PATTERN FORMATION IN NATURE
The rough text of a lecture delivered at the Universities of Limerick, Belfast and Dublin for the Ireland Institute of Physics, February 2003
There is an accompanying presentation in Powerpoint which contains details of references to the work cited here
This [slide] is Fingal's Cave on the island of Staffa, near Mull in Scotland. It is named after the legendary Irish leader Finn MacCumhaill (c. AD 250), whose story was brought to Scotland's west coast by early Irish settlers. The cave has inspired artists (such as J. M. W. Turner) and composers - Felix Mendelsohn wrote his orchestral piece named after the cave in 1829. But it also made an impression on an awestruck Joseph Banks, president of the Royal Society, when he sailed to Staffa in 1772 during an expedition to Iceland. This is what he said:
"Compared to this what are the cathedrals or palaces built by men! Mere models or playthings, as diminutive as his works will always be when compared with those of nature. What now is the boast of the architect! Regularity, the only part in which he fancied himself to exceed his mistress, Nature, is here found in her possession, and here it has been for ages undescribed."
Banks had noticed that the entrance to the cave was flanked by these great pillars of rock. Close up, you can see the regularity that Banks spoke about: hexagonal cross-sections.
Now, this of course has its counterpart on the west coast of Ireland itself: the Giant's Causeway in County Antrim, built in legend by the giant Finn MacCool.
See how the pillars have this extraordinarily regular and geometric honeycomb structure. When we make an architectural pattern like this, it is through careful planning and construction, with each individual element cut to shape and laid in place. At Fingal's Cave and the Giant's Causeway, the forces of nature have conspired to produce such a pattern without, we must presume, any blueprint or foresight or design. This is an example of spontaneous pattern formation.
This sort of natural pattern was, however, perfectly familiar to scientists in the eighteenth century. In fact, it had been known for several millennia: ever since humans began to keep bees.
This [slide] is the bee's honeycomb. The ancient Greeks presumed that bees must have some intrinsic sense of geometry to build a structure like this. Pappus of Alexandria claimed that the bees had 'wisely selected' the cell shape that could hold most honey. Pliny tells us that some of the ancients devoted a lifetime's study to the mystery of how bees were able to build something so regular.
In the eighteenth century, the French physicist R. A. F. de Réaumur suggested that the hexagonal pattern is the one that uses the least area of wall, and thus the least amount of wax, to divide up a given area into equal cells. But each cell has an end cap too, and the most economical shape for this is no easy thing to work out. De Réaumur asked the Swiss mathematician Samuel Koenig to solve that puzzle, and to do it Koenig needed to use calculus, only recently devised by Newton and Leibniz. He found that the best shape-the rhombic dodecahedron-was precisely the one the bees used. Now, it was too much for these men to suppose that the bees already knew the calculus before Newton, and so Bernard Fontenelle, secretary of the French Academy of Sciences, asserted that this must be the work of God: that the bees were "blindly using the highest mathematics by divine guidance and command".
So you see, this is where a consideration of natural pattern formation tended inevitably to lead. These patterns aren't always as geometrically perfect as the honeycomb or the Giant's Causeway, but they have a clear regularity of structure to them. When we make regular structures, we have to put that regularity in by hand (examples: Kilim quilt, tiered fields, patterned borders). Pattern, in other words, seems to be the fingerprint of intelligent design. To eighteenth-century scientists, that intelligence could come only from God.
Now, this is by no means the end of the story of the bee's honeycomb, and I want to follow it a little further since it is a tale with a peculiarly Irish flavour. A honeycomb is a kind of two-dimensional foam: here [slide], for example, is what a layer of bubbles looks like. Mathematicians and scientists have long wondered about the equivalent question in three dimensions: what shape of cell should you choose in order to make a three-dimensional foam that has the minimum wall area? The eighteenth-century English clergyman Stephen Hales claimed that the cells in this ideal foam should be shaped like a rhombic dodecahedron, of which the end caps of the honeycomb cells are a fragment. But in 1887 the Irish physicist Lord Kelvin showed that a 14-sided polyhedron, the truncated octahedron, did better: [slide]. It was believed for a long time that Kelvin's solution couldn't be bettered, although there was no mathematical proof of that. But in 1993, Dennis Weaire and Robert Phelan at Trinity College in Dublin showed that there is a regular foam-that is, one whose cellular units repeat again and again, like a crystal-that has a slightly smaller surface area than Kelvin's. Here it is, and it is not exactly so elegant: this is the repeating unit, and it is made up of eight cells in all: six with 14 faces and two with 12.
