Broken Symmetry : Nature's Numbers Chapter 6

Something in the human mind is attracted to symmetry. Symmetry appeals to our visual sense, and thereby plays a role in our sense of beauty. However, perfect symmetry is repetitive and predictable, and our minds also like surprises, so we often consider imperfect symmetry to be more beautiful than exact mathematical symmetry. Nature, too, seems to be attracted to symmetry, for many of the most striking patterns in the natural world are symmetric. And nature also seems to be dissatisfied with too much symmetry, for nearly all the symmetric patterns in nature are less symmetric than the causes that give rise to them.




This may seem a strange thing to say; you may recall that the great physicist Pierre Curie, who with his wife, Marie, discovered radioactivity, stated the general principle that "effects are as symmetric as their causes." However, the world is full of effects that are not as symmetric as their causes, and the reason for this is a phenomenon known as "spontaneous symmetry breaking."

Symmetry is a mathematical concept as well as an aesthetic one, and it allows us to classify different types of regular pattern and distinguish between them. Symmetry breaking is a more dynamic idea, describing changes in pattern. Before we can understand where nature's patterns come from and how they can change, we must find a language in which to describe what they are.




What is symmetry? Let's work our way to the general from the particular. One of the most familiar symmetric forms is the one inside which you spend your life. The human body is "bilaterally symmetric," meaning that its left half is (near enough) the same as its right half. As noted, the bilateral symmetry of the human form is only approximate: the heart is not central, nor are the two sides of the face identical. But the overall form is very close to one that has perfect symmetry, and in order to describe the mathematics of symmetry we can imagine an idealized human figure whose left side is exactly the same as its right side. But exactly the same? Not entirely. The two sides of the figure occupy different regions of space; moreover, the left side is a reversal of the right-its mirror image.

As soon as we use words like "image," we are already thinking of how one shape corresponds to the other-of how you might move one shape to bring it into coincidence with the other. Bilateral symmetry means that if you reflect the left half in a mirror, then you obtain the right half. Reflection is a mathematical concept, but it is not a shape, a number, or a formula. It is a transformation-that is, a rule for moving things around.

There are many possible transformations, but most are not symmetries. To relate the halves correctly, the mirror must be placed on the symmetry axis, which divides the figure into its two related halves. Reflection then leaves the human form invariant-that is, unchanged in appearance. So we have found a precise mathematical characterization of bilateral symmetry-a shape is bilaterally symmetric if it is invariant by reflection. More generally, a symmetry of an object or system is any transformation that leaves it invariant. This description is a wonderful example of what I earlier called the "thingification of processes": the process "move like this" becomes a thing-a symmetry. This simple but elegant characterization opens the door to an immense area of mathematics.

There are many different kinds of symmetry. The most important ones are reflections, rotations, and translations-or, less formally, flips, turns, and slides. If you take an object in the plane, pick it up, and flip it over onto its back, you get the same effect as if you had reflected it in a suitable mirror. To find where the mirror should go, choose some point on the original object and look at where that point ends up when the object is flipped. The mirror must go halfway between the point and its image, at right angles to the line that joins them (see figure 3). Reflections can also be carried out in threedimensional space, but now the mirror is of a more familiar kind-namely, a flat surface.

To rotate an object in the plane, you choose a point, called the center, and turn the object about that center, as a wheel turns about its hub. The number of degrees through which you turn the object determines the "size" of the rotation. For example, imagine a flower with four identical equally spaced petals. If you rotate the flower 90°, it looks exactly the same, so the transformation "rotate through a right angle" is a symmetry of the flower. Rotations can occur in three-dimensional space too, but now you have to choose a line, the axis, and spin objects on that axis as the Earth spins on its axis. Again, you can rotate objects through different angles about the same axis.

FIGURE I.

Where is the mirror? Given an object and a mirror image of that object, choose any point of the object and the corresponding point of the image, Join them by a line, The mirror must be at right angles to the midpoint of that line.

Translations are transformations that slide objects along without rotating them. Think of a tiled bathroom wall. If you take a tile and slide it horizontally just the right distance, it will fit on top of a neighboring tile. That distance is the width of a tile. If you slide it two widths of a tile, or three, or any whole number, it also fits the pattern. The same is true if you slide it in a vertical direction, or even if you use a combination of horizontal and vertical slides. In fact, you can do more than just sliding one tile-you can slide the entire pattern of tiles. Again, the pattern fits neatly on top of its original position only when you use a combination of horizontal and vertical slides through distances that are whole number multiples of the width of a tile.

Reflections capture symmetries in which the left half of a pattern is the same as the right half, like the human body. Rotations capture symmetries in which the same units repeat around circles, like the petals of a flower. Translations capture symmetries in which units are repeated, like a regular array of tiles; the bees' honeycomb, with its hexagonal "tiles," is an excellent naturally occurring example.

Where do the symmetries of natural patterns come from?

Think of a still pond, so flat that it can be thought of as a mathematical plane, and large enough that it might as well be a plane for all that the edges matter. Toss a pebble into the pond. You see patterns, ripples, circular waves seemingly moving outward away from the point of impact of the pebble. We've all seen this, and nobody is greatly surprised. After all, we saw the cause: it was the pebble. If you don't throw pebbles in, or anything else that might disturb the surface, then you won't get waves. All you'll get is a still, flat, planar pond.

Ripples on a pond are examples of broken symmetry. An ideal mathematical plane has a huge amount of symmetry: every part of it is identical to every other part. You can translate the plane through any distance in any direction, rotate it through any angle about any center, reflect it in any mirror line, and it still looks exactly the same. The pattern of circular ripples, in contrast, has less symmetry. It is symmetric only with respect to rotations about the point of impact of the pebble, and reflections in mirror lines that run through that point. No translations, no other rotations, no other reflections. The pebble breaks the symmetry of the plane, in the sense that after the pebble has disturbed the pond, many of its symmetries are lost. But not all, and that's why we see a pattern.

However, none of this is surprising, because of the pebble, In fact, since the impact of the pebble creates a special point, different from all the others, the symmetries of the ripples are exactly what you would expect. They are precisely the symmetries that do not move that special point. So the symmetry of the pond is not spontaneously broken when the ripples appear, because you can detect the stone that causes the translational symmetries to be lost.

You would be more surprised-a lot more surprised-if a perfectly flat pond suddenly developed a series of concentric circular ripples without there being any obvious cause, You would imagine that perhaps a fish beneath the surface had disturbed it, or that something had fallen in and you had not seen it because it was moving too fast. So strong is the ingrained assumption that patterns must have evident causes that when in 1958 the Russian chemist B. P. Belousov discovered a chemical reaction that spontaneously formed patterns, apparently out of nothing, his colleagues refused to believe him. They assumed that he had made a mistake. They didn't bother checking his work: he was so obviously wrong that checking his work would be a waste of time.

Which was a pity, because he was right. The particular pattern that Belousov discovered existed not in space but in time: his reaction oscillated through a periodic sequence of chemical changes. By 1963, another Russian chemist, A. M. Zhabotinskii, had modified Belousov's reaction so that it formed patterns in space as well. In their honor, any similar chemical reaction is given the generic name "Belousov-Zhabotinskii [or B-ZJ reaction." The chemicals used nowadays are different and simpler, thanks to some refinements made by the British reproductive biologist

Jack Cohen and the American mathematical biologist Arthur Winfree, and the experiment is now so simple that it can be done by anyone with access to the necessary chemicals. These are slightly esoteric, but there are only four of them.' In the absence of the appropriate apparatus, I'll tell you what happens if you do the experiment. The chemicals are all liquids: you mix them together in the right order and pour them into a flat dish. The mixture turns blue, then red: let it stand for a while. For ten or sometimes even twenty minutes, nothing happens; it's just like gazing at a featureless flat pond-except that it is the color of the liquid that is featureless, a uniform red. This uniformity is not surprising; after all, you blended the liquids. Then you notice a few tiny blue spots appearing-and that is a surprise. They spread, forming circular blue disks. Inside each disk, a red spot appears, turning the disk into a blue ring with a red center. Both the blue ring and the red disk grow, and when the red disk gets big enough, a blue spot appears inside it. The process continues, forming an ever-growing series of "target patterns"-concentric rings of red and blue. These target patterns have exactly the same symmetries as the rings of ripples on a pond; but this time you can't see any pebble. It is a strange and mysterious process in which paUern-order-appears to arise of its own accord from the disordered, randomly mixed liquid. No wonder the chemists didn't believe Belousov. But that's not the end of the B-Z reaction's party tricks. If you tilt the dish slightly and then put it back where it was, or dip a hot wire into it, you can break the rings and turn them 'The precise recipe is given in the Notes to The Collapse of Chaos, by Jack Cohen and Ian Stewart. into rotating red and blue spirals, If Belousov had claimed that, you would have seen steam coming out of his colleagues' ears,

