The Bactra Review: Occasional and eclectic book reviews by Cosma Shalizi   67

The Self-Made Tapestry

Pattern Formation in Nature

by Philip Ball

Oxford UP, 1999

The Blind Snowflake-Maker

Nobody looks at a snowflake and thinks of a snowflake-maker, of someone who deliberately set every bit of ice just so. Nor, looking at a sand-dune, do we imagine a dune-maker. Most of us have even gotten over vortex-makers, though really spectacular ones (like hurricanes) tempt us. We accept as a fact of life that pattern happens. But why, or rather when and how? It's also a fact of life that uncoordinated forces usually just make a mess --- why do they sometimes lead to regular, even elegant patterns instead? In physical contexts, this is called "pattern formation" or "self-organization." The former term is tied to nuts-and-bolts questions --- what distinguishes pattern-forming situations from the mess-making ones? what are the mechanisms through which patterns form? when are these mechanisms at work? how do they connect to fundamental physics? --- rather than the more nebulous ones about Emergence and Order and Meaning associated (perhaps wrongly) with "self-organization," so we'll stick to the former. (People also speak of biological development as "pattern formation," but here we'll use the slightly old-fashioned name "morphogenesis" to avoid confusion.) Pattern formation has, mostly since the 1960s, become a large, growing and increasingly important part of modern physical science, one relevant to everyday phenomena and, even without the nebulosities, tied to Deep Issues. Despite this, despite even the opportunity for really neat pictures, there hasn't been a single decent popular book on it, one which explains the science adequately in lay terms and leaves out the metaphysico-theologo-cosmolonigology. Ball's book (need I say it?) fills this distressing gap, and does so almost perfectly.

The science, to start with, is all up-to-date and correct. The descriptions of particular mechanisms and processes are clear, making effective use of diagrams and analogy with everyday physical experience, with (despite fears raised by the title) next to no reliance on metaphor. Though Ball describes a great many cellular automata rules which have been used to model these processes (with enough detail to implement them; a collection of such programs would've made a nice accompaniment to the book), his emphasis is refreshingly on things in Nature or the lab, and well-developed physical theories, rather than on computer models which are merely reminiscent of them. More abstract concepts, like symmetry, symmetry-breaking and minimization principles are also handled well, but very briskly --- the emphasis is on the physics (and the chemistry!), and curious readers are referred to Weyl's classic on symmetry, and Stewart and Golubitsky's classic-to-be on broken symmetry. (All this is exemplary.)

The coverage of particular pattern-formers, while not quite comprehensive, comes very close. Only two --- Rayleigh-Bernard convection, and the Belusov-Zhabotinsky reagent --- are commonly discussed in the non-specialized literature. They're here, but so is lots of other stuff. Chapter two covers surface and interface phenomena, from soap-bubbles through heads of beer to the (often literally) convoluted structures built by surfactant molecules in three dimensions, and no less strange effects achieved in Langmuir films, effectively two-dimensional soaps. (This chapter also covers honeycombs, and introduces the natural selection vs. pattern formation theme, of which much more below.) Chapter three is on chemical oscillators and waves (including the aforementioned BZ reagent, and the melancholy story of Belusov), and their ties to other "excitable media," like some kinds of combustion, the growth of certain species of bacteria in culture media, heart muscle and nerve cells, and even one phase in the life-cycle of the remarkable cellular slime mold Dictyostelium discoideum. Chapter four brings together more biological instances --- mammalian coat-markings and molluscan shells (both interpreted as "activator-inhibitor" systems, using ideas originally going back to Alan Turing), the wings of butterflies and the body-plans of fruit flies. Chapter five discusses "dendritic growths" --- mineral dendrites, snowflakes, yet other kinds of bacterial colonies, and other things which grow outward by spikes and branches. These are very similar to crack and fracture patterns, the subject of chapter six, and may be related to river-basins. This brings us to the subject of fluid flow (chapter seven), where we get to look at convective flows (like Rayleigh-Bernard convection) and eddies and turbulent flows (which are, famously, a fabulous mess). Chapter eight is on granular materials like sand, rice, and mixed nuts, which can also flow, and finally chapter nine discusses spatial ecology, or anyhow models of it. I am greedy enough of to want more --- liquid crystals, self-assembly, and "birds and bees" topics like flocking and the construction of termite mounds --- but not too much more.

