Marcus du Sautoy — ‘Symmetry and the making

of a mathematician’ Date: 17th October 2013, 2:30PM I actually grew up around here in Oxfordshire

and I went to a school about 20 miles away. And actually, when I went up to school about

the age of 11, last thing on my mind was becoming a mathematician. I had this kind of dream

that when I grew up, I wanted to be a spy, which had been partly fuelled by my mom, who

had been in the Foreign Office. And when she had children she was moved sideways in deciding

— wasn’t as exciting –, but she told me and my sister that she had been allowed to

keep the black gun that every member of the Foreign Office gets given. She was very imaginative

and creative, but me and my sister immediately assumed she must have been a spy. We used

to spend all our time trying to look for this gun in the house. And we could never find

it, because it didn’t exist. But I had this kind of dream. It sounded really exciting

kind of life — to be in the Foreign Office. So when I went up to my secondary school,

I thought, ‘Foreign Office, that’s all about language.’ so I signed up for every foreign

language course my school did. They did French obviously. It was probably one of the few

state schools at that time doing Latin. They did German. And actually, at the same time,

the BBC were doing a course teaching Russian. I thought Russian would be very good for a

spy. You can tell how old I am. I grew up in the Cold War era. And my French teacher

helped me out with the Russian course. But actually, as I began learning all these languages,

I became more and more frustrated because they had all these kinds of strange, weird

words that you just had to learn how to spell. All these irregular verbs which just didn’t

seem to make any sense at all. And the Russian course was a total disaster because I couldn’t

get pass the word ‘hello’ which had so many consonants and no vowels in it. Are there

any Russian speakers here? How do you say ‘hello’ in Russian? [здороваться.]

Zdravstiye. Yes, exactly, did you hear any vowels? I didn’t. It was name of the course,

as well. It was depressing. I couldn’t even say the name of the course. So I got very

disillusioned actually, with this dream of going off, becoming a spy, and joining the

Foreign Office. But it was around that time, second year in secondary school, when my maths

teacher suddenly went in the middle of the lesson: ‘Du Satoy! I want to see you after

the class!’ And I thought, ‘Gosh, I’m in trouble.’ And at the end of the class I went up to see

him and he took me round the back of the maths block, and I thought, ‘Oh man, I’m in real

trouble now.’ But then he got out his break-time cigar — he wasn’t allowed to smoke in the

common room — and he said, ‘I think you should find out what mathematics is really about.’

And I was curious because I thought we were doing maths in the class. And he said, ‘No.

Maths has nothing to do with what we are doing in the classroom. In the classroom, we were

doing these long divisions, percentages, and so forth, but maths is actually something

much more exciting.’ And he recommended a few books to me that he thought would open

up this world of mathematics for me. And I took this home. That weekend my dad brought

me up to Oxford, because he’d been told about this bookshop that was quite good, called

Blackwell’s. And we arrived at the front and thought, ‘Gosh, this is a tiny little shop!

This is pretty hopeless.’ But then we went inside and, as you can see, it’s basically

like the Tardis. It’s very small on the outside, but when you get in… And then we came down

to the Norrington Room, which is where we were told the science books were. And my dad

took this list of books — those exact books are still there — and he went and got the

books. And I sort of wondered around, watching undergraduates. And they were leaning up against

the bookshelves, reading these books as if they were novels. I took one of the books

down and it was just a total secret code. I couldn’t understand a word of it. I took

the books away. It was intriguing. They were giving some free journals away. And I picked

up one of these mathematical journals. It was called ‘Inventiones Mathematicae’, which

is one of the top journals. And they just gave away free samples. And I still have that

copy that I took away that day. And it was just totally — even now, many of the articles

in it are impenetrable — but it was intriguing. And one of the books we bought that day — I

still have it — was called The Language of Mathematics. And it cost £1.25. I defy you

to find a book here costing £1.25. I was very intrigued by this book because, first

of all, I never thought of mathematics as language before. But as I began to read this

book, I began to understand what a powerful language it is. It is an amazing language

for all sciences. I mean, you physicists are using this language all the time to understand

the world around you. And it was also a very exciting language because it didn’t have any

irregular verbs. Everything made total sense. That’s not to say it didn’t have interesting

twists and turns and surprises — and that’s what makes mathematics exciting for me. It’s

that it’s incredibly logical. It appealed to that sense of logic and reason that I wanted.