This kind of behavior is not just a chemical conjuring trick. The regular beating of your heart relies on exactly the same patterns, but in that case they are patterns in waves of electrical activity, Your heart is not just a lump of undifferentiated muscle tissue, and it doesn't automatically contract all at once, Instead, it is composed of millions of tiny muscle fibers, each one of them a single cell. The fibers contract in response to electrical and chemical signals, and they pass those signals on to their neighbors, The problem is to make sure that they all contract roughly in synchrony, so that the heart beats as a whole, To achieve the necessary degree of synchronization, your brain sends electrical signals to your heart. These signals trigger electrical changes in some of the muscle fibers, which then affect the muscle fibers next to them-so that ripples of activity spread, just like the ripples on a pond or the blue disks in the B-Z reaction. As long as the waves form complete rings, the heart's muscle fibers contract in synchrony and the heart beats normally. But if the waves become spirals-as they can do in diseased hearts-the result is an incoherent set of local contractions, and the heart fibrillates. If fibrillation goes unchecked for more than a few minutes, it results in death. So every single one of us has a vested interest in circular and spiral wave patterns.

However in the heart, as in the pond, we can see a specific cause for the wave patterns: the signals from the brain. In the B-Z reaction, we cannot: the symmetry breaks spontaneously-" of its own accord"-without any external stimulus.

The term "spontaneous" does not imply that there is no cause, however: it indicates that the cause can be as tiny and as insignificant as you please. Mathematically, the crucial point is that the uniform distribution of chemicals-the featureless red liquid-is unstable. If the chemicals cease to be equally mixed, then the delicate balance that keeps the solution red is upset, and the resulting chemical changes trigger the formation of a blue spot. From that moment on, the whole process becomes much more comprehensible, because now the blue spot acts like a chemical "pebble," creating sequential ripples of chemical activity. But-at least, as far as the mathematics goes-the imperfection in the symmetry of the liquid which triggers the blue spot can be vanishingly small, provided it is not zero. In a real liquid, there are always tiny bits of dust, or bubbles-or even just molecules undergoing the vibrations we call "heat"-to disturb the perfect symmetry.

That's all it takes. An infinitesimal cause produces a large-scale effect, and that effect is a symmetric pattern. Nature's symmetries can be found on every scale, from the structure of subatomic particles to that of the entire universe.

Many chemical molecules are symmetric. The methane molecule is a tetrahedron-a triangular-sided pyramid-with one carbon atom at its center and four hydrogen atoms at its corners. Benzene has the sixfold symmetry of a regular hexagon. The fashionable molecule buckminsterfullerene is a truncated icosahedral cage of sixty carbon atoms. (An icosahedron is a regular solid with twenty triangular faces; "truncated" means that the corners are cut off.) Its symmetry lends it a remarkable stability, which has opened up new possibilities for organic chemistry.

On a slightly larger scale than molecules, we find symmetries in cellular structure; at the heart of cellular replication lies a tiny piece of mechanical engineering. Deep within each living cell, there is a rather shapeless structure known as the centrosome, which sprouts long thin microtubules, basic components of the cell's internal "skeleton," like a diminutive sea urchin. Centro somes were first discovered in 1887 and play an important role in organizing cell division. However, in one respect the structure of the centrosome is astonishingly symmetric. Inside it are two structures, known as centrioles, positioned at right angles to each other. Each centriole is cylindrical, made from twenty-seven microtubules fused together along their lengths in threes, and arranged with perfect ninefold symmetry. The microtubules themselves also have an astonishingly regular symmetric form. They are hollow tubes, made from a perfect regular checkerboard pattern of units that contain two distinct proteins, alpha- and betatubulin. One day, perhaps, we will understand why nature chose these symmetric forms. But it is amazing to see symmetric structures at the core of a living cell.

Viruses are often symmetric, too, the commonest shapes being helices and icosahedrons. The helix is the form of the influenza virus, for instance. Nature prefers the icosahedron above all other viral forms: examples include herpes, chickenpox, human wart, canine infectious hepatitis, turnip yellow mosaic, adenovirus, and many others. The adenovirus is another striking example of the artistry of molecular engineering. It is made from 252 virtually identical subunits, with 21 of them, fitted together like billiard balls before the break, making up each triangular face. (Subunits along the edges lie on more than one face and corner units lie on three, which is why 20 x 21 is not equal to 252.)

Nature exhibits symmetries on larger scales, too. A developing frog embryo begins life as a spherical cell, then loses symmetry step by step as it divides, until it has become a blastula, thousands of tiny cells whose overall form is again spherical. Then the blastula begins to engulf part of itself, in the process known as gastrulation. During the early stages of this collapse, the embryo has rotational symmetry about an axis, whose position is often determined by the initial distribution of yolk in the egg, or sometimes by the point of sperm entry. Later this symmetry is broken, and only a single mirror symmetry is retained, leading to the bilateral symmetry of the adult.

Volcanoes are conical, stars are spherical, galaxies are spiral or elliptical. According to some cosmologists, the universe itself resembles nothing so much as a gigantic expanding ball. Any understanding of nature must include an understanding of these prevalent patterns. It must explain why they are so common, and why many different aspects of nature show the same patterns. Raindrops and stars are spheres, whirlpools and galaxies are spirals, honeycombs and the Devil's Causeway are arrays of hexagons. There has to be a general principle underlying such patterns; it is not enough just to study each example in isolation and explain it in terms of its own internal mechanisms.

Symmetry breaking is just such a principle. But in order for symmetry to break, it has to be present to start with. At first this would seem to replace one problem of pattern formation with another: before we can explain the circular rings on the pond, in other words, we have to explain the pond. But there is a crucial difference between the rings and the pond. The symmetry of the pond is so extensiveevery point on its surface being equivalent to every otherthat we do not recognize it as being a pattern. Instead, we see it as bland uniformity. It is very easy to explain bland uniformity: it is what happens to systems when there is no reason for their component parts to differ from each other. It is, so to speak, nature's default option. If something is symmetric, its component features are replaceable or interchangeable. One corner of a square looks pretty much the same as any other, so we can interchange the corners without altering the square's appearance. One atom of hydrogen in methane looks pretty much like any other, so we can interchange those atoms. One region of stars in a galaxy looks pretty much like any other, so we can interchange parts of two different spiral arms without making an important difference.

In short, nature is symmetric because we live in a massproduced universe-analogous to the surface of a pond. Every electron is exactly the same as every other electron, every proton is exactly the same as every other proton, every region of empty space is exactly the same as every other region of empty space, every instant of time is exactly the same as every other instant of time. And not only are the structure of space, time, and matter the same everywhere: so are the laws that govern them. Albert Einstein made such "invariance principles" the cornerstone of his approach to physics; he based his thinking on the idea that no particular point in spacetime is special. Among other things, this led him to the principle of relativity, one of the greatest physical discoveries ever made.

This is all very well, but it produces a deep paradox. If the laws of physics are the same at all places and at all times, why is there any "interesting" structure in the universe at all? Should it not be homogeneous and changeless? If every place in the universe were interchangeable with every other place, then all places would be indistinguishable; and the same would hold for all times. But they are not. The problem is, if anything, made worse by the cosmological theory that the universe began as a single point, which exploded from nothingness billions of years ago in the big bang. At the instant of the universe's formation, all places and all times were not only indistinguishable but identical. So why are they different now?

The answer is the failure of Curie's Principle, noted at the start of this chapter. Unless that principle is hedged around with some very subtle caveats about arbitrarily tiny causes, it offers a misleading intuition about how a symmetric system should behave. Its prediction that adult frogs should be bilaterally symmetric (because embryonic frogs are bilaterally symmetric, and according to Curie's Principle the symmetry cannot change) appears at first sight to be a great success; but the same argument applied at the spherical blastula stage predicts with equal force that an adult frog should be a sphere.

A much better principle is the exact opposite, the principle of spontaneous symmetry breaking. Symmetric causes often produce less symmetric effects. The evolving universe can break the initial symmetries of the big bang. The spherical blastula can develop into the bilateral frog. The 252 perfectly interchangeable units of adenovirus can arrange themselves into an icosahedron-an arrangement in which some units will occupy special places, such as corners. A set of twentyseven perfectly ordinary microtubules can get together to create a centriole.