Ball's general approach, in these chapters, is to open with some striking phenomenon, e.g., mineral dendrites, which look so organic that they're often mistaken for fossil plants, say a little bit about why it is puzzling (the dendrites are made of crystals, normally by-words for rectilinear regularity), and then go on the mechanisms which produce them, or anyhow are thought to do so. (He is good at indicating just how much is conjectural in each case.) The mechanics are often reasonably involved mathematically, but Ball does an excellent job of getting the "essential physics" (i.e. the hand-waving bits) across. Dendrites, for instance, grow by a process of accretion, with randomly moving particles attaching themselves to the cluster at its edges.

It isn't hard to see why this model produces very imperfect, irregular clusters, since aggregation takes place following random diffusion. But why are the clusters branched? We could perhaps imagine instead the formation of a dense mass with a highly irregular edge, like a spreading ink blot. Why is this not what happens?

The answer is that the model possesses an instability that amplifies any small bumps or irregularities, causing them to extend into fingers rather than becoming smoothed out again.... [I]t's not hard to imagine that a particle taking a tortuous, meandering path through the surrounding medium is likely to encounter one of the branch tips before being able to penetrate very far down the channels between them. So once they are formed, the branches tend to grow from their tips while the gaps in between them get even less accessible to new particles.

Preferential growth at a tip ensures that any tiny bumps formed by chance at the cluster surface will have a tendency to grow faster than flat parts of the surface, because there is a better chance that a randomly diffusing particle will hit it. And crucially, this growth advantage is self-enhancing --- the more the bump develops, the greater the chance of new particles striking and sticking to it. The probability of this is always greatest at the very tip of the bump, since this is the most exposed part. So the slightest small bump soon grows into a sharp finger. Because irregularities are springing up by chance all over the surface all the time, the deposit becomes increasingly branched, with each new tip constantly sprouting extra appendages. [p. 114]

(As Ball goes on to say, it's crucial that the particles move randomly. A low density of diffusing particles is also crucial, otherwise the law of large numbers comes into play and swamps the irregular growth.) Having explained the key mechanism in such a way, he proceeds to other phenomena where a similar process is at work --- deposition of metals about an electrode, forcing one fluid into another, more viscous one, the accretion of snowflakes about an initial seed of dust. Along the way, he explains how the mechanism must be changed to accommodate the peculiarities of these different systems; also the physical reasons for their having such similar dynamics. This raises the question of why there are such similarities in the behavior of things at least as different as chalk and cheese. Like Ball, we'll save that question for the very end, after we've looked at how, if at all, pattern formation connects to biology --- one of the promised Deep Issues.

I began by talking about how the idea of a snowflake-maker is one which is repugnant to our understandings. On the other hand, we are reluctant to not infer a watch-maker from a watch, and historically nothing has been easier than to proceed from an organism to its Maker. We find, let us say, a fluke (Bilharzia mansoni) in our neighbor's intestine. We observe the elegant simplicity of its oral and ventral suckers, the precision with which its genital pore finds the veins leading to the kidneys and the efficiency with which it then expels eggs (up to three hundred a day), the speed with which its cercariae penetrate human skin to colonize a new host. We conclude that here, manifestly, is a work from the Hand of a Great Artificer --- in a word, we infer a Fluke-Maker from flukes. As David Hume pointed out, there is nothing to say that there is only one Fluke-Maker, or that the Fluke-Maker is the same as the Tapeworm-Maker or the Celery-Maker or (the real point) the Maker of People. But still, what, after all, is the alternative?

