a gunshot was heard

ringing out across the 13th arrondissement in Paris.

(Gunshot)

A peasant, who was walking to market that morning,

ran towards where the gunshot had come from,

and found a young man writhing in agony on the floor,

clearly shot by a dueling wound.

The young man's name was Evariste Galois.

He was a well-known revolutionary in Paris at the time.

Galois was taken to the local hospital

where he died the next day in the arms of his brother.

And the last words he said to his brother were,

"Don't cry for me, Alfred.

I need all the courage I can muster

to die at the age of 20."

It wasn't, in fact, revolutionary politics

for which Galois was famous.

But a few years earlier, while still at school,

he'd actually cracked one of the big mathematical

problems at the time.

And he wrote to the academicians in Paris,

trying to explain his theory.

But the academicians couldn't understand anything that he wrote.

(Laughter)

This is how he wrote most of his mathematics.

So, the night before that duel, he realized

this possibly is his last chance

to try and explain his great breakthrough.

So he stayed up the whole night, writing away,

trying to explain his ideas.

And as the dawn came up and he went to meet his destiny,

he left this pile of papers on the table for the next generation.

Maybe the fact that he stayed up all night doing mathematics

was the fact that he was such a bad shot that morning and got killed.

But contained inside those documents

was a new language, a language to understand

one of the most fundamental concepts

of science -- namely symmetry.

Now, symmetry is almost nature's language.

It helps us to understand so many

different bits of the scientific world.

For example, molecular structure.

What crystals are possible,

we can understand through the mathematics of symmetry.

In microbiology you really don't want to get a symmetrical object,

because they are generally rather nasty.

The swine flu virus, at the moment, is a symmetrical object.

And it uses the efficiency of symmetry

to be able to propagate itself so well.

But on a larger scale of biology, actually symmetry is very important,

because it actually communicates genetic information.

I've taken two pictures here and I've made them artificially symmetrical.

And if I ask you which of these you find more beautiful,

you're probably drawn to the lower two.

Because it is hard to make symmetry.

And if you can make yourself symmetrical, you're sending out a sign

that you've got good genes, you've got a good upbringing

and therefore you'll make a good mate.

So symmetry is a language which can help to communicate

genetic information.

Symmetry can also help us to explain

what's happening in the Large Hadron Collider in CERN.

Or what's not happening in the Large Hadron Collider in CERN.

To be able to make predictions about the fundamental particles

we might see there,

it seems that they are all facets of some strange symmetrical shape

in a higher dimensional space.

And I think Galileo summed up, very nicely,

the power of mathematics

to understand the scientific world around us.

He wrote, "The universe cannot be read

until we have learnt the language

and become familiar with the characters in which it is written.

It is written in mathematical language,

and the letters are triangles, circles and other geometric figures,

without which means it is humanly impossible

to comprehend a single word."

But it's not just scientists who are interested in symmetry.

Artists too love to play around with symmetry.

They also have a slightly more ambiguous relationship with it.

Here is Thomas Mann talking about symmetry in "The Magic Mountain."

He has a character describing the snowflake,

and he says he "shuddered at its perfect precision,

found it deathly, the very marrow of death."

But what artists like to do is to set up expectations

of symmetry and then break them.

And a beautiful example of this

I found, actually, when I visited a colleague of mine

in Japan, Professor Kurokawa.

And he took me up to the temples in Nikko.

And just after this photo was taken we walked up the stairs.

And the gateway you see behind

has eight columns, with beautiful symmetrical designs on them.

Seven of them are exactly the same,

and the eighth one is turned upside down.

And I said to Professor Kurokawa,

"Wow, the architects must have really been kicking themselves

when they realized that they'd made a mistake and put this one upside down."

And he said, "No, no, no. It was a very deliberate act."

And he referred me to this lovely quote from the Japanese

"Essays in Idleness" from the 14th century,

in which the essayist wrote, "In everything,

uniformity is undesirable.

Leaving something incomplete makes it interesting,

and gives one the feeling that there is room for growth."

Even when building the Imperial Palace,

they always leave one place unfinished.

But if I had to choose one building in the world

to be cast out on a desert island, to live the rest of my life,

being an addict of symmetry, I would probably choose the Alhambra in Granada.

This is a palace celebrating symmetry.

Recently I took my family --

we do these rather kind of nerdy mathematical trips, which my family love.

This is my son Tamer. You can see

he's really enjoying our mathematical trip to the Alhambra.

But I wanted to try and enrich him.

I think one of the problems about school mathematics

is it doesn't look at how mathematics is embedded

in the world we live in.

So, I wanted to open his eyes up to

how much symmetry is running through the Alhambra.

You see it already. Immediately you go in,

the reflective symmetry in the water.

