Symmetry: The Thread of Reality
Ian Stewart

Symmetry may seem to be just an unimportant repetition of structure, but its influence on the scientific vision of the universe is profound. Albert Einstein based all of his revolutionary theories of physics on the principle that the universe is symmetrical-that the laws of physics are the same at each point of space and each instant of time. Because the laws of physics describe how events occurring at one place and time influence events at other places and times, this simple requirement binds the universe together into a coherent whole. Paradoxically, as Einstein discovered, it implies that we cannot sensibly talk of absolute space and time. What is observed depends upon who observes it-in ways that are governed by those same underlying symmetry principles.

It is easy to describe particular kinds of symmetry-for example, an object has reflectional symmetry if it looks the same when viewed in a mirror, and it has rotational symmetry if it looks the same when rotated. Respective examples are the external shape of the human body, and the ripples that form on a pond when you throw a stone into it. But what is symmetry itself? The best answer that we yet have is a mathematical one: Symmetry is "invariance under transformations." A transformation is a method of changing something, a rule for moving it or otherwise altering its structure. Invariance is a simpler concept; it just means that the end result looks the same as the starting point.

Rotation through some chosen angle is a transformation, and so is reflection in some chosen mirror, so these special examples of symmetry fit neatly into the general formulation. A pattern of square tiles has yet another type of symmetry. If the pattern is moved sideways (a transformation) through a distance that is a whole-number multiple of the width of a tile, then the result looks the same (is invariant). In general, the range of possible symmetry transformations is enormous, and therefore, so is the range of possible symmetrical patterns.

Over the last one hundred and fifty years, mathematicians and physicists have invented a deep and powerful "calculus of symmetry." It is known as group theory, because it deals not just with single transformations, but with whole collections of them-"groups." By applying this theory, they have been able to prove striking general facts-for example, that there are precisely seventeen different symmetry types of wallpaper pattern  [Ref : Video: BB17 OU Just17] (that is, repeating patterns that fill a plane), and precisely two hundred and thirty different types of crystal symmetry. And they have also begun to use group theory to understand how the symmetries of the universe affect how nature behaves.

Throughout the natural world, we see intriguing symmetrical patterns: the spiral sweep of a snail's shell; the neatly arranged petals of a flower; the gleaming crescent of a new moon. The same patterns occur in many different settings. The spiral form of a shell recurs in the whirlpool of a hurricane and the stately rotation of a galaxy; raindrops and stars are spherical; and hamsters, herons, horses, and humans are bilaterally symmetrical.
Symmetries arise on every conceivable scale, from the innermost components of the atom to the entire universe. The four fundamental forces of nature (gravity, electromagnetism, and the strong and weak nuclear forces) are now thought to be different aspects of a single unified force in a more symmetrical, high-energy universe. The "ripples at the edge of time"-irregularities in the cosmic background radiation -recently observed by the COBE (Cosmic Background Explorer) satellite help to explain how an initially symmetrical big bang can create the structured universe in which we now find ourselves.

Symmetrical structures on the microscopic level are implicated in living processes. Deep within each living cell there is a structure known as the centrosome, which plays an important role in organising cell division. Inside the centrosome are two centrioles, positioned at right angles to each other. Each centriole is cylindrical, made from twenty-seven tiny tubes (microtubules) fused together along their lengths in threes, and arranged with perfect ninefold symmetry. The microtubules themselves also have an astonishing degree of symmetry; they are hollow tubes made from a perfect regular checkerboard pattern of units that contain two distinct proteins. So even at the heart of organic life we find the perfect mathematical regularities of symmetry.

