Instant Genius - Hidden geometry, with Jordan Ellenberg

Episode Date: June 27, 2021

Mathematician Jordan Ellenberg tells us about his book, Shape, and why geometry is about so much more than triangles and circles. Once you’ve mastered the basics with Instant Genius, dive deeper with Instant Genius Extra, where you’ll find longer, richer discussions about the most exciting ideas in the world of science and technology. Only available on Apple Podcasts. Produced by the team behind BBC Science Focus Magazine. Visit our website: sciencefocus.com Hosted on Acast. See acast.com/privacy for more information. Learn more about your ad choices. Visit podcastchoices.com/adchoices

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I'm Sarah Rigby, online assistant at BBC Science Focus magazine. In this episode, I talk to Jordan Ellenberg. He's a mathematician and author of the book, Shape, the Hidden Geometry of Absolutely Everything. So when most people think about geometry, they usually think about triangles and circles and drawing a shape with a ruler and a pair of compasses. But in your book, you talk about lots of different things like artificial intelligence and games of chess and physics and things like that. So what exactly is geometry? Well, geometry is a very general notion. And really always has been, even in the days when, even in the days of Euclid, even in the days when geometry and the books consistent of triangles and circles only, right? I mean, geometry comes from a Greek word meaning measuring the world.

And I think that still is a pretty good description of what it is about. I would say any time that we talk about notions of distance, notions of closeness, notions of far away, which we would talk about in the context of our families, like close relative versus a distant relative, or we would talk about even when talking about the language we use, when we say that two words are nearby and close in meaning, we are using geometric thinking. We're using geometric arguments. anytime we say, well, even with arguments, right? We say there's two sides of an argument.

Okay, it's not literally a physical thing that has two sides. And yet somehow this geometric terminology has some purchase on it. It's built into the way we think about things. Right. Okay. So now I'd like to talk about one topic from your book, which is topology. And you talk about topology around the context of the discussion of how many holes there are in a straw. So first of all, could you just tell me what topology is, please? So topology is a form of geometry that really starts with a French mathematician named Henri Poincaray around the turn of the century. And I suppose nowadays I have to say that I mean the turn between the 19th and the 20th century, right? Of the students I teach sort of don't know which one I'm talking about.

So yes, I mean like around 1900, Puancares. And he calls it, by the way, he's an incredibly quotable mathematician. And I'll give some of his slogans as we go. But he didn't give it a great name. He called it Analysis Situs. So fortunately, this other name, Topology, which was a bit older, came from this guy, Johann Listing, who was this kind of itinerant and often in debt scientist who sort of like roamed around, like measuring diabetic people's urine and studying volcanoes and drawing

hundreds and hundreds of knots in his notebooks. He actually invented the word and it sort of caught on to, it caught on later to attach to Poncarais theory. I am going to answer this question. What is Poincaray's slogan? Among his many great slogans, he said, mathematics is the art of calling different things by the same name. This is very deep, and it's something we do all the time. And the idea is that what makes topology topology is that usually in geometry we're like, well, there's different shapes, like a square and a circle.

Puancaret would say, well, there's a sense in which it actually makes sense to think of a square in the circle as the same. Because if you had, like, say, some wire bent into the shape of a square, you could sort of mush it with your hands. And, of course, you can't see what I'm mushing as you listen to this. But imagine me mushing with my hands. You could sort of bend to the corners and sort of flatten them and make it into a circle. Or you could make it into an octagon. Or you could make it into a triangle. All those things from the point of its apology are the same.

So then you say, like, well, what's not the same? Well, for instance, if you broke the square, that's not the same. breaking is more violent than deforming and mooshing. So topology is the form of geometry where we call two things the same when one can be deformed, which is the more mathematical way of saying mooshed into another. Square, circle, triangle, all the same. A line segment, a different thing. Like a letter Y is like a third thing that's not the same as either of those two.

How many holes are there in a straw would you say? Now, this is a deep question, which doesn't sound like a deep question. But here's how you know it's a deep question. If you ask people this, you say how many holes are there on a straw, people will be like, well, the answer is obvious. That's the first thing that happens. But if you do this with a group, the second thing that happens is that everyone realizes that the answer they think is obvious is not shared by everyone in the room. And then people start to fight. And then it gets heated.