But it's on the walls where all the exciting things are happening.

The Moorish artists were denied the possibility

to draw things with souls.

So they explored a more geometric art.

And so what is symmetry?

The Alhambra somehow asks all of these questions.

What is symmetry? When [there] are two of these walls,

do they have the same symmetries?

Can we say whether they discovered

all of the symmetries in the Alhambra?

And it was Galois who produced a language

to be able to answer some of these questions.

For Galois, symmetry -- unlike for Thomas Mann,

which was something still and deathly --

for Galois, symmetry was all about motion.

What can you do to a symmetrical object,

move it in some way, so it looks the same

as before you moved it?

I like to describe it as the magic trick moves.

What can you do to something? You close your eyes.

I do something, put it back down again.

It looks like it did before it started.

So, for example, the walls in the Alhambra --

I can take all of these tiles, and fix them at the yellow place,

rotate them by 90 degrees,

put them all back down again and they fit perfectly down there.

And if you open your eyes again, you wouldn't know that they'd moved.

But it's the motion that really characterizes the symmetry

inside the Alhambra.

But it's also about producing a language to describe this.

And the power of mathematics is often

to change one thing into another, to change geometry into language.

So I'm going to take you through, perhaps push you a little bit mathematically --

so brace yourselves --

push you a little bit to understand how this language works,

which enables us to capture what is symmetry.

So, let's take these two symmetrical objects here.

Let's take the twisted six-pointed starfish.

What can I do to the starfish which makes it look the same?

Well, there I rotated it by a sixth of a turn,

and still it looks like it did before I started.

I could rotate it by a third of a turn,

or a half a turn,

or put it back down on its image, or two thirds of a turn.

And a fifth symmetry, I can rotate it by five sixths of a turn.

And those are things that I can do to the symmetrical object

that make it look like it did before I started.

Now, for Galois, there was actually a sixth symmetry.

Can anybody think what else I could do to this

which would leave it like I did before I started?

I can't flip it because I've put a little twist on it, haven't I?

It's got no reflective symmetry.

But what I could do is just leave it where it is,

pick it up, and put it down again.

And for Galois this was like the zeroth symmetry.

Actually, the invention of the number zero

was a very modern concept, seventh century A.D., by the Indians.

It seems mad to talk about nothing.

And this is the same idea. This is a symmetrical --

so everything has symmetry, where you just leave it where it is.

So, this object has six symmetries.

And what about the triangle?

Well, I can rotate by a third of a turn clockwise

or a third of a turn anticlockwise.

But now this has some reflectional symmetry.

I can reflect it in the line through X,

or the line through Y,

or the line through Z.

Five symmetries and then of course the zeroth symmetry

where I just pick it up and leave it where it is.

So both of these objects have six symmetries.

Now, I'm a great believer that mathematics is not a spectator sport,

and you have to do some mathematics

in order to really understand it.

So here is a little question for you.

And I'm going to give a prize at the end of my talk

for the person who gets closest to the answer.

The Rubik's Cube.

How many symmetries does a Rubik's Cube have?

How many things can I do to this object

and put it down so it still looks like a cube?

Okay? So I want you to think about that problem as we go on,

and count how many symmetries there are.

And there will be a prize for the person who gets closest at the end.

But let's go back down to symmetries that I got for these two objects.

What Galois realized: it isn't just the individual symmetries,

but how they interact with each other

which really characterizes the symmetry of an object.

If I do one magic trick move followed by another,

the combination is a third magic trick move.

And here we see Galois starting to develop

a language to see the substance

of the things unseen, the sort of abstract idea

of the symmetry underlying this physical object.

For example, what if I turn the starfish

by a sixth of a turn,

and then a third of a turn?

So I've given names. The capital letters, A, B, C, D, E, F,

are the names for the rotations.

B, for example, rotates the little yellow dot

to the B on the starfish. And so on.

So what if I do B, which is a sixth of a turn,

followed by C, which is a third of a turn?

Well let's do that. A sixth of a turn,

followed by a third of a turn,

the combined effect is as if I had just rotated it by half a turn in one go.

So the little table here records

how the algebra of these symmetries work.

I do one followed by another, the answer is

it's rotation D, half a turn.

What I if I did it in the other order? Would it make any difference?

Let's see. Let's do the third of the turn first, and then the sixth of a turn.

Of course, it doesn't make any difference.

It still ends up at half a turn.

And there is some symmetry here in the way the symmetries interact with each other.

But this is completely different to the symmetries of the triangle.

Let's see what happens if we do two symmetries

with the triangle, one after the other.

Let's do a rotation by a third of a turn anticlockwise,

and reflect in the line through X.

Well, the combined effect is as if I had just done the reflection in the line through Z

to start with.