 Okay, so I'd like to introduce Elisabeth Matsumoto, or Sabete, as she calls herself. So Sabete is an assistant professor in the Department of Physics at Georgia Tech. Her group focuses on the geometry and topology of soft materials and the effects of elasticity on the emergent structural and mechanical properties of complex systems. So that's one side of her scientific life, and the other side is working in mathematical art, where she's made a big splash, and she's already talked about some of her textile design to the teacher group. Otherwise, she works on virtual reality, 3D printing, jewelry designs, and various other types of textile designs. So today, her talk, so as you saw, there are a few technical issues, so it's not quite the talk that she was hoping to give, but it'll be great nonetheless. So please welcome Sabete Matsumoto. All right, thank you. Can you guys hear me in the back? Awesome. Thanks. Okay, so I guess you saw that there were a bunch of technical difficulties. So basically I have the highest end virtual reality system on the market, which means that their software has not kept up with their hardware, and I'm getting a problem where my computer is running right, but the cord is incapable of sending enough data to the headset for it to run properly. So it's kind of broken and kind of working, so the computers find the headset in there is broken. So normally what I do when I give this talk is I have a ton of volunteers come up from the audience and I explain how you guys sort of use your bodies to navigate around in hyperbolic space, but since there's nothing to see in the headset, I'm going to have Henry do it blind, so the headset is still detecting his position and all of the phenomena you will be able to see by looking at the screen. He however might bash into a chalkboard or fall off the stage. Hopefully that won't happen, but yeah, just be prepared. And the downside of that is that I also usually leave this up for everyone to come play with for a while afterwards, and unfortunately that means that it's not going to work right now. I will play around with it when we get after this and see if I can download a new version of the beta. So this only works with the beta version of the software, which is kind of silly because this has been on the market since last December or something like that, and they still haven't gotten the rack together. So bear with me for that. So I guess I'm going to be talking to you a bit about hyperbolic space today. So I guess show of hands how many of you are familiar with hyperbolic space. So this is kind of a more public lecture to get people who are not super familiar with it on board. So I'm going to talk a bit about hyperbolic space first, and then we're going to switch to the VR setup so that we can kind of show consequences of what happens when you move and walk around in a curved space. I guess I should get going because we're starting a little bit late. So this is not working, so can you grab me? So I'm going to start by telling you a bit about curvature, and I'm going to start from the point of view of thinking about clothing. So how many of you have ever made some art of clothing, sewed anything, fixed anything, put a button on a shirt? Okay. Okay. So I got a big show of hands for button and some people around who have maybe made some clothes or maybe fixed something. Okay. Thank you. Okay. So we're going to start by talking a bit about how you make a shirt. So the thing about clothing that's hard is that every human body is curved. Like, I don't care how skinny you are or how bony you are or whatever. You still have curves, and fabric is flat. Fabric is, if you're not dealing with like a knitted fabric, it is not stretchy. And you have to go from these flat sheets to something that you can wrap around a curved body. And so we do this using seams. So starting with this is a dress shirt, which I guess being at a mass conference in Hot Park City, probably not very many people in the audience are wearing this kind of shirt. But probably most of us have something that looks at least equivalent in our closet somewhere. So if we wanted to make this, I mean, you could imagine cutting out, you know, two pieces like this and sewing them together. But it turns out that the actual pattern for making a dress shirt looks like this. So each of these, there's something like 57 different pieces of fabric and 78 seams that go into just a simple dress shirt. And you'll notice, if you look really closely, there are almost no straight lines in this entire pattern, except for the places where it says fold. And that's just because we've got bilateral symmetry, there's no point in cutting a left side and cutting a right side that are mirror images of one another. You just fold and cut one side. And so the reason that this has all sorts of interesting curves and different shapes in the seams is to accommodate the curvature of the human body. And not only curvature, like you could imagine having like a shirt where you can't do anything but stand like this, but you do need to be able to move around. You need to be able to articulate your shoulders, bend your elbows, things like that. So all of these seams are designed to accommodate that part of the body. So this is sort of breaking it down into the couple of fundamentals of things that people in sewing do. So if you'd like to have positive curvature, you make a dart. So a dart, basically, you take a sheet of fabric, you fold it in half, and so a little diagonal seam through the fold. And then when you unfold it, you get a cone. So this would be a singular point of positive curvature. And these are used at places like busts and wastes. So this is basically taking a plane and trying to remove some area from it to give it some curvature. On the other side, you can imagine adding