# What is symmetry?

Symmetry underpins much of modern physics and maths but what exactly is it?
24 August 2015

## Interview with

Professor Mark Ronan, UCL

## Butterfly Symmetry is something that we're all very familiar with, whether it's checking that eyeliner is even, making sure photos are blu-tacked straight on the wall or even just noticing the symmetry of butterflies' wings. However, there's much more to symmetry than meets the eye, and it has formed the mathematical backbone for all of our modern physical theories. But what does symmetry mean to a mathematician? Kat Arney spoke to Mark Ronan to find out more...

Kat - So, as a mathematician, how do you see symmetry?

Mark - We come at it from different directions, but it can all be explained by taking an object like a square and looking at the different symmetries that it has.

Not just a left, right symmetry or a top, bottom symmetry, or a diagonal symmetry where you can reflect it across the diagonal, but there are actually 8 different symmetries for a square.

Kat - So, that means that you could say, put a mirror in the middle of it and it would still look like a square, or still look the same from all these kind of different angles.

Mark - That's right.  And the whole idea is to take all of them together and that is what mathematician's call group.

Kat - So, that's a number of different kind of equal things, equal views of something.

Mark - Yes, you can look at it that way. The idea is that you can do one and then another one. The concept of a group is absolutely fundamental. That's what physicists use when they're talking about symmetry. They're really talking about a group of symmetries.

The word 'group' was originally introduced in French - it's got an 'e' on the end, but it's the same word - by Evariste Galois, a young mathematician who never reached his 21st birthday.

Kat - In terms of his contribution to maths and symmetry, what was Galois thinking of ?

Mark - What he was doing with mathematics was, he was trying to solve a problem about equations.

Kat - What was that problem?

Mark - The problem was how to distinguish equations that have solutions that you can extract in a relatively simple manner and equations that have solutions you can't extract in a simple manner.

Kat - So, these are kind of beautiful solvable equations and then more complicated, maybe insoluble equations.

Mark - A good word because actually, mathematicians use that world 'solvable'. They're solvable and other ones that aren't solvable in the same sense.

Kat - In terms of symmetry, which ones are the symmetrical ones?

Mark - The more symmetry they have, the harder they are to deal with.

Kat - So, is this because you kind of look at the equation and it all sort of balances out. They're all more equivalent, more symmetrical, and it's harder to pick out, "Okay, we can start here. This is an answer."

Mark - That's absolutely right. You can't quite grapple with the thing very well because everything looks the same from different multiple points of view.

Kat - A lot of what we talk about in physics is all based on equations, coming up with equations that describe the physical laws of nature. In terms of symmetry, how symmetrical are the equations that we have in physics?

Mark - Well, it's not so much actually solving equations. Symmetry produces patterns. Physicists study patterns of different elementary particles for example.

When I was a teenager, there was a whole zoo full of these things. They managed to create order out of it and they did it by using symmetry.

In fact, Herman Weyl who was a famous mathematician, also a very good physicist said, "Symmetry, as wide or narrow as you may define its meaning is one idea by which man, through the ages has tried to comprehend and create order, beauty and perfection."

But they came up with some very clever ideas using symmetry in a serious way. I don't just mean in a silly way.

Kat - In a sort of a matchy-matchy, this particle is the mirror image of this one.

Mark - It's not just mirror image. The symmetries were more complicated than that when they start to look at the standard model of physics at a very most basic level where you're looking inside an atom.

Kat - So broadly, when we're thinking about equations that govern the universe, govern how it works, should we expect them to ultimately have symmetry within them?

Mark - I think we probably expect some symmetry because if we don't have any symmetry at all, we're not going to make sense of things.

Physicists really demand a lot of symmetry in order to make their theories comprehensible.