 I'd like to call up Patrick Honor, who's been a friend of the museum since before there was a museum. He's a wonderful teacher, not just we think so, but the President thinks so. He's a recipient of the Presidential Teaching Award. He's also the recipient of one of our Rosenthal Prizes for Excellence in Math Teaching. And we're very pleased to have Trustee Saul Rosenthal with us tonight as well. Thank you for your support of that prize and of the museum overall. And so now, let me turn it over to Patrick, who will do a good job introducing our speaker tonight. Well, thank you, Cindy. And good evening, everyone. My name is Patrick Honor. I teach mathematics at Brooklyn Technical High School, one of the largest and best high schools in the country. And I am really excited to be here to welcome you to tonight's math encounter. I'm a huge fan of the Museum of Mathematics, and not just because I won the Rosenthal Prize a few years ago. I've been a regular visitor here since it opened in 2012, and I love it as a mathematician, as a teacher, and now as a parent. In fact, my son Leo on Monday, as he often does, declared, I want to go to the math museum. And so he came. And given that he is in the 99th percentile of car obsession, he was very excited with the Twisted Throughway exhibit that is now open. I heard about that for several days. Even before the museum opened, it positioned itself as a center of mathematical activity and outreach, both in New York City and across the country. And math encounters, this wonderful public lecture series that's free, has been a big part of that. You know, the luxury of being able to hop on the train and come see and hear from mathematical luminaries like Stephen Strogatz, Terry Tao, and John Allen Paulos, is not lost on me. And even better, so many of my students, both current students and former students, come to these events to learn new things, to get new perspectives on their futures, and to socialize around mathematics. It's really a wonderful and unique opportunity for all of us. So my great admiration for the museum, the math encounters series, and the work of Cindy Glenn and everyone behind the success of the Museum of Mathematics makes being here tonight very special for me. And on top of all that, I get to introduce Henry Segerman, a mathematician whose work has blown my mind for the last five years. I first met Henry in 2011 at the Bridges Math and Art Conference in Coimbra, Portugal. I can still see the 3D printed space filling graph that Henry exhibited in the art exhibit there. And in fact, I got to kind of relive that this week when a new number file video was released that Henry stars in talking about space filling curves. When I first saw that, I just couldn't believe that someone had brought that into existence, this mathematical object that I couldn't have even really conceived or pictured in my mind. It blew my mind then, and it amazed and inspired me, and I think in the five years since, all of us who follow Henry's work have just come to expect to be amazed and inspired by what he does. Henry's work in mathematical 3D printing is truly groundbreaking. He builds things that make mathematics clearer, more tangible and more accessible. He builds beautiful things. He builds things that inspire questions. And he builds things that help us see things we otherwise wouldn't be able to see. And sometimes he even builds things that shouldn't exist. And so you'll see some of those things on the table later. His work has drawn the attention of both research mathematicians and math enthusiasts and has led to the publication of a beautiful book visualizing mathematics with 3D printing. It's really, I think, the first of its kind, maybe one of a kind, a book that contains both beautiful expository and visuals about mathematics but also includes 3D designs that you can print and experience mathematics in a way that you've never experienced it before by holding it and playing with it and moving it around. And I think it speaks to how dedicated Henry is to making mathematics accessible to everyone in that he makes those designs freely available for everyone, whether you buy the book or not. You can just download them and print them yourself. If that weren't enough, Henry has also engaged in eye-opening work in spherical video and virtual reality. Again, he's tapping into new technologies to help people understand experience and appreciate mathematics in new ways. It's kind of hard to keep up with all of his amazing work. Like, I personally feel like I'm just starting to get the hang of 3D printing things with my students and Henry's already 3D printing or he's 5D printing or something in virtual reality. Sometimes I wish he would slow down a little, but it is fun to try to keep up. I think something that unifies Henry's work as mathematician and artist and programmer is that he builds things, beautiful things, exciting things, and inspiring things. What inspires me most as a mathematician and as a teacher is that the tool Henry uses to build things is mathematics. One of my primary goals as a teacher is to get my students to understand that mathematics is a power to create. Henry's work helps me and my students understand that. And I think it's exciting that Henry's work is really just beginning. Henry Segerman studied mathematics at Oxford and Stanford and is currently an assistant professor at Oklahoma State University. His research interests include three manifolds and triangulations, hyperbolic geometry, and appropriate tonight's talk, Mathematical Visualization in 3D Printing. He's here tonight to cast some light on the fourth dimension for us. So please join me in welcoming Henry Segerman. So thank you, Patrick, and thank you all for being here. So I'm going to attempt to cast some light on the fourth dimension and tell you what that means. Before I get into that, let me plug the book one more time. It's already been plugged a couple of times, but let me just tell you a couple of things about it. So there's a website associated with the book 3Dprintmath.com. If you go to the websites, then there's a section for each of the chapters and then in each of these sections, there's a page for each of the figures. So if you want to know about this cube, you can go here and there's a 3D version that you can move around and this works well on the iPad and so on. So you can have the book in one hand, an iPad in the other, and get to experience these things, or you can download them as Patrick said. Okay, enough with the plug for now. So okay, what is four-dimensional space? So if I were a physicist, I might think of the fourth dimension as being time, but I'm really thinking of four spatial dimensions. And so how do we understand this? What does this mean? Let's go back to two-dimensions and how do we describe two-dimensional space? So here's a picture, I'm sure we've all seen many times before. Here's a piece of graph paper. I got an x-axis, a y-axis, and I can talk about this point here with two numbers. I can say how far I go along in the x-direction, three units, how far I go up in the y-direction, two units, and I can talk about this as the point three, two. Okay, great. Let's get rid of the graph paper. So what if I want to go to three-dimensions? Well, I add a new direction, which is at right angles to the previous two, and now I've got three numbers that are going to describe points in my space, x, y, and z, and the z tells me how far to go in that direction, which is at right angles to the previous two. Well, if I want to go to four-dimensions, I add another direction, which is at right angles to the previous three, and I add another number, w, which tells me how far to go in this direction, which is at right angles to the previous three. Some of you may be a little uneasy at the moment, saying, well, what is going on here? You can't find a direction which is at right angles to the other three. This is a lie. What are you trying to do? Okay, well, nobody complained when I was here. This line here is not at right angles to these ones here. Even though I drew this nice little angley thing here, this angle here