 I'm very delighted to bring up a dear friend of mine, Julia Kemp, who happens to be the director of the Center for Data Science at NYU, and she'll kick off tonight's program. Thank you. Thank you. You realize what an eminent speaker you have is that the introduction to the introduction, right? I'm delighted to introduce Henry Kohn, who will be speaking about higher dimensions and also data science, which is, I guess, the connection, which is why I'm here. So I have been asked a lot of times, what is data science? And it's a project, it's a work in project to define that science. But my thinking at the moment is that data science is something like biology. So when you think about biology, it is chemistry, it's a little bit of physics. But the difference is that it's dealing with something that exists, a living organism. Data science is mathematics, statistics, some computer science, but it's dealing with big data. And big data, in particular, in higher dimensions, which brings me to Henry's topic. Henry has obtained his PhD at Harvard under Noam Elkes and has then moved on to Microsoft Research, where he has spent the last, in fact, all his post-PhD years. First in Seattle, where he was part of the theory group, and in the end he was the head of the cryptography group. And then he was one of the founding members of Microsoft Research in Boston, where he is a principal researcher at this moment. Henry is on the MoMath Advisory Council. And he has been honored by a large string of awards. He was a speaker at the International Congress of Mathematicians in 2010, which is a big honor. He is a member of the American, of the AMS. And he received, this year actually, he received the Levy Conant Prize of the AMS. And without further ado, I pass the word to Henry. Thanks, and I'm happy to be here. Let me say something about this talk. I know it's a very broad audience. I recognize some of you as professional mathematicians. I'm sure some of you are very much not professional mathematicians. I'd like to make this talk interesting to and accessible to everybody. In particular, feel free to ask questions at any point during the talk. Don't feel the need to hold questions till the end. If things go off on too much of a tangent, we may have to defer till after the talk. But don't hesitate to ask. So what we'll be talking about today is high-dimensional geometry. And this is something I've loved since I was a kid. I was fascinated by thinking about higher dimensions and what they could mean. And when I was a kid, I thought that this was just sort of an arcane intellectual hobby. I thought that higher dimensions had very little relevance to the real world. It was just something fun for mathematicians to think about hypothetically. And so I was amazed to learn later that, in fact, higher dimensions play a really fundamental role in pretty much everything that involves data. In all sorts of statistics, computer science, all sorts of different areas. So what I want to talk about with you today is high dimensions. First of all, how do we think about them? How do we try to get a feeling for what they mean? And secondly, why should we care? And this is a really vast topic. We can't possibly do it justice in an hour. But we'll do what we can given the time constraints. So let's start by thinking about a problem I love, one that's very close to my heart, that at first doesn't sound like it has anything to do with applications or high dimensions. And this is the sphere packing problem. So the classical problem here is, imagine you've got a bunch of identical spheres. So for example, cannonballs or oranges or something. You want to cram them into space as densely as possible. So you can fit as many as you can into a given amount of space. In other words, we want to maximize the packing density. So from a practical point of view, this is not a very exciting problem. The thing is, it's obvious how to do it. You do it just like this. If you look at any pile of cannonballs in a war memorial or any grocery store in the fruit section, everybody does it this way. Cuz it's the obvious way to stack things. So you could ask, is this in fact the best possible way to do it? And for many hundreds of years, this was a famous unsolved problem in math called the Kepler Conjecture. And it was finally solved by Tom Hales, who published his solution in 2005. And he had an unbelievably complicated proof. He had hundreds of pages of intricate human reasoning, combined with massive computer calculations. But he finally sort of definitively resolved this problem. And the answer was, there's nothing better than the obvious way of stacking oranges. So at first, this sounds sort of like a parody of pure mathematics. You take something that's a kind of cool geometric problem, but one where everybody can see at a glance what the answer is. And then mathematicians devote their lives to trying to prove it axiomatically. And it's even worse than that. I listed two dates up here. So what happened was Hales' first attempt was in fact correct. But it was very hard to verify. In fact, when he submitted the paper for publication, a team of a dozen referees spent years working on this and eventually gave up and basically said, we've checked a lot of the paper. It all looks right, everything holds together. The argument looks good overall. But we have other things we want to do with our life besides checking Hales' work. And this was a very depressing situation when basically the referees gave up and said, we don't have the time or energy to check this definitively. So it looked at first like it was gonna go down in history with a big question mark, which would have been awkward. Hales declared that he was gonna resolve this in the best possible way. He was gonna work out a proof so incredibly detailed at the level of formal mathematical logic that even a computer could check it systematically and verify conclusively that not one case was overlooked. And he and a team of about two dozen collaborators spent the next 12 years doing that. And in 2017, that paper was published. So at this point, we can definitively say Hales was right. His proof is absolutely rigorous. It totally resolves this question. But still, as I was saying, it sounds like a parody of pure mathematics. So you could ask, why should you care about this spear packing problem aside from as a sort of intellectual exercise? And there are a lot of reasons you could give for this. You might say, well, we like granular materials. And even though granular materials aren't really made of perfect spears, that's the sort of simplest toy case. If you can't understand that, you're never going to understand real world materials. Or you might say, we care about packing complicated objects into suit cases. And if we can't even pack spears, how can we possibly pack clothing or something? And these are all valid points. But today, we'll see a much deeper reason that it turns out sphere packing plays an absolutely fundamental role in information theory. And the three-dimensional case is not the most important, but still it's something humanities gotta understand. So this problem, even though it doesn't sound at first like it has the remotest connection with the topic of the talk, is going to be really closely connected. So taking sphere packing a little bit further, one thing you can do as a mathematician is play with parameters and say, okay, let's fiddle with different aspects of this problem and see what happens if we change the dimension. So for example, we could reasonably say three-dimensional sphere packing was pretty hard. What if we dork in a plane? So in a plane, the analog of spheres are circles. So this is asking, how densely can you pack circles in two-dimensions? And once again, it's pretty easy to guess the answer. You arrange six around one to form a perfect repeating hexagonal pattern. This one's not as hard to prove. A mathematician named