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  • On the 30th of May, 1832,

  • 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.