is, I don't know, 50 degrees. This is flat on the screen. This angle is not 90 degrees at all. I just tricked you by drawing these little symbols here. But everybody was fine with this. Well, if you were fine with this, you should be fine with this as well. Okay, I mean, what's happening here? Like, we're happy with this picture because we evolved in a three-dimensional world, and when we were running away from Lyons on the Savannah, we needed to be able to interpret what was going on. It was coming into our eyes, our retinas are two-dimensional, and so we're seeing two-dimensional things, really, but that we interpret what we're seeing out in the world as a three-dimensional thing, even though what we're actually detecting is getting squished down on the two-dimensional plane of our retinas. So it's the same thing here. This is a two-dimensional picture drawn on the screen, but we're interpreting it as a three-dimensional thing squished flat. So in the same way, this is a four-dimensional thing, which is squished flat. It's just a little bit more squished than the three-dimensional one. But yeah, from a mathematical perspective, this is no problem. I've got four numbers, instead of three, I can still figure out distances, angles. I can do all the same things. It's really no problem. Maybe it's a problem for physicists, not a problem for mathematicians. Okay, so let's have an example object we can think of. An example thing we're gonna put in four dimensions, a hypercube, also known as a tesseract, also known as an octochoron. I'm told that the three rooms that we're all currently sitting in have those three names, so three names for a hypercube. Okay, so let's build a hypercube. So we're gonna start very simple with a point. I take a copy of that point, shift it across, connect up the two points, and I make a line segment. Okay, let's do it again. So I take my line segment, and I make a copy. I shift it at right angles to the line that the line segment is sitting in, and I connect up the corners, then I make a square. Great, let's do it again. Take a square, make a copy, shift it at right angles, I'm lying again, right? This is not shifted at right angles. I just moved it diagonally on the screen, but everybody's happy, sure. If you shift it at right angles to the plane that contains the square, connect up the corners and you make a cube. Again, take a copy of the cube, move it at right angles to the three-dimensional space that the cube is sitting in, if you can find a fourth direction. Connect up the corners, and there's a picture of a hypercube. When this isn't a fantastic picture of a hypercube, it's not somehow telling you about lengths or distances or, but you can get some things out of this. So for example, you can count how many corners does this thing have. So, well, so the cube has eight corners and I took two of them, so the hypercube must have 16 corners. Actually, there's a nice pattern there. So this line segment has two corners and then you double it to get the square. That has four corners. Cube has eight, two, four, eight, 16. It's doubling every time. We can also talk about this sort of number of sides of an object. So a square famously has four sides. Four, in fact, line segments. It's got four things, which were the one that you had before. The cube has six sides, which are squares. So six of the things that you had before make up the cube. The hypercube, it's a little hard to see. This four sides, six sides, this has eight sides, each of which is a cube, the thing that it came from. This actually is still there as well. The line segment has two sides, each of which is a point. And the point has, okay, it sort of stops working at the point. Okay, so I'm claiming that there are eight cubes here. This is a little hard to see. You can see the original two. Maybe if you sort of look at this area here, I'm trying to circle around one of the other cubes. It's kind of hard to see, right? It's not so great. So this gives us some sense about what a hypercube is, but we would like to do better. So how can we see four-dimensional things? Unfortunately, I'm sorry that I'm here to say you can't see four-dimensional things. It's impossible. There are some people who claim that they can see four, that I truly see four-dimensional things. I'm not sure if they actually can. Some of them I'm sure have given talks at no matter. But I'm unconvinced that you can actually do this. So what is the best that we can do? Well, there's a few different things you can do. One of the things you can do is you can cast shadows. So this is a picture here of a three-dimensional object, a cube, and a shadow of that cube on the table. The idea here is if you had a two-dimensional friend who lived in the table. So Flatlands is a novella which explored this idea perhaps for the first time. If you had a two-dimensional friend living in the table and you wanted to describe to her what is this cube thing that I've been hearing so much about, she can't go up into three dimensions and see this. But she can see this two-dimensional shadow here down in the table. And she can try and understand what's going on based on what we see in the shadow. So we're gonna do an audience participation event for a little, just in a second. And the reason that we're doing this where we're going with this is we're gonna take everything up a dimension. And we're gonna say, okay, I can't see this four-dimensional thing, but I can see a shadow. So when you take a shadow of a four-dimensional thing, you go down to three dimensions. We are the friends living in the table. The table happens to be three-dimensional. And we're gonna see the shadow, well, and make a 3D principle. So okay, so we're gonna do a audience participation event. So you should have under your, well, they used to be on your chairs. There are various things around. So there are these cubes on sticks. There are a lot of these. And there are also some of these strangely patterned spheres. And so I'll ask you to try and figure out what these are about. But here are some things to do with these. So make a square shadow. Oh, use your phone's flashlight. So where's my phone? So if you have a fancy phone, then you can use a flashlight to cast a shadow of the cube down on. So there should be some clipboards around. We also have some of these flashlights here, which we will want back at the end of the, say again. Oh, and the cubes as well. We want everything back. Don't take anything. But if you don't have a phone with a flashlight, then there should be somebody coming around with these. So you can play with these. And so there's a few questions here to try out. Can you make different shadows of different shapes? Can you see when the edges of the cube, the shadows appear to be parallel or not? And can you make the edges not crash into each other? So anybody has missing a phone flashlight? We have some flashlights coming down. Okay, so we're gonna talk about the cube a whole bunch in a second. I need to steal this back. So what were these things about? So these shadow spheres, I call them. So could we have the lights down please? You're on turn on. So okay, so here's this sphere. And what you're supposed to do is you're supposed to put the light right at the north pole of the sphere. And you get this beautiful design, which is perhaps familiar to people. Maybe. Anybody recognize what it is? Is that right? Something like, hold on. This is quite difficult to do, there we go. So if I do this, is that working? Okay, I'm too close, I can't really tell. Okay, okay, let's have the lights up again for the moment. So we'll come back to what was going on with these spheres soon enough. But let's talk about these cubes. So this picture here is called a parallel projection of the cube. So I'm projecting the cube, the shadow is making a projection of the cube onto the plane. The parallel there is talking about the light rays that are making it happen and they're parallel light rays. So, well, maybe it's coming from the sun or someplace very far away. In fact, it was coming from one of these little flashlights that I was just holding it far enough