And that heat, that electricity that people feel around this question. And you can go on the internet and watch like incredible video of people just like absolutely losing it over this and their arguments. that's the sign that there's actually something mathematically deep there. So how many holes are there in it? Well, first of all, where are you on this? Sarah, I want to let me... I would say there's one hole in a straw. So you're a one-holler.

Okay, so most people are one-holers or two-hollers. There are a few stalwart zero-holers out there. Right. So, I mean, there's those who say, like, well, look, there's one hole and it goes all the way through. And there's those who say, well, there's two holes, one at the top and one at the bottom. And obviously, there's betokin some kind of. basic disagreement about what a hole even means. Is there, you might ask, a way to give that word a meaning that has all the properties that you might want, a meaning that makes sense? Here's how

a topologist might think of it. They might say, well, we should think of two things as the same if they can be mooshed into each other. So that straw, the answer to the question of how many holes there are in a straw shouldn't change if the straw gets shorter and shorter and shorter and shorter. And as you can imagine, the straw going on shorter and shorter pretty soon it sort of stops looking like a straw at all and looks like like a little ring. And at some point, you might say, well, it feels like there's just one hole. Like if you punched a hole in a sheet of paper with a hole punch, you wouldn't say I punched two holes, one at the top, one at the bottom. The thickness of the paper wouldn't cause you to say that. So that's fundamentally the argument for one hole.

On the other hand, here's what a two-hole might say. I might have in mind something like a tall, thin vase. How many holes are there in that? Well, you'd say there's just one hole, the opening at the top. Now, I poke a hole on the bottom. Now topologically, it's the same as a straw. So now you've said, I had a thing that had one hole in it. I make a new hole, and now you're saying it still only has one?

Like, maybe there should be some principle that if I add a hole to something, the number of holes should go up by ones. There's some kind of tension and there's some kind of conflict here. According to, well, I would say according to Poncarae's theory, but it's really according to Emmy Nurtur's theory, which sort of it follows up and improves on Poincaray, here's what we would say in modern terms. We would say the reason that there's so much confusion here is because the hole at the top and the bottom of a straw, well, they're different holes, but one is the negative of the other. One is like the reflection of the other. So they're different, but they're

not independent from each other. They're aspects of the same. Listen, here's what my daughter said, ready, because one way we try to understand topological or any mathematical questions like this is make them a little harder and more complicated. Then we can really see the full spectrum of the question. Okay, so suppose we have, instead of a straw, and now I've got a sort of British eyes, my exposition, a pair of trousers. I'm not supposed to say pants. Although, if I understand correctly, I think pants and trousers would be topologically the same in a British context, right? Yeah, I suppose. Pants would work as well. If the legs of the trousers became much shorter, they would be topologically pants.

Exactly. In any event, so how many holes are there in a pair of trousers? That's a very similar question. And here, there's people who say one, there's people who say two, and there's people who say three. And to share a bit of wisdom for my 10-year-old daughter, she said, well, there's not really three holes because the waist hole, is just like the combination of the two leg holes. Right? You can kind of visualize that, right?

Those two leg holes coming together into this one thing. I think that's a pretty good answer that you might think there are three, but one of them is just the sum of the other two. So it's not really different. It sort of comes from those two. And that already is really eminotris, great insight, that a modern notion of hole, which it seems very geometric, it also has arithmetic in it,

the other big strand of math, because you can add holes together. Leg plus leg equals waste. Or if I, to be more precise, leg plus leg equals negative waste, okay, if we're going to be really pedantic about it. But the idea that holes themselves are things that have arithmetic and can be added and subtracted. This was absolutely new, and this is Nurtur's great insight. When might a mathematician use topology when you're not talking about straws and trousers?

