Drops, Dynamics, and Daisies : Nature's Numbers Chapter 9

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Chaos teaches us that systems obeying simple rules can behave in surprisingly complicated ways. There are important lessons here for everybody-managers who imagine that tightly controlled companies will automatically run smoothly, politicians who think that legislating against a problem will automatically eliminate it, and scientists who imagine that once they have modeled a system their work is complete.



But the world cannot be totally chaotic, otherwise we would not be able to survive in it. In fact, one of the reasons that chaos was not discovered sooner is that in many ways our world is simple. That simplicity tends to disappear when we look below the surface, but on the surface it is still there. Our use of language to describe our world rests upon the existence of underlying simplicities. For example, the statement "foxes chase rabbits" makes sense only because it captures a general pattern of animal interaction. Foxes do chase rabbits, in the sense that if a hungry fox sees a rabbit then it is likely to run after it.

However, if you start to look at the details, they rapidly become so complicated that the simplicity is lost. For example, in order to perform this simple act, the fox must recognize the rabbit as a rabbit. Then it must put its legs into gear to run after it. In order to comprehend these actions, we must understand vision, pattern recognition in the brain, and locomotion. In chapter 7, we investigated the third item, locomotion, and there we found the intricacies of physiology and neurology-bones, muscles, nerves, and brains. The action of muscles in turn depends on cell biology and chemistry; chemistry depends on quantum mechanics; and quantum mechanics may, in turn, depend on the much-sought Theory of Everything, in which all of the laws of physics come together in a single unified whole. If instead of locomotion we pursue the path opened by vision or pattern recognition, we again see the same kind of ever-branching complexity.



The task looks hopeless-except that the simplicities we start from exist, so either nature uses this enormously complex network of cause and effect or it sets things up so that most of the complexity doesn't matter. Until recently, the natural paths of investigation in science led deeper and deeper into the tree of complexity-what Jack Cohen and I have called the "reductionist nightmare."* We have learned a lot about nature by going that route-especially regarding how to manipulate it to our own ends. But we have lost sight of the big simplicities because we no longer see them as being simple at all. Recently, a radically different approach has been advocated, under the name complexity theory. Its central theme is that large-scale simplicities emerge from the complex interactions of large numbers of components.

In this final chapter, I want to show you three examples of simplicity emerging from complexity. They are not taken from the writings of the complexity theorists; instead I have chosen them from the mainstream of modern applied mathematics, the theory of dynamical systems. There are two reasons why I have done this. One is that I want to show that the central philosophy of complexity theory is popping up all over science, independently of any explicit movement to promote it. There is a quiet revolution simmering, and you can tell because the bubbles are starting to break the surface. The other is that each piece of work solves a long-standing puzzle about mathematical patterns in the natural world-and in so doing opens our eyes to features of nature that we would not otherwise have appreciated. The three topics are the shape of water drops, the dynamic behavior of animal populations, and the strange patterns in plant-petal numerology, whose solution I promised in the opening chapter.

To begin, let us return to the question of water dripping slowly from a tap. Such a simple, everyday phenomenon-yet it has already taught us about chaos. Now it will teach us something about complexity. This time we do not focus on the timing of successive drops. Instead, we look at what shape the drop takes up as it detaches from the end of the tap. Well, it's obvious, isn't it? It must be the classic "teardrop" shape, rather like a tadpole; round at the head and curving away to a sharp tail. After all, that's why we call such a shape a teardrop. But it's not obvious. In fact, it's not true. When I was first told of this problem, my main surprise was that the answer had not been found long ago. Literally miles of library shelves are filled with scientific studies of fluid flow; surely somebody took the trouble to look at the shape of a drop of water? Yet the early literature contains only one correct drawing, made over a century ago by the physicist Lord Rayleigh, and is so tiny that hardly anybody noticed it. In 1990, the mathematician Howell Peregrine and colleagues at Bristol University photographed the process and discovered that it is far more complicated-but also far more interesting- than anybody would ever imagine.

The formation of the detached drop begins with a bulging droplet hanging from a surface, the end of the tap. It develops a waist, which narrows, and the lower part of the droplet appears to be heading toward the classic teardrop shape. But instead of pinching off to form a short, sharp tail, the waist lengthens into a long thin cylindrical thread with an almost spherical drop hanging from its end. 

