 your phone, it may work, but this is one of the problems we're working with sort of new technology. Very recently, the latest versions of Apple iOS for security reasons have sort of broken the ability to detect the orientation of the phone, which is what you need to use to look around. So it may or may not work for you. Anyway, so that's one simulation. And so we're hoping to do the other first in geometries. This is a very recent project that I've been working on with David Bachman and Saul Schleimer, making these really beautiful fractal patterns. And so these are called Canon Thurston Maps, or these are sort of closely related to Canon Thurston Maps. And again, there's a simulation, so let's go and see that simulation. You can go to, if you turn your head sideways, you can see henrysag.github.io slash canon-thurston. So let's go and take a look at that. So this is, you can drive around this. In principle, you should be able to do this in VR, but I don't think we have actually tried yet. And you can, actually, let's do this. Let's increase the speed. There's all of these controls and things you can do, and then you can just sort of zoom into this thing forever and it just keeps going and going and going. Really, what's happening is that you're traveling through a three-dimensional hyperbolic manifold, and, well, let me tell you what's going on. So, it's sort of mesmerized. Well, and there's all of these different triangulations you can do, and they all look different. Let's change the colors to neon, and yeah, I mean, there's just endless fractals. One of the interesting things about this, unlike the Mandelbrot fractal, where you zoom into it and you keep sort of seeing different things, here you keep sort of seeing the same thing, and it's because you're looping back and forth through the same manifold lots and lots of different times in different ways. So, let me try and explain a little bit about what's going on here. So, this is, you can also get the software to do this, and this is showing you the view of a bunch of tetrahedra. So, maybe you can see, so there's a big triangle in front of us, and then that's one face of a tetrahedron, and you can see the other three edges there. And so, and there's lots of tetrahedra everywhere, in fact. So, this is a big tiling of H3 by these tetrahedra. And this is, it's sort of hard to believe, but it's possible to take the component of the figure eight knot, so that the worm drilled out through the ball manifold from before, and you can triangulate it with two tetrahedra. So, there's some way to cut that manifold up into two tetrahedra, only two. And then, you can put hyperbolic structure on them, and this is one of the two tetrahedra, and all of its neighbors are copies of the other tetrahedron. And so on and so forth, as you look along geodesics into this manifold, you're going through the same two tetrahedra in lots of different combinations, wrapping around and around and around. And so, this is colored by distance from the viewer, so just, you know, edges that are close to you are brighter than those that are further away. Now, let me change the coloring. So, now how is this colored? So, in addition to these tetrahedra, we need a surface that's made up of faces of the tetrahedra. So, you take two triangles, and they sort of glue together to make a nice surface inside of this three-dimensional manifold. And what we do is we keep track as our sort of, we look outwards, because light rays leave your eye and go forwards, rather than leave the world and go into your eyes, and that's how it works. Anyway, that's how it works here. A light ray leaves your eye, and you count how many times it crosses this surface that we've got, and we count it with sine. So, if you cross from the positive side, then you get plus one, and then if you count again, if you cross again another one, which has, from the positive side, you get plus two and so on, and then you can go plus and minus. And then we just color it with brightness by that plus minus, by that count that you get when you go some depth into it. And if I remove the edges, that's what you get. And so, this is a new algorithm to draw these Canon Thurston maps, those sort of older algorithms, which didn't sort of give you the whole picture. They were just sort of drawing the boundary of this surface made out of triangles. And with this new sort of pixel-based version, with this ray tracing, you get into all of these, these are sort of cusps, these are the corners of the trying of the tetrahedra, you get this very sort of bright contrast, and you get into all of these squiggles. Okay, so that's the new project, and we're hoping to go more into that in the future. So, let me tell you a little bit about spherical video. So, this is a frame from a spherical camera, or in the business world, which means in the real world, these are called 360 cameras, which is obviously wrong. It gives the sense that you can look all around you, but it's really capturing a sphere of data. And so, the way that they work is you have this little gadget and it has a fisheye lens on either side that allows you to see all around. So, this is a standing around the camera in at Stanford on a rainy day. Actually, let me play this as a video. So, I haven't done any math yet. Let me show you, so that's the sort of unwrapped equirectangular view, so that's, oh yeah, so what is going on here around this way is around the equator, and then up and down is going towards the North Pole and towards the South Pole. So, this is what it looks like in the way that it's supposed to be viewed. So, you know, it's like you're here and you're standing in the quad and there are three people juggling around you and you don't feel very safe. But, because these cameras aren't cheap, but we're very good jugglers, so we knew what we were doing. So anyway, yes, we stopped. So, we do a bunch of that. Okay, so this was all just what you see in the camera as usual, but then this is not. So, what is going on here? So, I've sort of doubled up the world. There's two copies of me on either side. Well, there's two copies of everything. The tripod, you can't see all of the legs, but it now has six legs because it's not a tripod anymore. Oh, you can see that the number there is the same as the number there, so it's been unwrapped. So, let me show you this over here in the unwrapped view. So, there we go. So, what's going on here? Get the fully work so you can hear what's going on as well. So, what's going on here is, you have all this data on a sphere, so what did I do to get this effect? First, I stereotypically projected it to the