away that you couldn't tell. So, okay, what do we see? So what does our two-dimensional friend see about this cube that we're trying to tell her about? So she might say something like, okay, I see, well, I can count how many corners there are. There are eight corners, fantastic. I can see that, oh, look, I've got an edge here and another edge here and they look like they're parallel. Are they actually parallel? And you three-dimensional person would say yes, they're actually parallel. Let's see, this edge is maybe, God, it's a little hard to tell. Oh, and I think this edge is the top edge here and this edge is probably that edge there. And so, yes, those are really parallel in the three-dimensions. That's an accurate thing that you saw from this two-dimensional shadow. And then she might say, oh, and this edge here and this edge here, they actually intersect. They crash into each other, right? And he's there, ah, yeah, actually no. That's not what's happening. So let's see, what is happening? Why are these two shadow edges crashing into each other? We can see it's this edge here. I think the light is somewhere up here and this edge here, the shadow is cutting across this edge here and so in the shadow on the table, they appear to be intersecting. But that's not really happening in the real three-dimensional object. So that's not so good. That's gonna be a problem for what I wanna do later, so we'll have to try and fix this. Okay, but sure, let's move everything up a dimension and this is a 3D print using, again, parallel projection of the hypercube. So I will hand these around so these will wander all over the audience. I should mention this is a sculpture called Hypercube B by Bathsheba Grossman. Bathsheba Grossman is really a pioneer in 3D printed mathematical artwork. She's been really on the forefront of it for many years. So okay, so what do we see? So okay, again, you can count the corners and you can see there are 16 corners, great. You can also say, all right, I see this edge here and this edge here and they look like they're parallel. Are they really parallel? And your four-dimensional friend, assuming you have a four-dimensional friend, or a friendly local mathematician, would say yes, they really are parallel. That's a true fact, a true fact, as opposed to a false fact about this hypercube. And so okay, there aren't any edges that look like they're crashing into each other. But we're still not particularly good, right? So look at this edge here and you see that it's going through this face here. So this edge and this face look like they're crashing into each other and know that isn't really happening in four dimensions. So this is some problem with the way that we're projecting that it's making it look like these things intersect when in reality they don't. Okay, what else can we do? Other than a parallel projection. So, well, here's another thing you can do. This is a perspective projection. So what have I done? I've moved the light source from very far away to close enough to be in this photograph. And so, well, this doesn't really seem like a win actually, because I mean, what's happening? I've still got this crashing between edges. That's still a problem I don't like. And I've lost the parallel property, right? This edge and this edge are parallel in the cube, but they're not parallel in the shadow. So this looks like just a loss. Well, let's switch it around a little bit more. Let's put the light directly above the top square of the cube. So this was the fourth puzzle on the audience of participation. Where can you hold the light so that in the shadow, none of the edges crash into each other? And this is how you do it. You put it quite close to and directly above one of the square faces, and then none of the edges crash into each other. However, I'm still not happy. So none of the edges are crashing into each other, but there's something else which is crashing, which is intersecting, which I don't like. So what is it? So, well, where are the six square faces of this cube? So, well, first of all, where is the bottom face, the bottom square of the cube in the shadow? Well, where is it in this picture? It's in the middle, right? It's this guy here. This square here is the shadow of the square on the bottom. What about the ones on the sides? There are four squares on the sides, and we can see those as these four trapezoids. They're called trapeziums in England. I always get confused, but I've remembered this time. These are trapezoids. So, okay, so that's five of them. Where's the sixth one? Where's the top square of the cube? It's, yes, it's gone. You're looking through it to see the rest of it. Yeah, yeah, it looks a bit like a perspective picture, actually, and there's some interesting, like, why is it that this does as well? This looks like a perspective drawing. Why is that? Anyway, I won't get into that, but it's in my book. So, okay, so, yes, so the top square is over the other ones. So there are no edges that are crashing into each other. But the top square is crashing over all of everything else. And as I say later on, I'm gonna need that not to happen. Really, what I want is there to be a one-to-one map, correspondence between points on the boundary of this cube and points down in the shadow. And I'm not getting that yet, so I'm not done. But okay, this is still pretty good. And I think in some ways this is better than the parallel projection because you can sort of see everything that's going on. There aren't things crashing through each other. So, okay, let's do this one dimension up. And this is what you get. This is Hypercube A by Bathsheba Grossman. Maybe I'll hand it over here so that you guys can have things and I'll alternate or something. So, okay, so what's going on here? Same sort of thing, right? So, okay, so somehow in four dimensions, I've got this Hypercube and there's a four-dimensional light somewhere and it's casting a shadow down onto a three-dimensional table which is where we live. So we can see the shadow and this is what we see. So, as with the cube, here we are, little square in the middle, little cube in the middle. So this one here is the cubical face of the Hypercube that is closest to the table that we live in. Around the cube, there are six, okay, what do you call this thing? It's like a truncated pyramid, right? This is a frustrum, frustrum, yes. So there are six of these distorted cubes, frustrums, truncated, take a pyramid, chop off the top, that's what you get. There were six of these arranged around this cube here. Those are the same sort of thing as the four trapezoids that we had for the cube. And then there's an eighth cube. So one in the middle, six around the outside, that's seven. Then there's the eighth cube, the top cube of the Hypercube. Where is it? It's covering all seven of us. So it's really very similar. You can, when you play with that going three to two and four to three, you can start to get the sense of what's going on. Okay, how do we do better than this? How do we avoid all of the crashing? So for that, we need to go to something called stereographic projection, and I'm gonna need the lights down again. So, all right. So I've got, here's a sphere with a pattern on it. Let's put this down. And I'm actually gonna go behind the screen. I've never used a projection screen, a real projection screen before, but it means I can do this. So there I am. So you can see the shadow of this sphere, and I'm gonna hold it here, and I'm gonna move the lights slowly towards the sphere. And we'll see what happens. There's a perfect square grid hidden inside of this curvy sphere. Okay, so that sounded good. It sounded like people appreciate it. So let me do the... Thank you, thank you. So let me do it again from the front of the screen so you can see what was really going on. So here we are. So here's the flashlight. These flashlights, by the way, you may have noticed, you can take the lens off, and then you can just get this point light source, which works really well for doing these kinds of things. So there it is. If I move it and put it by the North Pole, then you get this perfect square