 as it drawn on the surface of that torus. And then we did some optimization. We wanted to sort of fatten up the circles into tubes that are as fat as possible and then see how they bump up against each other and try and maximize the radius of these tubes. The idea here was that if we allow them to move around and fatten up as much as they can, then maybe when they're just made as tubes, they won't be able to move other than to rotate around their axes. If that were the case, then that's very easy to sort of keep them in place. We just have to add some gears to keep them at the relative rates, but they won't sort of move out of place. Unfortunately, that didn't happen. So, and you only find this out when you print the thing. So here's the three rings in the configuration they were supposed to be. And this is what happens when you actually let them move around a little bit. It turns out if you break the three-fold symmetry, then you can move them a little bit more. And you couldn't, in theory, make them slightly thicker if you broke the three-fold symmetry. So I didn't quite work out. But we're going to put teeth on them to make this work. And so maybe I won't say, no, maybe I will. I'll say a little bit about how the tooth design was made. So I've got these three rings and they're here sitting in space. And so, first of all, as I rotate them against each other, they're going to touch in various places. And those are the places where they touch are the places where I need gear teeth. So I'm going to take one of them away. These two points are the places where these two rings touch. And just to show that it's a symmetrical configuration, I'm going to tilt it up a little bit. So there it is, tilt it up. And you can see the two-fold axis of rotation here. There's two points where these touch. And so those are these two points. As I rotate this ring around, it's going to trace out three circles of places on that ring where it touches this other ring. If I add the third ring back in again, I'm going to have four circles on this ring where I have to put gears. I don't need to put gears anywhere else because they don't touch the other rings. Now, these ones in here, these ones in here are very close to each other. So you want them to do the same thing. If you tried to have sort of different gear teeth on this circle versus this circle, they'd be crashing into each other. Or it seemed likely we didn't want to do that. So we sort of treat them as a single unit. I treat these two as a single set of gear teeth. And this is going to be another set of gear teeth. And then I can use symmetry to move everything else around and make it work. So how do we do this? So there's actually this question of what should the shape of gear teeth be is a very old one. The two-dimensional case of what should the shape of gear teeth be was worked out by Euler in I think 1760 something. And there's various properties that you need to happen. You want sort of constant transferral of force. So you can imagine just a water wheel which a wheel with sticks coming out of it and another one and it just clacks along. And that will work in a sense but it's not a smooth transferral of force. You don't want this in the engine of your car. So the three-dimensional case is still an active area of research. We actually looked into what had been done and actually implemented some of the proposals and none of them had worked. So we sort of made something up. So these are sort of, this is one set of teeth that just sort of made something up. This is that if you imagine toroidal coordinates. So you've got the angle around here, the angle around this way and then the radius. So in toroidal coordinates, these are just pieces of planes. And then we use these things to carve out something on the other ring. So this is sort of what we do. We generate these curves and generate these teeth and then put them together. And that's the final thing. Of course, after we did this, we found much easier ways to solve this problem. So this is a fun one with three different screw gears that pairwise mesh into each other and all three of them can move. So hand that around as well. Oh, this one here is a sort of linear here. So I guess a rack is the technical term. So rather than rotational movement, this has translational movement. And again, you can have three gears that are meshing with each other. And we found that relatively recently, we found this version with four different linear gears. So they're arranged as the diagonals of a cube and somehow as tetrahedral symmetry. That's very fun to play with. So okay, so that was one whole project. Here's another one. This is joint work with Jeffrey Irving who's now working at Google, I think, on artificial intelligence. So this was the first project I've ever started as a result of looking in a drop box. And I found this cool thing. I said, who did that? And then I got talking with this guy. So this is about space filling curves. So this is a particular example of a space filling curve called the Ter-Dragon curve. You may be familiar with the Hilbert curve. I'll show that later on. And many of these space filling curves are generated as a sort of sequence of polygonal approximations. So we start with a straight line and then a little bit later we get a sort of zigzag. You can't really see it inside of here. But then as we go further on, each of the straight bits in this zigzag get more zigzags on them and more and more and more. And then usually you'd see this as an animation in time. Instead, this is an animation in space. And you get this sort of squiggly fractal object at the end. So here's the 3D print. Again, this is the Ter-Dragon curve. This is the Hilbert curve. You may be more familiar with this one. I think this is the most famous of these fractal curves constructed by David Hilbert. And I think maybe 1900 or so or 1899, something like that. This is the Sipinski arrowhead curve. So it's a curve that generates the Sipinski triangle. It's not a space filling curve. It fills the Sipinski triangle. This is the Dragon curve. As you get these very sort of interesting sort of sinuous surfaces, somebody described this to me as a skyscraper that went for a walk, sort of a nice image. It's also a skyscraper where if you're up on the penthouse