Well, the one we now subscribe to for people, flukes, tapeworms and celery is natural selection. It is, in Dawkins's famous phrase, a blind watchmaker, which is good (at least for us infidels), but it's still a watchmaker, shaping our ends (and means), rough-hew them though variation will. It's tempting to ask if there's a way of avoiding both fluke-makers and watch-makers. Might not something even blinder --- more mechanical, more mindless, more unclubbable --- than natural selection yet be able to create patterns and organization?

Enter snowflakes. Enter, also, D'Arcy Wentworth Thompson, who in 1917 published a book, On Growth and Form, which has haunted all discussion of these matters ever since. Thompson's aimed to show that huge chunks of biology are simply the consequences of physics and (less often) chemistry. When he wrote that "the form of an object is a `diagram of forces,' in this sense at least, that from it we can judge of or deduce the forces that are acting or have acted upon it," he meant forces. His accounts of the physics behind morphogenesis were ingenious, extremely elegant, very convincing and, significantly, aimed at very large features of the organism: the architecture of the skeleton, the curve of horns or shells, the outline of the organism as a whole. Most of us are resigned to abandoning biochemical details to crawling molecular chaos, but these are supposed to be more mysterious and inspiring affairs. Thompson tried to explain them using little that a second-year physics undergrad wouldn't know. (Thompson's anti-reductionist admirers seldom put it this way.) In particular, Thompson made a point of not invoking natural selection, indeed of leaving any kind of history out of the story. "A snow-crystal is the same today as when the first snows fell": so, too, the basic forces acting upon organisms, so why bring history into it? The early years of this century are littered with biologists with little use for natural selection; they are now almost all deservedly forgotten. Thompson owes his continuing influence to the fact that his alternative doesn't beg questions at every turn. (Also, of course, he wrote beautifully, better than the poets of his day.)

Since Thompson's day, then, there has been a tension in the study of morphogenesis between evolution and (other kinds of?) self-organization, and this is one of Ball's themes, though not the leading one. Partly it is an argument about logical and theoretical questions --- what is natural selection competent to explain? what features of organisms could not be modified by selection? to what extent is self-organization unavoidable? --- and partly it's about where the balance between self-organization and evolution lies in actually existing organisms.

The case for the self-organizers can be put very strongly, at least for multicellular organisms, for metazoans. These are not bloated sacks of protoplasm but (as the biologist say) "differentiated" --- there are different chemicals in different parts of the body. This in itself could be achieved without much effort; shake up a bunch of marbles in a box and there will be more of one color of marble in one corner, more of a different color in another corner. But mere chance won't give us enough differentiation, and it won't give us a reliable pattern of differentiation; it would be as though every fertilized egg was equally liable to turn into a frog, an oak, Bilharzia or something out of Lovecraft. So there has to be some particular differentiating influence. It cannot be the genes (on which natural selection acts), since genes only encode information about proteins, i.e. about what chemicals to make, not where to put them. So it would seem that differentiation, morphogenesis, must be due to some internal process, some reaction of the proteins and their associated chemicals which sorts out what goes where; but this is to say that there needs to be spontaneous pattern formation, that development must be self-organizing.

It may, admittedly, look like we're in trouble with some obvious facts, that this argument leaves genes and natural selection with no purchase at all on morphogenesis. But not even the most enthusiastic of the self-organizers, the ones with the least use for Darwin (e.g., Brian Goodwin) goes that far. It's very clear that developmental traits can be inherited. Individual mutations can cause the organism to grow six well-formed fingers on each hand (in human beings), or legs in place of antennae (in fruit flies); somehow changes in one protein have to ratchet up into large modifications of the organism. One way to achieve this is to recognize that the same mechanism can form many different patterns depending on its parameters, which, in the case of morphogenesis, will involve the chemical properties of the proteins and protein-products in the embryo, which are just what the genes control. The genes twiddle the knobs, so to speak, and then let self-organization do its voodoo.