I'll return to the bees shortly, and indeed to the Giant's Causeway. But at this point I need to say something more general about natural patterns.
First, I should perhaps try to define what I mean by a pattern in the first place. Or rather, to confess that I am not going to define it rigorously at all. We notice all kinds of patterns around us, not just in space but in time: the repeating patterns of night and day, of the four seasons, the patterns in a Bach fugue or in the rhythm of a trotting horse. We speak of patterns of behaviour, and it is sometimes said that the whole of science is about recognizing patterns in the behaviour of nature. I'm going to talk here primarily about patterns in space, and by that I generally mean that the space is filled with identical or near-identical elements that recur in some regular arrangement. But that can mean many things, and I can really do no more than explain my terms by example, and to assert that, even if we cannot say precisely what we mean by a pattern, we know one when we see one.
The commonality of patterns
And the most curious thing about natural patterns-indeed, the central consideration of their scientific study-is that these patterns are often the same, in systems that might seem to have nothing at all in common with one another. We have seen that already: the hexagons of Fingal's Cave put us in mind of the hexagons of the honeycomb, but why should this hexagonal pattern crop up in both cases?
We can find examples of hexagonal patterns elsewhere too: we've already seen the bubble raft, but here also is a pattern formed in a convecting fluid; and here's the pattern in a very unusual chemical reaction, where the blue and yellow correspond to regions of the mixture of chemicals that have quite different compositions. And this [slide] is one of the drawings of microscopic sea creatures called radiolarians made in the late nineteenth century by the biologist Ernst Haeckel.
Let me just show you a few other of the common patterning motifs of nature [slides]:
Stripe patterns: zebra, dunes, cloud streets
Spiral patterns: BZ, galaxy, snail shell (Note that here we've already deviated somewhat from my sketchy definition of a pattern as an arrangement of repeating elements. But I hope you'll agree that this warrants inclusion as a natural pattern. There is a very fuzzy boundary between a pattern and a form.)
Branching patterns: tree, river, lung, veins, cracks. Now these are very interesting. Are they pattern or form? Are they regular or disorderly? They don't have the geometric repetitive regularity of a honeycomb, but some of these structures do indeed turn out, on close inspection, to be made up of identical elements that recur again and again - but at different scales. We need only to make this kind of recursive branching pattern a little more orderly in order to arrive at one of the most familiar and alluring of natural patterns: the snowflake.
The big question is: do these patterns really have anything in common, or is the similarity in appearance just coincidence?
This question tantalized scientists like Johannes Kepler and Lord Kelvin, but the first person to really face it head-on was a Scottish zoologist named D'Arcy Wentworth Thompson (1860-1948) [slide]. In 1917 Thompson published his masterpiece, On Growth and Form, which collected together all that was then known about pattern and form in nature in a stunning synthesis of biology, natural history, mathematics, physics and engineering. Stephen Jay Gould has called this "the greatest work of prose in twentieth-century science". Peter Medawar went even further, saying that it is "beyond comparison the finest work of literature in all the annals of science that have been recorded in the English tongue." You'll gather from this that On Growth and Form is beautifully written, although you would need to be proficient in ancient Greek, Latin, French, German and Italian to appreciate it fully. But the book is also a profound scientific achievement, and is decades ahead of its time.
D'Arcy Thompson had a mission, and that mission was to stem the tide of uncritical and indeed sometimes unscientific Darwinism that was washing throughout every corner of biological science. By the beginning of the twentieth century, the central idea of Darwin's theory was finally becoming appreciated, and it was an idea so powerful that some biologists had seemingly come to the conclusion that they needed nothing else. For every question that one could ask about biological shape and form, there seemed to be a single answer: natural selection. Why does this creature or that plant look the way it does? Natural selection! The form has obviously been selected because it is the one that is most evolutionarily fit, the one that does the job best. It had become tempting to see nature as an infinite palette of possibilities, from which only the best were selected.