There is another important aspect of symmetry.Symmetrical objects are made of innumerable copies of identical pieces, so symmetry is intimately bound up with replication. Symmetries occur in the organic world because life is a self- replicating phenomenon. The symmetries of the inorganic world have a similarly "mass produced" origin. In particular, the laws of physics are the same in all places and at all times. Moreover, if you could instantly permute all the electrons in the universe-swapping all the electrons in your brain with randomly chosen electrons in a distant star, say-it would make no difference at all. All electrons are identical, so physics is symmetrical under the interchange of electrons. The same goes for all the other fundamental particles. It is not at all clear why we live in a mass-produced universe, but it is clear that we do, and that this produces an enormous number of potential symmetries. Perhaps, as Richard Feynman once suggested, all electrons are alike because all electrons are the self-same particle, whizzing backward and forward through time. (This strange idea came to him through his invention of "Feynman diagrams"-pictures of the motions of particles in space and time. Complex interactions of many electrons and their antiparticles often form a single zigzag curve in space-time, so they can be explained in terms of a single particle moving alternately forward and backward in time. When an electron moves backward in time, it turns into its antiparticle.) Or perhaps a version of the anthropic principle is in operation: Replicating creatures (especially creatures whose own internal organisation requires stable patterns of behaviour and structure) can arise only in mass- produced universes.

How do nature's symmetrical patterns arise? They can be explained as imperfect or incomplete traces of the symmetries of the laws of physics. Potentially, the universe has an enormous amount of symmetry-its laws are invariant under all motions of space and time and all interchanges of identical particles-but in practice, an effect known as "symmetry breaking" prevents the full range of symmetries from being realised simultaneously. For example, think of a crystal, made from a huge number of identical atoms. The laws of physics look the same if you swap the atoms around or move them through space and time. The most symmetrical configuration would be one for which all of the atoms are in the same place, but this is not physically realisable, because atoms cannot overlap. So some of the symmetry is "broken," or removed, by changing the configuration into one in which the atoms are displaced just enough to allow them to stay separate. The mathematical point is that the physically unrealisable state has a huge amount of symmetry, not all of which need be broken to separate out the atoms. So it is not surprising that some of that symmetry is still present in the state that actually occurs. This is where the symmetry of a crystal lattice comes from: the huge but unseen symmetries of the potential, broken by the requirements of the actual.

This insight has far-reaching consequences. It implies that when studying a scientific problem, we must consider not only what does happen, but what might have happened instead. It may seem perverse to increase the range of problems by thinking about things that don't happen, but situating the actual event inside a cloud of potential events has two advantages. First, we can then ask the question "Why does this particular behaviour occur?"-because implicitly, this question also asks why the remaining possibilities did not, and that means we have to think about all the possibilities that don't occur as well as the ones that do. For instance, we can't explain why pigs don't have wings without implicitly thinking about what would happen if they did. Second, the set of potential events may possess extra structure-such as symmetry-that is not visible in the lone state that is actually observed. For example, we might ask why the surface of a pond is flat (in the absence of wind or currents). We will not find the answer by studying flat ponds alone. Instead, we must disturb the surface of the pond, exploring the space of all potential ponds, to see what drives the surface back to flatness. In that way, we will discover that nonflat surfaces have more energy, and that frictional forces slowly dissipate the excess, driving the pond back to its minimal energy configuration, which is flat. As it happens, a flat surface has a lot of symmetry, and this, too, can best be explained by thinking about the "space" of all possible surfaces.

This, to me, is the deepest message of symmetry. Symmetry, by its very definition, is about what would happen to the universe if it were changed-transformed. Suppose every electron in your head were to be swapped with one in the burning core of the star Sirius. Suppose pigs had wings. Suppose the surfaces of ponds were shaped like Henry Moore sculptures. Nobody intends to perform actual experiments, but just thinking about the possibilities reveals fundamental aspects of the natural world. So the prosaic observation that there are patterns in the universe forces us to view reality as just one possible state of the universe from among an infinite range of potential states-a slender thread of the actual winding through the space of the potential.

IAN STEWART is one of the best-known mathematicians in the world. He is professor of mathematics at the University of Warwick, Coventry, where he works on nonlinear dynamics, chaos, and their applications, and has published over eighty research papers. He believes that mathematicians owe the public an explanation of what they are doing and why.
He has written or coauthored over sixty books, including mathematics textbooks, research monographs, puzzle books for children, books on personal computers, and three mathematical comic books which have appeared only in French. Recent popular science books include Does God Play Dice?; The Problems of Mathematics; Fearful Symmetry; Is God a Geometer?; and The Collapse of Chaos (with biologist Jack Cohen). He writes the "Mathematical Recreations" column of Scientific American; is mathematics consultant to New Scientist ; and contributes to magazines such as Discover, New Scientist, and The Sciences. Professor Stewart also writes science fiction short stories for Omni and Analog, one of which was recently made into a play for Czech Radio.