plane, and then I applied C squared, thinking of it as the complex plane, and then I wrapped it back up onto the sphere. And so, this is doing a complex transformation into this sphere of data, and you get this effect. So, if you show up to juggling, and you really want to do a pattern that involves six people, but only three people showed up, if this pattern has enough symmetry, you should choose. Let's see, what else do I have? Ah, yes, so I'll let this play for a minute. A past version of me. So, as I was saying, my name is Henry Segerman. This is a spherical droster video. So, you're sort of slowly zooming this way, or rather the frame here is coming over you this way, and over there there's a sort of weird pedal portal in the middle of my apartment. This is the future, so there's future versions of me over there, and this is the past. So, I'll hand you off to a past version of me to explain again what's going on. A past version of me. So, as I was saying, my name is Henry Segerman. This is a spherical droster video. So, slowly zooming this way, or rather the... So, how did I make this? Well, so there's a little bit of sort of camera trickery. I don't really have a portal. This is really a window. But, you know, so I could cut that out of the image and replace it with a translated version of the same, well, actually not the same image, but the image 30 seconds in the past. And so, how do you move it? So, the movement is using Mobius transformations, if people are familiar with those transformations of the Riemann sphere, or you don't even need to know about Mobius transformations, take the sphere projected to the plane, using stereographic projection, and then just scale the plane, multiply every point by two, wrap it back up onto the sphere, and the effect is that you've got closer to the North Pole, sorry, closer to the South Pole and further away from the North Pole. So, it looks like you've moved that direction, so this is a bunch of trickery. Let's see. And now, this was a collaboration with the Fine Heart. I'm gonna let this play for a minute or two and then I'll say something about it. So, the voices who are all singing at the same time, and they're offset in time from each other by 20 seconds. In this case, all three voices are the same. They're all set in time by 20 seconds, but also the geometry of the world is offset in time by 20 seconds each time. Like the Z squared, I always said the Z cubed, so you have three copies of everybody, three copies of Vi, but there's only two copies of me. Where is it? There's only one copy of the sheet music. So, the sheet music actually moves around the circle. This Vi is gonna pick it up a little bit after that by putting it down and pass it through and put it on this stand. There's only two copies of the hammer that's used to strike the triangle. So, somehow it's the various parts of moving around the three scenes with a whole bunch of moving trickery. There's a wonderful thing itself, which Vi put an incredible amount of work into. So, you should check it out for all of the gory details about sustained pedals and everything. Okay, so, and if you have access to a VR headset, there's a version for the VR headset where you're actually in there looking around a little and it's a better experience. You should do it. So, right, so I mentioned at the start, there's this illustrating mathematics semester at ISAM. So, let me advertise just a little bit what this is about. So, the idea is to bring together mathematicians and makers and artists who are interested in doing mathematical things and illustrating mathematical ideas with computational tools. So, we're here for the whole semester. We wanna produce some really good illustrations of mathematical things, not just for outreach purposes and sort of pedagogical purposes, but also drawing good pictures sometimes means that you get a proof, right? That you wouldn't or you get a conjecture that you wouldn't have got otherwise. This is basically how I do all of my papers is I just try and draw good pictures and figure things out. But yeah, we wanna communicate research, mathematics and spark collaborations. If you can't make it over for the entire semester, there are three week-long workshops. There's illustrating geometry and topology, illustrating number theory in algebra and illustrating dynamics and probability. And so, we're bringing lots of people in for those shorter workshops. So, well, there's lots of information here that maybe I won't bore you with. So, you can go on the website and look it up. There's funding for travel and accommodation that's available. You can apply on the ICEM website. So, that's pretty much my talk. I'm on Twitter. This is my website. It has links to all of the things. This is my book, which is, well, it's got lots of fun math in it and it's all illustrated with 3D printing. And come to my 3D printing workshop this evening. Thanks. Thanks very much, Henry. So, are there some time for some questions? Oh, like the monster group. Yeah. So, you notice I never did anything more than four dimensions. Yeah, it gets really hard. Yeah, so, right. So, I mean, in four dimensions, there's sort of maybe two basic strategies for reducing dimension. You either slice through something and you get a lower dimensional thing or you project. And I don't know of many good ways to do anything beyond, I mean, even five dimensional things. It seems just really, really hard. So, I mean, yeah, there's sort of a broader point that is like, we've got all this visual machinery in our heads for seeing things in the world and we want to try and apply that to tell us about our mathematics. But yeah, I don't know that anybody has really any good way to see, to get much. I mean, sometimes you see the projection of the E8 group and it's just this mess of lines and does anybody get anything out of that picture? I mean, it's nice that it exists, but I just don't know. Yeah, so, yeah, which is, is that an illustration of the thing? I mean, it is, but it's a two dimensional, I don't know. So, yeah, it's just really hard. I wish I knew I had a better answer for you. I sometimes think, actually, that sometimes we have this sort of, oh yeah, there's this interesting stuff that happens in three and four dimensions and then in some areas of mathematics sort of, everything gets too simple in five and above. And I suspect that we just don't know what the right questions are to ask because we're not living in six dimensions. And mathematicians who did would know what the right questions are to make it interesting. Other questions? Okay, well, if not, let's thank Henry again.