Axl 2A approved it in the late 19th century. It still takes a genuine idea to do. So okay, so going from three to two-dimensions made it a little bit easier, well, actually a lot easier. So you could ask, what about one-dimensional sphere packing? So here we've got to ask, what is a sphere? So remember that mathematically a sphere is specified by two things, a center and a radius. So in three dimensions that defines a sphere, all the points at this distance from the center. In two-dimensions you get a circle. In one dimension what happens, if you look at all the points at distance r from the center x, you just get two of them. And if you look at sort of connecting them to look at the points between, you just get an interval. And this makes sense. The same way a circle is a cross-section of a sphere, a line segment is a cross-section of a circle here. So the one-dimensional sphere packing problem amounts to how densely can you pack intervals on the line, line segments on the line. And this is stupid. You can cover the line completely with line segments. So okay, so we've got this sort of terrible progression. In one dimension the problem is almost vacuous, it's ridiculous. In two-dimensions there's a beautiful answer which we can understand but which takes some serious work. In three-dimensions, Haile's can understand it and Haile's computer can understand it, but the proof's never going to fit into my head unless we develop new techniques. So this sounds like a terrible progression where at this rate the difficulties are increasing exponentially and how can we possibly understand four-dimensions? So what I'll try to convince you of in this talk is two things. One is that high dimensions are actually the super practical aspect, the case that real-world engineers need to know about. And secondly, that we actually can understand them better than three-dimensions, which is crazy, but somehow it turns out that, for example, eight- and 24-dimensions are considerably easier than three. This is something that still amazes me and it's one of the sort of key things I want to understand better in my life is how can some dimensions, much larger than three, be so much easier to understand than three is? Questions about anything so far? Okay, so let's take a step back and look at history for a moment. As far as I know, the first person to really deal with this in the scientific literature was a scientist and mathematician named Thomas Harriet. He was in England in the 16th century and he's not nearly as well-known as he should be. If you judge him just by his scientific accomplishments, he ought to be fabulously well-known. He discovered sunspots before Galileo. He discovered Snell's Law of Refraction before Snell. He didn't publish these things, though, so in the end it didn't really count. In any case, he spent a lot of his life as a mathematical assistant to Sir Walter Raleigh and when Sir Walter Raleigh was dealing with the British Navy, one of the things he asked Harriet was, let's imagine sailors are stacking cannonballs on a deck of a ship and they form a little pyramid and they count how many there are along the base of the pyramid, but it would take too long to count the entire pyramid. How can they compute the number in the pyramid from the base? And so Harriet solved this problem and in the process he started thinking and wondering, is this really the best way you could possibly stack cannonballs? And then being a really creative guy, he started wondering, you know, what about atoms, these sort of new-fangled things? Is it possible that packings of spheres might relate to atoms and relate to questions like why do atoms form crystals? So he got really interested in it. He wrote to his colleague Johannes Kepler to say, hey, I've been thinking about spheres and atoms and I've got these conjectures about packing. And then Kepler wrote to his friends to say, ah, you know, I've been thinking about spheres and atoms and have these conjectures related to packing, which is how it became known as Kepler's conjecture. But in any case, Harriet was there first. Actually, the most famous thing about him is he was the first European scientist to visit the British colonies in North America. He came back and wrote a book, A Brief and True Report of the Newfound Land of Virginia, which is actually fascinating despite the title that sounds awfully defensive about whether it's true. You can find this online and it's full of all sorts of amazing things along the way he actually also brought back the habit of smoking to the British court, which is probably his greatest impact on history. So in any case, when Harriet was working on this, he was very interested in, like, literally the problem of how do you stack cannonballs. And he had these vague ideas about how it might have applications to physics, but on the other hand, if you traveled back in time to meet Harriet and said, hi, I'm from the future and I studied 24-dimensional sphere packing, he probably would have looked at you like you were crazy. I think the first person to really believe that high-dimensional sphere packing mattered for practical problems was Claude Shannon. So Shannon was one of the great mathematicians of the 20th century. In particular, he developed the basics of information theory in a really amazing paper he wrote in 1948 called the Mathematical Theory of Communication. And this really laid out the fundamentals of the field from scratch. It's really amazing. One of the things I like doing is reading old papers in mathematics and science, because it's fascinating to see how things developed. But this can be a little bit frustrating because things are often unreadable for various reasons. The historical context is sufficiently complex that it's hard to understand things in isolation. Shannon's paper is actually beautifully readable and you can find it online. I'll give you links and some material at the end of the talk. I highly recommend reading through it. It's full of fascinating stuff where he describes, for example, the entropy of English text and why this means you can do two-dimensional crossword puzzles, but not three-dimensional crossword puzzles. It's a fascinating thing to read. In particular, along the way, he realized that one of the fundamental problems in information theory basically amounted to spear packing in high dimensions. So in some sense, 1948 is the point at which suddenly questions like, how do you pack spears in 100 dimensions moved from sort of science fiction to practical engineering problem people really wanted to solve? And I'll explain this in much more detail later. So far, I haven't told you why this is true. So, okay, before we go on, let's review a little bit about what high dimensions mean. So, I mean, I know some of you are mathematicians, but some of you aren't, and let's all get on the same page here. So what do we mean by high dimensions? Basically, N dimensions just means N coordinates. So let's think about it systematically. If you're working in one dimension on the number line, you've got one coordinate. To specify a point on the number line, you just need to specify a single number. To specify a point in the plane, like two, three, you need two coordinates, how far over to the right and how far up. In three dimensions, you need three coordinates. Two, three, five is two units to the right, three units up, and five units out towards the audience. And similarly, two, three, five, seven is a point in four dimensions. And here, our spatial understanding totally breaks down. We don't have a fourth spatial direction to go in, but you can still deal with this as a mathematical abstraction. So basically, if you haven't dealt with this before, don't get too hung up on the sort of philosophy of, but what does it mean? Think about it operationally. Four dimensions just means four coordinates. Fifty dimensions just means 50 coordinates. In particular, I