FIGURE 4. The shapes taken by a falling drop of water as it becomes detached. 

Then the thread starts to narrow, right at the point where it meets the sphere, until it develops a sharp point. At this stage, the general shape is like a knitting needle that is just touching an orange. Then the orange falls away from the needle, pulsating slightly as it falls. But that's only half the story. Now the sharp end of the needle begins to round off, and tiny waves travel back up the needle toward its root, making it look like a string of pearls that become tinier and tinier. Finally, the hanging thread of water narrows to a sharp point at the top end, and it, too, detaches. As it falls, its top end rounds off and a complicated series of waves travels along it.

I hope you find this as astonishing as I do. I had never imagined that falling drops of water could be so busy. These observations make it clear why nobody had previously studied the problem in any great mathematical detail. It's too hard. When the drop detaches, there is a singularity in the problem-a place where the mathematics becomes very nasty. The singularity is the tip of the "needle." But why is there a singularity at all? Why does the drop detach in such a complex manner? In 1994, J. Eggers and T. F. Dupont showed that the scenario is a consequence of the equations of fluid motion. They simulated those equations on a computer and reproduced Peregrine's scenario.

It was a brilliant piece of work. But in some respects it does not provide a complete answer to my question. It is reassuring to learn that the equations of fluid flow do predict the correct scenario, but that in itself doesn't help me understand why that scenario happens. There is a big difference between calculating nature's numbers and getting your brain around the answer-as Majikthise and Vroomfondel discovered when the answer was "Forty-two."

Further insight into the mechanism of detaching drops has come about through the work of X. D. Shi, Michael Brenner, and Sidney Nagel, of the University of Chicago. The main character in the story was already present in Peregrine's work: it is a particular kind of solution to the equations of fluid flow called a "similarity solution." Such a solution has a certain kind of symmetry that makes it mathematically tractable: it repeats its structure on a smaller scale after a short interval of time. Shi's group took this idea further, asking how the shape of the detaching drop depends on the fluid's viscosity. They performed experiments using mixtures of water and glycerol to get different viscosities. They also carried out computer simulations and developed the theoretical approach via similarity solutions. What they discovered is that for more viscous fluids, a second narrowing of the thread occurs before the singularity forms and the drop detaches. You get something more like an orange suspended by a length of string from the tip of a knitting needle. At higher viscosities still, there is a third narrowing-an orange suspended by a length of cotton from a length of string from the tip of a knitting needle. And as the viscosity goes up, so the number of successive narrowings increases without limit-at least, if we ignore the limit imposed by the atomic structure of matter. Amazing!

The second example is about population dynamics. The use of that phrase reflects a long tradition of mathematical modeling in which the changes in populations of interacting creatures are represented by differential equations. My pig/truffle system was an example. However, there is a lack of biological realism in such models-and not just as regards my choice of creatures. In the real world, the mechanism that governs population sizes is not a "law of population," akin to

Newton's law of motion, There are all kinds of other effects: for example, random ones (can the pig dig out the truffle or is there a rock in the way?) or types of variability not included in the equations (some pigs habitually produce more piglets than others),

In 1994, Jacquie McGlade, David Rand, and Howard Wilson, of Warwick University, carried out a fascinating study that bears on the relation between more biologically realistic models and the traditional equations. It follows a strategy common in complexity theory: set up a computer simulation in which large numbers of "agents" interact according to biologically plausible (though much simplified) rules, and try to extract large-scale patterns from the results of that simulation. In this case, the simulation was carried out by means of a "cellular automaton," which you can think of as a kind of mathematical computer game. McGlade, Rand, and Wilson, lacking my bias in favor of pigs, considered the more traditional foxes and rabbits. The computer screen is divided into a grid of squares, and each square is assigned a color-say, red for a fox, gray for a rabbit, green for grass, black for bare rock. Then a system of rules is set up to model the main biological influences at work. Examples of such rules might be:

• If a rabbit is next to grass, it moves to the position of the grass and eats it. • If a fox is next to a rabbit, it moves to the position of the rabbit and eats it. • At each stage of the game, a rabbit breeds new rabbits with some chosen probability. • A fox that has not eaten for a certain number of moves will die.