Fine, but why patterns? Why not a structureless mess, in which all symmetries are broken? One of the strongest threads that runs through every study ever made of symmetry breaking is that the mathematics does not work this way. Symmetries break reluctantly. There is so much symmetry lying around in our mass-produced universe that there is seldom a good reason to break all of it. So rather a lot survives. Even those symmetries that do get broken are still present, in a sense, but now as potential rather than actual form. For example, when the 252 units of the adenovirus began to link up, anyone of them could have ended up in a particular comer. In that sense, they are interchangeable. But only one of them actually does end up there, and in that sense the symmetry is broken: they are no longer fully interchangeable. But some of the symmetry remains, and we see an icosahedron.

In this view, the symmetries we observe in nature are broken traces of the grand, universal symmetries of our massproduced universe. Potentially the universe could exist in any of a huge symmetric system of possible states, but actually it must select one of them. In so doing, it must trade some of its actual symmetry for unobservable, potential symmetry. But some of the actual symmetry may remain, and when it does we observe a pattern. Most of nature's symmetric patterns arise out of some version of this general mechanism.

In a negative sort of way, this rehabilitates Curie's Principle: if we permit tiny asymmetric disturbances, which can trigger an instability of the fully symmetric state, then our mathematical system is no longer perfectly symmetric. But the important point is that the tinest departure from symmetry in the cause can lead to a total loss of symmetry in the resulting effect-and there are always tiny departures. That makes Curie's principle useless for the prediction of symmetries. It is much more informative to model a real system after one with perfect symmetry, but to remember that such a model has many possible states, only one of which will be realized in practice. Small disturbances cause the real system to select states from the range available to the idealized perfect system. Today this approach to the behavior of symmetric systems provides one of the main sources of understanding of the general principles of pattern formation.

In particular, the mathematics of symmetry breaking unifies what at first sight appear to be very disparate phenomena. For example, think about the patterns in sand dunes mentioned in chapter 1. The desert can be modeled as a flat plane of sandy particles, the wind can be modeled as a fluid flowing across the plane. By thinking about the symmetries of such a system, and how they can break, many of the observed patterns of dunes can be deduced. For example, suppose the wind blows steadily in a fixed direction, so that the whole system is invariant under translations parallel to the wind. One way to break these translational symmetries is to create a periodic pattern of parallel stripes, at right angles to the wind direction. But this is the pattern that geologists call transverse dunes. If the pattern also becomes periodic in the direction along the stripes, then more symmetry breaks, and the wavy barchanoid ridges appear. And so on.

However, the mathematical principles of symmetry-breaking do not just work for sand dunes. They work for any system with the same symmetries-anything that flows across a planar surface creating patterns. You can apply the same basic model to a muddy river flowing across a coastal plain and depositing sediment, or the waters of a shallow sea in ebb and flow across the seabed-phenomena important in geology, because millions of years later the patterns that result have been frozen into the rock that the sandy seabed and the muddy delta became. The list of patterns is identical to that for dunes.

Or the fluid might be a liquid crystal, as found in digitalwatch displays, which consist of a lot of long thin molecules that arrange themselves in patterns under the influence of a magnetic or electric field. Again, you find the same patterns. Or there might not be a fluid at all: maybe what moves is a chemical, diffusing through tissue and laying down genetic instructions for patterns on the skin of a developing animal. Now the analogue of transverse dunes is the stripes of a tiger or a zebra, and that of barchanoid ridges is the spots on a leopard or a hyena.

The same abstract mathematics; different physical and biological realizations. Mathematics is the ultimate in technology transfer-but with mental technology, ways of thinking, being transferred, rather than machines. This universality of symmetry breaking explains why living systems and nonliving ones have many patterns in common. Life itself is a process of symmetry creation-of replication; the universe of biology is just as mass-produced as the universe of physics, and the organic world therefore exhibits many of the patterns found in the inorganic world. The most obvious symmetries of living organisms are those of form-icosahedral viruses, the spiral shell of Nautilus, the helical horns of gazelles, the remarkable rotational symmetries of starfish and jellyfish and flowers. But symmetries in the living world go beyond form into behavior-and not just the symmetric rhythms of locomotion I mentioned earlier. The territories of fish in Lake Huron are arranged just like the cells in a honeycomb-and for the same reasons. The territories, like the bee grubs, cannot all be in the same place-which is what perfect symmetry would imply. Instead, they pack themselves as tightly as they can without one being different from another, and the behavioral constraint by itself produces a hexagonally symmetric tiling. And that resembles yet another striking instance of mathematical technology transfer, for the same symmetry breaking mechanism arranges the atoms of a crystal into a regular lattice-a physical process that ultimately supports Kepler's theory of the snowflake.

One of the more puzzling types of symmetry in nature is mirror symmetry, symmetry with respect to a reflection. Mirror symmetries of three-dimensional objects cannot be realized by turning the objects in space-a left shoe cannot be turned into a right shoe by rotating it. However, the laws of physics are very nearly mirror-symmetric, the exceptions being certain interactions of subatomic particles. As a result, any molecule that is not mirror-symmetric potentially exists in two different forms-left- and right-handed, so to speak.

On Earth, life has selected a particular molecular handedness: for example, for amino acids. Where does this particular handedness of terrestrial life come from? It could have been just an accident-primeval chance propagated by the massproduction techniques of replication. If so, we might imagine that on some distant planet, creatures exist whose molecules are mirror images of ours. On the other hand, there may be a deep reason for life everywhere to choose the same direction. Physicists currently recognize four fundamental forces in nature: gravity, electromagnetism, and the strong and weak nuclear interactions. It is known that the weak force violates mirror symmetry-that is, it behaves differently in left- or right-handed versions of the same physical problem. As the Austrian-born physicist Wolfgang Pauli put it, "The Lord is a weak left-hander." One remarkable consequence of this violation of mirror symmetry is the fact that the energy levels of molecules and that of their mirror images are not exactly equal. The effect is extremely small: the difference in energy levels between one particular amino acid and its mirror image is roughly one part in 1017• This may seem very tiny-but we saw that symmetry breaking requires only a very tiny disturbance. In general, lower-energy forms of molecules should be favored in nature. For this amino acid, it can be calculated that with 98% probability the lower energy form will become dominant within a period of about a hundred thousand years. And indeed, the version of this amino acid which is found in living organisms is the lower-energy one.

In chapter 5, I mentioned the curious symmetry of Maxwell's equations relating electricity and magnetism. Roughly speaking, if you interchange all the symbols for the electric field with those for the magnetic field, you re-create the same equations. This symmetry lies behind Maxwell's unification of electrical and magnetic forces into a single electromagnetic force. There is an analogous symmetry-though an imperfect one-in the equations for the four basic forces of nature, suggesting an even grander unification: that all four forces are different aspects of the same thing. Physicists have already achieved a unification of the weak and electromagnetic forces. According to current theories, all four fundamental forces should become unified-that is, symmetrically related-at the very high energy levels prevailing in the early universe. This symmetry of the early universe is broken in our own universe. In short, there is an ideal mathematical universe in which all of the fundamental forces are related in a perfectly symmetric manner-but we don't live in it.

That means that our universe could have been different; it could have been any of the other universes that, potentially, could arise by breaking symmetry in a different way. That's quite a thought. But there is an even more intriguing thought: the same basic method of pattern formation, the same mechanism of symmetry breaking in a mass-produced universe, governs the cosmos, the atom, and us.

Chapter 5 : From Violins to Videos

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

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The Rhythm of Life : Nature's Numbers Chapter 7

Nature is nothing if not rhythmic, and its rhythms are many and varied. Our hearts and lungs follow rhythmic cycles whose timing is adapted to our body's needs. Many of nature's rhythms are like the heartbeat: they take care of themselves, running "in the background." Others are like breathing: there is a simple "default" pattern that operates as long as nothing unusual is happening, but there is also a more sophisticated control mechanism that can kick in when necessary and adapt those rhythms to immediate needs. Controllable rhythms of this kind are particularly common-and particularly interesting-in locomotion. In legged animals, the default patterns of motion that occur when conscious control is not operating are called gaits.