want to sort of derail one thing in case it's troubling you. I think one of the sort of iron laws of human society is as soon as you say the fourth dimension, somebody is going to say, you mean time? And this is entirely reasonable. The things physicists often deal with four-dimensional spacetime, with three spatial dimensions, and one dimension used to measure time. This is important, very practical. It's probably the most common use of the fourth dimension. But it's not the only thing we can do. Basically, anything you can measure with four numbers, you can consider a four-dimensional point, that there's no sort of reason that one of your dimensions has to be time. And in particular, please don't get hung up on the sort of physics of it, of whether we're talking about spacetime or sort of shortcuts between points far away in space or something. This is all sort of potentially valid physics, but it's only one narrow slice of what the fourth dimension can mean. Questions about anything so far? Okay, by the way, please ask questions at some point in the talk. If we make it through the entire talk with no questions, it will make me very sad, because I'll decide that nobody is actually listening. So if you care about my feelings, please interrupt eventually. Oh, yes? Inefficient packing? So one thing you could do is stack up everything at right angles. You could make a square of spheres and put another square directly on top of it, not nestled in carefully. So you could do things like that that are just wasting space. But you're right that the problem seems so intuitive that it's almost hard to think about how to do it wrong because it's so clear how to do it right. So I wish I could visualize high dimensions. In my fantasy, I could sort of close my eyes and think about a thousand dimensions and mathematics would unfold in front of me, and I would say, I see the answer. Now I just have to prove it. Unfortunately, I don't think anybody can visualize a thousand dimensions. Basically, our only hope is to turn it into algebra that basically a thousand dimensions just means a thousand coordinates. So basically everything that we want to understand about this, the way we can get at it is by restating it algebraically. So for example, how do you compute distances between two points in four dimensions? You take the sort of familiar distance formula from two or three dimensions. If you don't remember this from high school, the important thing is it's just a formula based on the Pythagorean theorem. And you just stick in some extra variables. And similarly, if you want to compute four-dimensional volume, do it just like two or three dimensions, only now you have four factors rather than two or three. So basically all this is extending things by analogy by extending the formulas to encompass as many variables as we have available. So a very reasonable thing to ask is why choose this formula? I mean it's not uniquely determined. There are any number of different formulas that you could put in with different ways of adding the extra variables. This is a particularly simple down-to-earth, familiar-looking one. It's not the only one, though. And the answer is you can do whatever you want. Yes? Oh, sorry. Exactly. So we're going to stick with Euclidean geometry here, but we could discuss any geometry. You can generalize these things in a lot of different ways that basically all sorts of different ways of measuring distance or volume correspond to perfectly good geometries. We'll stick with the simplest sort of most direct generalization of high school geometry. It's not because it's sacred. So how does this relate to data? So the point here is anything you can measure with a bunch of different numbers is secretly in high dimensions. So mathematicians refer to this as Rn for n-dimensional space, where R is the real numbers. So for example, if you go to the doctor and your nurse records your height, weight, and age, that's forming a point in three dimensions. If you're carrying out a scientific experiment where you make 20 measurements, you're creating a point in 20 dimensions. And you can get ridiculous numbers of dimensions. So for example, imagine you take a picture with a digital camera. Let's say a low-resolution digital camera, so it's 1 megapixel. So how many numbers does it take to measure this? You've got 1 megapixel, so 1 million pixels. Each of those pixels is described by three numbers, three different colors, red, blue, and green components. So you've got three numbers times a million pixels, or three million numbers to describe your photo. So that photo corresponds to a point in three million dimensions. And three million is nothing. There are climate models that have 10 billion variables because they've got all sorts of things about temperature, pressure, wind speed, chemical composition of the atmosphere at millions of different points around the globe. And you can get tens of billions of variables. You've got points in a very high-dimensional space. So here the point is, once you believe that the number of coordinates is simply the dimensionality, then coordinates are everywhere. Most data sets involve a ton of different measurements. And so most data sets really correspond to points in high dimensions. So from this perspective, high dimensions are not an anomaly. High dimensions are the default. Almost everything in life is high-dimensional, except a handful of things that happen to be very low-dimensional that we focus on because we like simplicity. So one way of looking at this is this all goes back to Descartes. What Descartes realized when he developed analytic geometry was anytime you're doing math with two or three variables, anytime you write down an equation and a few variables, you're secretly describing geometry in two or three dimensions. Yes? It can matter in certain sorts of algorithmic questions, but in terms of the underlying geometry, you can pick whatever order you want. So it doesn't matter there what number you order the X season. Yep. But you're right, that's an interesting question. And for algorithmic purposes, your algorithms may work more or less efficiently depending on how you order things. So sometimes it matters, but not for the underlying geometry. So here Descartes realized that for two or three variables it's equivalent to two or three-dimensional geometry. And we know from high school that understanding there's this equivalence between algebra and geometry can be really helpful that sometimes complicated algebra turns into beautiful geometry, sometimes beautiful geometry, sometimes complicated geometry turns into simple algebra. And it's really important to go back and forth. And the sort of 21st century version of Descartes is if you have a ton of variables you're secretly doing high-dimensional geometry and you should expect it to be equally useful. Questions about anything so far? Yes. Wait a second. The one megapixel... Yep. Sorry, I think that's five dimensions. Meaning there's an X and a Y and then there's the dimension of the red which has some number of values in it. The blue and whatever your RG coordinates are. Meaning that's a five-dimensional problem with many values in each. So here you're right that each individual pixel is much lower than three million dimensions but to describe the image as a whole you've got to describe all the pixels. So basically each individual pixel is much lower dimensional but the image as a whole constitutes a point in some vast dimensional space. You're describing just the size of the space not really the number of dimensions just the magnitude of the entire image really. Exactly, so it's the magnitude of the image but the thing is the larger the image the more numbers you need to describe it and therefore the higher dimensional the underlying spaces. But you're right that it's basically a measure of information here. Yes? Exactly. So basically you have a three million