McGlade's group played a more complicated game than this, but you get the idea. Each move in the game takes the current configuration of rabbits, foxes, grass, and rock, and applies the rules to generate the next configuration-tossing computer "dice" when random choices are required. The process continues for several thousand moves, an "artifical ecology" that plays out the game of life on a computer screen. This artificial ecology resembles a dynamical system, in that it repeatedly applies the same bunch of rules; but it also includes random effects, which places the model in a different mathematical category altogether: that of stochastic cellular automata-computer games with chance.

Precisely because the ecology is an artificial one, you can perform experiments that are impossible, or too expensive, to perform in a real ecology. You can, for example, watch how the rabbit population in a given region changes over time, and get the exact numbers. This is where McGlade's group made a dramatic and surprising discovery. They realized that if you look at too tiny a region, what you see is largely random. For example, what happens on a single square looks extremely complicated. On the other hand, if you look at too large a region, all you see is the statistics of the population, averaged out. On intermediate scales, though, you may see something less dull. So they developed a technique for finding the size of region that would provide the largest amount of interesting information. They then observed a region of that size and recorded the changing rabbit population. Using methods developed in chaos theory, they asked whether that series of numbers was deterministic or random, and if deterministic, what its attractor looked like. This may seem a strange thing to do, inasmuch as we know that the rules for the simulation build in a great deal of randomness, but they did it anyway.

What they found was startling. Some 94 percent of the dynamics of the rabbit population on this intermediate scale can be accounted for by deterministic motion on a chaotic attractor in a four-dimensional phase space. In short, a differential equation with only four variables captures the important features of the dynamics of the rabbit population with only a 6-percent error-despite the far greater complexities of the computer-game model. This discovery implies that models with small numbers of variables may be more "realistic" than many biologists have hitherto assumed. Its deeper implication is that simple large-scale features can and do emerge from the fine structure of complex ecological games. My third and final example of a mathematical regularity of nature that emerges from complexity rather than having been "built in with the rules" is the number of petals of flowers. I mentioned in chapter 1 that the majority of plants have a number of petals taken from the series 3, 5, 8, 13, 21, 34, 55, 89. The view of conventional biologists is that the flower's genes specify all such information, and that's really all there is to it. However, just because living organisms have complicated DNA sequences that determine which proteins they are made of, and so on, it doesn't follow that genes determine everything. And even if they do, they may do so only indirectly. For example, genes tell plants how to make chlorophyll, but they don't tell the plants what color the chlorophyll has to be. If it's chlorophyll, it's green-there's no choice. So some features of the morphology of living creatures are genetic in origin and some are a consequence of physics, chemistry, and the dynamics of growth. One way to tell the difference is that genetic influences have enormous flexibility, but physics, chemistry, and dynamics produce mathematical regularities.

The numbers that arise in plants-not just for petals but for all sorts of other features-display mathematical regularities, They form the beginning of the so-called Fibonacci series, in which each number is the sum of the two that precede it. Petals aren't the only places you find Fibonacci numbers, either. If you look at a giant sunflower, you find a remarkable pattern of florets-tiny flowers that eventually become seeds-in its head. The florets are arranged in two intersecting families of spirals, one winding clockwise, the other counterclockwise. In some species the number of clockwise spirals is thirty-four, and the number of counterclockwise spirals is fifty-five. Both are Fibonacci numbers, occurring consecutively in the series. The precise numbers depend on the species of sunflower, but you often get 34 and 55, or 55 and 89, or even 89 and 144, the next Fibonacci number still. Pineapples have eight rows of scales-the diamond-shaped markings-sloping to the left, and thirteen sloping to the right.

