The Importance of Symmetry

05 November 2015
Posted by James Farr.

Symmetry is one of the most underappreciated concepts around today, at least in my opinion. To most, it seems superficial, but in reality it has been a guiding light for physicists since the turn of the 20th Century, and is at the root of all of our modern theories. It inspired a complete sea change in the way we think of the world and yet it is often reduced to the trivial matter of hanging a picture straight, or making sure that eyeliner is applied evenly. Hopefully this article will change how you think about symmetry...

What is symmetry?

As with all maths, the best place to start is with a definition. A good one can make life incredibly simple, whereas a bad one can scupper your approach irreconcilably. Without further ado, then:

A "thing" is symmetrical if there is something you can do to it such that it ends up looking exactly as it did at the start.

As you can tell from the above, a 'good' mathematical definition does not necessarily equate to an eloquent one. To clear things up, an example or two always helps.

Possibly the simplest way to visualise symmetry is via a good old square. You can rotate it by 0, 90, 180 or 270 degrees and it looks the same as it did at the start. Equally, you can reflect it across 4 different axes so we say that the square has 8 symmetries - 8 different transformations which don't change how it looks.

Symmetry of a square

Now for something slightly more abstract. Take the equation x+y=2. Whereas the symmetry of a square is instantly recognisable, this time it's slightly more subtle. Rather than a physical rotation or reflection, this time consider the transformation of swapping x and y over. So where there's an x we write y and vice versa. This gives us the equation y+x=2. But that's exactly the same equation as it doesn't matter what order you add things in. 1+2 is the same thing as 2+1, right? So we've got a transformation - swapping x and y - which leaves our equation unchanged. i.e. the equation is symmetrical! Specifically we say that the equation is symmetrical with respect to swapping x and y.

But why should we care about this? Well in this case it isn't terribly helpful - it's a simple equation which has fairly straightforward solutions. Unfortunately maths gets somewhat more complex than that, and then you can't just think up solutions off the cuff.

The real importance of symmetry was first discovered by a young Frenchman - Evariste Galois - who was a revolutionary both in terms of his maths and his politics. Involved in the fight against the aristocracy of the French Revolution, he was tragically killed in a duel over a failed romance at the age of 20. His entire collected works fit onto less than 60 sides of paper, but even so, they have proved to be enormously important. He was a founding father of 'Group Theory' - a mathematical theory which is crucial to our understanding of symmetry. Rather than looking at individual symmetries of an object, this theory looks at all of the symmetries in one go.

Returning to the square, instead of saying that it has 8 separate symmetries, a Group Theorist would consider all of the symmetries in one group. This group as a whole then captures the essence of 'square-ness' - what it means for something to be a square - rather than just one facet of it, in some sense encoding the fundamental structure of the square.

This arises from the restrictive nature of symmetry - specifying that something must be symmetric in a certain way dictates how it should look mathematically. Thinking of this visually, if I ask 100 people to draw a shape on some paper, then everyone could come up with something different: squares, triangles, abstract squiggles, they're all allowed. However, if I dictate that the shape has to look *exactly* the same no matter how much you rotate the piece of paper by then what are you still allowed to draw? There's only one option - a circle. If you draw any other shape, then you can find an angle of rotation which makes it change its appearance. Imposing one symmetry condition, then, has reduced the number of possibilities from infinity to 1.

Admittedly, the condition that I specified - complete rotational symmetry - is a strong one, but just as the group of 8 symmetries captures 'square-ness', this single symmetry is what captures 'circle-ness'. Through a singly symmetry, then, we have defined the circle in a concise, elegant and, most importantly, correct way.

Within the idea of restriction lies the real beauty of symmetry. By restricting how a physical system can behave, you force it to create recognisable patterns, such as the circle in the above example. This process works both ways: you can impose symmetry conditions to restrict the output, or, if you spot patterns, then you can infer underlying symmetry.

Symmetry and general relativity

The first major use of this link was down to perhaps the most famous scientist of all time - Albert Einstein. While the idea that Einstein failed his school-level mathematics classes is a myth, he himself admitted that his language skills were very poor. In fact, he did not talk until the age of four! As a result, he often thought 'in images', inventing ways to visualise concepts which he had trouble articulating in a similar way to deaf people. In some ways, this unusual method of thinking turned out to be his greatest strength.