dimensional space of all possible one megapixel images. Yep. Each individual image might have a much simpler description it might be very compressible or it might be very complicated. So let's do some visualization exercises. I said nobody can visualize a thousand dimensions but there were a lot of fun ways to try to visualize four dimensions. So first of all let's think about hyper cubes the four-dimensional analog of cubes. This is a standard picture of hyper cubes. I've drawn it here in four-point perspective. If you like sort of technical drawing a fun sort of exercises to do things in four-point perspective in four dimensions but it's probably easier to see with sort of the miracle of zone tools. So here you've got a hyper cube represented as a big blue cube with a smaller blue cube inside connected vertex to vertex. So this is a sort of standard picture of a hyper cube. Let's think about what this means. So to think about this picture of a hyper cube let's drop down a dimension and imagine you wanted to draw a cube. So how would you draw a cube? There are two sort of very natural ways to draw a cube. One thing you could do is just draw the standard picture where part of the cube one of the corners and three edges are hidden behind the other ones. That's probably the fairest representation of what a cube looks like from a distance but it's kind of got things behind other ones. It's got its limitations as a picture. Another way you could draw a cube is let's think about it this way. Imagine you have a cube made of glass and what you do is you hold this cube up to your eye looking at one of the square faces. What do you see? So what you see is a big square next to your eye. That's the sort of nearby square. Off in the distance through the glass cube you see a smaller square off in the distance which is the opposite face of the cube and you see four more glass squares sort of viewed at an angle so they seem to form trapezoids. So another way of thinking about it is imagine looking into a pit. A pit is a cube just with one side open. So looking down into a pit you can see the entire thing with no overlap. So here you can think of this as a perspective picture of a cube sort of cut down from three to two dimensions looking at it really up close so that you can see the whole thing without overlap. So this is exactly the same thing in four dimensions. Imagine you have a glass hypercube you hold it up to your four-dimensional eye. What do you see? You see a nearby face of the hypercube which is a cube. That's the big blue cube on the outside. Off in the distance you see the opposite face of the hypercube also a cube. That's the smaller cube inside. In between you see basically the sides of the pit. The things that are trapezoids here. So each of those consists of a blue square a further away smaller blue square and some diagonals connecting them. And you can see this is sort of a distorted view of a cube. So here the same way in this picture you can see that a cube is made up of six squares the outer square the inner square and four connecting squares. Here we can see the hypercube is made up of eight cubes. The outer cube the inner cube and six connecting cubes. Questions about the hypercube picture? Or anything for that matter? So you can look at perspective pictures and the hypercube is pretty nice because there are a lot of pieces here but it's pretty well analogous to the thing one-dimension lower. You can sort of pick it apart and try to understand it. You can also look at things in other ways. So for example instead of a perspective picture you can ask, what about a shadow? It's really fundamentally the same thing. You can think of a perspective picture as it's sort of equivalent to the shadow, well it's not quite a shadow but more or less a shadow that something leaves on your retina. So you could try to understand things by their shadows. The problem with projections is projections rapidly get complicated. So for example here's a projection of something called a regular 120 cell from four dimensions down to, well here, two dimensions. And this is sort of the four-dimensional equivalent of a dodecahedron and maybe you're really good at picking structure out of pictures like this but for me when I look at this picture I say wow that's a lot of edges. It looks very symmetrical but it does not jump out at me as a four-dimensional image in my mind's eye. So to sort of help with practicing visualization skills I wanted to spend a few minutes doing an audience participation exercise. Where did ah can you help him? Thanks. So here there are some 3D models. Let's do them one every other person because I don't think we have quite enough for everybody but everybody or their neighbor will have one. So what these are going to be is these are going to be projections of a four-dimensional object called a regular 24 cell. So in particular the same way a hypercube is made up of cubes a regular 24 cell is made up of octahedra. Once everybody has one we'll play around with it and take a look at it but let me hold off on that for a second. We were thinking of leaving them on the seats but decided that would lead to too many 24 cells that got sat on and wouldn't that be a reasonable projection? Well that's true then they would be projected to two dimensions. So on the other hand that would just make it harder to understand we would sort of lose more geometry. So the question is whether insects with sort of faceted eyes see in more dimensions. I love that question. I've never thought about this I want to say yes and I want to think more about this because that's really cool. Okay so what is this? So the same way here we had a cube where we have an outer cube it's subdivided into smaller cubes the middle one is undistorted but the ones near the faces are all distorted. This is going to be the same thing with a octahedra. So first of all if you look at the oops if we look at the outer boundary of this the outer boundary is an octahedron in other words you've got a sort of square pyramid on top and down beneath it you've got a square pyramid on the other side sort of glued along their base so if you sort of hold it vertically with one vertex at the top let me use the bigger one one vertex at the top and one at the bottom the sort of top hemisphere is a pyramid the bottom hemisphere is a pyramid together they form something more symmetrical than the pyramid namely an octahedron. So if you look at what's inside you can see some of the things inside really look like octahedra so for example if you hold it with one vertex at the top and look just below that you can see just below it is a little square with another vertex below that and you can see the thing right below the top vertex is sort of an elongated octahedron but part of what's cool about this is everything here is an octahedron you've just got to look at it in different ways so for example part of what I love about octahedra I mean I love all polytopes they're all great but octahedra are particularly beautiful because you can look at them in two ways there's the way that we were discussing where you have two pyramids sort of back to back another way of looking at it is suppose you put this down in your palm so that it's flat with a triangle on the top so what have you got you've got a triangle on the top pointed sort of in this case it's pointed at my ring and the base that it's resting on is a triangle pointing in the other direction so basically what this looks like if you lay it flat is you've got a triangle on top you've got a triangle down beneath it partly obscured and then you take the convex hull in other words you connect these this is something called a triangular antiprism antiprism a prism is when you stack two things lined up with each other an antiprism is facing the opposite direction here the fact that the top triangle is pointing up and the bottom triangle is pointing down makes it an