Leonardo Fibonacci, in about 1200, invented his series in a problem about the growth of a population of rabbits. It wasn't as realistic a model of rabbit-population dynamics as the "game of life" model I've just discussed, but it was a very interesting piece of mathematics nevertheless, because it was the first model of its kind and because mathematicians find Fibonacci numbers fascinating and beautiful in their own right. The key question for this chapter is this: If genetics can choose to give a flower any number of petals it likes, or a pine cone any number of scales that it likes, why do we observe such a preponderance of Fibonacci numbers? The answer, presumably, has to be that the numbers arise through some mechanism that is more mathematical than arbitrary genetic instructions. The most likely candidate is some kind of dynamic constraint on plant development, which naturally leads to Fibonacci numbers. Of course, appearances may be deceptive, it could be all in the genes. But if so, I'd like to know how the Fibonacci numbers got turned into DNA codes, and why it was those numbers. Maybe evolution started with the mathematical patterns that occurred naturally, and fine-tuned them by natural selection. I suspect a lot of that has happened-tigers' stripes, butterflies' wings. That would explain why geneticists are convinced the patterns are genetic and mathematicians keep insisting they are mathematical.

The arrangement of leaves, petals, and the like in plants has a huge and distinguished literature. But early approaches are purely descriptive-they don't explain how the numbers relate to plant growth, they just sort out the geometry of the arrangements. The most dramatic insight yet comes from some very recent work of the French mathematical physicists Stt'iphane Douady and Yves Couder. They devised a theory of the dynamics of plant growth and used computer models and laboratory experiments to show that it accounts for the Fibonacci pattern.

The basic idea is an old one. If you look at the tip of the shoot of a growing plant, you can detect the bits and pieces from which all the main features of the plant-leaves, petals, sepals, florets, or whatever-develop. At the center of the tip is a circular region of tissue with no special features, called the apex. Around the apex, one by one, tiny lumps form, called primordia. Each primordium migrates away from the apex-or, more accurately, the apex grows away from the lump-and eventually the lump develops into a leaf, petal, or the like. Moreover, the general arrangement of those features is laid down right at the start, as the primordia form. So basically all you have to do is explain why you see spiral shapes and Fibonacci numbers in the primordia.

The first step is to realize that the spirals most apparent to the eye are not fundamental. The most important spiral is formed by considering the primordia in their order of appearance. Primordia that appear earlier migrate farther, so you can deduce the order of appearance from the distance away from the apex. What you find is that successive primordia are spaced rather sparsely along a tightly wound spiral, called the generative spiral. The human eye picks out the Fibonacci spirals because they are formed from primordia that appear near each other in space; but it is the sequence in time that really matters.

The essential quantitative feature is the angle between successive primordia. Imagine drawing lines from the centers of successive primordia to the center of the apex and measuring the angle between them. Successive angles are pretty much equal; their common value is called the divergence angle. In other words, the primordia are equally spaced-in an angular sense-along the generative spiral. Moreover, the divergence angle is usually very close to 137.5°, a fact first emphasized in 1837 by the crystallographer Auguste Bravais and his brother Louis. To see why that number is significant, take two consecutive numbers in the Fibonacci series: for example, 34 and 55. Now form the corresponding fraction 34/55 and multiply by 360°, to get 222.5°. Since this is more than 180°, we should measure it in the opposite direction round the circle-or, equivalently, subtract it from 360°. The result is 137.5°, the value observed by the Bravais brothers. The ratio of consecutive Fibonacci numbers gets closer and closer to the number 0.618034. For instance, 34/55 = 0.6182 which is already quite close. The limiting value is exactly (vf5-1)/2, the so-called golden number, often denoted by the Greek letter phi. Nature has left a clue for mathematical detectives: the angle between successive primordia is the "golden angle" of 360. In 1907, G. Van Iterson followed up this clue and worked out what happens when you plot successive points on a tightly wound spiral separated by angles of 137.5°. Because of the way neighboring points align, the human eye picks out two families of interpenetrating spirals-one winding clockwise and the other counterclockwise. And because of the relation between Fibonacci numbers and the golden number, the numbers of spirals in the two families are consecutive Fibonacci numbers. Which Fibonacci numbers depends on the tightness of the spiral. How does that explain the numbers of petals? Basically, you get one petal at the outer edge of each spiral in just one of the families.

At any rate, it all boils down to explaining why successive primordia are separated by the golden angle: then everything else follows.

Douady and Couder found a dynamic explanation for the golden angle. They built their ideas upon an important insight of H. Vogel, dating from 1979. His theory is again a descriptive one-it concentrates on the geometry of the arrangement rather than on the dynamics that caused it. He performed numerical experiments which strongly suggested that if successive primordia are placed along the generative spiral using the golden angle, they will pack together most efficiently. 