This is a simplistic way of portraying Darwinism, and it would certainly be simplistic to say that this is how biologists see the living world today. But in Thompson's time there was a sense in which Darwinism was the deus ex machina, the magical force that could be invoked to explain everything. Thompson pointed out that. No matter how ingenious and inventive evolution was, in the end an organism had to work. Its components had to fulfil their roles. This meant that they were subject to constraints imposed by physical law: the same constraints encountered by engineers. The shape of a bone or a cell or a shell can't be arbitrary, just as the shape of a bridge or a skyscraper can't be arbitrary. Form, to put it in the slogan of the Bauhaus, follows function.
That's why, for example, the iguanodon and the kangaroo have a long heavy tail: as a counter-balance for the long heavy body and neck. It's why the metacarpal bone of a vulture's wing resembles the Warren's truss, the engineering structure familiar from cranes: this is the ideal way to stiffen a long strut without making it too heavy.
Thompson showed that Haeckel's radiolarians, with their delicate mineral exoskeletons, can be related to foams, and explained on the assumption that the mineral is precipitated at the intersection of bubble-like organic vesicles.
What's more, Thompson, said, biological form and pattern is not just a question of static mechanical engineering: these things have to grow. A bridge doesn't have to expand in all directions, but a plant does. 'Everything is what it is', he said, 'because it got that way.' That's why, after all, the book was called On Growth and Form. The shell of the Nautilus mollusc is a logarithmic spiral because that is the shape you get from steady growth along the leading edge. It is a shape that retains exactly the same form while getting bigger and bigger through growth at only one place.
'On Growth and Form' puts the basic issue of pattern formation in a nutshell. That's to say, some patterns are determined by 'form': they are static equilibrium patterns, which have to be mechanically stable. A foam is like this, to a first approximation. But most patterns involve growth or movement: they are dynamic, non-equilibrium structures. In particular, all of biology happens out of equilibrium, because that's a defining characteristic of life. Equilibrium is death. Organisms grow by taking in energy and matter from their environment, organizing it, and excreting the waste. I'll come back to this at the end.
The engineer's perspective on biology that D'Arcy Thompson provided was sorely needed. He was one of the first to call clearly for biology to use mathematics. Today many biologists do that without a second thought. But many others don't. There is still some suspicion and dare I say fear of maths in biology: so-called 'theoretical biology' has been deemed a faintly disreputable subject. I think that even the most entrenched molecular biologist is starting to realise that now, faced with an increasing mountain of increasingly quantitative data on genes and their interactions, this has to change.
But it's instructive to see how Thompson wasn't always right. He was desperately keen to rationalize the hexagonal regularity of the bee's honeycomb by appealing to simple physical forces: to surface tension pulling the soft wax into shape just like a bubble raft. Biology, however, is more complicated. We now know that bees do make the comb like so many construction workers erecting a building: they place each piece of wax carefully, using special organs to measure the angles relative to a plumb line defined by gravity. They use another set of organs to engineer the thickness of the cell walls very accurately, to a tolerance of two thousandths of a millimetre. They even manage to construct the correct tilt angle of the cell channel, which slants down at 13 degrees below the horizontal so that the honey doesn't run out. When it comes to the living world, we'd be unwise to assume that all patterns are the blind, spontaneous creations of physics.
All the same, we can explain some of nature's most striking biological patterns without recourse to this sort of painstaking assembly. Many animals show prominent surface markings [slide], which are used for all kinds of purposes: species recognition, camouflage, warning signals, mimicry, sexual selection. Some of these patterns are quite complex, but many are based on two simple motifs: spots and stripes.
These two patterns can be generated in inorganic mixtures of chemical reagents, and it's thought that chemical processes of this sort lie at the root of the biological patterns too.
