Symmetry, reality's riddle - Marcus du Sautoy

On the 30th of May 1832, a gunshot was heard ringing out across the 13th arrondissement in Paris. 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 Évariste 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. 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."

That 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 of 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. This is how he wrote most of his mathematics. So the night before that duel, he realized that this possibly was his last chance to try and explain his great breakthrough.

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 he 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 and what crystals are possible can be understood through the mathematics of symmetry. In microbiology, you really don't want to get a symmetrical object because they're 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 at a larger scale of biology, actually, symmetry is very important because it 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'll probably be drawn to the lower two. That's because it's hard to make symmetry. 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 there 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 learned a 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 love to play around with symmetry too. They also have a slightly more ambiguous relationship with it. Here's 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." He added, "Deathly, the very marrow of death."

What I just like to do is to set up expectations of symmetry and then break them. A beautiful example of this, I found actually when I visited a colleague of mine in Japan, Professor Kurokawa. He took me up to the temples in Nikko, and just after this photo was taken, we walked up the stairs. 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.

I said to Professor Kurokawa, "Wow, the architects must have been really kicking themselves when they realized that they made the mistake and put this one upside down." He said, "No, no, no, it was a very deliberate act." He referred me to this lovely quote from the Japanese essays in idleness from the 14th century, in which the author says: "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 loves. This is my son, Tamir. 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 that it doesn't look at how mathematics is embedded in the world we live in. So I wanted to open up his eyes to how much symmetry is running through the Alhambra.

You see it all really 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 would deny the possibility to draw things with souls, so they explored a more geometric art.

So what is symmetry? And the Alhambra somehow asks all of these questions. What is symmetry? When two of these walls do they have the same symmetries? Can we say whether they discovered all of the symmetries in the Alhambra? It was Galois who produced a language to be able to answer some of these questions—the Galois symmetry, unlike for Thomas Mann, which was something still and deadly, the 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 move: what can you do to something, you close your eyes, do something, put it back down again, and it looks like it did before it started.

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. If you open your eyes again, you wouldn't know that they've 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. The power of mathematics is often to change one thing into another, to change geometry into language, to take you through, perhaps push you a little bit mathematically. So brace yourselves, I'll push you a little bit to understand how this language works, which enables us to capture what is symmetry.

Let's take these two symmetrical objects here. Let's take the twisted six-pointed starfish. What can I do to this 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 can rotate by a third of a turn or a half a turn and put it back down on its image, or 2/3 of a turn.

And a fifth symmetry—I can rotate it by five-sixths of a turn. Those are things that I can do to the symmetrical object which make it look like it did before I started. Now for Galois, there was actually a sixth symmetry. Can anybody think of what else I could do to this which would leave it like it did before I started? I can't flip it because I put a little twist on it; 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. For Galois, this was like the zeroth symmetry. Actually, the invention of this number zero was a very modern concept, 7th century AD by the Indians. It seems mad to talk about nothing, and this is the same idea. This is a symmetrical to everything—it 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 anti-clockwise. 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 zero 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's a little question for you. I'm getting a look at 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, I want you to think about that problem as we go on and count how many symmetries there are, and there'll be a prize to the person who gets closest at the end. But let's go back to symmetries that I got for these two objects. What Galois realized is that 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. 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 do 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. So 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, S, is as if I just rotated it by half a turn in one go. So the little table here records how the algebra of these symmetries works. I do one followed by another—the answer is its rotation, D, half a turn.

What if I did it in the other order? Would it make any difference? Well, let's see. Let's do the third of a 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. There's some symmetry here in the way the symmetries interact with each other.

But this is completely different from the symmetries of the triangle. Let's see what happens if we do two symmetries with a triangle, one after the other. Do a rotation by a third of a turn anti-clockwise and reflect in the line through X. Well, the combined effect is as if I'd just done the reflection in the line through Z to start with.

Now, let's do it in a different order. Let's do the reflection in X first, followed by the rotation by a third of a turn anti-clockwise. The combined effect: the triangle ends up somewhere completely different. It's as if it wasn't reflected in the line through Y. Now it matters what order you do the operations in, and this enables us to distinguish why the symmetries of these objects—they both have six symmetries.

So why shouldn't we say they have the same symmetries? But the way the symmetries interact enables us—we've now got a language to distinguish why these trees are fundamentally different. You could try this when you go down the pub later on: take a beer mat and rotate it by a third or quarter of a turn, then flip it, and then do it in the other order. The picture will be facing in the opposite direction.

Now, Galois produced some laws for how these tables, how symmetries interact—it's almost like little Sudoku tables. You don't see any symmetry twice in any row or column. Using those rules, he was able to say that there are, in fact, only two objects with six symmetries, and they'll be the same as the symmetries of the triangle or the symmetries of the six-pointed starfish.