Until the development of high-speed photography, it was virtually impossible to find out exactly how an animal's legs moved as it ran or galloped: the motion is too fast for the human eye to discern. Legend has it that the photographic technique grew out of a bet on a horse. In the 1870s, the railroad tycoon Leland Stanford bet twenty-five thousand dollars that at some times a trotting horse has all four feet completely off the ground. To settle the issue, a photographer, who was born Edward Muggeridge but changed his name to Eadweard Muybridge, photographed the different phases of the gait of the horse, by placing a line of cameras with tripwires for the horse to trot past. Stanford, it is said, won his bet. Whatever the truth of the story, we do know that Muybridge went on to pioneer the scientific study of gaits. He also adapted a mechanical device known as the zoetrope to display them as "moving pictures," a road that in short order led to Hollywood. So Muybridge founded both a science and an art.

Most of this chapter is about gait analysis, a branch of mathematical biology that grew up around the questions "How do animals move?" and "Why do they move like that?" To introduce a little more variety, the rest is about rhythmic patterns that occur in entire animal populations, one dramatic example being the synchronized flashing of some species of fireflies, which is seen in some regions of the Far East, including Thailand. Although biological interactions that take place in individual animals are very different from those that take place in populations of animals, there is an underlying mathematical unity, and one of the messages of this chapter is that the same general mathematical concepts can apply on many different levels and to many different things. Nature respects this unity, and makes good use of it.




The organizing principle behind many such biological cycles is the mathematical concept of an oscillator-a unit whose natural dynamic causes it to repeat the same cycle of behavior over and over again. Biology hooks together huge "circuits" of oscillators, which interact with each other to create complex patterns of behavior. Such "coupled oscillator networks" are the unifying theme of this chapter.

Why do systems oscillate at all? The answer is that this is the simplest thing you can do if you don't want, or are not allowed, to remain still. Why does a caged tiger pace up and down? Its motion results from a combination of two constraints. First, it feels restless and does not wish to sit still. Second, it is confined within the cage and cannot simply disappear over the nearest hill. The simplest thing you can do when you have to move but can't escape altogether is to oscillate. Of course, there is nothing that forces the oscillation to repeat a regular rhythm; the tiger is free to follow an irregular path around the cage. But the simplest option-and therefore the one most likely to arise both in mathematics and in nature--is to find some series of motions that works, and repeat it over and over again. And that is what we mean by a periodic oscillation. In chapter 5, I described the vibration of a violin string. That, too, moves in a periodic oscillation, and it does so for the same reasons as the tiger. It can't remain still because it has been plucked, and it can't get away altogether because its ends are pinned down and its total energy cannot increase.

Many oscillations arise out of steady states. As conditions change, a system that has a steady state may lose it and begin to wobble periodically. In 1942, the German mathematician Eberhard Hopf found a general mathematical condition that guarantees such behavior: in his honor, this scenario is known as Hopf bifurcation. The idea is to approximate the dynamics of the original system in a particularly simple way, and to see whether a periodic wobble arises in this simplified system. Hopf proved that if the simplified system wobbles, then so does the original system. The great advantage of this method is that the mathematical calculations are carried out only for the simplified system, where they are relatively straightforward, whereas the result of those calculations tells us how the original system behaves. It is difficult to tackle the original system directly, and Hopf's approach sidesteps the difficulties in a very effective manner.

The word "bifurcation" is used because of a particular mental image of what is happening, in which the periodic oscillations "grow out from" the original steady state like a ripple on a pond growing out from its center. The physical interpretation of this mental picture is that the oscillations are very small to start with, and steadily become larger. The speed with which they grow is unimportant here.

For example, the sounds made by a clarinet depend on Hopf bifurcation. As the clarinetist blows air into the instrument, the reed-which was stationary-starts to vibrate. If the air flows gently, the vibration is small and produces a soft note. If the musician blows harder, the vibration grows and the note becomes louder. The important thing is that the musician does not have to blow in an oscillatory way (that is, in a rapid series of short puffs) to make the reed oscillate. This is typical of Hopf bifurcation: if the simplified system passes Hopf's mathematical test, then the real system will begin to oscillate of its own accord. In this case, the simplified system can be interpreted as a fictitious mathematical clarinet with a rather simple reed, although such an interpretation is not actually needed to carry out the calculations.

Hopf bifurcation can be seen as a special type of symmetry breaking. Unlike the examples of symmetry breaking described in the previous chapter, the symmetries that break relate not to space but to time. Time is a single variable, so mathematically it corresponds to a line-the time axis. There are only two types of line symmetry: translations and reflections. What does it mean for a system to be symmetric under time translation? It means that if you observe the motion of the system and then wait for some fixed interval and observe the motion of the system again, you will see exactly the same behavior. That is a description of periodic oscillations: if you wait for an interval equal to the period, you see exactly the same thing. So periodic oscillations have time-translation symmetry.

What about reflectional symmetries of time? Those correspond to reversing the direction in which time flows, a more subtle and philosophically difficult concept. Time reversal is peripheral to this chapter, but it is an extremely interesting question, which deserves to be discussed somewhere, so why not here? The law of motion is symmetric under time reversal.

If you make a film of any "legal" physical motion (one that obeys the laws), and run the movie backward, what you see is also a legal motion. However, the legal motions common in our world often look bizarre when run backward. Raindrops falling from the sky to create puddles are an everyday sight; puddles that spit raindrops skyward and vanish are not. The source of the difference lies in the initial conditions. Most initial conditions break time-reversal symmetry. For example, suppose we decide to start with raindrops falling downward. This is not a time-symmetric state: its time reversal would have raindrops falling upward. Even though the laws are time-reversible, the motion they produce need not be, because once the time-reversal symmetry has been broken by the choice of initial conditions, it remains broken.

Back to the oscillators. I've now explained that periodic oscillations possess time-translation symmetry, but I haven't yet told you what symmetry is broken to create that pattern. The answer is "all time translations." A state that is invariant under these symmetries must look exactly the same at all instants of time--not just intervals of one period. That is, it must be a steady state. So when a system whose state is steady begins to oscillate periodically, its time-translational symmetries decrease from all translations to only translations by a fixed interval. This all sounds rather theoretical. However, the realization that Hopf bifurcation is really a case of temporal symmetry breaking has led to an extensive theory of Hopf bifurcation in systems that have other symmetries as well-especially spatial ones. The mathematical machinery does not depend on particular interpretations and can easily work with several different kinds of symmetry at once. One of the success stories of this approach is a general classification of the patterns that typically set in when a symmetric network of oscillators undergoes a Hopf bifurcation, and one of the areas to which it has recently been applied is animal locomotion.

Two biologically distinct but mathematically similar types of oscillator are involved in locomotion. The most obvious oscillators are the animal's limbs, which can be thought of as mechanical systems-linked assemblies of bones, pivoting at the joints, pulled this way and that by contracting muscles. The main oscillators that concern us here, however, are to be found in the creature's nervous system, the neural circuitry that generates the rhythmic electrical signals that in tum stimulate and control the limbs' activity. Biologists call such a circuit a CPG, which stands for "central pattern generator." Correspondingly, a student of mine took to referring to a limb by the acronym LEG, allegedly for "locomotive excitation generator." Animals have two, four, six, eight, or more LEGs, but we know very little directly about the ePGs that control them, for reasons I shall shortly explain. A lot of what we do know has been arrived at by working backward-or forward, if you like-from mathematical models.

Some animals possess only one gait-only one rhythmic default pattern for moving their limbs. The elephant, for example, can only walk. When it wants to move faster, it ambles-but an amble is just a fast walk, and the patterns of leg movement are the same. Other animals possess many different gaits; take the horse, for example. At low speeds, horses walk; at higher speeds, they trot; and at top speed they gallop. Some insert yet another type of motion, a canter, between a trot and a gallop. The differences are fundamental: a trot isn't just a fast walk but a different kind of movement altogether. In 1965, the American zoologist Milton Hildebrand noticed that most gaits possess a degree of symmetry. That is, when an animal bounds, say, both front legs move together and both back legs move together; the bounding gait preserves the animal's bilateral symmetry. Other symmetries are more subtle: for example, the left half of a camel may follow the same sequence of movements as the right, but half a period out of phase-that is, after a time delay equal to half the period. So the pace gait has its own characteristic symmetry: "reflect left and right, and shift the phase by half a period." You use exactly this type of symmetry breaking to move yourself around: despite your bilateral symmetry, you don't move both legs simultaneously! There's an obvious advantage to bipeds in not doing so: if they move both legs slowly at the same time they fall over.

