Philip Ball PATTERN FORMATION IN NATURE: PHYSICAL CONSTRAINTS AND SELf-ORGANISING CHARACTERISTICS Pattern formations are apparent in natural systems ranging from clouds to animal markings, and from sand dunes to shells of microscopic marine organisms. Despite the astonishing range and COPYRIGHTEDvariety of such structures, MATERIAL many have comparable features. In this article, Philip Ball reviews some of the common patterns found in nature. He explains how they are typically formed through simple, local interactions between many components of a system – a form of physical computation that gives rise to self-organisation and emergent structures and behaviours. The Giant’s Causeway in County Antrim, Ireland. 22 —c01.indd 22 23/01/2012 09:56 When the naturalist Joseph Banks first Such regularities are not just a feature pattern and form in nature in a synthesis of encountered Fingal’s Cave on the Scottish of the insensate or instinctive natural world, biology, natural history, mathematics, physics island of Staffa, he was astonished by the but may also be found in human social and engineering. Thompson pointed out that, quasi-geometric, prismatic pillars of rock that systems that are seemingly subject to the in biology at least, and often in the non-living flank the entrance. As Banks put it: whims of free will; for example, in the evenly world, pattern formation is not a static thing, spaced waves of congestion that might but arises from growth: everything is what it is Compared to this what are the cathedrals appear in moving traffic, or quasi-periodic because it got that way. or palaces built by men! Mere models or cycles in economic systems. Understanding The answer to the riddle of pattern lies in playthings, as diminutive as his works how spontaneous pattern formation occurs how it got to be that way. This is less obvious will always be when compared with those is therefore an endeavour that unites many than it sounds: a bridge or a paddy field or a of nature. What now is the boast of the disparate fields of science, from zoology microchip is ‘explained’ by how it looks, not architect! Regularity, the only part in to fracture mechanics, and from chemical how it was made. Thompson objected to the which he fancied himself to exceed his kinetics to sociology. Many patterns in nature fact that Darwinian, adaptive explanations of mistress, Nature, is here found in her have a universal aspect that does not respect form and pattern in living systems tended to possession, and here it has been for ages the traditional divisions between the natural take that approach too. The zebra’s stripes undescribed.1 sciences, or even between the living and the might be explained in one sense by invoking non-living world. Rather, natural patterns the adaptive benefit of their camouflage This structure has a counterpart on the seem to come from a relatively limited palette, (although in fact the effectiveness of the coast of Ireland: the Giant’s Causeway in even in systems that might seem to have stripes for concealment is still debated), but County Antrim, where again one can see nothing at all in common with one another. this does not account for how any particular the extraordinarily regular and geometric The hexagonal columns of Fingal’s Cave may zebra acquires these pigmented markings honeycomb structure of the fractured igneous put us in mind of other natural hexagonal on its hide as it grows from an embryo. rock. forms. Do these patterns really have anything Thompson argued that evolutionary biology When we make an architectural pattern in common, or is the similarity in appearance needs to take into account both the limitations like this, it is through careful planning and just coincidence? imposed and the possibilities provided by construction, with each individual element The first person to confront this question purely physical forces (including chemical cut to shape and laid in place. Our experience in a systematic way was the Scottish zoologist processes) acting on a growing organism. from human technologies thus suggests that D’Arcy Wentworth Thompson in his book On It is now understood that the common making a pattern requires a patterner. But at Growth and Form (1917),3 which collected features of many natural patterns result from Fingal’s Cave and the Giant’s Causeway, the together all that was then known about mathematical analogies and equivalences forces of nature have conspired to produce a pattern without, we must presume, any blueprint or foresight or design. This is an example of spontaneous pattern formation.2 In earlier times, such regularity in nature was taken as evidence of God’s guiding hand. We now know that no intelligent agency is needed to create the patterns that appear profusely in both the living and the inorganic natural world. These organised arrays of elements arise spontaneously from the interactions between their many component parts, whether these are chemical reagents that react and diffuse, small