Well, we do talk about straws and trousers. Oh, okay, but other than, actually, one thing that's really interesting, this is not in the book, actually. When you write a book like this, there's so many different directions you can go and you're basically limited by time and wanting to have a book that you can bring home from the bookshop without throwing out your back. There's a wonderful new trend among people in data science of doing what's called topological data analysis, where they say, if I'm trying to sort of study some data set that's like very high dimensional, you know, the things have like, you're studying things with have like many features. many might be 10, many might be 100, many might be 1,000, and you're just trying to get some sense of like, you know, all the users in a social network or something like that. You know, maybe you can't precisely define a distance between two people

or something like that, the way you might have more classical geometry. But maybe there is some sense in which the set of all people forms a shape, which has a topological nature. Maybe it has a hole in it, for instance. Okay, let me give you an example. of this is a bit low rent, but have you ever heard of what's called the Horseshoe Theory in Politics? I don't know if this tracks in UK politics.

It's something that people talk about in the United States. The idea that the far left and the far right are, rather than being maximally far apart, actually have certain features in common or are kind of close. Is that something people talk about in England? I think that's why people say that. So if you think that's true, let's try to visualize that. It's like saying instead of being a line segment where you have, well, Again, geometric metaphor, right?

The far left on the left and the far right and they'd be very far apart. You imagine it like a horseshoe, those two ends bending towards each other, maybe even meeting at some point where the far left and the far right cannot be distinguished. And if that's how you think politics work, which, don't get me wrong, is a very controversial notion, you are making a topological statement. You're saying politics is more like a circle, more like something with a hole in it and less like a line segment, something that doesn't have a hole in it. Okay, that's not, that particular question is not one that sort of these nouveau methods

of topological data analysis would be adequate to capture. But it's that kind of thing that you ask these topological questions about literally how many holes are there in your data set. That turns out actually to carry some interesting information. Right, I see. Another thing you talk about in your book is the geometry of space type. How can space and time have geometry, especially in areas where there's nothing there?

Yeah, well, this is a wonderful story. and it goes back in some sense to Euclid himself, who is this kind of the progenitor of the kind of geometry that people do customarily. Like, he's this great mathematician of North Africa of around 300 CE who sort of compiles all the mathematical knowledge of his time. We don't actually know how much was Euclid original

and how much he was kind of an anthologizer. We don't know anything about Euclid the person. We do know there was such a person, but that's about all we know. But certainly, at the very least, his genius was organizational, that it was really there first where we see this idea that we're going to start from these axioms of how geometry must be. And from these very simple statements which you can hardly doubt, you build up step by step everything else about geometry, all these facts about triangles and line segments and circles and tangents and intersections and everything from these very short deductive steps. just from these five axioms of Euclid. I mean, this move has been like incredibly inspiring to like people across history,

this idea that you can build up this knowledge, um, starting from so little. But even I said, starting from these five axioms that you can hardly doubt, but there was always a sticking point. One of the axioms was more complicated than the others. It sort of stuck out like a sore note.

People didn't like it. They said, wouldn't it be great if that one followed from the other four so that we didn't have to assume, it, we could actually prove it. And people tried and they tried and they tried and they kept failing for thousands of years. And eventually people started to look to consider playing the other side of the game. What if that wasn't true? Or what if we could create some other kind of geometry, which had Euclid's first four axioms, but where that other axiom was different.

Now, that's a weird thing to do because I want to be straight with you. This axiom is obviously true. I'm going to tell you what it is and let's visualize it. It says that in one form, if you have a line and then you have a point that's not on that line, that there's a line through the other point, which is parallel to the first line, and that there's only one. Well, if you visualize that, you'll say, like, well, that's just true. I actually don't have a problem with that. It's not controversial. But it's the habit of geometers to be like, but is there a proof of it? Well, I am going to get to physics, by the way. I didn't forget your question. It may seem like I did. So in the 19th century, what happens is people start to sort of take seriously the idea of what would geometry

look like if that were not true, maybe we can derive some kind of contradiction in Joe that that can't be. And not only did they not do that, they developed entirely novel geometries, which were like the geometry we know, but differed in that this axiom was false. Now, this may seem like the most sterile, possible intellectual exercise and an excuse to, like, defund every math department and every university, right? Like, why are they studying, okay, what would things be like if something that's manifestly true were instead false. But it is the mathematical habit to say not only sort of like what is, but what could be. We study it all.