The seven most common quadrupedal gaits are the trot, pace, bound, walk, rotary gallop, transverse gallop, and can ter, In the trot, the legs are in effect linked in diagonal pairs. First the front left and back right hit the ground together, then the front right and back left. In the bound, the front legs hit the ground together, then the back legs, The pace links the movements fore and aft: the two left legs hit the ground, then the two right. The walk involves a more complex but equally rhythmic pattern: front left, back right, front right, back left, then repeat. In the rotary gallop, the front legs hit the ground almost together, but with the right (say) very slightly later than the left; then the back legs hit the ground almost together, but this time with the left very slightly later than the right. The transverse gallop is similar, but the sequence is reversed for the rear legs. The canter is even more curious: first front left, then back right, then the other two legs simultaneously, There is also a rarer gait, the pronk, in which all four legs move simultaneously.

The pronk is uncommon, outside of cartoons, but is sometimes seen in young deer. The pace is observed in camels, the bound in dogs; cheetahs use the rotary gallop to travel at top speed, Horses are among the more versatile quadrupeds, using the walk, trot, transverse gallop, and canter, depending on circumstances.

The ability to switch gaits comes from the dynamics of CPGs. The basic idea behind CPG models is that the rhythms and the phase relations of animal gaits are determined by the natural oscillation patterns of relatively simple neural circuits. What might such a circuit look like? Trying to locate a specific piece of neural circuitry in an animal's body is like searching for a particular grain of sand in a desert: to map out the nervous system of all but the simplest of animals is well beyond the capabilities even of today's science. So we have to sneak up on the problem of ePG design in a less direct manner.

One approach is to work out the simplest type of circuit that might produce all the distinct but related symmetry patterns of gaits. At first, this looks like a tall order, and we might be forgiven if we tried to concoct some elaborate structure with switches that effected the change from one gait to another, like a car gearbox. But the theory of Hopf bifurcation tells us that there is a simpler and more natural way. It turns out that the symmetry patterns observed in gaits are strongly reminiscent of those found in symmetric networks of oscillators. Such networks naturally possess an entire repertoire of symmetry-breaking oscillations, and can switch between them in a natural manner. You don't need a complicated gearbox. For example, a network representing the ePG of a biped requires only two identical oscillators, one for each leg. The mathematics shows that if two identical oscillators are coupled- connected so that the state of each affects that of the other-then there are precisely two typical oscillation patterns. One is the in-phase pattern, in which both oscillators behave identically. The other is the out-oJ-phase pattern, in which both oscillators behave identically except for a halfperiod phase difference. Suppose that this signal from the ePG is used to drive the muscles that control a biped's legs, by assigning one leg to each oscillator. The resulting gaits inherit the same two patterns. For the in-phase oscillation of the network, both legs move together: the animal performs a two-legged hopping motion, like a kangaroo. In contrast, the out-of-phase motion of the ePG produces a gait resembling the human walk. These two gaits are the ones most commonly observed in bipeds, (Bipeds can, of course, do other things; for example, they can hop along on one leg-but in that case they effectively turn themselves into one-legged animals.)

What about quadrupeds? The simplest model is now a system of four coupled oscillators-one for each leg. Now the mathematics predicts a greater variety of patterns, and nearly all of them correspond to observed gaits. The most symmetric gait, the pronk, corresponds to all four oscillators being synchronized- that is, to unbroken symmetry. The next most symmetric gaits-the bound, the pace, and the trot-correspond to grouping the oscillators as two out-of-phase pairs: front/back, left/right, or diagonally. The walk is a circulating figure-eight pattern and, again, occurs naturally in the mathematics. The two kinds of gallop are more subtle. The rotary gallop is a mixture of pace and bound, and the transverse gallop is a mixture of bound and trot. The canter is even more subtle and not as well understood.

The theory extends readily to six-legged creatures such as insects. For example, the typical gait of a cockroach-and, indeed, of most insects-is the tripod, in which the middle leg on one side moves in phase with the front and back legs on the other side, and then the other three legs move together, half a period out of phase with the first set. This is one of the natural patterns for six oscillators connected in a ring.

The symmetry-breaking theory also explains how animals can change gait without having a gearbox: a single network of oscillators can adopt different patterns under different conditions. The possible transitions between gaits are also organized by symmetry. The faster the animal moves, the less symmetry its gait has: more speed breaks more symmetry. But an explanation of why they change gait requires more detailed information on physiology. In 1981, D. F. Hoyt and R. C. Taylor discovered that when horses are permitted to select their own speeds, depending on terrain, they choose whichever gait minimizes their oxygen consumption.

I've gone into quite a lot of detail about the mathematics of gaits because it is an unusual application of modern mathematical techniques in an area that at first sight seems totally unrelated. To end this chapter, I want to show you another application of the same general ideas, except that in this case it is biologically important that symmetry not be broken.

One of the most spectacular displays in the whole of nature occurs in Southeast Asia, where huge swarms of fireflies flash in synchrony. In his 1935 article" Synchronous Flashing of Fireflies" in the journal Science, the American biologist Hugh Smith provides a compelling description of the phenomenon:

Imagine a tree thirty-five to forty feet high. apparently with a firefly on every leaf. and all the fireflies flashing in perfect unison at the rate of about three times in two seconds. the tree being in complete darkness between flashes. Imagine a tenth of a mile of river front with an unbroken line of mangrove trees with fireflies on every leaf flashing in synchronism, the insects on the trees at the ends of the line acting in perfect unison with those between. Then. if one's imagination is sufficiently vivid. he may form some conception of this amazing spectacle.

Why do the flashes synchronize? In 1990, Renato Mirollo and Steven Strogatz showed that synchrony is the rule for mathematical models in which every firefly interacts with every other. Again, the idea is to model the insects as a population of oscillators coupled together-this time by visual signals. The chemical cycle used by each firefly to create a flash of light is represented as an oscillator. The population of fireflies is represented by a network of such oscillators with fully symmetric coupling-that is, each oscillator affects all of the others in exactly the same manner. The most unusual feature of this model, which was introduced by the American biologist Charles Peskin in 1975, is that the oscillators are pulsecoupled. That is, an oscillator affects its neighbors only at the instant when it creates a flash of light.

The mathematical difficulty is to disentangle all these interactions, so that their combined effect stands out clearly. Mirollo and Strogatz proved that no matter what the initial conditions are, eventually all the oscillators become synchronized. The proof is based on the idea of absorption, which happens when two oscillators with different phases "lock together" and thereafter stay in phase with each other. Because the coupling is fully symmetric, once a group of oscillators has locked together, it cannot unlock. A geometric and analytic proof shows that a sequence of these absorptions must occur, which eventually locks all the oscillators together.

The big message in both locomotion and synchronization is that nature's rhythms are often linked to symmetry, and that the patterns that occur can be classified mathematically by invoking the general principles of symmetry breaking. The principles of symmetry breaking do not answer every question about the natural world, but they do provide a unifying framework, and often suggest interesting new questions. In particular, they both pose and answer the question, Why these patterns but not others?

The lesser message is that mathematics can illuminate many aspects of nature that we do not normally think of as being mathematical. This is a message that goes back to the Scottish zoologist D' Arcy Thompson, whose classic but maverick book On Growth and Form set out, in 1917, an enormous variety of more or less plausible evidence for the role of mathematics in the generation of biological form and behavior.

In an age when most biologists seem to think that the only interesting thing about an animal is its DNA sequence, it is a message that needs to be repeated, loudly and often.

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

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Do Dice Play God : Nature's Numbers Chapter 8

The intellectual legacy of Isaac Newton was a vision of the clockwork universe, set in motion at the instant of creation but thereafter running in prescribed grooves, like a well-oiled machine. It was an image of a totally deterministic worldone leaving no room for the operation of chance, one whose future was completely determined by its present. As the great mathematical astronomer Pierre-Simon de Laplace eloquently put it in 1812 in his Analytic Theory of Probabilities:



An intellect which at any given moment knew all the forces that animate Nature and the mutual positions of the beings that comprise it, if this intellect were vast enough to submit its data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom: for such an intellect nothing could be uncertain, and the future just like the past would be present before its eyes. This same vision of a world whose future is totally predictable lies behind one of the most memorable incidents in Douglas Adams's 1979 science-fiction novel The Hitchhiker's Guide to the Galaxy, in which the philosophers Majikthise and Vroomfondel instruct the supercomputer "Deep Thought" to calculate the answer to the Great Question of Life, the Uni- verse, and Everything. Aficionados will recall that after five million years the computer answered, "Forty-two," at which point the philosophers realized that while the answer was clear and precise, the question had not been. Similarly, the fault in Laplace's vision lies. not in his answer-that the universe is in principle predictable, which is an accurate statement of a particular mathematical feature of Newton's law of motion-but in his interpretation of that fact, which is a serious misunderstanding based on asking the wrong question. By asking a more appropriate question, mathematicians and physicists have now come to understand that determinism and predictability are not synonymous.