curvature by taking a wedge and inserting it into the fabric. So I guess I have something of a tulip skirt on, so that's sort of the type of thing that would have extra area around the hem of the skirt rather than at the waist of the skirt. And so this is a point here where you have a singular point of negative Gaussian curvature. So here I've taken a plane and I've added some area to it. So this is how in sewing I can accommodate for negative Gaussian curvature. So here's some nice fabric that has hexagonal symmetry. So what I can do is I can cut out a sixth of it and sew it together into a cone. And you'll notice that not only do I have a conical shape, I've got a bit of positive curvature, but because I removed exactly one sixth of it, I have a point here that has five-fold rotational symmetry. So this is perfectly five-fold rotational symmetry here. If I take this wedge now and instead add it into another piece of fabric, I end up with a point of seven-fold symmetry, except the fabric doesn't lay in a way that has this symmetry unless you perfectly arrange it so that it does. So this is how one might start building something like a hyperbolic plane. So you've got points that have negative curvature and you want to have this negative curvature everywhere. So my friends, Andrea Shuey, who is a clothing designer, and Robin Selinger, who is a theoretical physicist, decided that they would take this idea, I gave a sort of similar talk in Santa Barbara, I don't know, three or four years ago, and they decided they would turn this into a dress. So they wanted to come up with a way of making a dress that would sort of fit any woman's body, providing you had sort of a parametrization for it, and not just skinny women or not, you know, just sort of like the clothing model type women, but any woman with any shape. And the idea they had was that they would take regular pentagons, hexagons, and heptagons, and sew them together in a patchwork so that the pentagons are places where you have positive curvature and the heptagons are places you have negative curvature. And they turned this into the bodice of a dress. Speaking of dresses, this is a beautiful couture French wedding dress. And these are some human intestines. So what do you guys think these have in common? Folds? Anything else? Big surface? No to cylinders, but yes to everything else. So these do have a lot of folds and ruffles, and both of them, it turns out, have negative curvature. And not only do they have negative curvature, but the mechanism by which, the physical mechanism by which they have negative curvature is the same in both situations. Let's take the dress to start with. So in this case, imagine you've got a piece of tool, which is the stretchy mesh stuff in the skirt, and you want to sew it to what's called boning, which is just a stiff layer. So the stiff layer, you can't extend, but the tool is pretty stretchy. So you stretch the tool as far as it possibly goes, and you sew it to the boning, and then you let it relax. And when it relaxes, you end up with this roughly periodic structure down here. So for human intestines, so it turns out, many biologists have said that gastrulation is the most important step of life. And gastrulation is the point at which you go from a three ball to a solid torus, so this is when you're an embryo. So you're basically getting a digestive tract. So you need a way of separating inside and outside. You need a way of digesting food. So this is the topological change that allows you to become an animal. Basically, you can ingest and excrete food. And so when once you've undergone gastrulation, you have a neural tube that starts to form, and that's what's going to become your spinal cord. And sort of nervous system precursor. And there's a membrane that goes between your gut tube, which is going to become your intestines, and your neural tube. And what happens as you grow is that your gut tube is stiffer, and it also grows a lot faster than the membrane connecting it to your neural tube. And so what happens is that exactly this is going on. So your gut tube acts like the boning. It grows faster, and it's stiffer. And the membrane that connects it to the neural tube acts as the tool, which is flexible, and you end up having an elastic instability, which gives you the sort of periodic ruffling pattern. And that's why this person's intestines and my intestines and your intestines all basically have the same type of ruffles. It's not as if someone took an 18 foot long tube and just started shoving it randomly into our stomachs. We actually have sort of a physical reason why our insides look the way they do. So we see these sorts of ideas of curvature all throughout nature. I think I'd better hurry up. So we see this at the microscopic level. This is inside the retina of a tree shrew. This is the gyroid minimal surface. This is something that is used as a UV filter. You see it in soap films. So this is again a gyroid but done at the centimeter scale. And then there's a question as to is the universe curved or not curved? It seems like it's flat, but we only have data that goes out as far as the visible universe, so we're not completely sure yet. We also see this in nature. So here, a nudibranch is using the fact that it has negative curves, negatively curved mantle to swim. So it's sending traveling waves down its sides and that propels it forwards. These coral and kale use the increased surface area as a way to increase their exposure to nutrients. And I just put the Kella lily in there because it's my favorite flower. So we can look at this again from a slightly more rigorous mathematical point of view. So these are, I guess, a bunch of platonic solids. And these are all of the platonic solids that you get made exclusively from triangles. So we've got a tetrahedron, which