [46]. We are optimistic that our approach can be used, at least in principle, to find soft-particle systems that self-assemble into these structures.…”

“…We choose the side lengths L x and L y so that the resulting (now periodic) profile is a good approximant for the true QC, with the values used chosen following the approach of Refs. [43,44]. Further details are given on this below in Sec.…”

Localised Radial Patterns on the Free Surface of a Ferrofluid

“…The function u Q is a special case of the arbitrarily rotated hexagon lattices considered in [11] and forms a quasipattern with D 12 symmetry, as plotted in Figure 3(c). The quasipattern u Q has the following Bessel expansion…”

“…Other more complicated patterns can be considered beyond the four that we concentrate on in this paper; for instance one could choose squares or other cases of rotated hexagons considered in [11], including other quasipattern or superlattice structures.…”