Appropriately enough, the discovery of chemical pattern formation stemmed from studies into the chemistry of living organisms, specifically metabolism. In the 1950s a Soviet biochemist named Boris Belousov devised a chemical reaction that he believed would mimic the breakdown of glucose by enzymes. His reaction mixture changed between yellow and colourless as the reaction proceeded. But it didn't just do this once: it kept on changing back and forth. In other words, the reaction seemed to go first in one direction and then in reverse. It seemed to oscillate.
Other chemists didn't like this at all, because it seemed to violate the Second Law of Thermodynamics, which assigned a preferred direction to every spontaneous process. So Belousov found that he couldn't publish his results anywhere prominent, and had in the end to sneak them out in a conference proceedings. It wasn't until Belousov's reaction was revisited by Anatoly Zhabotinsky at Moscow State University in the 1960s that anyone began to take them seriously. Zhabotinsky found a way to make the colour change more apparent: from red to blue. The so-called Belousov-Zhabotinsky (or BZ) reaction basically involves the bromination of malonic acid by potassium bromate in acid, catalysed by cerium ions. An indicator compound called ferroin turns red in the presence of cerium(III) and blue in the presence of cerium(IV). The oscillating colour change shows that the cerium oscillates between these two oxidation states [slide].
If the mixture went on doing this indefinitely, it would indeed violate thermodynamics. But it doesn't. After a while, the oscillations go away and the mixture settles into an equilibrium state in which malonic acid is brominated. But it takes a long while to get there, and seems to keep changing its mind en route. What really happens is that there are two competing reactions taking place, with one or the other dominant at any one time. Each reaction contains the seeds of its own destruction, exhausting itself and creating the conditions necessary for the other reaction to take over. Crucially, the BZ reaction autocatalytic, rather like a chain reaction. This helps to make the colour switch rather sudden.
Thermodynamics tells us nothing about the progress of the reaction, but only about its end state. So the fact that the oscillations eventually die away means that the Second Law is preserved. The oscillations are maintained only so long as the system is out of equilibrium. We can keep it this way, however, by continually feeding in fresh reagents and carrying away the end products. This can be done in a vessel called a continuous stirred-tank reactor or CTSR. Then the oscillations can be sustained indefinitely.
The CTSR is rather like a living organism, constantly devouring raw materials and spitting out wastes. While this flow of matter persists, the system can maintain its out-of-equilibrium pattern. In fact, metabolism is indeed like this, involving oscillatory biochemical processes. Belousov was more right than he realised.
If the mixture is stirred, the colour change takes place more or less everywhere at once. But if the mixture is left unstirred in a Petri dish-or better still, if it is infused into a gel to slow down the rate at which the reagents diffuse-then differences in composition can arise from place to place. Because of the autocatalysis, a small initial difference in concentrations at two points can become blown up into a substantially different outcome: the reaction can be proceeding on one branch (red, say) in one place while it is on the other branch (blue) elsewhere. This means that the reaction mixture becomes inhomogeneous. What's more, the oscillations produce a series of chemical waves emanating from some point, creating a series of concentric target-like circles, like ripples on a pond [slide]. The target patterns can be mutated into spirals by disturbing the wavefronts: for example, if they hit some obstacle at some point in the medium [slide].
All this became apparent in the 1960s and 70s, when the significance of Belousov's results were appreciated thanks to Zhabotinsky's efforts. It was then that the link was made to theoretical work in 1910 by a mathematician called Alfred Lotka, who showed how a chemical reaction might sustain oscillations. This work led William Bray in California to actually find a chemical reaction that showed Lotka's oscillations. But no one believed Bray either, and the chemistry community forgot his and Lotka's contributions. In other fields that was not the case, as we'll see shortly.
The details of the BZ reaction are very complex: the reaction schemes used to describe the process theoretically invoke many different intermediate chemical species. But there's a much simpler way to describe these patterns that doesn't need to take account at all of the specific chemical agents involved. Instead, it simply assumes that each point in the medium can exist in one of three different states: receptive, excited and refractory. The medium can thus be divided up into lots of tiny cells, each of which is in one of these three states. This is called an excitable medium.