Yet that doesn’t mean it’s boring. I mean, you might thing that when something is logical,

it all plays out and you have no involvement in it. But actually, it’s got twists and turns

and surprises and a real story to it. And so I took this book away and began to read.

And actually, one of the languages that I really fell in love with that’ described in

this book is the language for symmetry. It’s a language called ‘group theory’ that was

developed by one of the most romantic figures in the history of mathematics — Évariste

Galois, who was a French revolutionary at the beginning of the 19th century and was

killed in a duel by the age of 20. But before then he invented this language, to be able

to understand the subject of symmetry. And actually, as I read this book, I began to

realise that symmetry is, in some ways, its own language. It’s one of nature’s most fundamental

languages. If you look into something like the garden and you see a bumblebee. Bumblebee

has a very bad vision, but what it can pick out, very clearly, are shapes with symmetry,

because that is likely to be a flower, which will be a sustenance for it. The flower, in

its own turn, needs the bee to visit it, so it needs to form a shape that could possibly

attract the bee. And actually, there’s been some evidence that the more symmetrical the

flower the sweeter the nectar inside the flower. So symmetry is almost like a language that

these two can use to communicate and actually come together. Even humans use symmetry as

a way of communicating information. A lot of research has shown that the faces we find

most beautiful are those which are most symmetrical. Why do we associate beauty with a symmetrical

face? Well, symmetry is actually quite hard to achieve in the natural world, because it

gets broken if there is a small disturbance. If you can achieve symmetry — if your face

is very symmetrical — it is quite a powerful sign that you have a good genetic heritage,

good upbringing, that you would make a good mate. It’s actually communicating genetic

information if you’ve got a symmetrical face. There seems to be evidence that’s what we

are drawn to. So symmetry really is an incredibly powerful language in the natural world, for

people to communicate that kind of information. It communicates structure. And we become very

sensitive to symmetry. If you’re in a jungle, and there’s lot of chaotic leaves and things,

and then suddenly you see something with symmetry, it is likely to be an animal — which is either

going to eat you or you are going to eat it. So you better pay attention! There is a lovely

quote by Galileo which sums up this: ‘Mathematics and symmetry are languages that help us navigate

the world around us.’ What I’ve ended up doing as a research mathematician is studying Group

Theory. I spend a lot of time trying to discover new symmetries. It is really something that

we’ve been doing since really ever since civilizations have started crafting the world around them.

You can see this if you look at the first symmetrical objects that humans started making.

If you go to the British Museum in London, there’s this wonderful game called the Game

of Ur, which is an early forerunner of backgammon. And it’s got some dice. Now dice, you want

to have it symmetrical if you want a fair dice. Actually the first dice were the knucklebones

of sheep, which land on four sides naturally, but not fairly. So people realised, ‘O.K.,

we need to carve this into a symmetrical shape.’ So actually the first dice in history were

not cubes we use in games like Monopoly, but little tetrahedrons. Tetrahedron is a triangular

base pyramid. I always carry a tetrahedron around with me. In the game, they — because,

of course, when that lands it’s got a point sticking up — so they would colour two of

the corners and you would throw lots of tetrahedrons and then count the number of coloured spots

pointing up — and that would be your move in the game. And that’s 2500 B.C. If you go

to the Ashmolean Museum, there are some Neolithic stones from Scotland. They are fist-size and

they’ve made patches on the side and tried to arrange patches in very symmetrical way.

So you see four patches, but also six, eight, 12, 20, and actually, already in 2500 B.C.

they are exploring what’s possible in the world of symmetry. And these are — any Dungeons

and Dragons players here will have used dice with all of these different faces on. But

I think the power of mathematics — you know, there is an artist exploring what is possible.

But the mathematician, the power of mathematical language is to know when you’ve discovered

everything. It’s a very powerful language to have a 100% certainty that there are no

other shapes out there. So actually, the culmination of Euclid’s Elements is the proof of the fact

that those five Platonic solids are the only shapes you can make, where all of the faces

have the same face, and are all symmetrically arranged. So you can’t find a sixth one. Plato

wrote about these shapes — which is why they are called Platonic solids — and he believed

that these shapes were somehow the building blocks of the whole of the natural world.