FIGURE 5. Successive dots arranged at angles of 137.5° to each other along a tightly wound spiral (not shown) naturally fall into two families of loosely wound spirals that are immediately apparent to the eye. Here there are 8 spirals in one direction and 13 in the other-consecutive Fibonacci numbers.

For instance, suppose that, instead of the golden angle, you try a divergence angle of 90°, which divides 360° exactly. Then successive primordia are arranged along four radial lines forming a cross. In fact, if you use a divergence angle that is a rational multiple of 360°, you always get a system of radial lines. So there are gaps between the lines and the primordia don't pack efficiently. Conclusion: to fill the space efficiently, you need a divergence angle that is an irrational multiple of 3600-a multiple by a number that is not an exact fraction. But which irrational number? Numbers are either irrational or not, but-like equality in George Orwell's Animal Farmsome are more irrational than others. Number theorists have long known that the most irrational number is the golden number. It is "badly approximable" by rational numbers, and if you quantify how badly, it's the worst of them all. Which, turning the argument on its head, means that a golden divergence angle should pack the primordia most closely. Vogel's computer experiments confirm this expectation but do not prove it with full logical rigor.

The most remarkable thing Douady and Couder did was to obtain the golden angle as a consequence of simple dynamics rather than to postulate it directly on grounds of efficient packing. They assumed that successive elements of some kind-representing primordia-form at equally spaced intervals of time somewhere on the rim of a small circle, representing the apex; and that these elements then migrate radially at some specified initial velocity. In addition, they assume that the elements repel each other-like equal electric charges, or magnets with the same polarity. This ensures that the radial motion keeps going and that each new element appears as far as possible from its immediate predecessors. It's a good bet that such a system will satisfy Vogel's criterion of efficient packing, so you would expect the golden angle to show up of its own accord. And it does.

Douady and Couder performed an experiment-not with plants, but using a circular dish full of silicone oil placed in a vertical magnetic field. They let tiny drops of magnetic fluid fall at regular intervals of time into the center of the dish. The drops were polarized by the magnetic field and repelled each other. They were given a boost in the radial direction by making the magnetic field stronger at the edge of the dish than it was in the middle. The patterns that appeared depended on how big the intervals between drops were; but a very prevalent pattern was one in which successive drops lay on a spiral with divergence angle very close to the golden angle, giving a sunflower-seed pattern of interlaced spirals. Douady and Couder also carried out computer calculations, with similar results. By both methods, they found that the divergence angle depends on the interval between drops according to a complicated branching pattern of wiggly curves. Each section of a curve between successive wiggles corresponds to a particular pair of numbers of spirals. The main branch is very close to a divergence angle of 137.5°, and along it you find all possible pairs of consecutive Fibonacci numbers, one after the other in numerical sequence. The gaps between branches represent "bifurcations," where the dynamics undergoes significant changes.

Of course, nobody is suggesting that botany is quite as perfectly mathematical as this model. In particular, in many plants the rate of appearance of primordia can speed up or slow down. In fact, changes in morphology-whether a given primordium becomes a leaf or a petal, say-often accompany such variations. So maybe what the genes do is affect the timing of the appearance of the primordia. But plants don't need their genes to tell them how to space their primordia: that's done by the dynamics. It's a partnership of physics and genetics, and you need both to understand what's happening.

Three examples, from very different parts of science. Each, in its own way, an eye-opener. Each a case study in the origins of nature's numbers-the deep mathematical regularities that can be detected in natural forms. And there is a common thread, an even deeper message, buried within them. Not that nature is complicated. No, nature is, in its own subtle way, simple. However, those simplicities do not present themselves to us directly. Instead, nature leaves clues for the mathematical detectives to puzzle over. It's a fascinating game, even to a spectator. And it's an absolutely irresistible one if you are a mathematical Sherlock Holmes.

Chapter 6 : Broken Symmetry
Chapter 7 : The Rhythm of Life
Chapter 8 : Do dice Play God 
Chapter 9 : Drops Dynamics and Daisies