A receptive cell is liable to become excited, once it receives an appropriate stimulus. A refractory cell is one that has become excited and exhausted itself, and must now remain quiescent for a certain period before it regains its receptiveness. So the cells can cycle from receptive to excited, refractory and back to receptive. Whether a receptive cell becomes excited or not depends on what state its neighbours are in: if enough of them are excited, that cell becomes excited too. But an excited cell must eventually become refractory. These, then, comprise the set of simple rules that determine the state of each cell, based on the state of their neighbours. This model of an excitable medium is called a cellular automaton.
We might imagine the excited state, for instance, to correspond to the 'red' branch of the BZ reaction, and the receptive and refractory states to correspond to the blue branch. Computer simulations of the cellular automaton model then show that it can develop target and spiral wave patterns, just like the BZ mixture.
There is another way of describing the BZ process too. Its behaviour depends on the rate at which the chemical species react, but also on the rate at which they diffuse through the medium so that regions depleted in one or other reagent can be replenished. So chemical waves are a consequence of a reaction-diffusion process. The precise behaviour depends on the relative rates of reaction and diffusion: the medium is only excitable for certain ratios of these rates.
Expressing the pattern-forming BZ reaction in these general terms has the attraction that it allows us to see what are the general requirements for pattern formation in such systems-for example, the competition between reaction, that is consumption, and diffusion, that is, replenishment-and to make connections with other kinds of chemical pattern formation. The banded patterns seen in some minerals [slide] and the formation of chemical striations called Liesegang rings, which D'Arcy Thompson noted but could not explain, are examples of reaction-diffusion processes. And in 1952, the mathematician Alan Turing identified another hypothetical chemical reaction, now seen to be another reaction-diffusion process, that could lead to stationary patterns.
You see, the chemical waves in the BZ reaction are always moving. But Turing showed-without of course knowing anything about Belousov's work-that under certain conditions, a chemical mixture might produce a static pattern, even though all the molecules in the mixture were continually diffusing through the medium. Turing was hampered from taking his analysis very far by the fact that the computer, whose theoretical principles he helped to define, had not yet been made. But in the 1970s the theoretical biologist Hans Meinhardt in Tubingen helped to clarify the generic characteristics of Turing's stationary chemical patterns. He and his colleague Alfred Gierer showed that one needed two ingredients in such a system: an activator and an inhibitor. The activator is a species that catalyses its own formation, while the inhibitor disrupts this autocatalysis. Both species diffuse and slowly degrade. Meinhardt and Gierer showed that, for a certain ratio of diffusion rates, this activator-inhibitor system develops Turing patterns. Specifically, the activator must assert itself over only short distances, while the effect of the inhibitor is longer-ranged. What this means is that islands of activator then appear, separated from one another by a roughly equal distance.
It turns out that this activator-inhibitor scheme creates two kinds of generic pattern: spots and stripes [slide]. This was known theoretically from the 1980s, but it wasn't until 1990 that researchers at the University of Bordeaux found a real chemical system, called the CIMA reaction, that was capable of developing stationary Turing patterns. The following year, Harry Swinney at Texas succeeded in growing large patches of chemical Turing patterns [slide].
Spots and stripes, then, are the generic motifs of stationary chemical patterns. Could these be responsible for the markings on animals like the leopard and the zebra? That seems quite likely. It's thought that pigment-inducing chemicals called morphogens might diffuse through the skin of the developing embryos and lay down their markings in the womb. The mathematical biologist Jim Murray has shown that many animal patterns can be rationalized by the formation of Turing-like structures on bodies or limbs of different shape and size. Even the reticulated-net markings of the giraffe have been reproduced using theoretical schemes like this. But no one has yet been able to identify the morphogens involved.
Many different patterns seen on sea shells have been mimicked by activator-inhibitor schemes devised by Hans Meinhardt. You can see that some of these are quite different from the standard spots and stripes, because the patterns evolve along the continually growing edge of the shells. But the best case for a chemical Turing pattern causing an animal marking pattern has been made for the angelfish, where the pattern is not simply laid down at birth but continues to evolve as the animal grows. For example, as the Pomocanthus semicirculatus grows, its stripes don't grow with it but stay spaced about the same distance apart: the fish just has more of them. And the stripes on the Pomocanthus imperator shift and merge as the animal gets bigger. Two researchers at Kyoto University in Japan have been able to mimic these changes using an activator-inhibitor model.