In Ancient Greek chemistry, the atoms of the natural world were Earth, Wind, Fire, Air,

and Water. And so he associated each of these Platonic shapes with one of those elements.

So, this was the shape of Fire [tetrahedron], the spikiest of them. All the way to the icositetrahedron,

which has got twenty triangular faces, and that’s the most spherical one, representing

Water. That may seem a bit crazy from our modern perspective — this idea of shapes

and these symmetries being the building blocks of nature, but actually Plato got to something

quite fundamental, because some of these shapes really are the heart of many things in the

scientific world. The physicists amongst you will know that the fundamental particles make

up the natural world. Around the 1950s and 1960s it looked like this kind of menagerie

of particles being thrown up. It didn’t make any sense at all, until someone spotted an

underlying symmetry, which showed that all of these strange articles are facets of some

strange symmetrical object in very high dimensional space. And using that you can then make predictions

about what you are missing. So symmetry is a very strong motivator for making sense of

what could have looked like some sort of strange zoo of particles. Chemists as well, of course

crystallography. The strength of the diamond comes from the fact that carbon bonds in a

sort of tetrahedral design and it is a very, very strong shape. Diamond’s strength comes

from symmetry. And we can classify crystal structures and why they behave similarly using

the mathematics of symmetry. Biology too. Anyone’s who’s got a virus like I have at

the moment is full of symmetrical objects. Because many different viruses use a symmetrical

shape as their structure so the proteins bind together. Partly because this is a very efficient

structure — because you only need a very small program to construct the whole thing,

because there are very simple rules across the whole of the shape. Viruses have very

small amount of RNA or DNA at their heart, so they need an efficient program. There’s

also lot of strength involved in that shape, also. The artistic world as well is also very

fascinated by symmetry. In music, Bach uses a lot of symmetrical games in order to generate

themes and variations — the Goldberg Variations for example. It’s really an exercise in symmetry.

You can see all the different games Bach is playing, covering all the different possibilities.

In architecture too, there is so much playing around with symmetry. I suppose if I was going

to be cast out to one building in the whole of the world, I’d probably chose the Alhambra

in Granada, which I think is a palace celebrating symmetry. Has anyone been to Alhambra in Granada?

[Yes.] It’s just stunning. You must go there if you get a chance, because the Moorish artists

were exploring all the different possible symmetries. Against, here it is the artist

exploring possibilities. And it wasn’t until Galois discovery of the language of Group

Theory that by the end of the 19th century we were able to say that just as there were

five Platonic solids, it turned out to be only 17 basic symmetrical designs that you

can do on the walls in Alhambra. I was fascinated on a trip I made with my family. We go on

these very nerdy, mathematical half-term trips. So we went around, trying to discover whether

they had found all 17 different symmetry groups in the Alhambra. And I think there’s just

one they missed, that I found quite difficult to comprehend. If we repainted some of the

tiles then they got it, but we weren’t allowed to do that, which is a shame. In a way, that

is what I do as a practising mathematician. I came up to Oxford. Frighteningly I worked

it out. I did the Maths. 30 years ago when I came up here as an undergraduate. And by

the end of my undergraduate time, I really wanted to become a research mathematician,

studying the world of symmetry. And there’d just been the culmination of the most extraordinary

project, spanning for about 150 years, which was the creation of a periodic table of symmetry.

Because symmetry can be broken down into atomic symmetrical objects. For example, a fifteen-sided

figure — actually, the symmetries of that can be built from the symmetries of a pentagon

and the symmetries of a triangle. If you want to do a turn of a fifteen-sided figure, you

can combine pentagons and triangles. But there are actually much more interesting atomic

symmetries out there. And there’d just been this amazing project which culminated in this

book here, which, for the group theorists like myself, contains building blocks from

which you can make all sorts of symmetrical objects. This was a project that was done

in Cambridge actually. At the end of my undergraduate degree, I went up to Cambridge to talk to

the group there to see whether I might join them to carry out my research. And I went