particles or molecules that cohere into clusters, propagating cracks, wind-blown sand grains or flowing liquids. Such patterns are said to be self-organised. Their scientific study comprises one aspect of research into so- called complex systems, which typically show emergent behaviours that cannot be deduced Understanding how spontaneous pattern or predicted by a focus on the properties of the individual elements. formation occurs is therefore an endeavour that unites many disparate fields of science, from zoology to fracture mechanics, and from chemical kinetics to sociology. left: Close-up of the prismatic cross sections of rock columns in the Giant’s Causeway. right: The crack network of a representative section, showing the quasi-hexagonal pattern. 23 —c01.indd 23 23/01/2012 09:56 in the rules governing their formation – about the progress of a chemical reaction, but whether these rules are expressed in terms pronounces only on its final, stable state. The of continuum equations describing, say, oscillations occur only so long as the system diffusion and transport of material, or as is out of equilibrium. It can be maintained in local interactions between the components. that state indefinitely, however, by continually Both descriptions tend to give rise to feeding in fresh reagents and carrying away solutions that involve symmetry-breaking the end products. of an initially uniform or random system, If the mixture is stirred, the colour change creating spatial and/or temporal segregation takes place more or less everywhere at once. of the constituents. These solutions are often But if the reaction proceeds in an undisturbed modified by specific circumstances such thin layer of liquid – or better still, a gel, to as boundary conditions, the appearance of slow diffusion and suppress fluid-dynamic defects, and path-dependent hysteresis in disturbances – parts of it can undergo the transitions between patterns. At root, however, switch at different times. This does not just they expose self-organisation as an inherent produce a patchwork, but regular patterns property of complex systems. of concentric bands and spirals: chemical waves that spread through the medium, like Chemical Patterns the expanding ripples of a stone thrown into An explanation for the patterning apparent in a pond. Where two wave fronts meet, they animal markings – noted, but barely explored, annihilate each other. by D’Arcy Thompson – has emerged from The oscillations arise because the reaction the study of so-called oscillating chemical has two possible states or branches. Each reactions, in which a mixture of chemical branch involves an autocatalytic process: reagents seems first to react in one direction the reaction products promote the formation and then to reverse and reform the original of more of themselves. This is a positive mixture.4 The classic example is the Belousov- feedback process, and it means that each Zhabotinsky (BZ) reaction discovered in branch eventually exhausts itself as the the 1960s, in which the mixture alternates feedback speeds up the reaction and depletes between red and blue owing to changes in the ingredients. When that happens, the the charge state of the iron ions that catalyse system becomes primed to switch to the the reactions. If these oscillations went on other branch. The oscillating process depends indefinitely, the system would violate the on two general factors: the ingredients are second law of thermodynamics. But eventually reacting to produce the coloured products, they die out and the mixture settles into an and the molecules involved are moving by equilibrium state. Thermodynamics is silent diffusion through the liquid. How quickly the system switches between the red and blue states then depends on how quickly diffusion brings in fresh ingredients to replenish those that have been lost by reaction. So the patterns here come from a balance between reaction, which destroys the ingredients, and diffusion, which replenishes them. This kind of process is known as a reaction-diffusion system. Notice that the pattern comes about through a competition between these two opposing processes. This is one of the universal principles of pattern formation: it requires a balance of opposite tendencies. In 1952 Alan Turing identified another kind of chemical reaction-diffusion process, which produces not moving chemical waves but stationary patterns.5 Turing’s theory was prompted by the question of how pattern formation occurs in the earliest stages of top: A quasi-hexagonal pattern is evident embryo formation on a fertilised egg. This is a in the markings on a giraffe. problem of symmetry-breaking: a spherically centre: A hexagonal mesh in the shells of symmetrical ball of identical cells becomes microscopic sea creatures (radiolarians). differentiated into those that will form the bottom: Snapshot of chemical-wave patterns in the oscillating Belousov-Zhabotinsky reaction. head, the limbs, the spine and so forth.