grid that was hiding inside of the sphere. Okay, we can get the lights up again, please. Here's the slide that makes it a lot easier to see what's going on. So I've got the sphere and I've got the light, so stereographic projection is another kind of, another one of these projections. There's ways of casting shadows. So as with all the others, this is a kind of map. It's mapping from some object in three dimensions to the plane. So here's the way it works. So I've got my light source at the North Pole of the sphere. And if I draw a ray from the light to a point on the sphere down onto the plane, it hits the sphere somewhere, it hits the plane somewhere. That is stereographic projection. Is where does it hit on the sphere? That maps to where does it hit on the plane? And stereographic projection was known to the ancient Greeks. Some of the earliest maps of the world were produced in stereographic projection. It's a way of mapping the spherical world onto a flat piece of paper or parchment. So okay, what else? So notice that there's this grid of squares. I had to stop. I made a six by six grid of squares. I had to stop because as you get close to the top, the edges have to get thinner and thinner. At some point you run out of molecules or something and you have to stop because these would get thinner and thinner and thinner. So this is an infinite grid. I could just keep going, squares and squares and squares as far as you want. And then if I kept making the corresponding squares on the sphere, they will get smaller and smaller and smaller up towards the North Pole of the sphere, which is sort of infinitely far away, right? If you imagine just keep going out in this direction, infinitely many squares, you would creep up to the top, you would never quite get to the North Pole itself. So every point of the sphere other than the North Pole appears in the shadow, the North Pole itself does not. So the plane plus one point gives you the entire sphere. You may have noticed as I was holding the lights and trying to get the grid to match up that it's hard to get it to look really precise, right? Tiny movements here produce a huge difference in what the shadow looks like. So you may wonder, how did I take this photograph, right? I'm there with my lights, trying to get it in the right place. I've got my camera, I'm looking through the viewfinder while I cheated. So this flashlight is taped onto a rod, which goes up here and then there's a cross beam and a couple of clamp stands. So I could get it exactly the right height and in the right position, the hand is purely decorative. It's just there to make it look like I'm holding it. Right, yeah. Okay, so that's stereographic projection. Now, how am I going to use this to get a cube onto the plane? That's what I wanted to use this for. Because a cube is not a sphere and this map goes from spheres to the plane. So first I have to get the cube onto the sphere. So this picture on the right is a little hard to interpret. But what's going on is I've got a cube and this is a sort of roundy sphere, a beach ball sphere and the cube is right in the middle of the sphere and then I've got one of my flashlights and you can see the flashlight is projecting rays from the center of the sphere that hit the cube and then they hit the sphere and actually they hit the shell, the beach ball cube that I've drawn. And so you can see the shadow here. This is a sort of schematic picture. I've got a cube inside of a sphere and then I sort of inflate the cube to make this beach ball shaped cube and that's where I make my sort of roundy version, my spherical version of the cube. And that's what I'm going to use because then now it's on the sphere. Well, so I take this shadow and then I 3D print it and then I cast a shadow from that 3D printed shadow and this is what you get. So okay, so I've got a shadow and I mean, you could object, right? I was trying to make a better shadow of a cube. These aren't even straight lines anymore. But very important, okay, where are the six squares? So there's the one at the bottom, right? Which fits inside of, so the bottom face goes to this square here. Rather than trapezoids, we've got these kind of distorted squares here. Those are the four around the side. Where is the top face? So well, what do you have to do? You look at a light ray that goes from the top, it hits the sphere somewhere. So if I come out at a really shallow angle, then I hit the sphere sort of above this corner of the cube and then it goes out here. So the top square is outside of all of the other five. So now finally this is a one-to-one map. There's a correspondence between the points on the boundary of the cube and the points on the plane. So everything matches up apart from that point at the North Pole, we don't quite get the North Pole. We'll get everything else and it matches up beautifully. There are some other reasons to like stereographic projection, which maybe I won't go into. Let's see, there's another view. And if you do this one dimension up, this is what you get. So this is, it looks kind of similar to the previous perspective projection of the cube, except that things are sort of rounded. So I'll hand this, let's do one over here this time. Again, you've got the one in the middle and then the six rounded, frustrum-ish, rounded cubes, whatever. And then the eighth one is outside and contains the rest of the universe. So here's a 3D render of the same thing, this time with these, I've drawn in the two-dimensional surfaces of the cube as well. So unfortunately, 3D printing technology isn't there yet in terms of having translucent surfaces. It's just not possible to do yet. So transparency doesn't exist really in a usable way in 3D printing yet, when it does, of course, I will print this, but for the moment, we have to stick with computer rendering. Okay, so this is a hypercube and here you can see the surface. So this very rounded square is one of the six squares on the boundary of the top cube of the hypercube and we're inside of that top cube, or that we're inside of the shadow of that top cube. It contains everything else. Here are some more pictures of some other things you can do with stereographic projection. I won't go into all of these things. I'll just mention some other wonderful properties in terms of geometry that stereographic projection has. It preserves angles. Angles on the sphere between lines are reflected or identical, the angles between those lines on the shadow are the same as on the sphere. Also circles mapped to circles, so there's this pattern of circles up here coming from a pattern of circles on the sphere. Okay, so I went past something quite quickly a second ago. I said, let's do the same thing and here's the hypercube that you get. What did I mean by the same thing? So what was the process? Well, I took my cube or a hypercube. First thing I do is I blow it up into a beach ball cube or a beach ball hypercube. So what does that mean? I'm radially projecting it onto a sphere. So we know what the sphere is in three-dimensional space, but I haven't told you what the sphere is in four-dimensional space. Once we've got it onto the sphere, then we can do stereographic projection that's this light at the top of the sphere projects onto the table and the mathematics works pretty much exactly the same going from three to two as it does from four to three, but what is that sphere? So let's try and visualize the sphere in four-dimensional space. So okay, what is a sphere? A sphere is a set of points at some fixed distance from some central point. So in three-dimensional space, we're very familiar here's a sphere or this is a sphere with a particular pattern. So it's actually an octagonal pattern on the sphere. So I've got an equator to this sphere and I've got four triangles in the southern hemisphere and I've got four triangles in the northern hemisphere and this is another of the stereographic projection pictures so you can see what the shadows look like. So in the southern hemisphere below the equator, I've got four shadows of triangles and I hope you'll allow me to say that this is a triangle even