and talked to John Conway, who was the first author on the list. He pulled out this thing,

slammed it on the table, and said, ‘We’re very obsessed with symmetry here. What’s your

name?”Marcus du Sautoy’, I said. He said, ‘Well, you can only join our group if you

drop the ‘du’.’ I was like, ‘What’s that about, my French aristocracy out the window?’ And

he said, ‘No. All the five authors for this periodic table — all five authors have six

letters in their surname, so Sautoy is fine.’ And then he said, ‘Initials?’ ‘My initials

are M.P.F.’ So he said, ‘You have to drop the F.’ This was because all authors only

had two initials. I said, ‘M.P. Sautoy is fine, I want to join your group, and can compromise

on that.’ ‘There is still one more thing’, he said. Because John Conway was the guy who

wanted to bring all this together into this periodic table. And his PhD student at the

time was Rob Curtis. And they were working together. And then, there was this guy who

just kept hanging around their office called Simon Norton, who enjoyed doing — and was

quite crucial in some of the later bits — so he joined third. Parker was very good at computing

aspect of it — he joined fourth. And then Wilson was the last student. They’d all joined

in alphabetical order. So if I was going to be on the bottom here, John Conway said I

had to change my name to Zoutoy with the ‘Z’. At which point I decided this was going far

too far. So I came back to Oxford and started my graduate work here. And I was kind of nervous.

I thought, ‘Perhaps maths was finished. Perhaps symmetry was finished because we now have

this periodic table.’ But of course, mathematics is the most unfinished subject, and every

discovery you make just releases more and more questions. Fermat’s Last Theorem, I think

most of the public thinks it was the last theorem, and we’d finish maths. But that’s

far from truth. There are so many things that we don’t know. So I spend my time in trying

to make the molecules which you can make out of these atoms. So I try to piece together

new symmetrical objects. And I made some discoveries, some new objects, which have some extraordinary

links to Number Theory, to do with counting points on elliptic curves. Which is actually

quite related to Fermat’s Last Theorem. These are kind of beautiful objects. I wouldn’t

say they are going to be useful — they might be useful for something, you never know with

mathematics. But that’s not why I created them. I created them because they tell an

extraordinary story of how symmetry can be related to something completely different.

But I did think I should have some use. I discovered quite a lot of these — actually

infinitely many of them — and so I’ve set up a project which you can help me with. People

love to get their names on things like asteroids or stars or craters on the Moon, or new species

named after them. I think my predecessor Richard Dawkins has a species named after him, for

example. But species die out. And stars blow up. And moons get cracked. But mathematics

lasts forever. That’s the beauty of mathematics. It gives you a certain bit of immortality.

So I’ve got all of these groups and they don’t have names on them. I have a project which

we run at the Mathematical Institute. If you go to our website where you can get one of

these symmetrical objects — uniquely defined for you. You can chose four numbers. Perhaps

you want to give it as a present for somebody. So I will construct one of these objects for

you and you get a certificate which defines what these objects are. And all the money

that gets raised goes to a charity that I support in Guatemala, which gets street kids

off the streets into education. And it’s a very empowering charity because, provided

they stay in education, their families get health care and housing support. I think it’s

a good cause. These groups are beautiful but hopefully helping some kids in Guatemala.

If you want to know more about this story, the second book that I wrote, Finding Moonshine,

is a story of symmetry and my story, a year in life of a working mathematician, starting

on my 40th birthday. The bit about Blackwell’s is, I think, on page 3. I’m happy to answer

any questions you have. Questions: 1. Why is math so ‘unreasonably effective’

at explaining nature? There’s quite a lot of elements to your question.

This paper by Vigner is kind of expression how extraordinary powerful mathematics is

in describing the universe. Why should mathematics be able to unite things like the fundamental

particles. There is no reason, a priori, why the fundamental particles should have such

a beautiful story which unites them all. And when you discover that, it make you jaw-drop.

