Turing patterns beyond hexagons and stripes
The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a
... hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern's wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns. Turing patterns in reaction-diffusion systems have been proposed as a mechanism for morphogenesis, 1-3 and the Turing instability may play a major role in the generation of skin patterns in a number of animals. 3-5 Early research focused on Turing patterns that arise spontaneously from random initial conditions, which are typically stripes or hexagonal arrangements of spots. Additional patternssquares and superlattices composed of several simple lattices-have been found in hydrodynamics. These results prompted a search for conditions under which such patterns appear in reaction-diffusion systems with Turing instability. Here we examine two overlapping classes of these patterns: superlattices and squares.