antiprism and this is the same thing as a regular octahedron so in particular if you hold this flat on your hand and look just inside the top just below the top you see let me show you here I've tried coloring it in we'll see how it works in the document camera you look at this you see on the top you've got these this triangle right beneath it you've got another smaller triangle facing the opposite direction and in between the sort of top triangle and the next layer you've got a sort of squashed octahedron and so if you look at this 24 cell every sort of void in the interior every gap in it is actually an octahedron that may have been distorted depending on the perspective so the same way in this picture of a hypercube we've taken it and we've sort of broken it up into cubes we've got a four dimensional solid here that's completely tiled with octahedron which is actually a sort of remarkable fact this is one of my sort of five favorite geometric objects the thing is that it's something that exists only in four dimensions where this sort of thing doesn't work in three it doesn't work in five it's sort of a miracle of four dimensions that everything fits together here so I really enjoy playing around with models like this unfortunately, MoMath will need the small plastic ones back but at the end of the talk I'll give you a link to an STL file you can use yourself on a 3D printer or you can order one from Shapeways for 6 bucks so basically what I love about models like this is the hypercube is not so complicated when I close my eyes I can visualize this model of the hypercube in my head the 24 cell when I close my eyes somehow I can't quite see the detail of the 24 cell it gets a little bit blurry but on the other hand it's a fun visual aid because you can look at it and you can say oh I see an octahedron there I see an octahedron there look how they're meeting together any questions about anything so far? yes what are your other four favorites? Icosahedron E8 root system leech lattice and one that I'm saving in reserve in case I think of something cool okay so we can do projections and think about things in terms of shadows or perspective pictures the other thing we can do is think about cross sections so for example imagine you're slicing a cube you've got a cubic loaf of bread so an obvious thing to do is to cut it like a normal human being where you cut square slices out of your cubic loaf of bread on the other hand if you really want to annoy everybody else who uses this loaf of bread you could do something like cut it from the corner and you could say okay I'm going to start at the corner I'm going to cut a slice right by the corner with a little triangle and then as I cut deeper and deeper through the loaf of bread I'm going to cut bigger and bigger triangular slices until they get so big that you reach three more corners of the loaf and at that point you've made the biggest triangular cut you can in this sequence and at that point you're going to start getting hexagons but not regular hexagons your hexagons are going to be sort of blunted triangles where you tried to cut a really big triangle but you're missing the corners so as you slice further and further through the loaf of bread your hexagon will become more and more regular as you cut down deeper the long sides will get shorter the short sides will get longer in the center of the loaf of bread you'll get a perfect hexagon this is one of the things I love it doesn't seem if you're not familiar with this sort of thing like a hexagon should be obtained by slicing a cube but you totally can in fact if you noticed at the entryway to MoMath the sort of atrium is a cube of glass with a red hexagon sliced through it so the hexagons there are already at MoMath but in any case you can get cool cross sections of cubes so you could ask what about a hypercube imagine that you wanted to cut slices of a hypercube from a corner so the same way when you slice a three-dimensional cube you get a two-dimensional slice when you slice a four-dimensional hypercube you're going to get a three-dimensional slice so what does it look like so when you slice through the corner you're going to get a three-dimensional triangle in other words you're going to get a tetrahedron so you get a small tetrahedron by slicing through your hypercube and now as you slice deeper into the hypercube your tetrahedron gets bigger so you get a bigger slice still a tetrahedron like we got bigger triangles until the tetrahedron gets so big that it hits the corners of the hypercube and we snip off the corners of the tetrahedron to get a truncated tetrahedron where now it would have been a really big tetrahedron but the hypercube doesn't go out that far so what you're getting is you're getting this truncated tetrahedron as you slice further the truncations get bigger and bigger you're slicing off more and more of the corners of the tetrahedron until finally the truncations meet in the middle if you truncate it so much that this truncation meets that one you get knocked a hedron in the middle of your hypercube and then the octahedron turns into a truncated tetrahedron in the opposite direction and you sort of back out the other side of the hypercube so one way of... Yes? To slice three-dimensional bread we use two-dimensional flat knife so to slice four-dimensional hypercube we use three-dimensional knives Exactly, you would use an object in four dimensions to slice it which is flat, in other words three-dimensional flat and beyond Exactly, that's the beyond part Yes So basically you can use this to help visualize things in a lot of ways so for example if you wanted to visualize a cube but you only have two spatial dimensions but fortunately you have a time dimension also you could make a time-lapse movie of slicing through it and your movie would be bigger and bigger, corners get blunted hexagon, triangle, etc and so you could imagine representing the cube as a series of cross-sections same thing with the hypercube so this gives a really radically different way of looking at it I mean, I love this picture you can actually think about it this way a sort of fun exercise is to try to understand what all of this means in terms of this picture but frankly when I look at this picture, I don't instantly think ah, you know octahedral slice through the metal it takes some real thought incidentally, if you've seen all of this before and you're sort of sitting there twiddling your thumbs, a good exercise to think about is you can get a cube, a tetrahedron and an octahedron by slicing a hypercube can you get a dodecahedron or an octahedron anyway, I'll leave you to think about that don't get too distracted though that's a sort of side question for experts to amuse themselves so let me talk about briefly one more historical anecdote I said nobody can visualize a thousand dimensions I know of one case in history of somebody who I think legitimately could visualize four dimensions better than anyone else so the story goes back to somebody named Charles Howard Hinton it wasn't him, but he instigated it he was a really fascinating guy he lived in England and then America in the late 19th century he shows up everywhere he invented the word Tesseract which is really cool if you're a Madeline Lengel fan he invented the automatic pitching machine after he left England in scandal and moved to the US and took up baseball he was a fascinating guy in particular he loved the fourth dimension and what happened was he was courting Mary Boole who was the daughter of the famous logician George Boole so Hinton wanted to spend as much time as he could with Boole's family so he was looking for a good excuse and his excuse was I'm going to teach the younger children about the fourth dimension Hinton had a bunch of colored cubes and he brought them over to the Boole household and sort of laid them out and did visualization exercises for hours on end with the younger