There are elements of the natural world, of course, which don’t quite have such a beautiful

reasoning. If you look at the animal kingdom, you can use mathematics to understand it,

but it is a much messier system. It’s the idea that so much of the natural world that

is not messy and has this structure to it. Mathematics, of course, grew out of trying

to discover the natural world. We were trying to understand and navigate change in the environment

around us, going back to Ancient Egypt. And then that mathematics grew in a life of its

own. And suddenly you’re studying mathematics for its own sake. I spend my time in this

mathematical world exploring that, not necessarily thinking of impact it may have on the physical

universe. But because it started as a powerful language to describe the physical universe,

often you can generate some material which looks quite abstract, but then suddenly maps

back down again. And it’s perhaps not so surprising that these imaginative games that we play

with mathematics will eventually have some impact back down. Why do I get drawn to certain

bits of mathematics than others? That must have something to do with my engagement with

my environment. Why will I find symmetry, as opposed to some other bit of mathematics,

so appealing. Perhaps it’s because it is so prevalent that I’m drawn to symmetry. Now

there is this other aspect of your question about creativity and discovery. Because I

think that’s a real tension that always exists for a mathematician. I certainly feel that

with these symmetrical objects. I created them. It felt like an act of artistic creation

in a way. And it was motivated by a sense of aesthetics, of the drama, of what they

tell you about these structures. Yet, on the other hand, once they’d been found, I really

felt that they had been discovered, that they were there for anybody to discover them. A

mathematician is very much involved in making choices. And that’s what people don’t appreciate.

I’m not just making true theorems. I could get a computer to churn out true theorems

in the same way as you can get a computer to churn out music. But to make really good

music, to make really moving mathematics, that’s about making a choice. And that choice

can often be one that is about creation rather than discovery. 2. How do we know you’re not a spy? I’ve been in a way. I feel like I almost did

fulfil that dream. I learnt this secret language. This secret language, which allows you to

spy on the natural world around you. So I can pick those books out now, and what’d looked

like code is now a language that I understand. 3. Will you be doing any more documentaries

for the BBC? I have a series that I’m doing with Dara Ó

Briain and will be filming next week. It’s going into its third series and it’s called

The School of Hard Sums. That’s in progress at the moment. [On your own?] At the moment?

I’ve got a couple of things I pitched to the BBC, so let’s see how they go. They’re in

a bit of transition at the moment, because of the BBC 4. But the project I’m working

on at the moment is a play actually. So I wrote a play which I’m performing in. We just

had it last night in London at the Science Museum, and we’ll be doing a run of ten shows

in Manchester next week. And I’m hoping to bring it to Oxford at some stage. I spent

lot of the last year involved with live artistic projects, including something I did at the

Royal Opera House, about the magic flute. So I’m quite enjoying that. It’s fun to have

an audience, as opposed to a cameraman or camerawoman. 4. How did you get involved with the BBC and

broadcasting in the first place? In a way, it’s a product of the Oxford system.

Because, as an undergraduate here, in college, you spend most of your time interacting with

people who aren’t mathematicians or physicists. I would sit around in the Wadham College with

my mates and they were all doing things like literary criticism, philosophy. And you’d

have to justify why your subject is as interesting as doing Derrida. That training was what put

me in a good position for this. Quite randomly, when I was Fellow at All Souls, I sat next

to the Features editor of The Times, at dinner, and he said, ‘So what do you do?’ And when

I kind of described what I do mathematically, he said, “Oh, that sounds so sexy. Write me

an article!’ So I woke up next morning, found his card in my pocket, and, at that stage

I was a post-doc and felt nervous going in front of my community to try and explain.

And I thought I would be judged really harshly so I didn’t do it. There’s an old adage in

Oxford that the fellows change in the college, but guests remain the same. And so three years

later, at the same event — the annual dinner — the Features editor of The Times was there.

And I was still Fellow. And he said, ‘You never wrote me that article!’ And I was amazed

that he even remembered. And I felt a bit braver then and felt I wanted to pay back

people like my teacher, who’d inspired me about mathematics with his bold move to pull

me out of the class, and the people who’d written those books. Somebody spent some time

to write that language of mathematics. So I thought I owed it. So I wrote this article

for The Times. And that was the beginning. As soon as you put your head above the parapet,

things flow naturally. And there aren’t many mathematicians who are doing it. Quite a lot

of physicists and biologists do it. But media-savvy mathematician was a unique point in the Venn

Diagram.