Broken Symmetry : Nature's Numbers Chapter 6

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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