Patterns are all around us. But what is the mathematics behind them? Fractals, space-filling and dynamics. The science of pattern formation studies the statistical outcomes of self-organisation and the origins of similar patterns.
The Swift–Hohenberg equation is a partial differential equation noted for its pattern-forming behaviour:
where u = u(x, t) or u = u(x, y, t) is a scalar function defined on the line or the plane, r is a real bifurcation parameter, and N(u) is some smooth nonlinearity and was derived from the equations for thermal convection Swift and Hohenberg (1977), Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319–328.
In Biology, genetic factors and the flow of hormones in the stage of cell differentiation lead (the morphogenic field) leads to the wonderful patterns seen in animal markings, insect segmentation, plant phyllotaxis and even predator-prey equations’ trajectories. One of the best understood examples of pattern formation is the patterning along the future head to tail (antero-posterior) axis of the fruit fly Drosophila melanogaster. The growth of bacterial colonies also reveal beautiful patterns whose complexity depends on growth conditions. Vegetation patterns are modulated by aridity (dryness) and slope and arise as the consequence of ressource redistribution and concentration. In Chemistry, Turing patterns and Liesegang Rings result from the Belousov-Zhabotinsky reaction.
In physicsm fluid flows can arise from Bénard cells, and hydrodynamic convection leading to stripes and rolls in clouds and ripples in icicles. Paths of least action lead to numerous dendrites – in burning air during lightning, liquid crystals, sea foams and even Lichtenberg Figures in soil and humans after lightning strikes.