In our daily lives, we encounter innumerable cases where Laplacian determinism seems to be a highly inappropriate model. We walk safely down steps a thousand times, until one day we turn our ankle and break it. We go to a tennis match, and it is rained off by an unexpected thunderstorm. We place a bet on the favorite in a horse race, and it falls at the last fence when it is six lengths ahead of the field. It's not so much a universe in which-as Albert Einstein memorably refused to believe-God plays dice: it seems more a universe in which dice play God.



Is our world deterministic, as Laplace claimed, or is it governed by chance, as it so often seems to be? And if Laplace is really right, why does so much of our experience indicate that he is wrong? One of the most exciting new areas of mathematics, nonlinear dynamics-popularly known as chaos theoryclaims to have many of the answers. Whether or not it does, it is certainly creating a revolution in the way we think about order and disorder, law and chance, predictability and randomness.

According to modern physics, nature is ruled by chance on its smallest scales of space and time. For instance, whether a radioactive atom-of uranium, say-does or does not decay at any given instant is purely a matter of chance. There is no physical difference whatsoever between a uranium atom that is about to decay and one that is not about to decay. None. Absolutely none.

There are at least two contexts in which to discuss these issues: quantum mechanics and classical mechanics. Most of this chapter is about classical mechanics, but for a moment let us consider the quantum-mechanical context. It was this view of quantum indeterminacy that prompted Einstein's famous statement (in a letter to his colleague Max Born) that "you believe in a God who plays dice, and 1 in complete law and order." To my mind, there is something distinctly fishy about the orthodox physical view of quantum indeterminacy, and 1 appear not to be alone, because, increasingly, many physicists are beginning to wonder whether Einstein was right all along and something is missing from conventional quantum mechanics-perhaps "hidden variables," whose values tell an atom when to decay. (I hasten to add that this is not the conventional view.) One of the best known of them, the Princeton physicist David Bohm, devised a modification of quantum mechanics that is fully deterministic but entirely consistent with all the puzzling phenomena that have been used to support the conventional view of quantum indeterminacy. Bohm's ideas have problems of their own, in particular a kind of "action at a distance" that is no less disturbing than quantum indeterminacy.

However, even if quantum mechanics is correct about indeterminacy on the smallest scales, on macroscopic scales of space and time the universe obeys deterministic laws, This results from an effect called decoherence, which causes sufficiently large quantum systems to lose nearly all of their indeterminacy and behave much more like Newtonian systems. In effect, this reinstates classical mechanics for most humanscale purposes. Horses, the weather, and Einstein's celebrated dice are not unpredictable because of quantum mechanics.

On the contrary, they are unpredictable within a Newtonian model, too. This is perhaps not so surprising when it come to horses-living creatures have their own hidden variables, such as what kind of hay they had for breakfast. But it was definitely a surprise to those meteorologists who had been developing massive computer simulations of weather in the hope of predicting it for months ahead. And it is really rather startling when it comes to dice, even though humanity perversely uses dice as one of its favorite symbols for chance. Dice are just cubes, and a tumbling cube should be no less predictable than an orbiting planet: after all, both objects obey the same laws of mechanical motion. They're different shapes, but equally regular and mathematical ones.

To see how unpredictability can be reconciled with determinism, think about a much less ambitious system than the entire universe-namely, drops of water dripping from a tap. * This is a deterministic system: in principle, the flow of water into the apparatus is steady and uniform, and what happens to it when it emerges is totally prescribed by the laws of fluid motion. Yet a simple but effective experiment demonstrates that this evidently deterministic system can be made to behave unpredictably; and this leads us to some mathematical "lateral thinking," which explains why such a paradox is possible.

If you turn on a tap very gently and wait a few seconds for the flow to settle down, you can usually produce a regular series of drops of water, falling at equally spaced times in a regular rhythm. It would be hard to find anything more predictable than this. But if you slowly turn the tap to increase the flow, you can set it so that the sequence of drops falls in a very irregular manner, one that sounds random. It may take a little experimentation to succeed, and it helps if the tap turns smoothly. Don't turn it so far that the water falls in an unbroken stream; what you want is a medium-fast trickle. If you get it set just right, you can listen for many minutes without any obvious pattern becoming apparent.

In 1978, a bunch of iconoclastic young graduate students at the University of California at Santa Cruz formed the Dynamical Systems Collective. When they began thinking about this water-drop system, they realized that it's not as random as it appears to be. They recorded the dripping noises with a microphone and analyzed the sequence of intervals between each drop and the next. What they found was shortterm predictability. If I tell you the timing of three successive drops, then you can predict when the next drop will fall. For example, if the last three intervals between drops have been 0.63 seconds, 1.17 seconds, and 0.44 seconds, then you can be sure that the next drop will fall after a further 0.82 seconds. (These numbers are for illustrative purposes only.) In fact, if you know the timing of the first three drops exactly, then you can predict the entire future of the system.

So why is Laplace wrong? The point is that we can never measure the initial state of a system exactly. The most precise measurements yet made in any physical system are correct to about ten or twelve decimal places. But Laplace's statement is correct only if we can make measurements to infinite precision, infinitely many decimal places-and of course there's no way to do that. People knew about this problem of measurement error in Laplace's day, but they generally assumed that provided you made the initial measurements to, say, ten decimal places, then all subsequent prediction would also be accurate to ten decimal places. The error would not disappear, but neither would it grow.

Unfortunately, it does grow, and this prevents us from stringing together a series of short-term predictions to get one that is valid in the long term. For example, suppose I know the timing of the first three water drops to an accuracy of ten decimal places. Then I can predict the timing of the next drop to nine decimal places, the drop after that to eight decimal places, and so on. At each step, the error grows by a factor of about ten, so I lose confidence in one further decimal place. Therefore, ten steps into the future, I really have no idea at all what the timing of the next drop will be. (Again, the precise figures will probably be different: it may take half a dozen drops to lose one decimal place in accuracy, but even then it takes only sixty drops until the same problem arises.)

This amplification of error is the logical crack through which Laplace's perfect determinism disappears. Nothing short of total perfection of measurement will do. If we could measure the timing to a hundred decimal places, our predictions would fail a mere hundred drops into the future (or six hundred, using the more optimistic estimate). This phenomenon is called "sensitivity to initial conditions," or more informally "the butterfly effect." (When a butterfly in Tokyo flaps its wings, the result may be a hurricane in Florida a month later.) It is intimately associated with a high degree of irregularity of behavior. Anything truly regular is by definition fairly predictable. But sensitivity to initial conditions renders behavior unpredictable-hence irregular. For this reason, a system that displays sensitivity to initial conditions is said to be chaotic. Chaotic behavior obeys deterministic laws, but it is so irregular that to the untrained eye it looks pretty much random. Chaos is not just complicated, patternless behavior; it is far more subtle. Chaos is apparently complicated, apparently patternless behavior that actually has a simple, deterministic explanation.

The discovery of chaos was made by many people, too numerous to list here. It came about because of the conjunction of three separate developments. One was a change of scientific focus, away from simple patterns such as repetitive cycles, toward more complex kinds of behavior. The second was the computer, which made it possible to find approximate solutions to dynamical equations easily and rapidly. The third was a new mathematical viewpoint on dynamics-a geometric rather than a numerical viewpoint. The first provided motivation, the second provided technique, and the third provided understanding.

The geometrization of dynamics began about a hundred years ago, when the French mathematician Henri Poincare-a maverick if ever there was one, but one so brilliant that his views became orthodoxies almost overnight-invented the concept of a phase space. This is an imaginary mathematical space that represents all possible motions of a given dynamical system. To pick a nonmechanical example, consider the population dynamics of a predator-prey ecological system.

The predators are pigs and the prey are those exotically pungent fungi, truffles. The variables upon which we focus attention are the sizes of the two populations-the number of pigs (relative to some reference value such as one million) and the number of truffles (ditto). This choice effectively makes the variables continuous-that is, they take real-number values with decimal places, not just whole-number values. For example, if the reference number of pigs is one million, then a population of 17,439 pigs corresponds to the value 0.017439. Now, the natural growth of truffles depends on how many truffles there are and the rate at which pigs eat them: the growth of the pig population depends on how many pigs there are and how many truffles they eat. So the rate of change of each variable depends on both variables, an observation that can be turned into a system of differential equations for the population dynamics. I won't write them down, because it's not the equations that matter here: it's what you do with them.