This is a pretty convincing line of argument; at least, I'd like to think so, since it convinced me for years. No longer; let me try to say why with a fairly concrete example. There is, as I mentioned, a very nice theory about the markings of mammalian coats, developed by Hans Meinhardt and his collaborators. It explains them through spontaneous pattern formation, using some ideas invented by Alan Turing in the 1950s, and, depending on the parameters of these "activator-inhibitor" systems, they can give you tiger-stripes, or leopard-spots, or even giraffe-blotches, with controllable size, density, wiggliness, etc. Suppose we established that this is really the mechanism at work; could we then finally close the books on how the leopard got its spots? No: it is a conspicuous fact (or, rather, an inconspicuous one) that tigers are tiger-striped, while leopards and leopard-spotted, and there is nothing in the activator-inhibitor story to say why this should be so. It's not enough for a pattern to form, it must also function. Pattern formation is in general unable to create adaptations, to produce something which fits with the organism's environment. The case of tiger-stripes and leopard-spots could be resolved by appealing to parameter-tuning, as above, but this doesn't go far enough. There are, after all, many, many different pattern-forming mechanisms, producing many different sorts of patterns. Assuming an organism employs such a mechanism, why that one rather than another, producing very different patterns? The only answer, so far as I can see, is that, in the organism's environment, the patterns it makes are more adaptive than those of rival mechanism. (Lipids are just as capable of forming open "plumber's nightmare" surfaces as closed vesicles; cell walls are vesicles because it helps an organism if it doesn't leak all over creation.) So even the kind of self-organization which happens in morphogenesis is under evolutionary control.

If, that is, there is any self-organization happening at all. We've been tacitly arguing as though the embryo really was a featureless sack of protoplasm in an equally featureless environment; an assumption, admittedly, that physicists find it very natural to make. Nonetheless, it is not so; as Ball nicely puts it (p. 101) "broken symmetry is passed from generation to generation," often by reasonably complicated means. Mother Nature was not, alas, schooled at the Bauhaus or MIT, and so cares less about elegance than cost-effectiveness, and is perfectly happy to do something in a complicated, messy way if it's cheap at the price. (This is part of why charts of metabolic reaction pathways cover whole walls in biology labs.) There is no reason to think that it's generally cheaper or more reliable to use self-organization than not, and so no reason to expect to encounter it very often in morphogenesis. This is in line with a final conspicuous absence. The experimental study of biological development is more than a hundred years old now, and advances in it fill scores of fat journals every month. In all this vast wealth of detail, there is not a single case where a kind of self-organization proposed by theorists has been confirmed (though there are a few likely-looking candidates), and many cases where self-organization is definitely known not to take place. As the old joke says, "If it's slimy it's biology, if it stinks it's chemistry, and if it doesn't work it's physics."

This doesn't mean, of course, that the whole exercise has been a waste of time, much less that there is no role for theory in developmental biology, that the current find-a-gene-sequence-it-and-move-on mania is the last word on the subject. Between the DNA and the extra fingers there are a whole host of biophysical problems from the shapes of molecules to the mechanical properties of muscles which we need to solve before our knowledge of morphogenesis will be reasonably complete. As a theoretical physicist interested in biology and anxious about long-term employment, I find this comforting.