Einstein's most well known theories - those of Special and General Relativity - were first inspired by what appeared to be a mistake in the equations that describe light. These equations seemed to imply that the speed of light is always the same for every observer, irrespective of their speed or direction of travel. This is a very strange thought. Normally we consider all speeds to be relative to each other, but this seemed to suggest that our intuition would, in this case, be misleading.

Imagine that you have a device which measures the speed of photons, the particles which make up light. If someone shone a light at you, you would see that all of the photons would be moving at about 300 million metres per second (m/s), as would they. That gets a big thumbs up from Einstein. Things get tricky when that someone with the light starts to move towards you, though. If they ran at, say, a 1000 m/s towards you, then you would expect that the photons would then appear to be going at 300 million + 1000 m/s, right? Wrong. According to Einstein, the photons would appear to be going at *exactly* the same speed as they were before, 300 million m/s, both to you and to the person with the light.

Speed of light

The link to symmetry here is not immediately clear. However, this idea of an 'invariant' or 'conserved quantity' - something which remains the same no matter how you look at it - is fundamental to symmetry. We can tell that an object is symmetrical if it looks the same when we change our viewpoint: if you rotate your head by 90 degrees when looking at a square then it looks the same. This is caused by one of the symmetries of the square. Similarly, if we're looking at a beam of light, and we decide to 'change our viewpoint' by moving in some direction and at some speed, then Einstein says that the speed of light must always stay the same. This must then be caused by some kind of underlying symmetry.

The type of symmetry which causes this is called a 'Lorentz Symmetry'. The exact details of this are horribly complex, but we can still appreciate Einstein's genius without them. Whereas many lesser minds would have seen the constancy of the speed of light as an error to be ironed out, Einstein's mind took it as an undisputable, fundamental fact of nature, as the equations which produced it were too beautiful not to be true. He then had to rethink our understanding of space and time to incorporate it. The theories of relativity which sprung from this realisation explained how and why gravity works, as well as quantifying it more accurately than any previous theories.

On a day to day basis, this is fundamental to the idea of GPS, as it allows us to measure time differences much more accurately. But to reduce such a ground-breaking theory to a relatively trivial application simply does not do it justice. Einstein's way of thinking marked a change in the entire philosophy of Physics. Whereas previously, physicists would observe data and make theories to explain it, Einstein championed the formulation of theories on the basis of pure thought, with symmetry as a guiding light.

These theories of Relativity work well on large scales - galaxies moving through the universe, planets orbiting the sun or even just humans standing on Earth. If we zoom in further though, we enter a whole new world governed by (yet more) strange and counterintuitive laws. This is the world of quantum mechanics, which describes the fundamental building blocks of the universe.

The Standard Model

In Einstein's day, we knew about atoms, the nucleus at the centre of each atom and the electrons surrounding it. The protons and neutrons which make up the nucleus were discovered in the 20s and 30s respectively, but that was by no means the end of the story.

As technology developed after WWII, scientists built increasingly powerful particle accelerators, smashing things together at ever increasing speeds largely out of curiosity. As they did so, they produced more particles, which looked kind of like protons and neutrons. More and more of these particles were being churned out, sometimes at a rate of one per week, each one being added to the fast-growing 'particle zoo'. To use a technical term, this created a mess: scientists believed that each particle found must also be a new fundamental one to sit alongside the proton and the neutron et al.

Such disorder grated horribly - physicists wanted desperately to arrange and categorise their discoveries into a physics equivalent of the periodic table of chemistry. For all the disorder, though, there were some hints at patterns, with different particles often sharing some properties but not others. What caused these seeming coincidences remained a mystery until 1964, when two separate groups of physicists both postulated the existence of 'quarks': the building blocks of particles like protons and neutrons.

Once again, the link to symmetry is not explicitly obvious. This time, physicists had really exploited the idea of symmetry groups, like those which Galois had introduced. They studied the patterns amongst the 'zoo' of particles, linking these with certain symmetry groups as they went. This revealed the underlying structure of the particles, which physicists could then define in terms of fundamental particles and certain laws which determine how these particles interact. The result of this is the Standard Model, which has very complex symmetries formed of a number of different groups combined.