More recently, a group in Taiwan has shown how the formation of Turing patterns on curved surfaces can account for the marking patterns seen on the wing covers of ladybirds [slide] [S. S. Liaw et al., PRE 64, 041909 (2001)). It's natural also to ask whether the same process is at play in the complicated markings of butterflies, which are often more complex than mere spots and stripes. For example, these coloured markings sometimes reflect the vein structure of the wing. Single eyespots are common, as opposed to extended arrays of spots. It seems likely that the general idea of diffusing morphogens is at work here, but the details are more complicated than in the case of 'pure' Turing patterns: to explain some of these markings, for example, it might be necessary to construct activator-inhibitor models in which the diffusing species have several sources and sinks at certain points on the wing or around the periphery of the wing panels. On the other hand, some of the specific genes and proteins responsible for these markings have been isolated: for example, a gene called Distal-less is responsible for eyespot markings in some species. In general, butterflies seem to draw on a certain range of different motifs that diffusing morphogens can generate. In other words, there is probably an interplay here between natural selection and the limited range of possible patterns that activator-inhibitor mechanisms provide. Clearly, evolution is very ingenious at making use of this restricted palette, using it to create all kinds of effects such as camouflage and mimicry.
That inventiveness is illustrated here [slide of 'butterfly alphabet']. Interestingly, a couple of Hungarian researchers have recently shown how, by manipulating chemical Turing patterns, they can write out the entire alphabet in activator-inhibitor patterns [slide].
The stripes of the angelfish or the zebra might put you in mind of something else [slide]. These are patterns in windblown sand; you see something very similar, of course, in sand lapped by waves at the water's edge. Notice that there's a characteristic way in which these stripes end or merge with one another, which we see both in the animal markings and the ripples. In fact, Hans Meinhardt has argued that the formation of sand ripples can be itself regarded as a kind of activator-inhibitor process. What happens is that the wind picks up the sand grains and transports them in a certain direction. If they hit a protrusion as they blow over the ground surface, they may stick. The higher the protrusion, the more it accumulates sand: there is positive feedback here, or autocatalysis, characteristic of the process we've called activation. But that extraction of sand grains from the air at a bump means that they are less likely to be deposited only a short distance from the bump: each bump inhibits the formation of another bump close by. So the result is a succession of ripples spaced an even distance apart.
Well, this is basically what goes on, but the real process that creates sand ripples and dunes is more complicated. In fact we have to take account of the way the grains get deposited: they don't just stick where they strike, but perform a series of small hops called saltations before coming to rest. This creates the characteristic profile of ripples, with a shallow windward slope and a steep leeward slope [slide].
And there are in fact several patter-forming processes going on in the desert, which create structures of different sizes. Sand ripples are small - typically only a few centimetres high. Dunes are much bigger: they can be a few metres tall, and the ridges are typically tens to hundreds of metres apart. Some sand features are even bigger: dune-like structures called draas may be up to several kilometres apart.
The formation of dunes from wind-blown sand has been studied a lot. Dunes aren't just stripey: they can form various shapes, such as crescents (called barchan dunes) and stars. What's more, the ridges don't always lie across the wind direction; sometimes they can be parallel to it. Bradley Werner at the Scripps Oceanographic Institute in California has devised a computer model of how dunes form in which he makes certain assumptions about how sand is picked up and laid down by the wind. He has shown that different wind conditions create different dune patterns. If it blows steadily in one direction, you tend to get transverse dunes, but if it is variable, the dunes can be longitudinal; and other conditions lead to barchan dunes [slide].
Very recently (last month), Werner and Kessler explained a different kind of geological patterning in grainy media: the formation of patterned ground. This happens in cold regions such as Spitsbergen here in Norway [slide], where stones can gather spontaneously into rings and polygons. Some explanations for this have invoked convection in the frozen ground as it thaws - convection is known to produce patterns, as we saw earlier. But the model of Werner and Kessler doesn't need convection. Instead, it looks at the effects of frost heaving, by which the expansion of soil as it freezes can sift large stones up to the surface. It seems that a combination of this process and the effect of gravity when a hillside slopes can result in a variety of different self-organized structures, in which the stones become focused into mounds (spots), stripes, rings and polygons.