though it's got a sort of curved edge, again distortions, whatever. I've got four triangles in the southern hemisphere but what is this thing here? Well, that is also a triangle, right? It's the shadow of this triangle up here, right? This triangle up here is the same as this triangle down here. That's clear just looking at the three-dimensional ordinary sphere. The shadows look kind of different but I wanna hopefully convince you to accept that those are really just the same thing. Look up here, it's the same thing. Down here, they're kind of weird and distorted but they're also the same thing. Okay, now we go to four-dimensions. I can't show you the four-dimensional sphere. All I can do is show you the shadow. Here's the shadow. So over here, we had the sphere is the same thing as the whole of the two-dimensional plane plus one point. The sphere in four dimensions is the same thing as the whole of the table, the whole of the three-dimensional space plus one extra point infinitely far away. And I've drawn some sort of landmarks to help you orient yourself in the same way. So over here, I had an equator that was a circle. A circle, by the way, is a sphere in two-dimensional space. It's the set of points that are at some fixed distance from the center. So the equator over here is a circle. The equator for the sphere in four dimensions is an ordinary sphere. Everything goes up a dimension. So in here, inside of this sphere, is the southern hemipersphere, southern hyperhemisphere, whatever. The southern hyperhemisphere of the sphere in four-dimensional space is inside of this thing. And the northern hyperhemisphere is everything outside. So over here, we had four triangles in the southern hemisphere. Over here, we have eight tetrahedra. So a tetrahedron is a sort of three-sided pyramid. It's got four sides, each of which is a triangle. I've actually got one of them over here. Not a good idea to step on. It lands this way with a sharp pointy thing upwards. But it's got four triangles. So you can see one of the tetrahedra here. So here's a triangle, and then here's another triangle on the back of it, and then another triangle and the fourth triangle. So this in front of us is one of the eight tetrahedra. There's four more up here. There's four more down there. So there are eight tetrahedra in the southern hyperhemisphere. And then these things around the outside are exactly the same. They look really different, but it's just because of the distortion of the projection. As with the Mercado projection, when you're making a map, it distorts the sizes and shapes of things. It's impossible not to, you have to. So these ones around the outside are the same as the ones around the inside. And if you're feeling uneasy about it, look back over here and convince yourself that the ones on the inside, the southern hemisphere, are the same as the ones on the northern hemisphere. And if you're uneasy about that, look back at the sphere and say, well, look, they're the same. Okay, so that may hopefully give some idea of what the sphere in four-dimensional space is. Okay, let's move on to look at some other things that we can draw in higher dimensions and try and understand them. So in two-dimensional space, so I'm gonna talk about these regular polytopes. So in two-dimensions, the polytopes are the polygons and three-dimensions are the polyhedra, and these things go up in all dimensions. There are versions of these regular shapes. So in two-dimensions, you've got your triangle, square, pentagon, and so on. So this is infinite sequence of these two-dimensional polygons. In three-dimensions, there are only five. There are only the five platonic solids. So there's the tetrahedron I just mentioned. Here's the cube, everybody's very familiar with. The octahedron, the dodecahedron, and the icosahedron. If you've ever played Dungeons and Dragons, the six standard dice types that you use to roll to get various random numbers to play the game are five of these plus an extra one because they wanted a 10-sided dice because I don't know why you want a 10-sided dice. Maybe it's because we have 10 fingers or something. But all of the other ones are these very nice regular shapes. So two-dimensions, you've got an infinite sequence. Three-dimensions, for some reason, there's only five. There are actually some other infinite sequences of these polytopes. But unlike the polygons, which all sit in two-dimensions, the other infinite sequences jump between dimensions. So here's a sequence that we saw already. This was me building up a hypercube, starting from a point, doubling it, making a line segment, then a square, then a cube, then this is a hypercube, or a four-dimensional hypercube. Then this carries on five-dimensional hypercube, six-dimensional hypercube. You can just continue this. The other ones here are a similar sort of process, but a little bit different. They're all the same in dimensions zero and one, and then they start diverging. So what about this one up here? So this makes the triangle and then the tetrahedron. So what do you do? So suppose we're starting with a line segment. Rather than doubling the entire line segment, I just add one extra point. I add one point, which is off of the line which contains this line segment, and I connect things up and I get a triangle. And then I add one point, which is off of the plane that contains the triangle. I connect things up, I get a tetrahedron. And then this thing is called a four-symplex, or pentachoron, or various names. This is the tetrahedron in four dimensions. And then down here, rather than adding just one point off what you've got already, you add two points either side. So here I take a line segment, add a point below and above, and I get a diamond, also known as a square. And then I do it again. I add a point in front of this square and behind, connect up, and I get an octahedron. This is something called a 16-cell and so on. So there are these four infinite families, and then these weird exceptions. So the dodecahedron, the icosahedron, those are the only other things in three dimensions, and then there's these three things which I'll come to in four dimensions, and then that's it. Above four dimensions, there's nothing else. All you get are the ones coming from these things. So somehow three and four dimensions are very special, and these things exist for very special reasons, which are kind of confusing. Here's an interesting question to ask your math teachers. Why does the dodecahedron exist? Like why is it there? I mean, all of these other things, like this is like, yeah, sure, you just keep doubling and connecting it. What is this thing doing there? What is that? Okay, what are these other things? So you could make dice, so that's why it exists, yes. So in four dimensions, as well as these other ones we've seen, there's this thing called the 24 cell, which is just weird, like there's nothing like it in any other dimension. Then there's these 120 cell, which is somehow really complicated, and the 600 cell, which is in some sense even more really complicated. Incidentally, what are these names? So 24 cell, 120 cell. So it's the same naming scheme as the three-dimensional shapes. So dodecahedron, why is it called dodecahedron? Dodeca means 12, and hedron means faces. So it's a thing that has 12 faces. In, for four-dimensional things, well, so three-dimensional things, their faces, their sides are two-dimensional shapes, pentagons, triangles. When you go up to four dimensions, the sides are three-dimensional things. So the sides of this 120 cell are dodecahedron, and it's cell like a honeycomb cell, like beehive, like you've got all of these things that are piecing together to make the boundary of this object. And there are 120 dodecahedron inside of this, and 600 tetrahedra, 600 cells inside of this. So it's the same naming scheme. Here's, well, okay, so these are renders of these two things. This is a 3D print of only half of the 120 cell. The reason being this would be enormous and really expensive to print. So this is half of it, and I've cut it off along the equator, right? Remember, the equator is a sphere. So