And of course, the punchline to this is that there's this crisis in physics near the end of the 19th and beginning of the 20th century where we sort of come face to face with the fact that like things don't quite make sense. Maxwell's equations that govern electricity and magnetism. They don't really behave properly with respect to the geometry that we do. know. And it took Einstein to kind of have the courage to be like, maybe that's because the geometry that we think we know is not actually a description of physical space. The space that we live in is not that described by Euclid, but is something much more non-Euclidean. So all this kind of

stuff that seemed completely abstract, in fact, was describing the space time where we actually live. What does it mean, by the way, you say what does it, I mean, but it's a good question. What does it mean for a space to have a certain geometry when, even if there's nothing there, even if it's empty space? I would say the answer is, let me bring it back to Poincaray's notion of calling different things by the same name. We say, now we're sort of going to try to dustily crack open our textbooks if you remember this word. We say the two figures are congruent. Do you remember that word? Vaguely. Yeah, so what does it mean to say the two triangles are congruent? There's two ways to say it. You could say it means, okay, all their sides and all their angles are the same. But a better

way to say it, a more physical way to say it is you could make one into the other by moving it around, maybe spinning it, maybe rotating it, maybe moving it around on the page. You can physically transform one into the other. That's a notion of symmetry, this notion of like, which things are alike if you're allowed to move around and like rigidly move the plane in a way that doesn't change shapes, just doing spins and reflections and moving things around, we think that physical law should not change under any transformation like that. We think that physical law should look the same if you turn your head and look in a different direction, or if you stand in your head and look at things upside down, or if you sort of get on a bus and are moving along at like 20 kilometers

an hour, none of those things should change the laws of physics. And it turns out that the different geometries have different kinds of symmetries. Different geometries have different kinds of transformations you're allowed to do that are supposed to leave everything in the same. And the laws of physics that we can observe turn out not to be invariant under the Euclidean symmetries. There's some, and this is all these kind of if you sort of have ever read about Einstein, you know, there's all these kind of things that like, you know, you get on a bus and the bus happens to be, you get on the wrong bus and you get on the one that goes half the speed of light instead of like the usual one that goes like 20 kilometers per hour. And then like things,

things start to change and things change their shape and like time slows down and all this stuff happens that you think is not supposed to happen. Well, if you understand correctly what geometry we actually live in, then everything works again. And the laws of physics are the same no matter what your vantage point. This is the so-called principle of relativity. And by the way, this relationship between what are the transformations under which the laws of physics stay unchanged, the so-called symmetries, the fundamental centrality of that question to sort of what a physical law even means, that's Emmy Nurtur again. I mean, so she's, this is Emmy Nurtur in the 20s sort of like really understanding the mathematical principles that underlie how we're supposed to think about

physics and what space is. Now I'd like to talk about artificial intelligence. You have a metaphor that I really liked in the book where you describe artificial intelligence in terms of mountaineering. Can you just give us a quick overview of that, please, and how geometry could be used in artificial intelligence. Yeah, absolutely. I mean, and I know I just want to, again, emphasize for the audience that this is all in the same book. I know it seems like we're talking about a lot of different stuff,

but that's the nature of geometry. It kind of like roams around and, like, touches on everything. So, you know, in some sense, everything we do in artificial intelligence is a subfield of the field of mathematics called optimization. And optimization is one of those fields. We gave it a good name. It's exactly what it sounds like. It means sort of trying to do things in the best way.

mountaineering is a form of optimization. So the sort of metaphor I use in the book is, you know, imagine you're trying to get to the top of the mountain. That's a very basic optimization problem where your objective is to get as high up as you can. Like higher is better. But the problem is maybe the landscape that you're in is like some very dense forest or something. So you can't just look and see where the peak is. You really can't see anything except for your immediate neighborhood. But there is like a very natural strategy that you can do if you are the mountaineer in this situation.

You can look down at your feet and try taking a step in a bunch of different directions. You know, there's like a little circle of possible directions you could go. And of those directions, some of them are going to be uphill and some of them are going to be downhill. You probably don't want to go downhill, right? You want to go up. And what's more among the uphill directions, there's probably one that's the steepest. And the strategy is you just choose that one. You choose the steepest upward direction you can find.

