 guy that I sketched with using sine and cosine and trig functions and stuff like that in Mathematica. He's obviously more detailed than this, but you can just go with me. So this is illustrating what he would have done. So he would have, actually, this is the opposite direction that he walked, but again, bear with me. So he'd go forward into the blue room, turn 90 degrees to the left, and in real life, he should have turned 90 degrees to the left. Then he'd walk into the pale green room, turn 90 degrees to the left, walked into the rainbow sherbet ice cream colored room, turned 90 degrees to the left, walked into the sort of salmon colored room, turned 90 degrees to the left. This is where he should have gotten back to where he started, but he didn't. So he has to finish his tour. He walks into the dark red room, turns 90 degrees to the left, walks into the dark green room, turns 90 degrees to the left. And now he's back where he started. And in real life, he did, in fact, make six right angles up here. So what's going on is a phenomenon that's called holinomy. So this is something that we're familiar with if we think about the surface of a sphere. So imagine I'm standing at the equator and my left hand is pointing along the, I guess, red arrow to the east. And my right arm is pointing along the blue arrow to the north. And instead of actually turning, I'm going to always face in the same direction and I'm going to walk in a path around like this. So first I go around the equator a quarter of the wave and up to the north pole and back down to the equator. And when I've completed this circuit and I get back to the place I started, I'm going to find that my arms have now rotated. So my left arm is now pointing north and my right arm is now pointing towards the west. So this phenomenon called holinomy is something that is sort of a signature of living in curved space is when you go around a closed loop, you come back as having been rotated. And this is something if you take differential geometry like towards the end of it, you work out Christoffel symbols and it's really nasty and you find this out using like rank three tensors and it's really not fun. But this is a really easy way to figure this out. So this is what Henry would have done had he not been turning. So imagine he's facing along the blue arrow and his right arm is pointing along the red arrow. So first he walks forward, then to the left, then backwards, then to the right, then forwards, then to the left. And you can see that when he comes back to where he started, he's rotated by 180 degrees. So this is sort of equivalent to him starting facing you and ended up facing 180 degrees from you. And so this is a phenomenon that we're going to explore a bit more in our other views of hyperbolic space. So what we're going to do is look at fully 3D hyperbolic space. And so this is going to be made up of cubes again, but instead of cubes meeting six around every vertical edge and four around every horizontal edge, they're going to meet six around every edge. And we can expand our version of Schlafly symbols to take into account three-dimensional tilings as well. So we start out with this tiling of the plan. This is 4, 4. We have squares that meet four around every vertex. When we move up a dimension, the first two numbers stay the same. So 4, 3 would be what shape cube? So these are squares that meet three around every vertex. So you can look at sort of the corner of the room to see that. And then the next number here is how many of those meet around every edge. So in the Minecraft tiling of space, you have four cubes that meet around every edge. In hyperbolic space, keep forgetting this doesn't work. In hyperbolic space, we can play around with these numbers again. So this is what would happen if you had cubes that meet five around every edge. And this is in the Poincaré ball model. And we could do this again for cubes that meet six around every edge. So this is a cube inside the Poincaré ball model. And we're going to start adding cubes around each of these edges. We're going to complete at the rest of the five around them. And we get this. And then we're going to add five around every edge there. And we get this. And this just doesn't make any sense to me. I mean, I do have a picture of something from this type of hyperbolic space tattooed on my shoulder, but I still can't make any sense of what's going on here. And so this makes a lot of sense as to why we want to move to virtual reality. So we don't need to look at this sort of mess of sort of mess of sort of squished cubes. And so this is just a model of what you're actually going to see in this space. This was designed by Royce Nelson. So this is again these truncated cubes. You can imagine if you put your eye right at the origin here and look radially outward. This is kind of like what you would see if you were actually in the virtual reality, but you don't get to walk around on this. Okay, so Henry, hopefully the headset works. We might have to keep you closer to the corner over here. I think it was just losing tracking. Okay, so before, I guess before you put this on, bear with me and the rest of the audience while I teach you the moves to the hall and only dance. I know you already know them, but it's way more fun for them if you pretend you don't. So okay, so take off the headset because otherwise they're going to know what happens in advance. Okay, so this is for all of you dance aficionados in the audience. This is like bad middle school dancing. So you're going to keep your feet planted and move your head around night at the Roxbury style. Okay, only the old people in the audience got that, sorry. So okay, so the first one is going to be the hula hoop. So you're going to plant your feet and you're going to make big circles with your head. Okay, you think you got that? Okay, and the next one is the TV set. So again, feet planted, hands up. Okay, you're going to go over to the left, down over to the right, up and keep doing this. Okay, and then the last one is the bicycle. So we're going to change up our stance a little bit, you know, a little bit fighting style. So now we're going to go forward, down, back, up, forward, down, back, up. Okay, I think you got those. Okay, so let's put you in the headset