I think this is an amazing development; it's almost like the concept of number being developed for symmetry in the frontier. I've got one, two, three people sitting on one, two, three chairs—the people and the chairs are very different, but the number, the abstract idea of the number, is the same.

We can see this now. We go back to the walls in the Alhambra. Here are two very different walls with very different geometric pictures, but using the language of Galois, we can understand that the underlying abstract symmetries of these things are actually the same.

For example, let's take this beautiful wall with the triangles, did a little twist on them. You can rotate them by a sixth of a turn if you ignore the colors; we're not matching up the colors, but the shapes match up. If I rotate by a sixth of a turn around the point where all the triangles meet, what about the center of a triangle? I can rotate by a third of a turn around the center of the triangle, and everything matches up.

Then there's an interesting place halfway along an edge where I can rotate by 180 degrees and all the tiles match up again. So rotate along halfway along the edge, and they all match up. Now let's move to the very different-looking wall in the Alhambra, and we find the same symmetries here and the same interaction.

So there was a sixth of a turn, a third of a turn where all the pieces meet, and then the half a turn halfway between the six-pointed stars. Although these walls look very different, Galois has produced a language to say that, in fact, the symmetries underlying these are exactly the same.

And it's a symmetry we call six, three, two. Here's another example in the Alhambra. This is a wall, a ceiling, and a floor. They all look very different, but this language allows us to say they are representations of the same symmetrical abstract object, which we call four, four, two.

Nothing to do with football, but because of the fact that there are two places where you can rotate by a quarter of a turn and one by half a turn. Now this part of the language is even more because Galois can say: Did the Moorish artists discover all of the possible symmetries on the walls in the Alhambra? It turns out they almost did.

You can prove using Galois's language that there are actually only 17 different symmetries that you can do on the walls in the Alhambra, and if you try and produce a different wall with its 18th one, it will have to have the same symmetries as one of these 17. But these are things that we can see.

And the power of Galois's mathematical language is it also allows us to create symmetrical objects in the unseen world beyond the two-dimensional, three-dimensional, all the way through to the four, five, or infinite-dimensional space. And that's where I work; I create mathematical objects—symmetrical objects—using Galois's language in very high-dimensional spaces.

So I think it's a great example of things unseen, which the power of mathematical language allows you to create. So like Galois, I stayed up all last night creating a new mathematical symmetrical object for you, and I've got a picture of it here. Well, unfortunately, it isn't really a picture. If I could have my board at the side here—great, excellent! Here we are.

This is—unfortunately I can't show you a picture of this symmetrical object, but here is the language which describes how the symmetries interact. Now this new symmetrical object does not have a name yet. Now, people like getting names on things—on sort of craters on the moon or new species of animals. So I'm going to give you the chance to get your name on a new symmetrical object which hasn't been named before.

This thing in species, die away, and moons kind of get hit by meteors and explode, but this mathematical object will live forever. It will make you immortal. In order to win your symmetrical object, what you have to do is to answer the question I asked you at the beginning: How many symmetries does the Rubik's Cube have?

Okay, I'm going to sort you out rather than you all shouting out. I want you to count how many digits there are in that number. If you've got it as a factorial, you have to expand the factorial.

Now, if you want to play, I want you to stand up. If you think you can, you've got an estimate for how many digits, right? We've already got one competitor here. Yeah, you all stay down; he wins it automatically. Okay, excellent. So we've got four here, five, six—great, excellent. After that, I should get us going.

Alright, anybody with five or less digits, you've got to sit down because you've underestimated—five or less digits. So hundreds, thousands, thousands—you've got to sit down. Sixty digits or more, you've got to sit down; you've overestimated. Twenty digits or less, sit down.

Oh, twenty! How many digits are there in your number? Two? So you sort of sat down earlier. Let's have the other ones who said—oh, they said the other ones who sat down during the twenty up again. Okay, if I told you twenty or less, stand up because we're this one.

I think there are a few here; you've just said the people who just last sat down. Okay, how many digits do you have in your number? Ah, ha, ha! How many? Twenty-one? Okay, good. How many do you have? Eighteen.

So it goes to this lady here. Twenty-one is the closest. Actually, the number of symmetries in the Rubik's Cube has 25 digits. So now I need to name this object. So what is your name? I need your surname. Groups—the symmetrical objects—generally spell it for me. G-H-E-Z.

Now, SO2 has already been used as you in the mathematical language, so you can't have that, so there we go. That's your new symmetrical object; you are now immortal. And if you'd like your own symmetrical object, I have a project—a way of raising money for a charity in Guatemala where I will stay up all night and devise an object for you, for a donation to this charity to help kids get into education in Guatemala.

And I think what drives me, as a mathematician, are those things which are not seen—the things that we haven't discovered—and it's all the unanswered questions which make mathematics a living subject. I always come back to this quote from the Japanese essays denying idleness: "In everything, uniformity is undesirable. Leaving something incomplete makes it interesting and gives one the feeling that there is room for growth."

Thank you.