When this theory was proposed, we hadn't yet discovered all of the particles that it predicted would exist. Over the past 40 years, then, this has provided an excellent test of the Standard Model, checking to see that each and every predicted particle can be observed in nature, and that they all have the correct properties. Fortunately, there were no contradictions.

An asymmetrical world

The development of Relativity and the formation of the Standard Model epitomise the philosophical effect that symmetry has had on physics. Einstein spotted a 'conserved quantity' in the speed of light and decided not to dismiss it, deducing the underlying symmetry of space and time as a result. The Standard Model on the other hand, was born from a search for symmetry - having already observed its power scientists hoped that it would restore some kind of order to their world. Two different approaches at either end of the symmetrical revolution, but both fundamentally symmetric in nature.

If our physical theories are so symmetrical, though, then why when we look out of the window are we not living in an entirely symmetrical world?

Symmetry building

The answer to that question is simple - randomness. Ever since the introduction of quantum theories, the universe has had a decidedly unpredictable side to it. Particles' properties are determined by probabilistic processes rather than definite, deterministic ones, spontaneously becoming certain when we choose to observe them. This causes what are incredibly symmetric physical laws to produce remarkably asymmetric objects.

For example, consider a pencil standing on its pointy end on a piece of paper. If you rotate the piece of paper, the pencil still looks the same. Similarly, there's no 'bias' as to which direction the pencil wants to fall in (ignoring any kinds of wind movement etc, the pencil would fall due to quantum fluctuations in the particles which form it, which are entirely random in nature). As a physicist would say, the system has 'no preferred direction'. I.e. it is symmetrical under rotations, in exactly the same way as a circle.

However after the pencil has inevitably fallen over, this symmetry breaks - if you then rotate the paper, the pencil changes how it looks, pointing in a certain 'preferred' direction which changes as the paper rotates.

Here we have a symmetrical system producing asymmetrical results, but what would happen if we repeated this over and over again? As the pencil's direction of fall is random, it would end up falling in just about every direction possible, all of the way round the circle. If you then take an average, the different preferred directions created by each fall all cancel out: each time the pencil falls in one direction, it will also fall in the exact opposite direction at some point in the future. On average, then, we reclaim the original symmetry of the system, with no preferred direction at all.

In a similar way, if you could take all of the Oak trees in existence and in some way 'average' them, then you would end up with something which looked very symmetrical indeed. Whilst this isn't something we can actually do, it reassures us that we haven't been barking up the wrong tree all along in our pursuit of symmetry.

Truth and Beauty

So where do we go from here? Has symmetry run its race? Fortunately, that doesn't seem to be the case. The major objective in physics at the moment is to find a 'Theory of Everything' - one theory to rule them all. We still have a long way to go to reach this goal, with even the recently revamped LHC insufficiently powerful of explicitly testing any contending theories at the moment. However, we are very sure that any successful candidate would have to be a highly symmetric theory, uniting all of the universes forces and particles in one big symmetry group and explaining the underlying structure of the universe. The trouble is, we've got very little idea as to which symmetry group this ought to be.


For now then, symmetry continues to rule the roost and looks set to guide our thinking for some time to come, even if we haven't quite mastered its subtleties just yet. No matter what the future holds, though, symmetry has certainly changed physics for the better. The style of thinking it inspired has left our technology struggling to keep up, despite the incredible advances of the 20th Century. It has given us an unprecedented comprehension not only of how our universe works, but more importantly of *why* it functions in the way it does.

From Einstein onwards, the beauty and elegance which are inherent to symmetry-based theories have been used very successfully as indicators of truth, with humans following their unmistakable scent wherever possible. Einstein himself was completely besotted with this concept, regarding beauty and truth as one and the same. While this opinion is not necessarily universal, it certainly has its merits and certainly from a philosophical point of view, it provides a romantic motivation for future work. As Einstein himself said, "The pursuit of truth and beauty is a sphere of activity in which we are permitted to remain children all our lives."

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