You take a step that way, and now you're in a new spot, and now you do it again. Now you say, from here, what's the steepest upward step I can take? Might be the same direction I was going, or I might shift a little bit. And you just keep on doing this. That is a method, that is sort of sounds very primitive, but it is a mathematical tool called gradient descent. And literally every single machine learning system we have is built on this. So how would an artificial intelligence learn to do something using gradient descent? Right. So what it is trying to do, any machine learning system is trying to learn,

you might call it if you were fancy an algorithm, if you were being less fancy, you might call it a strategy. Let's say a strategy for predicting the next movie you're going to watch or a strategy for looking at an image on a screen and telling you like whether it's, it's a cat or a dog. You could think of the space of all strategies as like a landscape, a landscape you're trying to explore, just like the mountaineer is. Now, it's a much bigger landscape, right? I mean, when you think, it's even hard to get in your mind,

what could the space of all possible strategies for a problem look like? But, you know, the good thing about geometry is like we're not scared by big spaces. It can be like a million dimensional space. Like, that's okay. You know, there's a famous interchange where Jeffrey Hinton, who is one of the leaders in the field of neural nets that sort of predominates modern machine learning, somebody said, okay, but how do you visualize a 14-dimensional space?

This happened to be what he was talking about in a lecture at a time. And he's like, well, you visualize a three-dimensional space and then you very loudly say, 14. That's it. That's what we do. And it's kind of a miracle that we work. So the fact that we can explore some, and I literally mean, for instance, in GPT3, which is a popular sort of language-generating machine learning tool, I think it's 175 billion dimensional space.

And it's sort of a miracle that if we're going to try to talk about exploring 175 billion dimensional space, we can execute a strategy that we develop by thinking about three-dimensional space, and it still basically works. That's the miracle of geometry, that we start with our sort of physical intuitions that come from, the fact that we're beings that live in space, three-dimensional space, very puny space, the fact that these ideas actually scale up pretty well is a miracle and a very fortunate one, otherwise we couldn't do anything at all. So to your question, what you do is you start with sort of some completely random strategy

for telling a cat from a dog, let's say, which is probably going to be pretty bad, right? And then you say, of all the ways I could modify it, let me look at all the tiny ways tweaks I could make to it, which one improves the strategy the most? Well, what does improve mean? Usually you have some, you have to see the algorithm with some predigested list of, like, here's a thousand cats and here's a thousand dogs. And I'm going to tell you which ones are cats and which ones are dogs. Without that, you can't even get started, which makes sense. I mean, because if you think about it, unless you had some existing information, like a baby, right?

A baby is a very good learning algorithm. But a baby is not just going to figure out what's a cat and what's a dog. A baby has this, like, large person who's constantly pointing. at stuff in its perceptual environment and going cat, like dog, and it needs to see a certain number of those before the baby can do it by himself, right? Okay, so like a machine learning algorithm is much like a baby, but less cute. But it's the same idea. You give it sort of some population called training data of cats and dogs that it has seen before. And then you sort of tweak your strategy a little bit, try all the different tweaks you could think of and see which one best improves

its performance on the cats and dogs that you gave it to train on, and then do that again, and then do that again, and then do that again, and keep doing it until no small change you can make makes the algorithm any better. That is a very simple-minded strategy, and in some sense, the great theoretical mystery in current machine learning is, why does it work so well? But it does work really well. Great, thank you. And now I'd like to talk about the geometry of games. You mentioned a few different games in your book and how geometry can be used to play them. And there's one in particular called NIM. And NIM was the game that my math team. choose to make us play in the last lesson before the end of term. And then at any point, we could go up

and challenge him. And if we beat him, he'd give us a chocolate bar. But no one ever beat him. And having read your book, I now know it's because he set us up to fail. He set up the game to make it impossible for us to beat him. So can you tell us a bit more about NIM and how it's possible to set up the game so that no one can beat you? Do you think he even had a chocolate bar? I think he had one for show. Right. So NIM is a game and there's many different versions of it. And I don't know which version your teacher played, but a typical version would be there's several piles of stones and the game is very simple. You take turns and the only thing you're allowed to do is take stones from a pile. You can take any number of stones, but they all have to be from one pile.