Grainy materials have a strong propensity to form patterns. I've got an example here [demo of stratified landslides in sand/sugar].
This is in fact not at all trivial to explain. Basically it seems to result from the fact that the two types of grain have different angles of repose - that's the steepest angle the slope can form without incurring a landslide. But it's also crucial that the grains here have different sizes: you need both of these things to get stripes. And even that's not the whole story, because the precise spacing of the stripes depends on how fast you pour: so there are dynamical factors involved here: how the grains tumble, and what happens when they collide.
People have been pouring grains and powders for decades - centuries even - and so it's rather astonishing that this pattern-forming process apparently wasn't noticed until 1995. In fact, I can't resist showing this picture from Per Bak's book 'How Nature Works', which is all about self-organized criticality. I don't have time to go into it for those not familiar with it, but suffice to say that for sand piles, avalanches happen on all scales as you pour more grains on top, from just a few grains rolling to the whole slope sliding down. So the idea is that there is no intrinsic size scale in the system. So what, then, are these regularly spaced stripes that you can see in this sand pile?
Harry Swinney and his group in Texas found another pattern-forming process in granular media several years ago, when they started shaking shallow layers of tiny metal spheres in a vacuum. They found that the layer of grains became organized into these wave patterns, which showed the now-familiar structures of spots and stripes. Some of the spots are regularly packed in hexagonal lattices, some in square lattices, and some are disordered. Each of these spots is in fact a kind of dimple of rising and falling grains, captured at one point it its trajectory by strobe lighting in these photos. In fact, Swinney's group found that these little waves can occur in isolation - here's one [slide] - which they called oscillons. These lone waves can move around in the granular layer, and when they encounter one another they sometimes stick together, like atoms forming molecules [slide].
I want finally to move onto a very particular kind of grainy medium: one in which the grains consist of living organisms. Here [slide] are some examples of patterns that form in colonies of bacteria and other single-celled organisms being grown in a Petri dish.
i. Slime mold Dictyostelium discoideum: these cells aggregate
when under stress (lack of food, say), and in the initial stages of
that process they do so by forming these target and spiral waves,
just like a BZ reaction mixture.
ii. Patterns formed by E. coli under stress (e.g. lack of nutrients,
coldness, too much oxygen). These are stationary patterns, unlike
the travelling waves of Dictyostelium.
iv. Chiral patterns in Bacillus.
What's going on here? Unlike particles of sand, bacteria have their own source of propulsion: they have little flagella that whip around and can act as corkscrew-like propellers to drive the cells in a certain direction. When they are under stress, these cells find that they can increase their survival prospects if they act collectively. This means they have to start talking, and they do that by emitting chemicals that act as attractants, rather like animals emitting pheromones to attract a sexual partner. The cells are genetically programmed to start moving up the gradient in the chemical attractant, towards the source. This is called chemotaxis, and it gives rise to this amazing variety of patterns.
Well, bacteria are pretty primitive things, but higher organisms can show all kinds of organization too. These [slide] are spatial patterns showing the distributions of predators and their prey in a computer model of the interactios between them. Strictly speaking, the model simulates interactions between parasitoids and their hosts. Parasitoids are particularly nasty parasites that lay their eggs in their host's body, so that when the eggs hatch, the larvae devour and kill the host. This kind of thing goes on all the time in the insect world. The parasitoids are attracted to the hosts, but there is effectively a kind of repulsion between parasitoids, because if there are too many of them in one place then the hosts die out and there's nothing for the parasitoids to live on. This interplay of attraction and repulsion causes the kinds of complicated distributions seen here.
We can see some striking patterns in even more advanced organisms: the swarms that are formed by schooling fish and by birds. The interesting thing about these structures is that they are dynamic: the fish are all moving, yet they manage to maintain their collective form. We've all seen birds swooping and diving together at dusk, and it has been a long-standing puzzle how they manage to do it. How do they stay in formation? Are they following a leader? That's what some ornithologists used to think, but no one could ever identify which bird the leader was. And in fact it became clear that the birds' reaction times just weren't fast enough for them all to be responding to a single leader.