now. And okay, so now I'm going to reset you so that you're facing forward. And I'm going to switch everything over so they can see it. Okay, can you, are you good? Okay, can you turn like 45 degrees to either side? Keep going. Okay, perfect. Okay, so I'm going to change his view so that it actually is, it looks more like cubes that you're seeing. So he's looking straightforward into a magenta cube, look up a little bit. That's a red cube, down is a cyan cube, to your right is a blue cube, and to the left is a yellow cube. Okay, so face forward again, I'm going to tilt you just a little bit. So you're a square, yeah. Okay, so now pick your favorite one of those moves. Okay, so let's start with the Hula Hoop. And everyone, he's going to do this a couple of times and then stop and face straightforward. Okay, and stop and face straightforward. Okay, and what you'll notice is that the blue cube that was on his right side has now sort of started migrating towards the front. You want to do that again and see if we can bring the blue cube around in front of you. Okay, stop. Do that once more. Okay, stop. Perfect. Okay, so now he's facing square onto the blue cube. Okay, so pick another move. No, let's try the TV. Okay, so let's try that. So let's see what happens. All right, and stop. Okay, so the green cube that was on his right has now rotated up so that it's sort of at a 45-degree angle up here. And let's try the last move. But first, I'm going to ask the audience, what do you think is going to happen when he does this move? I mean, I see vertically stuff. Is he going to rotate? I guess so if this is the axis, is he going to rotate this way or is he going to rotate this way? So I guess from your point of view, this is clockwise and this is anti-clockwise. Am I right about that? So which way do you think clockwise or anti-clockwise? So you think he's going to go this way? Okay, let's... There is not consensus in the audience, but guess what? This is experimental math and we can actually test it out. So let's try it out and see what happens. So in fact, he is taking the stuff from the top and dragging it down towards the front so that is going for you clockwise. Okay, so what I'm going to do now is... This is going to involve me having to direct your head because I'm going to stick you inside. Yeah, you know the story. Okay, so what I'm going to do is change his view back to the sort of original view of the cubes, the truncated cubes here. And then what I'm going to do next is I'm going to remove the pillars that are connecting the triangular windows to one another. Okay, so this kind of looks like now he's facing sort of in outer space filled with all of these jewels. And these things look kind of like Eicosahedra, right? You know, they look like closed... closed polyhedron. So why don't you stick your head inside one of these windows and I'm going to have to direct him to do that. So, okay, so walk this way and put your head down and down. Perfect, okay. So now he's inside one of these. Thank you, Henry. Now he's inside one of these and it really doesn't look like a closed polyhedron from the inside. So it turns out that what he's looking at from the inside, this is actually an entire Euclidean plane. So you can sort of keep looking in there or pull your head out, whatever you'd like to do. So the idea here... Oh yeah, perfect to do that. If you can actually sit down a little bit. So what you can see here is that each of these is an equilateral triangle. So we've got a cube and we truncate the cube corner that gives us an equilateral triangle. There are six cubes that meet around every edge. That means six equilateral triangles meet around every vertex. So this is a three, six tiling of space and we showed at the beginning that that is a tiling of the Euclidean plane. So we can see this here. So he's looking at one of the vertices and there are indeed six different triangles that meet around these vertices. So I guess there's one last question and normally I ask other people this, but I don't have to ask you. I'm going to put you in here with an infinite number of monkeys. So normally I check to make sure that no one has like a monkey phobia because that would be pretty not fun. So you're not going to be able to see anything. You're actually staring at a monkey head. So normally when this happens just by the pure fact that there are infinite number of monkeys, someone is inevitably staring straight at a monkey's butt. So infinite number of monkeys means an infinite number of monkey butt. And it turns out that these monkeys, what they're doing is they are assembled into sort of an infinite structure. So they're all doing some sort of crazy yoga move like this, where their arms are pointing through two sides of the cube. Their head is at the top. One foot is at the bottom. Their leg is at this way and their tail is out the back. And this tiling, you go forward and then rotate by 90 degrees about that axis and continue going. And this is something, this is a manifestation of the quaternions. And this is something that you can't do in Euclidean space. So I apologize for all of the technical problems here. And if anyone has a very old version of iPhone software, there is a version that works on your phone. There's a current problem with it not talking to the gyroscope in the newer versions. But if you happen to have an older version, or if you happen to have a computer browser, you can go to h3.hypernom.com or h2xe.hypernom.com. And you can play around with this software yourself. So thank you so much for putting up with all of these technical issues. And I'd be happy to take any questions. Thank you very much, Sabata. So before I take any questions, I just want to remind something I forgot to say at the beginning. So at 5.30 today, there's going to be an informal presentation from a representative from the NSF who will speak about the types of grants that are out there. And I think those would be interesting to many of you. So it will be in here at 5.30. Okay, so any questions for Sabata? I was so clear there's nothing you'd want to ask. Okay, well, so she'll be here. And again, please come to the 3D printing workshop this evening and you can ask questions informally. And to our interpretive dancer, Henry, give us a hand.