level, you have these very large windows. And as you get lower down, you get smaller and smaller, exponentially smaller windows. This is a version of the same curve. This is a render generated by Jeffrey Irving. So there's a sort of combinatorial self-similarity between different levels in these surfaces. And this is showing the geometric self-similarity. So different patches that have the same color are actually geometrically similar to each other. OK, another project. This was joint work with Marco Mala. He makes mobiles. He's an artist. He makes big mobiles, big metal things that you'd see in fancy hotel lobbies. And so there's various sort of interesting balancing things going on here that you can use a little bit of mathematics to do. This is interesting sort of one dimensional mobiles. All of the weight of the entire rest of the mobile is balanced by this one stick out here. So this thing here is actually, you've seen something very similar to this. If I turn this upside down, you get this sort of picture, this classic picture of you've got a pile of dominoes and how far can you push them over? And you start talking about the harmonic series. So there's a very similar sort of thing here with this upside down. Here are a few more mobiles. This is a binary tree, a ternary tree, a quaternary tree. It's good for computer science. If you look at the bottom of the ternary tree, you see this. You see the Sapinski triangle again. This is a fun project. This is another one with Jeffrey Irving. Let me find this one if I can. Now here we go. So this is another model of the hyperbolic plane, this sort of floppy fabric-like model. So if you try and lay this flat on the table, you will fail because it's got too much negative curvature. So there's this sort of really interesting mathematical question here. And by the way, this was printed in one go. I did not sort of print 100 tiny little triangles and then systems get afterwards. It was printed in one go. And so the question is, back to this sort of, how do you get the geometry? How did I choose the geometry of where those triangles were arranged in space? I know sort of combinatorially how the triangles should be arranged, but how do I actually get them positioned in space? So this one is a sort of subdivided version of a simpler thing. So this one here, all the triangles are equilateral and there's seven around each vertex. So it's sort of the anti-icosahedron. Rather than having five triangles around each vertex, it has seven. This one that's going around, this is sort of like a geodesic dome. You make a geodesic dome from an icosahedron by subdividing and reducing the angle defects in each vertex. Same thing here. So let's look at this one. This is a simpler problem. How do I arrange this in the printer? So this is what we did. And there's sort of an animation here. So we start out with the triangles arranged in the Poincaré disk model with seven triangles around each vertex. And so we can do that sort of using a parameterization, if you will. Now the triangles, if I arrange them with seven around each vertex, the combinatorics is correct. They're not crashing into each other, but the lengths are wrong. In the same way that the angels and demons and Esha's pictures get smaller as you go out, our triangles also get smaller. But we want all of the triangle edges' lengths to be the same length. And so then we take an iterative method. We put springs on all of the edges, and we shake, and we simulate what happens, and they sort of buckle up out of the plane. And eventually the lengths all converge, the correct lengths. And then we get this thing at the end. So there it is. And then there's a final manual step, which is we've got this mesh of triangles, and so I wrote a very simple script that took manually created hinge parts and put them in the appropriate places. And this is how we generated the geometry. So there's actually an interesting open problem here that I think Thurston also thought about this at some point. So if you have this configuration of 70 collateral triangles around each vertex, how far out can you build this into three-dimensional space without collision? So if your triangles have any thickness, if they have any volume, then you can easily see that there has to be a bound. Because as you build outwards, the volume that you're allowed to put this thing inside of goes up as the cube of the radius, because there's just how far out can you build a strip from the center. But the number of triangles goes up exponentially, and since exponential is gonna beat any polynomial, you're very quickly gonna run out of space. But the open problem is, what if they don't have any volume, if they're just two-dimensional triangles? Is it possible they can go very, very far out and somehow lots and lots of parallel triangles stacking up in some way? So nobody knows. We think that the conjectures of how far you can get, maybe, so this is I think maybe four or five levels out, sort of rings going around the center. And then the conjecture, nobody thinks you can get above 10, but nobody knows. So it's an interesting open problem. This is a square torus. Sorry, not a square torus, this is a flat torus. So we're gonna find it here. So the torus famously has Euclidean geometry. This one really does have Euclidean geometry. You can unfold it and tile a plane. There's some Orxetic stuff. Maybe I won't get into this. Well, maybe I'll just show it. This is some very recent work with Sabetta Matsumoto. So sort of things kind of like the Hoberman's sphere is that we're interested in objects that tile space while changing, changing shape and size. And these are related things by somebody called, a colleague called Tenali Nuitoniami. It's a sort of size changing sort of cubic lattice and some other things. Here's a famous topological joke. Topologists can't tell the difference between this donut over here and this donut over here. I can't tell a difference, certainly. Apparently they're useful for different things. I'm gonna skip over this stuff because I don't really want to get into it and this stuff. Let me show you a couple of other things so I should plug my book given that I'm here. Visualizing Mathematics with QED Penning, it's a popular mathematics book, hopefully accessible to most everybody. Most