The Secret Mathematicians: Jorge Luis Borges - Marcus Du Sautoy

Now, I'm going to move to a more challenging part of the creative world for finding sort of connections with mathematics, which is the world of literature. And actually, my choice of my fourth secret mathematician from the world of literature was also quite obsessed with high dimensional shapes. But I'm still unclear whether he was knowingly obsessed with them or whether actually he sort of explored these shapes just for their own sort of artistic interests. But actually, they have a lot of resonance to me as a mathematician. Actually, my choice is Jorge Luis Borges. I'm sorry, I've got the names the wrong way around on my slide. I chose him actually just recently if you want to hear a little bit more about him for this radio program on BBC4 called Great Lives, where I get to choose somebody I'm interested in. And then a biographer came on and filled in some of the pieces. So I was quite interested in what sort of mathematics books Borges had in his library because he was clearly, he's an Argentinian writer who writes lovely stories. They're very short, 10 pages long, but they are real sort of gems. And inside those stories, he's continually obsessed with the ideas of paradox, of infinity, of the nature of space. And there's one short story in particular, which if you're going to read any Borges, I would recommend you to read, which is called The Library of Babel. And inside this story, it's about a librarian. In fact, Borges was a librarian himself in Argentina. This librarian is stuck inside this very strange kind of shaped library and he spends the short story trying to explore what the shape of this library is. The story opens with a description of the library from what he can see at the beginning of the story. He writes, the universe, which others call the library, is composed of an indefinite and perhaps infinite number of hexagonal galleries. From any one of the hexagons, one can see internally the upper and lower floors. So this library is laid out a little bit like a beehive, these hexagonal rooms. But what he's intrigued in is about is, do these hexagonal rooms just go on forever? Sort of, or do they go up and down forever? So he starts to explore the library, trying to understand the nature of the shape of this library. And what he is obsessed with, can it be infinite? Or if it isn't infinite, is there some sort of way out of it? And by the end, he comes to a kind of revelation. He says, I venture to suggest this solution to the ancient problem. The library is unlimited and cyclical. If an eternal traveller were to cross it in any direction, then after centuries, he would see the same volumes repeated in the same disorder. What's intriguing to me is that he's actually come to the same conclusion that many of us have come to about the shape of our library, which most of us call the universe. So actually, it's been a challenge for many centuries, particularly the 20th century, to understand, well, what is the shape of our universe? Does it have a shape? If you ask somebody, what's the shape of your universe, it's quite a tricky question. Well, the ancient Greeks actually did think it had a shape. They thought it was this dodecahedron. But now that doesn't really make much sense to us because, well, what happens when you hit that wall? Is there something on the other side? I mean, surely we're not living in the Truman Show. There isn't a camera crew on the other side filming us all in this kind of weird game show. So 20th century scientists and mathematicians have been trying to answer, well, yeah, what's the shape of our universe? As we go out into outer space, well, it could just be infinite. It could just go on forever. But it also could be finite. But how could it be finite? Because if it hasn't got a wall, we don't think it has a wall, how can it be finite? And actually, we've come to this kind of conclusion that, well, it could be finite because it could be kind of closed up and cyclical. So we've been trying to explore, like the librarian, well, OK, well, how could that work? So a rather nice example of this is to take a sort of two-dimensional universe. We live in a three-dimensional one, but let's start with a two-dimensional universe. So some of you might be old enough to have played this game. I certainly am old enough. It's a game called Asteroids. If I showed my kids this, they'd laugh that this was a game that you ever played. But the rules of this game are you've got the universe absolutely captured on the screen here. The whole universe is there and it's finite. But this has certain rules. So if the spaceship goes off the top of the screen, it doesn't bounce back again. It just reappears at the bottom. And if you go off to the left, then you reappear again at the right. So for a spaceship inside this universe, actually, it feels like it just goes on forever. But after a while, it starts to realise, I've seen that star before. So actually, what sort of shape does this universe have? We can talk about the shape of this universe because we can embed it in a three-dimensional universe and wrap it up. Because if you think, the top and the bottom are joined up and the left and the right are joined up. So actually, this universe has the shape of a torus. So let's do this. Let's join the top and the bottom of the screen up. We make a cylinder. So it's just going round the cylinder. But the two ends of the cylinder are also joined up. So if I join the two ends of the cylinder, the left hand and the right hand side of the screen, we get this shape called the torus, a doughnut, this shape with a hole in it. So this finite universe, which didn't have any boundaries, it was unlimited, can have a shape. And so Borges is very interested in, well, what about a three-dimensional universe, this library that he talks about in the story? Well, that's something we've been trying to answer. How do you wrap up a three-dimensional universe such that it's finite but doesn't have any kind of walls you bounce off? Well, we can do a sort of version of asteroids. Suppose that this is our universe here and there's nothing outside of here. So when you go off to the right-hand side of this lecture theatre, you come inside the left-hand side. And when you go out the top of the lecture theatre, you actually just reappear at the bottom of the lecture theatre. Well, that's the game of asteroids. But we've got another direction that we can move in. So we could go out the back through the screen here. And what would happen is that you'd reappear at the back of the lecture theatre. So here we are. This is a three-dimensional universe, which is finite. There's nothing else beyond this. You thought you'd be able to get out of this lecture. But actually, you know, you're stuck here in here with me. And every time you go out, you find yourself just reappearing. And it's very strange. What does this universe look like? Because there's light coming out the back of my head. It's going through the screen here and reappearing from the back of the lecture theatre. So I can actually see copies of myself going off to infinity. And this is what the universe would look like if you had something which was joined up in this way. So this is a three-dimensional universe. And if we had a four-dimensional world to wrap it up in, a bit like those cubes that Dali crucified Christ on, if we could wrap this up, we would get a four-dimensional version of the bagel, a four-dimensional Taurus. And actually, one of the great breakthroughs in mathematics, which has happened in the last decade, was made by a Russian mathematician called Grigori Palman, who actually classified what all the possible shapes our universe could be if it is finite and wrapped up. But what's exciting for me is Borges is exploring, with his own particular language, a way of investigating that problem about how can you wrap up the universe and be finite yet unlimited.