These equations determine-in principle-how any initial population values will change over time. For example, if we start with 17,439 pigs and 788,444 truffles, then you plug in the initial values 0.017439 for the pig variable and 0.788444 for the truffle variable, and the equations implicitly tell you how those numbers will change. The difficulty is to make the implicit become explicit: to solve the equations. But in what sense? The natural reflex of a classical mathematician would be to look for a formula telling us exactly what the pig population and the truffle population will be at any instant. Unfortunately, such "explicit solutions" are so rare that it is scarcely worth the effort of looking for them unless the equations have a very special and limited form. An alternative is to find approximate solutions on a computer; but that tells us only what will happen for those particular initial values, and most often we want to know what will happen for a lot of different initial values.

Poincare's idea is to draw a picture that shows what happens for all initial values. The state of the system-the sizes of the two populations at some instant of time-can be represented as a point in the plane, using the old trick of coordinates. For example, we might represent the pig population by the horizontal coordinate and the truffle population by the vertical one. The initial state described above corresponds to the point with horizontal coordinate 0.017439 and vertical coordinate 0.788444. Now let time flow. The two coordinates change from one instant to the next, according to the rule expressed by the differential equation, so the corresponding point moves. A moving point traces out a curve; and that curve is a visual representation of the future behavior of the entire system. In fact, by looking at the curve, you can "see" important features of the dynamics without worrying about the actual numerical values ofthe coordinates.

For example, if the curve closes up into a loop, then the two populations are following a periodic cycle, repeating the same values over and over again-just as a car on a racetrack keeps going past the same spectator every lap. If the curve homes in toward some particular point and stops, then the populations settle down to a steady state, in which neither changes-like a car that runs out of fuel. By a fortunate coincidence, cycles and steady states are of considerable ecological significance-in particular, they set both upper and lower limits to populations sizes. So the features that the eye detects most easily are precisely the ones that really matter. Moreover, a lot of irrelevant detail can be ignored: for example, we can see that there is a closed loop without having to work out its precise shape (which represents the combined "waveforms" of the two population cycles).

What happens if we try a different pair of initial values? We get a second curve. Each pair of initial values defines a new curve; and we can capture all possible behaviors of the system, for all initial values, by drawing a complete set of such curves. This set of curves resembles the flow lines of an imaginary mathematical fluid, swirling around in the plane.

We call the plane the phase space of the system, and the set of swirling curves is the system's phase portrait. Instead of the symbol-based idea of a differential equation with various initial conditions, we have a geometric, visual scheme of points flowing through pig/truffle space. This differs from an ordinary plane only in that many of its points are potential rather than actual: their coordinates correspond to numbers of pigs and truffles that could occur under appropriate initial conditions, but may not occur in a particular case. So as well as the mental shift from symbols to geometry, there is a philosophical shift from the actual to the potential.

The same kind of geometric picture can be imagined for any dynamical system. There is a phase space, whose coordinates are the values of all the variables; and there is a phase portrait, a system of swirling curves that represents all possible behaviors starting from all possible initial conditions, and that are prescribed by the differential equations. This idea constitutes a major advance, because instead of worrying about the precise numerical details of solutions to the equations, we can focus upon the broad sweep of the phase portrait, and bring humanity's greatest asset, its amazing image processing abilities, to bear. The image of a phase space as a way of organizing the total range of potential behaviors, from among which nature selects the behavior actually observed, has become very widespread in science.

The upshot of Poincare's gre'at innovation is that dynamics can be visualized in terms of geometric shapes called attractors. If you start a dynamical system from some initial point and watch what it does in the long run, you often find that it ends up wandering around on some well-defined shape in phase space. For example, the curve may spiral in toward a closed loop and then go around and around the loop forever. Moreover, different choices of initial conditions may lead to the same final shape. If so, that shape is known as an attractor. The long-term dynamics of a system is governed by its attractors, and the shape of the attractor determines what type of dynamics occurs.

For example, a system that settles down to a steady state has an attractor that is just a point. A system that settles down to repeating the same behavior periodically has an attractor that is a closed loop. That is, closed loop attractors correspond to oscillators. Recall the description of a vibrating violin string from chapter 5; the string undergoes a sequence of motions that eventually puts it back where it started, ready to repeat the sequence over and over forever. I'm not suggesting that the violin string moves in a physical loop. But my description of it is a closed loop in a metaphorical sense: the motion takes a round trip through the dynamic landscape of phase space.

Chaos has its own rather weird geometry: it is associated with curious fractal shapes called strange attractors. The butterfly effect implies that the detailed motion on a strange attractor can't be determined in advance. But this doesn't alter the fact that it is an attractor. Think of releasing a Ping-Pong ball into a stormy sea. Whether you drop it from the air or release it from underwater, it moves toward the surface. Once on the surface, it follows a very complicated path in the surging waves, but however complex that path is, the ball stays on-or at least very near-the surface. In this image, the surface of the sea is an attractor. So, chaos notwithstanding, no matter what the starting point may be, the system will end up very close to its attractor. Chaos is well established as a mathematical phenomenon, but how can we detect it in the real world? We must perform experiments-and there is a problem. The traditional role of experiments in science is to test theoretical predictions, but if the butterfly effect is in operation-as it is for any chaotic system- how can we hope to test a prediction? Isn't chaos inherently untestable, and therefore unscientific?

The answer is a resounding no, because the word "prediction" has two meanings. One is "foretelling the future," and the butterfly effect prevents this when chaos is present. But the other is "describing in advance what the outcome of an experiment will be." Think about tossing a coin a hundred times. In order to predict-in the fortune-teller's sense-what happens, you must list in advance the result of each of the tosses. But you can make scientific predictions, such as "roughly half the coins will show heads," without foretelling the future in detail-even when, as here, the system is random. Nobody suggests that statistics is unscientific because it deals with unpredictable events, and therefore chaos should be treated in the same manner. You can make all sorts of predictions about a chaotic system; in fact, you can make enough predictions to distinguish deterministic chaos from true randomness. One thing that you can often predict is the shape of the attractor, which is not altered by the butterfly effect. All the butterfly effect does is to make the system follow different paths on the same attractor. In consequence, the general shape of the attractor can often be inferred from experimental observations.

The discovery of chaos has revealed a fundamental misunderstanding in our views of the relation between rules and the behavior they produce-between cause and effect. We used to think that deterministic causes must produce regular effects, but now we see that they can produce highly irregular effects that can easily be mistaken for randomness. We used to think that simple causes must produce simple effects (implying that complex effects must have complex causes), but now we know that simple causes can produce complex effects. We realize that knowing the rules is not the same as being able to predict future behavior.

How does this discrepancy between cause and effect arise? Why do the same rules sometimes produce obvious patterns and sometimes produce chaos? The answer is to be found in every kitchen, in the employment of that simple mechanical device, an eggbeater. The motion of the two beaters is simple and predictable, just as Laplace would have expected: each beater rotates steadily. The motion of the sugar and the egg white in the bowl, however, is far more complex. The two ingredients get mixed up-that's what eggbeaters are for. But the two rotary beaters don't get mixed up-you don't have to disentangle them from each other when you've finished. Why is the motion of the incipient meringue so different from that of the beaters? Mixing is a far more complicated, dynamic process than we tend to think. Imagine trying to predict where a particular grain of sugar will end up! As the mixture passes between the pair of beaters, it is pulled apart, to left and right, and two sugar grains that start very close together soon get a long way apart and follow independent paths. This is, in fact, the butterfly effect in action-tiny changes in initial conditions have big effects. So mixing is a chaotic process. Conversely, every chaotic process involves a kind of mathematical mixing in Poincare's imaginary phase space. This is why tides are predictable but weather is not. Both involve the same kind of mathematics, but the dynamics of tides does not get phase space mixed up, whereas that of the weather does. It's not what you do, it's the way that you do it.

Chaos is overturning our comfortable assumptions about how the world works. It tells us that the universe is far stranger than we think. It casts doubt on many traditional methods of science: merely knowing the laws of nature is no longer enough. On the other hand, it tells us that some things that we thought were just random may actually be consequences of simple laws. Nature's chaos is bound by rules. In the past, science tended to ignore events or phenomena that seemed random, on the grounds that since they had no obvious patterns they could not be governed by simple laws. Not so. There are simple laws right under our noses-laws governing disease epidemics, or heart attacks, or plagues of locusts. If we learn those laws, we may be able to prevent the disasters that follow in their wake.