of the figures in the book are photographs of 3D printed objects and you can go to the website, maybe I won't try this because my wifi isn't working. Well, okay, let's see what happens. So the website for the book has a page for each of the chapters. Yeah, this isn't working. I'll just describe it. There's a page for each of the chapters and in each of those there's a page for each of the figures. And if you go to that page, there's a 3D model if your wifi is working that you can rotate around and see, you can also download the 3D files and put them yourselves and so on. Okay, there's one other thing that I want to show you and plug, let's see. So, oh, I should mention dice. This is not really 3D printing but it's the same sort of 3D geometry so these are joint work with Robert Fathower. We produced the first commercially available 60-sided die. We have these skew dice. So these are dice that are not cubicle but they are fair, or at least theoretically they're fair, no actual physical object is fair. There's a much smaller symmetry group on this object but it is transitive on the faces so it takes any face to any other face which is all you need for it to be fair. Let's see, we made the first commercially available 120-sided die. There's actually some interesting mathematics not only in the shape but also the numbering. So this was joint work with Bob Bosch who's in operations research at Oberlin College. So of course opposite sides on this die add up to 121 as you would expect but also there's sort of other kinds of balancing. So in fact, if you look around each of the vertices the sum of the numbers around each vertex is the same as the degree of the vertex times the average number on the die. So there's sort of even balancing around both the, well all the degree 10 vertices and the degree six vertices and the degree four vertices. So there's lots of sort of balancing. It was a very difficult problem to try and find these configurations. Okay, there's a bunch of references that I'm gonna ignore. I'm gonna show you one more thing so, so along with the 3D printing workshops that have been running 8 to 9 p.m. in the evenings, tonight is your last chance to learn how to do some 3D printing because we're going somewhere else. We've also been doing these, I'll make this big. So this is another very recent project with Cibera Matsumoto, Viha and Andrea Hawksley on virtual reality for non-Euclidean spaces. So this is me wandering around in virtual reality and you'll notice that on the floor here I've got this grid of four squares and in Euclidean space I'm walking around this grid of squares. In this virtual space I'm walking around a grid of cubes but this is a hyperbolic space and so well it turns out that I have to walk around through six straight lines and then turn by 90 degrees and straight, walk straight and turn 90 degrees. So I'm actually walking around a hexagon in hyperbolic space while I'm walking around a square of a Euclidean space. So there's all sorts of interesting illustrations, polonomy and the relationship between what's coming from parallel transport and so on. Okay, I think I will stop there and thank you for your attention. Thanks very much, Henry. Are there any questions? Yeah. Right, right. Yeah, so the printers we have here at PCMI which by the way are solely for the use of PCMI they come out once a year without this huge bucket of filament you should use it. So most printers that you find in universities or at least not in the fancy engineering department they're called Fuse Deposition Modeling. So it's basically you have a nozzle that's controlled by various server motors and it moves to the right place and it squirts out plastic. So I get most of my models made by an online company called Shapeways. So the deal is you upload the 3D model, you click add to cart and then it arrives in the mail a week later. So it's a lot easier than trying to get the 3D printer to work. They use a different printer which is much more expensive. So the way that it works is you've got a tank and it sort of lays down a very thin layer of plastic dust and it's heated just below the melting point of the dust and then the laser comes along and zaps the dust where you want it to be melted. So it tips it just over the melting point so the dust melts onto its neighbors and then another layer of dust zap, another layer of dust zap and you get a much finer result. You also don't have any problems with sort of so one of the problems with these FDM printers is with overhangs. If you have something like a bridge like this then as you're building up layer by layer you start wanting to build these things but you're sort of falling into air. So you just sort of squirt and it falls down onto the floor. So you have to build up scaffolding underneath things which you have to clear out afterwards with this powder based printer. At the end everything is supported in the material that you didn't use. So at the end you get this block of dust and you have to dig inside of here and blow all of it away. But yeah, you get much nicer results. It's a bit more expensive per print. It's more expensive than the FDM printers. But of course if it's somebody else's printer you don't have to spend a huge amount of money getting it in the first place. So it swings around about. William Beck. So I mean I sort of relate it to this. I know that there are services that let you upload an MP3 of your significant other saying I love you or something. And then you take that waveform and you turn it into jewelry. So I know that exists. I haven't seen that many things with sort of visualization of music. I mean it's suddenly, it's an interesting area. I haven't heard of that particular thing but I'm sure there are many people who are interested in getting visual representations of different sound waves and so on. Oh, one more, one more. Yeah, I think so. I think those are all on thing of us. But it's okay. Yeah, I mean you have to sort of design it to the kind of printer that you have. So I think we've got a few minutes until the next group comes, or maybe not. No, no, we do, we do. Okay, so there's a little bit of time to come and play with this. Because just that sort of the point is like it's physical, you can touch it. So you should come up and play with it.