this is the part of the 120 cell that lives in the southern hypohemisphere. So here we go, I'll go over here. You can pass that around. Okay, so I'm gonna talk about a couple of projects that we worked on. So first one, puzzling the 120 cell. This is joint work with my collaborator, Saul Schleimer, who is a mathematician at the University of Warwick. And we wanted to understand what is going on in this hugely complicated mess of a thing. So I mean, just to relost some statistics, there are 120 dodecahedral cells and there are 720 pentagonal faces that connect those cells together, and then 1200 edges and 600 vertices. So this is a big complicated thing. How do we understand this? So here's one thing, one way to try and understand how you build this thing up. I'm looking at spherical layers. So here I've got a single dodecahedron, one central dodecahedron, which is sitting in the middle of the shadow that I'm casting. And this is a schematic picture. Here's the light source of the north pole of the sphere. Here's that dodecahedron down at the south pole. Arranged around that central dodecahedron, there are 12 others. That makes sense. You've got 12 faces. You should glue 12 more dodecahedrons there. And there arranged at an angle of, well, it's a fifth of the way up towards the north pole. And then 20 more at an angle of a third of the way up, 12 more at two-fifths of the way up. And here we've got to the equator. Again, the equator is a sphere. And I've got 30 of these dodecahedra halfway between the south pole and north pole. And this pattern is mirrored in the last four layers. So 12, 20, 12, 30. And then you mirror that going backwards. And if you add up all of these things, anybody good at mental arithmetic? You do indeed get 120. I'm not gonna draw them on here because you would get very big very quickly. Okay, so that's one way to actually try and understand how do they fit together. Here's another way to do this. So I'm gonna make rings of 10 dodecahedron. So what is this ring of dodecahedron? So maybe the thing to think about is if you have a cube and you imagine, well, I've got a ring of four squares around it. So how do I construct such a ring if I'm only crawling around the outside? I start in a square and I go through a side of the square into another square and I go straight out the opposite side and I keep going. And if I keep going, eventually I will come back after going through four squares. And this is the same thing. I start in one of these dodecahedra, I go out of a face and into the next dodecahedron. I go straight through the dodecahedron out the opposite face and I keep going. And after I visit 10 dodecahedron, I come back and I've made a ring. So okay, if I do this again, starting in another dodecahedron sort of nearby or next to this ring, I get another ring which wraps around the first ring. And in fact, there are five of these rings wrapping around this central gray ring. So I've drawn one, two, three, four of them and there's sort of a death-style trench in here where the fifth ring goes. Let's put that in. And this is half of the 120 cell. So there are 10 dodecahedron each one of these rings and there are six rings here, one in the middle and five around it. And six times 10 is 60, which is half of 120. Let's make the other, so there's five more rings that wrap around this. So here's the seventh, eighth, ninth, 10th and 11th ring and we're in 12th ring. So I couldn't show you that because it would just be darkness. Okay, so Saul and I wanted to 3D print this because that's what you do. So we wanted to 3D print a ring with these five rings around it and we wanted them to be movable, right? I wanna see how they fit together. I wanna be able to pick it up and put them together again. And there's a problem when you're 3D printing something that if you want it to move, then you better, if you want two pieces to move, then they better be separated because if you print this, it'll just be a solid mass when it comes out of the printer. So, well, so I tried to arrange them in such a way that that would work and I could only actually manage to get two of these outer rings around the central one. There seemed to be no way to fit a third one in there. But that's okay, so we did it and this is what we got. So it's sort of a four-dimensional baby's rattle. So I guess we'll go, who did I do? Let's go over here this time. So you can try and put these. So this is, that's the central ring and then two of the outer rings around it and so I can sort of break it apart and put it back together again. And this works pretty well. But we still really wanted to do all five. So what do we do? So we cheated. So the reason that this is a problem with them getting glued together, why can't we just sort of print them separately? Is that they're linked together. But if I just sort of delete these things around the outside, these big expensive pieces here, to make this thing here, then I can print them separately and then put them together later. So we remove four of the biggest dodecahedra and we get this gently curving thing made out of dodecahedra, which we call a rib because it's gently curving and made out of dodecahedra. Just like ribs are. I guess they're white, anyway. Okay, so there's a rib and now I can put the second, third, fourth, fifth rib around there and then this is fine. Let's actually remove the central ring here and we get this cool shape here. This is what it looks like. We had inadvertently made a puzzle because I got these in the mail, printed them out as like, well now you need to put them together. So this is not a single object, so this is made out of five pieces and this is never particularly wise when I'm trying to attempt to solve it in real time on stage. This is not the easiest puzzle. There we go, there we go. So I will hand this around. If you break it, you have to put it back together again. No, it's fine, because I can put it together again quickly and in order to fit it in my suitcase, I need to break it apart anyway. Okay, so we made a puzzle. It just came out of all this math. Here's another way to cut something into pieces. Here's a straight piece made out of dodecahedra, which we call a spine, because spine's a straight and made out of dodecahedra. One, two, three, four, five of these ribs that fit around there, another five that fit around there and here's another puzzle. These 11 pieces fit together to make this dodecahedral symmetry object. So we ended settling on these six different kinds of piece and they make a sort of bewildering array of different things. Quick aside for people in the audience who happen to know about algebraic topology, I'll be back quick, don't worry. So the first two puzzles I showed you were all related to a sort of combinatorial version of the hop vibration. We were not expecting all this other stuff to come up, but it did anyway. Okay, back from the algebraic topology, back to my math. Here's a question for you. Two of these are photographs of the same object taken from different directions. Which two? I'll accept one guess because this is essentially impossible. Or zero guesses, zero guesses is fine. Say again, which two? The last two, no, sorry, not true. So I've given this talk, or very into this talk enough times, one person has got it, which I think is statistically what you would expect if it was completely random. So I lied, in fact, there were two pairs so they're the same as each other. This one down here, third on the bottom row is the same as this one up here at the top right. And this one down here on the bottom left is the same as this one up here, second in the top row. So how can I convince you of it? Well, I have it here. So this one here, you see there's this hole with a sort of hexagon with a hole in it. And if I just turn it like this, now it looks like this sort of square thing here. So I will hand this around so you can try and convince yourself that these are really the same. Okay, moral of the story is photographs are useless. But you can't even tell that it's the same thing seen from two different directions. So, A, buy my book, go to the website, right? What