And you take turns and whoever, and now I'm actually, there's, there's a version where do you play last player wins or last player loses? In one version you lose if you make the last play and another version you win if you make the last play. I think we played it so that you had to take the last stone to win. It's funny, people, there's a long tradition of referring it to it as an ancient Chinese game, whereas it seems to actually be from like Italy in the 16th century. At some point, someone sort of orientalistically thought it would sound cooler if they said it was like ancient than Chinese, which it is manifestly not.

This game has a geometry. That's not very visible. But the geometry is the geometry of a tree. And it turns out that actually a huge number of games, which look very different, NIM, checkers, chess, go. these games don't have so much in common with each other, but every one of them can be described as a tree. The tree is sort of depicting the decisions you make as you go along the game.

Each decision you make kind of branches the possible worlds we live in into, you know, maybe you have three choices. There's three branches that you may choose to go down from that branch, further branches, et cetera, et cetera, et cetera. It turns out that the geometry of the tree is what allows you, to analyze the game and say that any game like that, either the first player can always win or the second player can always win.

Or if it's a game that allows draws, it may be that two perfect players will always come to a draw. And I think that somehow comes as a surprise to people. Not so much about a game like Tick-Tac-Tow, or is it called knots and crosses for you? Is that what it's called? Yeah, knots and crosses.

Okay. Wait, so a game like that, I think people kind of, instinctively know if they have experience playing it, that the only way somebody's going to win is if the other person screws up, right? That's sort of like players, if they don't make a mistake, it will always come to a draw. I think people see a game like checkers or drafts, I should say, and chess and go. SMO with like a higher order that there's more art to them. And in a way that's true, but in another way, it's not true. There is a fact about chess where either in chess,

and again, this comes down to an analysis of the tree, which I read about in the book. It's either true that the first player can always win in chess or the second player can always win or perfect players will play to a draw every single time. And the nature of this fact is mathematical. It's really no different from like multiplying two very, very, very large numbers together. How you should feel about that is an interesting question.

To me, that doesn't diminish in the slightest the artistry of chess. or the fun of chess or the mystery of chess. But it is useful to know that, and maybe we'll come back to Poca-Rae. Topologically, chess is no different from Tic-Tac-Tot. It's just bigger. Okay, thank you. And just to finish up, what would you say are the three things I need to know about geometry to become an instant genius?

Well, first of all, you've got to know that I hate the word genius, even though it's the title of your podcast, so I hate to end on a sour note. But, you know, one of my slogans is that, you know, genius is a thing that happens, not a kind of person. So I think we can look for genius in the work, but I don't think being a genius is a proper goal for a person, instant or not. I will tell you three takeaways, though. One is that I think it's universally agreed that the reason we learn geometry is not so much to know facts about triangles and lines as it is to develop, habits of mind, to know what a proof is, not so much because we're going to be suddenly called upon in the dead of night to come up with one, but because we're so often presented

with very strongly stated arguments that purport to be proofs, but really are not. If we really know geometry and really know the language of proof, we know the difference between something that really is and something that really isn't, and there's so much out there that really isn't but pretends that it is. Thing two, I would say this slogan of Poincaray is. that have several said several times, to call different things by the same name. It's such a useful mathematical and geometric habit to look at two things which may be superficially different and understand that the differences between them are not so important for the question at hand.

Depending on what the question is, it might be important, and this might not. But to recognize different things, the same thing wearing different clothes is a vital geometric habit. And I think third, just to sort of recognize when we're thinking. geometrically, that it's absolutely ubiquitous and we do it across like every kind of cognitive domain, these notions of things being close and things being distant, that whenever we do that, we're sort of implicitly giving a geometric structure to whether it's our language, whether it's a game, whether it's space time itself, whether it's our family or our social network, whatever it may

be. It's worth actually sort of like letting your mind travel a little bit in that direction and really think about, like, you know, what is the map of this world that I'm talking about where I say these things are close and these things are far. Okay, so those are three things. Thank you for listening to this episode of Instant Genius. That was mathematician Jordan Ellenberg. If you want to know more about geometry, check out his book, Shape. Or, to hear him tell me more about geometry in the real world, including modeling the spread of COVID, head over to the Instant Genius Extra podcast. The June issue of BBC Science Focus magazine, is out now. Pick up a copy in store or visit sciencefocus.com.

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