Still, this rejection of the advances of the profession rankles; let us abandon it for the more pleasant topic of Ball's concluding chapter, that of general principles governing pattern formation. This is the question, already raised, of why it is that things which are so very different from each other physically proceed along very similar routes to very similar patterns. Ball retells the orthodox Instability Story, following Mark Cross and Pierre Hohenberg (with asides from the admirable, and admirably acerbic, Rolf Landauer). Briefly, it goes like this. We imagine our favorite pattern former before it has formed any pattern --- featureless, homogeneous, and consequently symmetrical (since there is nothing to distinguish one part of it from another). But this can only be on-average homogeneity; there will always be random fluctuations, more or less pronounced, with different spatial structures. In one range of the values of the parameters (representing, here, the properties of the materials and the environmental conditions), these fluctuations will all tend to decay and die off, regardless of their shape; we say that the featureless configuration is a stable state. In another parameter-region, certain kinds of fluctuations will actually be amplified, the system will assume the shape of one of the faster-growing fluctuations, and we go from the now-unstable featureless state to a newly-stable patterned state; the symmetry will be broken. (There will actually be many stable patterned states, as many as there are ways of breaking the original symmetry; which one is attained is in the lap of the gods.) Clearly, there will be a border between these two parameter-regions, parameter values at which one state becomes stable and the other unstable; at these points the system is said to "bifurcate." Two dynamical systems are said to belong to the same bifurcation class if, close to the threshold, they can both be approximated by equations of the same form; in the case of pattern-formers, this means that they'll have similar configurations, at least close to their thresholds. It is a remarkable fact that "generic" dynamical systems, ones without some mathematically peculiar properties, fall into a fairly small number of bifurcation classes. But this means that we should expect only a fairly small number of different types of patterns to form from spontaneous instabilities.

The Instability Story is a nice one, even in this Cliff Notes version, and better still in the original, with the math. (The approximations used close to bifurcation points, for instance, let us make very good fits to experimental data, with only one or two adjustable parameters.) We can be confident that it's true, as far as it goes, but we can also be confident that it doesn't go all the way, that it must be just part of a larger epic. Let's leave to one side the question of why dynamics which are generic in the mathematical sense should be common in the Realized World; we might as well take a number and get in line behind Plato to explain why math works at all. It's more troubling that the similarity of pattern persists far from the onset of instability, at which point bifurcation classes are irrelevant. Mathematical caveats (like the suitability of Fourier decomposition) keep the Instability Story from applying to things like dendritic growth (and so our original snowflakes). Other kinds of pattern, like the much-studied spiral waves of the Belusov-Zhabotinsky reagent, simply do not form via spontaneous instabilities. So if the Instability Story is not the whole story, what is?

One idea, popular in the '60s and '70s, was advanced by Ilya Prigogine and his minions (very well: followers). Pattern formation, said Prigogine, was a sequel to thermodynamics. In thermodynamic equilibrium, the stable state is that of minimum free energy. This can be symmetric or not, depending; the theory of symmetry-breaking was actually developed for handling the switch from one kind of equilibrium state to another. Near equilibrium, the stable state was supposed to be the one which minimized the rate at which free energy was dissipated into waste heat, a rate which could be calculated using methods which Lars Onsager developed in the 1930s, thereby initiating non-equilibrium thermodynamics. (Ball is the only popular writer on self-organization I know of who gets this point of priority correct.) Further from equilibrium, things were a bit more complicated (involving some second derivatives), but stability was still supposed to be a matter of minimizing a quantity related to the free energy. Like many attempts at writing sequels to classics, it did not go over well. The theory has been been useless experimentally; worse, a result of Landauer's (he calls it the "blow-torch theorem") shows that no stability-from-minimization story can be correct out of equilibrium.

Prigogine has gone from penning sequels to thermodynamics to attempting to vindicate Bergson, or perhaps Dr. Pangloss; there is no need to follow him in these excursions, at any rate not here. There is, however, another, more recent story which purports to explain How Nature Works, namely Per Bak's system of self-organized criticality. Ball examines it in chapter eight, on granular materials, its original setting being avalanches in sand-piles. He dismisses its grander pretensions (i.e. most of them) for good and cogent reasons, though not, to my mind, the most damning. There is really no very good explanation for the commonalities of far-from-threshold patterns at this point.