Boole children most of whom did not enjoy it and did not seem to learn anything from it but one of whom really did I'll tell you that story in a minute but first of all about Hinton's cubes you can find them described in a book with the ridiculously immodest title A New Era of Thought which he published in 1888 the book is fascinating but weird here is a figure from the book he describes his cubes in great detail this is model seven he gave them all sort of pseudo-latin names this one's called Pluvium and he named all of the faces edges and vertices ah this face is called Lotus this edge is called Lucas this vertex is called Ancilla and he gave elaborate descriptions Pluvium is supposed to be colored dark stone I don't even know what it is dark grey and you're supposed to color the different points silver, turquoise, fawn, gold quaker green, peacock blue, dull purple and light blue and then the edges are leaf green smoke, orange stone, dull blue dark pink etc so we give these elaborate descriptions and you color these cubes and things you can do with them and as far as I know nobody found this particularly illuminating I mean Hinton never had any great insights into the fourth dimension as far as I know he did great work coming up with cool words like Tesseract but he didn't make any noteworthy mathematical discoveries the one person who did was George Bool's daughter Alicia somehow this unlocked in her sort of uncanny ability to visualize the fourth dimension and she had no formal training in mathematics but she was really good at seeing in four dimensions and the really sad part of the story is it was the 19th century when George Bool's daughter Alicia could visualize the fourth dimension people didn't say wow you should become a mathematician instead they said that's nice honey and so she grew up married in actuary was a housewife didn't do anything with her talents for a few decades until mathematicians heard that she could visualize the fourth dimension and started coming to her house to ask her questions she would describe what she saw they would verify it with proofs and equations and then write papers sometimes joint papers sometimes they would say that's an amazing discovery you should write a paper on that this is from one of her papers in 1900 you can see an immense jump in sophistication that this looks an awful lot more scientific than when Hinton was doing but somehow Hinton was the impetus that sort of led to her being able to see things in any case I don't know what to make of this I don't think Hinton had the secret to the fourth dimension I don't think there is any such thing I think Alicia Bool-Stott was a genius but on the other hand I find it fascinating that at least one person had an amazing ability to sort of see and describe things that were completely correct without formal mathematical training any questions about anything so far okay so enough with visualization for the moment let's think about applications in particular I promised you a discussion of what packing had to do with information theory so here was Shannon's big realization he said imagine we're trying to describe a communication channel so what does a communication channel consist of you're going to send signals over your channel they might be sound waves they might be radio waves they might be electrical signals on a wire could be all sorts of things but the point is your channel characterizes them by a bunch of measurements namely whatever measurements you're receiving equipment is doing to determine the signal so these correspond to points in some space where the coordinates are just measurements so for example if you're sending radio signals the coordinates could be measuring the amplitude of your signal at different frequencies for example so the key point here is in most applications the number of measurements is huge this has nothing to do with ordinary spatial dimensions if you're measuring a radio signal there's no reason you should make only three measurements you could make a hundred measurements or a thousand measurements so in most applications you're going to make a lot of measurements and your channel is going to be represented by points in a high dimensional space so here the key insight is even though we ourselves are limited to three dimensions we're effectively sending signals that live in a much higher dimensional signal space meaning they're described by a lot of measurements yes they might or might not be digitized so here the signals might be sort of discrete binary values if you're sending things over the internet but for most physical channels they're most naturally modeled with real numbers so you've just got some numerical measurements not necessarily digitized exactly so it could be analog yep this can really describe either one so the big issue that Shannon had to deal with is noise the thing is in a world without noise where the signal that gets received on the other end is exactly what you were trying to transmit there would be no need for information theory, life would be simple nobody would even talk about communication channels because they would all be perfect the problem is in the real world there's noise any signal that gets distorted or disturbed in some way if you send a signal s and somebody receives r on the other end typically r is not actually equal to s nearby but not equal so think about a toy model of a channel real world channels may be a little bit more complicated than this but this is at least the sort of core model so imagine asking how much error is really plausible you can imagine surrounding each signal by an error sphere that says it might get perturbed but only a small distance so it's almost certainly going to be some point within this sphere and you can quibble with the precise details of the model but this is at least a sort of good first approximation so here we're envisioning each possible signal being surrounded by a sort of cloud of possible received signals forming an error sphere around it and now the key point is we want to agree ahead of time on which signals we're going to consider using we're going to sort of agree here are certain possible signals that we'll allow and we're going to try to choose the allowable signals to avoid ambiguity and the point is if two signals get too close to each other they could get confused in particular if these error sphere is overlap any point in the overlap could have come from here it also could have come from there so we've got ambiguity in particular if we want to avoid ambiguity and make sure any message can be uniquely understood then we've got to avoid overlap between these error spheres so now the problem is we've got this high dimensional signal space we're trying to choose signals within it in such a way that we never use two signals whose error spheres can overlap yes? Are there reasons to believe from actual situations that the error is constant? So the question is is there reason to believe the error is constant it won't literally be constant it could be a point in the interior of the error sphere and there are more or less complicated models so for ah, so ok so the question is could you have more complicated things like bursts of error or things like this the answer is yes this is sort of the simplest toy model but you're right that as you adapt this to real world situations you may need to complicate it in various ways most of the complications don't really hurt the underlying geometry but they do complicate it yes? Is there a reason to treat the model as a sphere rather than a hyper shape? So a sphere a sort of hyper ellipsoid is actually pretty natural there's this central limit theorem which sort of describes sort of universal behavior for noise in certain systems and it'll give something that can always be rescaled to a sphere so this is at least the most sort of basic universal case you can have additional complications so for example