SYMMETRY: The Thread of Reality

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.
BY-:Ian S.