Asymmetrical people make jealous lovers Asymmetry could account for a fifth of the variation in romantic jealousy from person to person, says a Canadian researcher. Just about everyone is lopsided to some extent. Hormone imbalances in the womb, for instance, can lead to one foot being bigger than the other. But in recent years, a series of animal and human studies have suggested that the implications of asymmetry go far beyond struggling to find shoes that fit both feet. It seems that people who are more symmetrical are not only healthier, more fertile and perhaps even smarter - they are also more attractive. This led William Brown at Dalhousie University in Halifax, Nova Scotia, to wonder about jealousy. "If jealousy is a strategy to retain your mate, then the individual more likely to be philandered on is more likely to be jealous," he speculated. And if people who are less symmetrical are less desirable, they are more likely to be cheated on. To test his theory, Brown looked at 50 men and women in various kinds of heterosexual relationships, comparing the sizes of paired features such as feet, ears and fingers. The volunteers then filled in a questionnaire already used in other studies to assess romantic jealousy. He found a strong link between asymmetry and romantic jealousy. Asymmetry could account for over a fifth of the variation in romantic jealousy from person to person, he says. To make sure less symmetrical people are not simply cursed with more jealous personalities, Brown also assessed their propensity to be jealous outside the relationship, in the workplace for example. But the less symmetrical people were no more likely to be jealous in general, he found, than more symmetrical folk. Brown's research was presented at the International Society for Human Ethology conference in Montreal. [10:30 22 August 02 Exclusive from New Scientist Print Edition ]Alison Motluk

Out of sight Love is often blind to tiny physical imperfections

CLAIMS that many animals-including humans-are influenced in their choice of mate by how symmetrical a potential partner appears have been dealt a blow. Experiments are now suggesting that the symmetry differences may be simply too small to be seen. The idea that creatures might advertise their "fitness" as mates through symmetry has been around since the 1930s. It rests on the notion that factors ranging from bad genes to coming off worse in too many fights will lead to asymmetries others can see. And over the past few years, experiments with animals and humans seemed to support the idea that symmetry in features such as plumage and facial characteristics influences mate selection ("Sex and the symmetrical body", New Scientist, 22 April 1995, p 40). But no one had actually checked that the ammals in these experiments can actually see the subtle differences in symmetry at the heart of the claims, typically of between 1 and 2 per cent. Now the first such expemrient has been carried out. The results are worrying for advocates of symmetry detection as a powerful factor in mate selection. John Swaddle, an ethologist at the University of Bristol, performed the experiment using wild starlings, which have excellent eyesight. He trained them to hit keys marked with bar patterns with varying levels of asymmetry in order to receive food rewards. This showed the starlings could easily detect asymmetries of between 5 and 10 per cent. But at between 1 and 2 per cent, their performance plummeted to no better than random guessing. "This suggests that the levels of asymmetry that birds encounter in nature will often be just too small to be detected," says Swaddle, who reports his findings in the current Proceedings of the Royal Society B (vol 266, p 1299). He says experiments that linked small asymmetries to fitness may have produced misleading correlations, and says the only direct evidence that birds use asymmetry as a cue to mate fitness involves levels of asymmetry of at least 10 per cent. "I think signalling by asymmetry will probably only occur when species show such very large asymmetries-and this doesn't occur that often." This raises questions about research suggesting that humans are influenced by visual asymmetry in their choice of partner Michael Burt of the Perception Laboratory at the University of St Andrews, who has carried out such experiments, concedes that no one has ever checked if humans have a threshold to asymmetry detection. He thinks a conclusive test of this would probably involve manipulating complex three-dimensional images: "It would be a very difficult experiment to do." Burt adds that the abilities of animals to detect asymmetry in bar patterns may not reflect their talent for spotting asymmetries in body shape. "The visibility of an asymmetry may well depend on its type," he says. Swaddle agrees, and is planning further experiments. "But I suspect that asymmetry is used as a visual cue less often than most people appear to presume," he warns. Robert Matthews [New Scientist 24 July 1999]

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