It's only in the past few years that enough understanding has accumulated about collective behaviour to explain swarming. All you need, it turns out, are rules governing the local behaviour of the birds. That's to say, each bird needs only to respond to its near neighbours, manouevering so as to stay close to them and to keep moving in the same average direction but to avoid collisions. Here's a snapshot of a computer model that invokes rules like this. If you set loose a group of virtual birds programmed to behave this way, they form a flock, moving in concert as if they were all being controlled by a single mind.
Programmes like this have been so successful in mimicking flocking motion that they have even been used in the movies: there's a swarm of bats in 'Batman Returns' that was created this way, and so was the herd of stampeding wildebeest in the Lion King.
Incidentally, Tamas Vicsek in Budapest has shown that the switch from disorderly, independent motion in a flock of particles to concerted, orderly movement can be shown to be formally analogous to the transition between a magnetically disordered and a ferromagnetic state in the XY model of magnetism. You can induce the switch in the flocking simulation by increasing the noise - the degree of randomness in the particle motions, which is equivalent to the temperature in the XY model. The two aren't exactly equivalent, of course, because one is an equilibrium process and one is a non-equilibrium process; but it's possible to show that the two are formally related.
Well, if birds form patterns then you might reasonably ask whether we do too. And of course we do. Dirk Helbing at Stuttgart (and now Dresden) has devised a model of pedestrian motion that captures the way people move around a space. This model shows all kinds of regularities emerging, even though each walker is programmed to follow his or her own agenda in moving from one place to another at a preferred speed. For example, people form spontaneously into counterflowing streams when they move in opposite directions down a corridor [slide]. And Helbing and his coworkers have been able to account for the shapes of the trails that people make when they walk over open spaces.
For example, when people cross regularly over a grassy space, you can see worn-down trails emerge in the grass. No one has planned these trails: they are a self-organized product of many footfalls. Here's one on the campus of the University of Stuttgart [slide].
The curious thing is that these trails are often not the most direct route between any two common entry or exit points. Yet people stick to them. Exactly why we follow these trails isn't clear, nor is it too important: perhaps it might be slightly easier to walk on the flattened ground than on the grass, or maybe we're just used to following a predefined path even if its clear that it is not an 'official' path. The fact is that we do it. In his model, Helbing assumed that people have a certain tendency to walk where others had walked before, that they wear down the grass at a steady rate when they do so, and that it grows back at a steady rate if a trail becomes disused. With these ingredients, his computer model showed how characteristic, gently curved trails emerge spontaneously. Again, notice that these trails don't connect the entry and exit points by the most direct path - they are a kind of compromise between the different routes that different people might take [slide].
Now, I haven't even got time to talk about the difficult issue of whether there are universal principles behind the formation of the many different patterns we see in nature. In brief, however, I can say simply that there are not, to the extent that there is nothing comparable to the minimization principles that govern equilibrium states. Non-equilibrium patterns are dictated by many factors, such as boundary conditions, fluctuations, the formation of defects, and the past history of the system. Often, what you end up with depends on precisely how you got there, so that some patterns can be frozen accidents of history. Let me say that there are several analogies between equilibrium phase transitions and non-equilibrium pattern-forming processes, but many important differences too.
And there are all kinds of stunning natural patterns I've not been able to cover, though I can't resist just showing you a few of them:
Viscous fingering, snowflakes
Golden spirals, phyllotaxis]
But I do, before I finish, have to complete the story of the Giant's Causeway. The most convincing explanation I've seen for this was published only last year. Whenever a layer of material cracks because it contracts everywhere at once, like a muddy lake bed drying out or a layer of hot rock cooling, it forms roughly polygonal cells [slide]. But it seems that as molten rock solidifies from the topmost layer downwards, the stresses redistribute these cracks so that they get progressively more geometrical and regular, eventually forming more or less perfect polygons that have predominantly six sides. I'm struck that D'Arcy Thompson wondered about Fingal's Cave and the Giant's Causeway; the fact that we're still trying to explain it now reassures me that there is still plenty to be uncovered about nature's patterns.