is this one, and octahedron and an icosahedron. So the tetrahedron has three triangles that meet at every vertex. The octahedron has four and the icosahedron has five. So we're going to use Schlafly symbols to describe these. So the first number tells you the number of sides in your regular polygon. So three is a triangle. And then the second number tells you how many of these meet around every vertex. So three-three is triangles that meet three around every vertex. And we're going to keep this in mind because we're going to come back to it later. So if we wanted to go to the next number, three-six, this is, I guess, a sculpture we have here. This is made by Henry and collaborators. So this is a tiling of the plane. This is the sort of penny-packing tiling of the plane. So there are triangles here that meet six around every vertex. And we can just continue this on and on. So we can do three-seven is triangles that meet seven around every vertex. So this is like taking that fabric I had made before that had one bit of seven-fold rotational symmetry, and now converting every single hexagon in that fabric to a heptagon doing exactly the same thing I did before. And that would create this fabric here. And you can keep going. This is three-eight. And if you kept going, this would no longer fit into the 3D printer. So this is about as far as could be done. So if we imagine looking at these as tiling of some sort of surface, the platonic solids are tiling of a sphere. So you can imagine kind of taking one and putting a beach ball inside of it and inflating it out. So this would be an octahedron that is sort of pushed onto the surface of a sphere. So we've got triangles that meet four around every vertex here. We have a plane would be the three-six. And then what is this? This is the three-seven. And this is something we're going to call the hyperbolic plane. And there are a lot of different models that people use to describe the hyperbolic plane. So this one here is the Poincare disc model. This is probably the most common of the models that you'll see. And so the idea with this is that geodesics, so straight lines, are arcs of circles that intersect the boundary, which here is infinity at right angles. And this is a conformal map, so circles stay circles and angles are preserved throughout. But you'll notice that this motif in the center, this is sort of close to a regular heptagon. As you move out, you get shapes that are more and more sort of squished. And this is something that we're all familiar with looking at things like the Mercator projection of a map. Is anytime you go from a curved surface onto a flat plane, you need to have some sort of distortion. So you can imagine having distortions that are based on size, which is what we're seeing here. You can have angular distortions, you can have all sorts of distortions, but you can't have a map that preserves area and angle when you make this projection. So this is the Poincare disc model. As I mentioned before, there's several other models that are quite useful, people like. So this is the Klein model. This is nice because geodesics are actually Euclidean straight lines in this map. But you'll notice out here, these circles are supposed to be circles and they've gotten squashed, so this isn't conformal. So this is a way we're trading off here. So we've got nice straight lines, but now we don't have circles, let's stay circles. So we're always going to play some sort of game when we want to look at hyperbolic space as a map onto a 2D plane. There's also the upper half plane model. This is a pretty nice model again because you have circles that intersect the boundary at right angles and these are your geodesics. So this is a model that's particularly easy to construct using a roller and compass. And it turns out that these are all related to one another. So this is a sculpture by Henry Segerman and Saul Schleimer. And the idea here is that you're using light as a way of projecting from this hemisphere model onto some surface. So when it is at the north pole of this, you get the Poincare disc model. When it's infinitely far overhead, you get the Klein model. And when you put it at the equator, you get the Hemisphere model. So these are all related to one another. But there's something that's a little bit, so sorry, this is something that you guys are probably familiar with. So Escher wanted a way of seeing infinity. And he did this after a discourse with Coxeter basically decided to use the hyperbolic plane as a way of visualizing infinity. And he did this using the Poincare disc model. This is circle limit four or angels and demons. So the idea is you've got three angels that are intermeshed with three demons and their wingtips here. The six wingtips are all live on a circle. And this is true for every set of three angels and three demons in this motif. So you're supposed to imagine that as you go out further and further, like this little demon here, who's, I guess, his left wing is like twice the size of his right wing. You're supposed to imagine that he's the same size and shape as this demon in the middle here. And so that's pretty hard for your brain to see, to be honest. So we're going to take a look at this. I'm going to sort of fade out the angels and demons if my clicker works. Oh, I think we're out of. And I'm going to leave behind just the circles that the wingtips trace out. So you can kind of see that these are hexagons and hexagons that meet one another at points in four at every vertex. So this is basically going to be the model of the first bit, the first simulation we are going to walk through. But I'm going to draw the dual of this. So instead of hexagons that meet four around every vertex, we're going to do squares that meet six around every vertex. So this is this is sort of a top down view of