is the point of trying to do 3D content with 2D pictures? You can't, the book is 2D, but the websites, the website is 3D, yes, good point. Unfortunately yet, there is some irony that is not lost on me writing a book about how awesome 3D printing it is. Second moral is you should come up and play with the stuff afterwards. Okay, second project that I wanna talk about, and maybe I will go through this a little bit quickly, we'll see. Which is joint work with Vi Hart, who you may know from her work in YouTube, and my brother Will Segerman, who made a monkey for me. This is called more fun than our happy cube of monkeys. Here it is. So let me tell you what is going on with this sculpture. So this is a sculpture that's showing a particular kind of symmetry. So let me tell you the mathematical way to think about symmetry or the modern mathematical way to think about symmetry. So here's a symmetrical object. So a symmetry is a motion, a movement of the object that leaves it looking the same after you've done with that motion. So this thing has five symmetries. So I can rotate it by a fifth of a turn, or two fifths of a turn, or three fifths of a turn, four fifths of a turn. It turns out you also wanna include the do nothing motion, you include the trivial motion. And you can add these together, right? So you can say if I do one fifth of a turn and then I followed up by doing two fifths of a turn, the combined thing you've done is like doing three fifths of a turn. So you can add these motions together and get other motions. This is one of the ways into group theory, abstract algebra, one of the really big areas in mathematics. Here's another example, how many symmetries does this have? How many ways are there of moving this somehow so that it looks the same afterwards? Yeah? Four. Four, there are definitely four rotations, but there are other things you can do as well. There are flips, how many flips are there? How many axes, yeah? There are eight total, very good. Yes, so there are rotations by 90 degrees, 180 degrees. I include the do nothing rotation. There are also four lines of mirror symmetry. Okay, so this thing here has a particular kind of symmetry that as far as we're aware has never appeared in any object in real life ever until we made this thing, which is trying to show a particular kind of symmetry. And in order to describe what the symmetry is, I need to tell you about monkey blocks. I'm sure you can see by heart's hand here in the images here. So what is a monkey block? Here's a cube and it's patterned in a particular way. So this is the unwrapped cube. If you fold that back up, then you get this thing here. So what are the faces of this cube? So there are two monkey paw faces. There's a left paw and a right paw. There are two monkey tail faces. There's a sort of question mark tail and an unquestion mark tail. And there are two monkey face faces. This one has the tongue sticking out to the left. This one has the tongue sticking out to the right. When you fold that back up into the cube, you get this thing here. And this thing itself has, there's nothing you can do. There's no motion that leaves it looking the same. Looks like there might be, turns out there isn't. But let me sort of note, so that the right paw here, there's a sort of you sort of push through the cube with this kind of screw motion, you get to the left paw on the other side. And that's the same with all of them. There's a sort of twisting thing going on. Okay. So how can you fit these together so that they match? All right, so what do I mean by match? I mean the left paw and the right paw paw match together like this. Like so they match up with each other. So here's one thing you can do. You can make a line of these blocks. So, well, so I told you that there's a paw on the front and a paw on the back and the paw on the back matches up with this paw here. So they just all stack together and this works out. So here's the question. What are the symmetries of this infinite line of monkey blocks? So maybe you can see one thing you can do this cube here and this cube here are the same and this one here and this one here are the same. So a symmetry, a motion that leaves it looking the same. I could just move everything four cubes this way and it would look the same. Well, I don't really like that one but there's something else you can do as well. So if I go forward one click and I also rotate 90 degrees to the left that also matches up. That's another symmetry of this infinite line. So I can get rid of this moving forward to the four along by sort of curling it around into a circle of four cubes. Going four along would be the same as just rotating the whole way around and doing nothing but I've still got this move forward one click and do this sort of screw motion. I've still got that. Okay, well, that's great and all. Oh, and what are the symmetries? So all you've got left is this sort of twisting motion as you rotate around. Now, okay, so you can make a line and you can twist that around into a circle. If you try and make a wall out of these cubes then it doesn't work. So there's a cube right in the middle and it's glued onto this one and this one is glued onto this one and you would like to be able to put a fourth cube there but there's no block that works, right? The two pores on opposite sides, it doesn't fit. If only I could sort of glue them together and close it up around there, then everything would fit for how am I gonna fit three cubes around an edge? Well, I can fit three cubes around an edge. Here it is. So here's this cube in the middle and then there's the cube up on top and the cube out here. There are three cubes around that edge there. So what I can do is I can glue eight of these monkey blocks together to make the eight cells of a hypercube and all the faces match up. So what are the symmetries of the eight monkey blocks when you glue them all together and you make a hypercube? So it's kind of complicated. There are these, so there are these rings of four cubes everywhere and you can still do all these twist motions. So okay, again, possibly for the people with PhDs here. The symmetries of this decorated hypercube match up with something called the eight element quaternion group. So the quaternions are like the complex numbers so the complex numbers, you know, i is this extra number so that when you square it, you get minus one. In the quaternions, you've got three of these things, different things when you square them, you get minus one, i, j and k and then I can make one of these groups, these collections of symmetries from these eight things here. So there's one which is the do nothing symmetry, don't do anything. There's i, j and k are sort of screw motions in three different orthogonal directions. There's the negatives of those which are the reverse screw motions and there's minus one which is, send every cube to its opposite. So this thing has this crazy symmetry which doesn't exist in three dimensional space. Somehow there's not enough space in three dimensional space to contain it. And these things happen to satisfy the defining relations of the quaternions. How do you make a sculpture that has this crazy symmetry? So it's the same symmetry as the monkey block. So what do I do to make a sculpture that has that symmetry? I need to put a design inside of the cube that has the same symmetries as this monkey block. Well, okay. The monkey block itself has no symmetry or what, the do nothing symmetry. Everything has the do nothing symmetry. But I need to put something with no symmetry inside of here and it turns out there was only one choice and it's a monkey. And I'm actually reasonably serious. It had to be a monkey. MoMath has quite a few monkeys. Here's one more. Let me try and explain to you why it had to be a monkey. So the thing itself can have no symmetry because if it had extra symmetry then when I put everything together the thing I'd end up with would have too much symmetry. I don't want that. So I have to have something that has no symmetry, some design, something figurative I guess because why don't I do something figurative for the first