cell phones you run into interference with metal objects and reflections off buildings and all sorts of complicating factors a sphere though is sort of the default behavior and other things are usually perturbations around that but that if you so if one of your measurements is very far off you would expect the others to be correctly exactly although usually you don't have one thing that's sort of much further off than the others but yeah you're right yep so here the idea is you want to choose what's called an error correcting code in other words you agree ahead of time here are the possible signals we're going to use you choose them to keep the error spheres from overlapping so you get unambiguous interpretation of signals but you want to cram in as many signals as possible into the available space so that you can communicate more quickly and basically the problem of how do you cram as many signals into the space as you can without leading to any ambiguity is exactly the sphere packing problem and what I love about this is that this is something very practical that engineers really care about in the real world despite the fact that the problem sounds ludicrous if you just meet someone on the street and say hi I study packing problems in 100 dimensions they'll think you're crazy but it's real so what do we know about sphere packing so let me say a little bit about what we know the answer sadly is not very much so first of all something amazing happens and that is that each dimension seems to behave a little bit differently and I think from the first three dimensions is that it's easy to guess the answer and it will all be in a nice sequence where you just sort of describe what is the best of all possible sphere packings and maybe mathematicians can't prove it because mathematicians can't prove a lot of things but on the other hand you found the answer and now for all practical purposes you're done the amazing thing is every dimension seems to have its own idiosyncrasies they all behave a little bit differently and we don't know how they work in general in particular we don't even know the most basic things so for example one thing you could ask is how structured should you expect an optimal packing to be should you expect it to be beautiful and symmetrical like a crystal or should you expect it to be somehow amorphous with things sort of glommed on in all sorts of pseudo random or random ways and the answer is we have no idea all we know is upper or lower bounds showing you how much can you definitely achieve and what's the most you could possibly achieve and these bounds are ridiculously far apart it's sort of humiliating dimensionless so for example in 36 dimensions which is not that high these bounds differ by a multiplicative factor of 52 if you take the best packing anyone knows in 36 dimensions and you ask how many more signals could you cram into it if you were as efficient as possible all we can say is if we take the spheres out you can't put any more than 52 times as many in as you removed which is crazy I mean if it were a matter of 10% I would say okay 36 is kind of high you know we can't expect to get it exactly but a factor of 52 means there's at least in principle the possibility of tremendous wasted space which if we could somehow reclaim it would be wonderful but the packing just doesn't seem to work very well so in high dimensions we really do not understand the packing problem well at all yes do we have a reason to believe the density increases with dimension? the density actually decreases so remember it started out with 100% the 90, 74 it seems to decrease almost always I'll show you one exception in a graph in a few minutes but that's a great question in any case this tells us that it'd better be decreasing to zero so let me tell you a proof the thing is they say that every math talk should include one proof and one joke preferably distinct so here's the proof I want to tell you how to get a good packing in high dimensions and if proofs aren't your thing don't be intimidated it'll be fun and it'll be short but what this will tell us is a very simple way to achieve density at least 2-n in n dimensions so here's the argument so it'll be so the proof works like this let's imagine you have any packing in n dimensions with spheres of some fixed radius and let's imagine we've crammed in so many spheres that there's no room for any more to be added so there might be lots of small gaps in our packing so let's make the gap big enough to put another sphere into the gap so okay that's reasonable we don't know what this packing is we don't know how dense it is but let's assume it's got no holes so now the argument says imagine if we double the size of all our spheres so we take these little blue spheres and we make them twice as big so what I claim is if you double all the sphere sizes you must cover space and what's the reason for that you can't have any uncovered space if you had any point that wasn't covered by the double radius balls it would be at least 2 radii away from any of the original centers and that would mean it would be far enough a way that you could have put another sphere there without overlap in the original packing so anything that's 2 radii away from the original packing is a hole big enough for another sphere if there are no holes there's nothing too radii away so you cover space completely so now how do volumes work in n dimensions remember that it scales with a power according to the dimension in one dimension twice as big twice as long in two dimensions twice as big four times the area in three dimensions twice as big eight times the volume in n dimensions twice as big two to the n times the volume in space when you make it twice as big you must have covered at least a 2 to the minus n fraction of space before that otherwise you couldn't get up to all of space by multiplying by 2 to the n so what we conclude is if you filled all the holes you must have achieved density at least 2 to the minus n any questions about this proof yes sorry I can't hear oh the minus is just from dividing that one divided by 2 to the n is 2 to the minus n so when I first saw this proof I had two thoughts I thought this is a beautiful clever argument I'm glad I learned it and I also thought this is a really pathetic argument for two reasons A, it tells us nothing about how to actually pack we've reached a conclusion but we never actually specified a packing all it said is if you actually fill in all the holes you must achieve at least this much but it didn't tell us what the actual packing is and also 2 to the n is pathetically small so my thought was okay this is a good job for phase 1 but the next proof I learned is going to give a much better density and is going to tell how to pack and the answer is no so here we've got a non-constructive proof it doesn't actually give us a packing and it turns out nobody in the world has any idea how to describe a packing that's anywhere near this good if you sort of describe a packing it invariably turns out exponentially worse than this which is crazy the thing is this bound that sounds terrifically low is actually vastly better than anybody knows how to achieve constructively I really don't know what this means I mean one possibility is it means high dimensional sphere packings may be so complicated that they are beyond human description I guess that's the pessimistic version the optimistic version is maybe we just haven't done it yet but it will be very simple and beautiful when we do I honestly have no idea yes what's the if you take a lattice packing what's the density of that so if you take a sort of square lattice or hyper cubic it turns out to be factorial in the denominator phenomenally bad worse than any exponential and that kills you and it's remarkably hard to avoid that factorial so let me describe the history slightly so we just proved a bound of 2 to the minus n and the famous