Already chaos has shown us new laws, even new types of laws. Chaos contains its own brand of new universal patterns. One of the first to be discovered occurs in the dripping tap. Remember that a tap can drip rhythmically or chaotically, depending on the speed of the flow. Actually, both the regularly dripping tap and the "random" one are following slightly different variants of the same mathematical prescription. But as the rate at which water passes through the tap increases, the type of dynamics changes. The attractor in phase space that represents the dynamics keeps changing-and it changes in a predictable but highly complex manner.

Start with a regularly dripping tap: a repetitive drip-dripdrip- drip rhythm, each drop just like the previous one. Then tum the tap slightly, so that the drips come slightly faster. Now the rhythm goes drip-DRIP-drip-DRIP, and repeats every two drops. Not only the size of the drop, which governs how loud the drip sounds, but also the timing changes slightly from one drop to the next.

If you allow the water to flow slightly faster still, you get a four-drop rhythm: drip-DRIP-drip-DRIP. A little faster still, and you produce an eight-drop rhythm: drip-DRIP-drip-DRIPdrip- DRIP-drip-DRIP. The length of the repetitive sequence of drops keeps on doubling. In a mathematical model, this process continues indefinitely, with rhythmic groups of 16, 32, 64 drops, and so on. But it takes tinier and tinier changes to the flow rate to produce each successive doubling of the period; and there is a flow rate by which the size of the group has doubled infinitely often. At this point, no sequence of drops repeats exactly the same pattern. This is chaos. We can express what is happening in Poincare's geometric language. The attractor for the tap begins as a closed loop, representing a periodic cycle. Think of the loop as an elastic band wrapped around your finger. As the flow rate increases, this loop splits into two nearby loops, like an elastic band wound twice around your finger. This band is twice as long as the original, which is why the period is twice as long. Then in exactly the same way, this already-doubled loop doubles again, all the way along its length, to create the period-four cycle, and so on. After infinitely many doublings, your finger is decorated with elastic spaghetti, a chaotic attractor.

This scenario for the creation of chaos is called a perioddoubling cascade. In 1975, the physicist Mitchell Feigenbaum discovered that a particular number, which can be measured in experiments, is associated with every period-doubling cascade. The number is roughly 4.669, and it ranks alongside 1t (pi) as one of those curious numbers that seem to have extraordinary significance in both mathematics and its relation to the natural world. Feigenbaum's number has a symbol, too: the Greek letter () (delta). The number 1t tells us how the circumference of a circle relates to its diameter. Analogously, Feigenbaum's number () tells us how the period of the drips relates to the rate of flow of the water. To be precise, the extra amount by which you need to turn on the tap decreases by a factor of 4.669 at each doubling of the period.

The number 1t is a quantitative signature for anything involving circles. In the same way, the Feigenbaum number () is a quantitative signature for any period-doubling cascade, no matter how it is produced or how it is realized experimentally. That very same number shows up in experiments on liquid helium, water, electronic circuits, pendulums, magnets, and vibrating train wheels. It is a new universal pattern in nature, one that we can see only through the eyes of chaos; a quantitative pattern, a number, emerges from a qualitative phenomenon. One of nature's numbers, indeed. The Feigenbaum number has opened the door to a new mathematical world, one we have only just begun to explore.

The precise pattern found by Feigenbaum, and other patterns like it, is a matter of fine detail. The basic point is that even when the consequences of natural laws seem to be patternless, the laws are still there and so are the patterns. Chaos is not random: it is apparently random behavior resulting from precise rules. Chaos is a cryptic form of order. Science has traditionally valued order, but we are beginning to appreciate the fact that chaos can offer science distinct advantages. Chaos makes it much easier to respond quickly to an outside stimulus. Think of tennis players waiting to receive a serve. Do they stand still? Do they move regularly from side to side? Of course not. They dance erratically from one foot to the other. In part, they are trying to confuse their opponents, but they are also getting ready to respond to any serve sent their way. In order to be able to move quickly in any particular direction, they make rapid movements in many different directions. A chaotic system can react to outside events much more quickly, and with much less effort, than a non chaotic one. This is important for engineering control problems. For example, we now know that some kinds of turbulence result from chaos-that's what makes turbulence look random. It may prove possible to make the airflow past an aircraft's skin much less turbulent, and hence less resistant to motion, by setting up control mechanisms that respond extremely rapidly to cancel out any small regions of incipient turbulence. Living creatures, too, must behave chaotically in order to respond rapidly to a changing environment.

This idea has been turned into an extremely useful practical technique by a group of mathematicians and physicists, among them William Ditto, Alan Garfinkel, and Jim Yorke: they call it chaotic control. Basically, the idea is to make the butterfly effect work for you. The fact that small changes in initial conditions create large changes in subsequent behavior can be an advantage; all you have to do is ensure that you get the large changes you want. Our understanding of how chaotic dynamics works makes it possible to devise control strategies that do precisely this. The method has had several successes. Space satellites use a fuel called hydrazine to make course corrections. One of the earliest successes of chaotic control was to divert a dead satellite from its orbit and send it out for an encounter with an asteroid, using only the tiny amount of hydrazine left on board. NASA arranged for the satellite to swing around the Moon five times, nudging it slightly each time with a tiny shot of hydrazine. Several such encounters were achieved, in an operation that successfully exploited the occurrence of chaos in the three-body problem (here, Earth/Moon/satellite) and the associated butterfly effect.

The same mathematical idea has been used to control a magnetic ribbon in a turbulent fluid-a prototype for controlling turbulent flow past a submarine or an aircraft. Chaotic control has been used to make erratically beating hearts return to a regular rhythm, presaging invention of the intelligent pacemaker. Very recently, it has been used both to set up and to prevent rhythmic waves of electrical activity in brain tissue, opening up the possibility of preventing epileptic attacks. Chaos is a growth industry. Every week sees new discoveries about the underlying mathematics of chaos, new applications of chaos to our understanding of the natural world, or new technological uses of chaos-including the chaotic dishwasher, a Japanese invention that uses two rotating arms, spinning chaotically, to get dishes cleaner using less energy; and a British machine that uses chaos-theoretic data analysis to improve quality control in spring manufacture.

Much, however, remains to be done. Perhaps the ultimate unsolved problem of chaos is the strange world of the quantum, where Lady Luck rules. Radioactive atoms decay "at random"; their only regularities are statistical. A large quantity of radioactive atoms has a well-defined half-life-a period of time during which half the atoms will decay. But we can't predict which half. Albert Einstein's protest, mentioned earlier, was aimed at just this question. Is there really no difference at all between a radioactive atom that is not going to decay, and one that's just about to? Then how does the atom know what to do?

Might the apparent randomness of quantum mechanics be fraudulent? Is it really deterministic chaos? Think of an atom as some kind of vibrating droplet of cosmic fluid. Radioactive atoms vibrate very energetically, and every so often a smaller drop can split off-decay. The vibrations are so rapid that we can't measure them in detail: we can only measure averaged quantities, such as energy levels. Now, classical mechanics tells us that a drop of real fluid can vibrate chaotically. When it does so, its motion is deterministic but unpredictable. Occasionally, "at random," the vibrations conspire to split off a tiny droplet. The butterfly effect makes it impossible to say in advance just when the drop will split; but that event has precise statistical features, including a well defined half-life. Could the apparently random decay of radioactive atoms be something similar, but on a microcosmic scale? After all, why are there any statistical regularities at all? Are they traces of an underlying determinism? Where else can statistical regularities come from? Unfortunately, nobody has yet made this seductive idea work-though it's similar in spirit to the fashionable theory of superstrings, in which a subatomic particle is a kind of hyped-up vibrating multidimensional loop. The main similar feature here is that both the vibrating loop and the vibrating drop introduce new "internal variables" into the physical picture. A significant difference is the way these two approaches handle quantum indeterminacy. Superstring theory, like conventional quantum mechanics, sees this indeterminacy as being genuinely random. In a system like the drop, however, the apparent indeterminacy is actually generated by a deterministic, but chaotic, dynamic. The trick-if only we knew how to do it-would be to invent some kind of structure that retains the successful features of superstring theory, while making some of the internal variables behave chaotically. It would be an appealing way to render the Deity's dice deterministic, and keep the shade of Einstein happy.

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

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