time in my life rather than doing math stuff? So I got my brother to design a monkey. Why does that have to be a monkey? Well, whatever this figure is it's got to connect onto its neighbors in the six cubes that are around it and not look horribly contorted at the same time. Spider monkeys not only do they have six limbs if you include the head and the tail but also they're very flexible and they don't look that strange when they're trying to grab onto their neighbor's foot on the other side or whatever it is they're trying to do. So okay, I've got this design and same as usual, what do I do? I've got this design on the hyper cubes. I readily project that onto the sphere in four dimensional space. I stereographically project that down onto the three dimensional table where we live so I can 3D print it. And this is what you get. Okay, so eight monkeys, why stop there? Oh, hold on, there's more. Hyper cube, why stop there? There's the 24 cell as well. Let's make one with 24 monkeys. Let's make one with 120 monkeys. There's this website down here. If you go to monkeys.hypernome.com then you get an animated version which is showing the symmetry and you can fly around. So every few seconds, so there's this ring of four lighter colored monkeys that are rotating around here. Every few seconds, this one turns into that one that's showing the symmetry. There's also this, another ring of four darker colored monkeys going down here. So this is sort of interesting. So these monkeys actually go through infinity. So if you follow this guy down here, he gets very, very big. This is sort of monkey gob that looks down upon you and it's kind of confusing. And then he comes back down through. Okay, so this is showing one of the symmetries of the hyper cube. Here's the version for the 24 cell. So this is a 24 cell of monkeys. There's another ring of this time six lighter colored monkeys in here. And then there's three other rings. So six times four, that gives you 24. If you happen to know about duality, you may be familiar with duality in three dimensions. There's a similar thing in four dimensions. The 24 cell is self-dual which means that you can fit another 24 monkeys in between the gaps left by the first 24 monkeys for 48 monkeys. And let's go all the way to 120 monkeys. So again, there's a ring, there are rings of different colored monkeys. So there's a ring of 10 light colored monkeys. And incidentally, these are exactly the same rings of 10 as I was making puzzle pieces out of before. So there's this ring of 10 monkeys here and then there's a whole cosmology, a whole pantheon of monkeys everywhere. The 600 doesn't work in the same way because you, let's talk later. I'm almost done with my talk but there's one more thing I wanna show you but just before we go, I'm on Twitter, this is the website for the book, I'm on YouTube, get stuff from Shapeways, the files are available down a little from thing of us. Let me show you one more thing. So, and this is also an audience participation thing. If you have a smartphone with a gyroscope, then go to hypernom.com, N-O-M. And this will just start immediately. You may need to lock the orientation, the screen orientation of your phone and I am going to switch over to my iPad. Okay, we seem to be in business, this is good. So this is just the feed from my iPad, this is just what it's showing. So, okay, what is going on here? So we are once again looking at the 120 cell. Well actually, so you get to this screen and then if you press the Dodecahedron one here, then you get the 120 cell. And as I move my iPad around, you can see that I'm moving in sort of strange way through the space. So, what is going on here? So again, very briefly, I'll do some technical jargon and then we'll come back. So the set of orientations of the iPad is, you can describe with three by three matrices. It's something called SO3. As a manifold, it's the same as real projector space, RP3, which is double covered by the sphere in four dimensional space. Okay, the upshot of all of that junk is that when you move the iPad, you're navigating through the sphere in four dimensional space. And so, well, so here's something to notice. So when I get close to one of these cells, it disappears, I'm eating it. This is called hyponom, hyporesin hyposphere, nom as in nom, nom, nom. This is four dimensional Pac-Man. I'm eating through these dodecahedial cells. I should mention this is joint work with Vi Hart and Andrew Hawksley with help from Emily Eiffler and Mark Tembosh. So okay, let me show you a couple of things and then I'll quit. So here's an interesting thing to notice. So let me just reset this. So we're back at the start. And so I mentioned the double cover. So what does that mean? So if I rotate this thing around 180 degrees, 360 degrees, it's a good thing this cable is long. We'll get it difficult otherwise. I'm not back where I started. My iPad is in the same position but where you are in the sphere in four dimensional space is not the same. But if I go another 180 degrees, 360 degrees, then I do get back where I started. You can see I've tunneled, I've made a circle, I've tunneled out a circle from inside of all of these cells. Okay, one last thing. This is a game. The game is to hit all of, is to eat all of the cells as quickly as possible. You're being timed. If you hit all of the cells, it tells you how long you took. And the last thing I'll mention is this was originally developed for virtual reality headsets. So really you should be playing this with a tape to your face. And you have to somehow try and eat everything. Okay, I'll stop there. Thank you very much. All right, we have time for a couple of questions. If anybody has any questions, they can raise their hand and I'll bring the microphone to you. There's one over there. Is there a publicly accessible way to use Hypernome with a VR headset? Yeah, so there may be some technical issues that I'm not particularly familiar with. Andrea Hawksley is the person who knows about this. But you should just be able to go to the website with the right web browser, with a headset plugged in, you just go to the website like you do for the phone and it's there. So yeah, question back there? Thank you, fascinating talk. So just wondering the steps that you had to do to get the projection of the four-dimension. These two, okay, so first you project out. Right, first we really project onto the sphere. So this doesn't work for like really complicated three-dimensional, four-dimensional things. And the way you do it is through some. Really what I'm doing is I'm showing you the boundary of a hypercube or a 120 cell. But that is really a three-dimensional thing. It's sitting on the surface of a sphere in four dimensions. Then so I get it out onto the sphere and then I stereographically project down onto the table, which if you're coming from four dimensions is three-dimensional space, which is where we are. Okay, thank you. More questions? So you said this is a representation of SO3, right? Well, the rotations of the iPad is SO3. Is SO3? Yeah, okay. Ways that you can rotate things, that's, yeah. So is there any way to then visualize the Lie algebra associated with it? Beyond my pay grade, maybe? I don't know, let's talk. You mentioned we live in three-dimensional space and I agree. Let's say there's, as you know, there's this view that, let's say there's a fourth dimension, which is time. And so has anyone tried to work with three-space plus one-time dimension to get a glimpse of a five-dimensional object? Oh, this is, yeah, well, so, I mean, I've been showing you one way to try and see things in higher dimensions. Using animation is a very popular way to do things. So you can imagine taking slices through a thing, but then which slice you're seeing moves along. So yeah, I actually haven't, I can't think off the top of my head of somebody trying to do five dimensions by doing this twice. But certainly trying to see four dimensions by moving slices through time, absolutely. Yeah, okay, thank you. And Henry, we'll stick around a little bit if you have more questions. So let's give Henry Sigerman one last, thank you.