mathematician Hermann Mankowski in 1905 proved a bound of 2 times 2 to the minus n and on the one hand he doubled the density of the densest packing in high dimensions which is pretty cool on the other hand the factor of 2 is small potatoes on a scale of exponentials but in any case he got this started and there were a lot of papers on this over the course of the 20th century by 1992 Keith Ball got an extra factor of n in high dimensions incidentally Keith is one of the two people in this area whose name I'm most envious of so ok in 1992 balls bound roughly 2n times 2 to the minus n and for 19 years nobody could beat this until a student of mine Stephanie Vance improved it just slightly in dimensions that are multiples of 4 if you remember the number e from calculus classes it's slightly less than 3 so 6 over e is slightly greater than 2 and a couple of years later Akshay Venkatesh who just got the Fields Medal this year improved it by a log log n factor which was phenomenal it was the first super linear improvement on 2 to the minus n but if you look at this we're struggling to get log log n factors on an exponential scale that's one exponential to go from 2 to the minus n to n that's three exponentials less than the sort of base case I don't know what this means once again we can take optimistic and pessimistic takes on this the optimistic take is pretty smart the fact that we're sort of struggling for log log ns probably means we're pretty close to the best answer the pessimistic case is humanity's pretty stupid we've probably got no idea what we're doing and I lean towards the latter although I don't know the best upper bound is also exponential but it's got an exponential gap away from this so in any case the state of the art is we know it's somewhere between these but we don't know where there's a plot of density and here you can see two things so I'm showing three things here the dimension is going up to 36 and it's a logarithmic plot so the fact that it's sort of decreasing linearly corresponds to exponential decay so there are three things in this plot the blue the blue curve is the best packing anyone knows and you can see the blue curve is very jagged and hard to predict doing a sort of simple smooth predictable pattern it zigs and zags and does crazy things that's why we can't understand it the sort of pinkish it came out as red here curve is balls bound which on this scale is indistinguishable from the other ones and the green curve is an upper bound that Noah Melchies and I proved 15 years ago you'll notice they seem to touch in 8 and 24 dimensions but basically what we see here is on the log plot the blue curve and the red curve are sort of the upper and lower bound the green curve and the red curve are the upper and lower bounds the truth has to be somewhere in this wedge in between but it's really hard to extrapolate from this data and come up with a good guess for what it should actually be doing so here I pointed out that in 8 dimensions and 24 dimensions the upper bound and the best being known seemed to touch and these are the most amazing dimensions of all in 8 dimensions there's something called the E8 lattice and in 24 dimensions there's something called the leech lattice not actually named after the animal but after a person John leech I don't know the history there maybe his ancestors were doctors I can't say but in any case E8 and the leech lattice are unbelievable packings they're the most symmetrical things you've ever seen they've got wonderful amazing structure and they're connected to everything in math and physics so they come up in all sorts of different areas when you arrive at an object like this which is incredibly dense incredibly symmetrical beautiful wonderful you say aha this must be the densest possible packing I mean look in 24 dimensions it's like the leech lattice is pulling the entire curve in this direction so basically in 2003 when Noam and I published our paper we noticed that these points were practically equal and the question was do the upper and lower bounds match and we could compute them to a bunch of decimal places and the more decimal places we computed them to the more they matched so we said aha it looks like this should solve the sphere packing problem in 8 and 24 dimensions but we had no proof and no explanation and this was a terrifically frustrating situation because mathematicians like proof and mathematicians like proof because we like understanding and saying well the numbers sure seem to agree because I'm computing decimal places and I can't find a difference is not an explanation is not understanding so from 2003 to 2016 it was terrifically frustrating because it looked like these bounds should solve the sphere packing problem but we couldn't figure out how and in 2016 in March March 14th actually Pi Day if you like that kind of thing a Ukrainian mathematician Marina Vyazovska came out with a paper that proved this conjecture and solved the 8 dimensional case by showing that these bounds really do match and this was an incredible breakthrough where she came up with these techniques from a totally different area of mathematics that somehow magically fed exactly what we were looking for and solved the 8 dimensional sphere packing problem and 24 had a few extra twists but she and several other mathematicians who've worked on this and I got together and extended this to 24 dimensions also so here in 8 and 24 dimensions the fact that these curves touches real we know the answer in these cases but in every other case you can see that the upper bound is very close very far from the best construction known and we've got no idea whether these are equal so somehow a miracle happens in 8 and 24 dimensions and you can see here that the miracle is on the sphere packing side that it's not that the bound is so great it dips down to the packing instead the packing is so great that it rises up to meet the bound I really cannot understand this in particular you can ask any number of naive questions if 8 is great and 24 is great why not 16 dimensions 16 dimensions is not great I can't explain why I would love to understand this better but there are still a lot of mysteries here and things that we don't understand so we're about out of time here but let me just summarize at the end by saying high dimensions are weird so we've seen some perspective pictures and cross sections and all sorts of things but basically the higher the dimensionality goes the weirder it gets and the harder it is to actually get a handle on it if you take even very simple concrete problems like how densely can you cram spheres into space in high dimensions it goes crazy we have no idea how dense you should get we have no idea how well structured nor it should be we don't understand this very well but the problems actually matter and even things like the sphere packing problem matter for information theory so the fact that modern technology works so well and that you don't worry about for example that when you transmit an email by your cell phone maybe the email will get corrupted that this all comes down to information theory so basically the takeaway message here is dimensions are weird and humanity is just beginning to understand them but they really matter for practical engineering so just to finish up I've put a link here to 3d.momath.org I guess the 3D is misleading it should probably be higher D but it is 3 so what this is I've gathered together some resources I've got links to papers and surveys I've got historical links so that you can look at Alicia Buolstotz papers at Harriet's book about Virginia things like this as well as some files for a 3D printer so that you can print your own model in any case so let's end here and please have fun with this if you have questions after the talk feel free to send me email to chat about things I'm always happy to talk with people about high dimensions now or later alright