As an undergraduate student at University College London (UCL) I took a course in GROUP THEORY, a branch of mathematics about which I knew very little at the time.
The lectures themselves were excellent – thanks largely to the lecturer, Professor Frank Johnson, who was a brilliant teacher. I quickly learned that GROUPS (the mathematical things which are studied in Group Theory) were very important, but I never really appreciated why and as a mathematician, I didn’t really care!
I could appreciate the beauty and elegance of the mathematics and that was enough for me – or so I thought at the time – hence I wasn’t really that bothered with why they were important or what they could be used for. Now I realise I was wrong and I regret that I didn’t use the opportunity I had at UCL to learn more about them. (At this point I would like to make it clear that Prof. Johnson is totally absolved of any blame for my shortcomings – my ignorance of these issues was entirely my own fault and despite his superb efforts to explain the wider significance of Group Theory!)
The reason for my change of heart regarding Group Theory was that I was fortunate enough to read some excellent books on the topic and these really opened my eyes to Group Theory, symmetry and their role in modern theoretical physics. My original plan was to write a blog about Group Theory and symmetry and try to explain why this is such an important and interesting topic in mathematics. Then I realised that I wouldn’t be able to do this topic justice and decided instead to just write a brief introduction to the topic, to hopefully whet your appetite in the process! So in this blog I will explain a little about why Group Theory is so useful and in subsequent blogs I’ll tell all about the wonderful books that I’ve read recently – and leave those authors to explain this topic in more detail – and far more eloquently than I could manage!
- Symmetry is everywhere
One of the reasons why groups are so important is that they give us a mathematical way of analysing SYMMETRY. Loosely speaking, something is symmetric if you can do something to it without actually changing the way it looks. For example, the letter ‘A’ is symmetric because if I look at it in a mirror, it still looks like the letter ‘A’ (this is called a REFLECTION SYMMETRY) whereas the mirror reflection of the letter ‘B’ would look like it had been written backwards (you can try this yourself; hold a book up in front of a mirror and look at the reflection – probably some letters in the title will look correct but others will look backwards – I just tried this with the book ‘Why Beauty is Truth’ by Professor Ian Stewart and the ‘W’ of ‘Why’ and the ‘T’ of ‘Truth’ looked the same in the mirror but the others looked backwards; hence ‘W’ and ‘T’ have a certain symmetry that the other letters do not). Another type of symmetry – ROTATIONAL SYMMETRY – is exhibited by regular shapes such as circles and squares; if I rotate a circle it looks exactly the same, hence this is also an example of symmetry.
All of us have an intuitive idea of symmetry – indeed there have been many scientific studies which highlight how our brain seeks out and recognises symmetry – and we can often recognise the aesthetic appeal of symmetric objects. Moreover, it also turns out that symmetry is present at the very heart of the fundamental theoretical physics that helps to explain the universe we live in, although this symmetry is often more abstract than the previous examples I have discussed. For example, suppose you carry out two identical experiments, one in a laboratory at UCL and the other in a car travelling at a constant speed in a straight line. Allowing for any experimental errors, you will get exactly the same results from each experiment and the reason for this is that the laws of physics are symmetric – I can change them in a certain way (by placing one experiment in a car) without changing the way it looks (in this case ‘the way it looks’ refers to the outcomes of the experiment – not the physical image of the experiment taking place – because that’s what I’m interested in).
And this is where Group Theory becomes important. Some symmetries are instantly recognisable; others are dramatically more abstract and far from obvious – and indeed quite removed from our everyday understanding of what symmetry is. Moreover, symmetry is, to many people, a visual property, something that we can see, recognise and appreciate for its aesthetic charm; not something that is naturally mathematical. Yet, if we are to understand the Universe around us we need to be able to use the power of mathematics and so we need to be able to talk about symmetry in a mathematical way. And this is exactly what Group Theory does. It gives us a mathematical description of symmetry and allows us to analyse symmetry in a way that wouldn’t be possible in any other way. And whilst I have long enjoyed Group Theory for its mathematical beauty, this is why I have now (and somewhat belatedly!) come to appreciate it for its importance to the whole Universe too; Group Theory helps us to understand symmetry and symmetry helps us to understand the Universe. It’s as simple as that!
Dr John McDarby is a Maths Lecturer at Bellerbys College.
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