the is like a floor plan per set, you could say, of the first simulation we're going to walk through. So all of these the all of these models are kind of views from from the outside of hyperbolic space. There are sort of ways that we've taken it and put it into our three so that we as natural flatlanders can see this. But what we'd really like to do is see what it would be like to actually live inside hyperbolic space. And the the basic idea that we're going to use is that in general relativity in our real world photons follow along geodesics. And so that's the mantra we're going to be using to visualize everything here. So we like every light ray to follow a geodesic in this curved space. And that's how we're going to visualize everything. So there are four elements we need to draw. All right, can you guys hear me now? OK, fantastic. OK, so there's four things we need in order to draw a visual world. We need a model of the space so ways to associate points in that space with a number in my computer. We need a way to draw those points on the screen. And this is going to be following the idea that light rays follow geodesics. We need a way to move around in that space. And so the reason we have this super fancy VR headset is because it's actually tracking your position as you walk around. So we actually are able to to move around via isometries in that space. And we also need a set of landmarks so that you can navigate through the space. Like, I could put you in there and have it be completely black and say, oh, well, I coded it up so that everything is hyperbolic in there. And you just look at me and be like, yes, so. So we're going to use a tiling of this space to give you some points to navigate by. So this first one we call H2 cross E. So it's the hyperbolic plane that's been extruded into the Z direction. And so basically, if we want to see along geodesics, you can imagine having your viewer at the origin here. And they want to look out into that space. They follow a geodesic until it hits some object. And then you render that object on the screen. You don't need to go back and render everything else because it's being blocked by something in here. So OK, so this is the first demo. So I'm going to borrow you, Henry, if that's OK. So I'm going to switch. All right, can everybody hear me now? Thank you. OK, so I'm going to actually, there's a twisty thing on the back. I can't see anything anyways. OK, so I'm going to sort of put you where I'd like you to stand. You just have to take my word for it. So OK, so I'm going to reset his view so that this will put his head in the center of one of these tiles. So these tiles are all cubes and they're cubes that meet six around every horizontal edge and four around every horizontal edge. So what I'm going to have Henry do is walk around the vertical edge that is right here. And he's going to have to walk around through six different rooms to get back to where he started. OK, so he's in kind of a dark green room. And he's looking at a red room. So he's going to walk into the red room. I will tell you when to stop. Another step and stop. OK, then turn 90 degrees to your left. OK, and walk forward. And I'm going to move these cords out of your way. OK, this would work if I was a bit taller. Keep walking a little bit more and stop. OK, so now he's in sort of a pinkish salmon colored room. He turns 90 degrees to the left again. He's going to walk into this sort of multi-colored and stop sort of ice cream colored room and turn 90 degrees to the left again. Walk forward into this, I guess the polynomial. Turn a little bit more. Yeah, OK, a little bit more. OK, there you go. So walk into the pale green room. And so, OK, stop. Turn 90 degrees to your left. Keep going. Keep going. There you go. Perfect. And so now he's stopped. Can I describe it? You know the drill. They don't. So OK, so now he's, you went too fast. You're supposed to have been. I needed to tell them where you were back where you started. OK, so now he's, well, I guess you should have been back where you started now. Yes. Yes, he should be turning by 90 degrees really each time. However, he kept walking. And so he, this is really a right-angled hexagon. This is a right-angled hexagon. He walked too far, so now he has to turn more than 90 degrees. So that's why that's throwing off this demo. If he could actually see what was going on, he would have stopped and not walked too far. So OK, so now you're pretty much square on. Turn a tiny bit more. OK, perfect. Now you're square on. OK, so this should have been where he started had he not walked a little bit too far. OK, so now walk straight forward into the blue room. Sorry, this is one past where he should have been when he started. OK, take your veering towards this way. You need to. OK, now you've totally lost. The headset has lost contact with the world. Do you want to go to the next demo or the animation? Just OK, just mime it for one sec. So he's going to turn 90 degrees. Oh, perfect. Wait, 90 degrees. And then just walk into the next. OK, OK, so basically the idea is that he should have started facing you guys here. He should have made six right-angled turns in R3. And when he's done making all of these turns looking back where he started, he should be facing this direction, facing away from me. Yes, question. He can. It looks pretty much like Euclidean space going up and down. So this is showing you that distances do fall off the way that they should because it is Euclidean vertically. It's hyperbolic in the x and y direction. OK, so I do actually have a video to show you what he should have done and seen. You can take this off. I'm going to skip the perspective one. Thank you, Henry. OK. OK, so what he was supposed to have seen. So this is a top-down view of what Henry would look like if he were a little.