Not that this is too surprising. The serious study of pattern formation is, as I said, only a few decades old. Hunting for ancestors and predecessors we find a number of late 19th century and early 20th century luminaries like Lord Rayleigh, Henri Poincaré and G. I. Taylor; but very little for the first two thirds, maybe even the first three quarters, of this century. Why not? Why so slow? One answer is of course quantum mechanics. Look at the options before a young physicist circa 1930. He could follow someone like Taylor, and wrack his brains trying to understand what happens to a fluid when you put it between two cylinders and spin one of them. Or he could follow Heisenberg or Oppenheimer, and figure out the most fundamental properties of matter, and use that knowledge to say why the stars shine and why materials constitute themselves the way they do. For anyone with ambition (to say nothing of our endemic hubris), this was hardly a choice, at least until quantum theory began to hit the point of diminishing returns. The other answer, the one Ball emphasizes, is that our progress in understanding pattern formation relies on tools from dynamics and statistical mechanics which were not themselves developed until the last few decades. The statistical-mechanical methods themselves are in part adaptations of tools from quantum mechanics...

This may be the Cunning of Reason, or dumb luck (is there a difference?); it also suggests a different way of looking at the puzzle of why so many different things produce similar patterns. In the words of an applied math professor I had as an undergraduate:

Feynman has this lovely passage about how it's so wonderful and mysterious that Nature uses the same mathematics over and over again. [In fact he has several; Ball quotes one.] I say that's total EXPLETIVE EXPLETIVE. It's just because the EXPLETIVE physicist don't know any EXPLETIVE math that they keep using the same EXPLETIVE equations over and EXPLETIVE over again.
(We never did figure out if Prof. N. had Tourette's syndrome, or was just on some kind of speed.) More printably, we are perhaps going at this from the wrong end. It's not that Nature has only a few basic forms available to her; it's that there aren't very many devices in the physicist's toolkit, so there are few forms we can handle. Science being the art of the soluble, we concentrate our attention on those forms, which, lo and behold, have much in common with each other. As we add more tools to our kit --- say, those of bifurcation theory and symmetry breaking --- we become able to handle more diverse phenomena. Mullah Nasruddin famously looked for things where the light was best --- but at least he wasn't shocked that everything he saw was so well lit!

Whether this rather deflating story is true, or there's some juicier account to be given of common patterns, I couldn't begin to say. Either way, it's clear that we are beginning to understand how patterns form in nature, without intent or design, in a way which fits very nicely with the rest of our mechanistic understanding of the physical world. This is an extremely exciting development, one with a claim on the attention and interest not just of scientists but the educated public at large. Ball's book is the first to really try to get across that understanding. Since it is also well-written, accurate, and well-illustrated, it is going to be extremely hard to improve upon, and deserves a wide audience for many years to come.

Disclaimer: I asked for, and got, a review copy of this book from the publisher, but I have no stake in its success.
Errata: p. 51n, decrease of free energy is strictly equivalent to increase of entropy only if the total (mechanical) energy is held constant; p. 57, for "1960s," read "1950s" (not even von Neumann did math from beyond the grave); p. 128, describing al-Khwarizmi as "Moorish" is a bit odd, since he was from (as his name indicates) Khwarazm in modern Uzbekistan, and while Europeans learned about 0 and place-value notation from his Kitab al-hisab al-Hind, i.e., Book on the Hindu Art of Reckoning, the title alone is enough to show that he invented neither; p. 202, for "mid-1900s", read "mid-1990s"; p. 241, for "1930s" read "1950s".
vii + 285 pp., 24 color plates, abundant black-and-white photos and diagrams, appendix of experiments, partial bibliography, index of personal names and topics.
Popular Science / Self-Organization, Complexity, etc.
In print as a hardback, US$37.50, ISBN 0-19-850244-3 [buy from Powell's] and as a paperback, US$19.95, ISBN 0-19-850243-5 [buy from Powell's], LoC QH491 B35
11 December 1998 / 21 January 1999
Thanks to Aryaman Shalizi for straightening out my thinking on morphogenesis, and comments on this review.