 So yes, there's a little bit of a clash with the movie, Night I See, but I'll hang around if people wanna get back from the movie and learn some about 3D printing and how to get started in it, then please come along. So yeah, that'll be seven to nine, well, maybe later if people are still showing up. In Coalition 3, which is the, there's a room just down the hall, which has the 3D printers in, and if you wanna know how to start doing it, then bring a laptop and I can get you set up with some software and we might even print something. So let's get started. So yeah, so it would be nice if this would work, there we go. So I'm gonna show you a tiny little bit of math of the kind of stuff that I do and Jen's wonderful diagrams here, I think are a good illustration that a lot of mathematicians, not all of them, there are mathematicians who don't use pictures at all. I don't understand how they can do anything but there are some who do. A lot of mathematicians do use pictures and this is the kind of stuff that my work looks like. And so there's a visual aspect to a lot of what we're doing, which has some connection to maybe an artistic side, a visual side. But I think there's a stronger connection between math and art, which is that both math and art are not afraid to ask questions about fictional worlds. But what we do is we set up some system and then we decide and try and figure out what the consequences are. And what that system is, it may have nothing obviously to do with the real world, but we don't care. Unlike, say, an engineer or most other scientists, we don't have to live in the real world. We can go off into our own world and do our own thing, which is what artists do as well. They go into fictional worlds and figure out what the consequences are. So moving sort of from math more into the sort of artistic side, this is a good sort of segue from the two. So this is a 3D print of the complement of the figure eight knots. It's a three manifold. It sounded like it might have come up in the previous lecture. But this is a 3D print that illustrates this. So for those of you who haven't heard of this kind of thing, the idea is that you've got some ball of clay maybe, and you've got a worm that eats its way through in a knotted path and leaves behind this ball with a knot taken out. And this is a three-dimensional manifold that has a boundary. And this is the kind of thing that a lot of low-dimensional topologists are interested in. And I guess I'll mention now, there's a lot of what I do is joint work with other people. I'm not gonna mention all of their names, but lots of, as with many things in life, it's more fun to do with other people. So okay, so that's a 3D print you can make. Let's start sort of maybe more basic. What I'm gonna do here is basically show you a whole bunch of 3D prints of interesting things. I'll hopefully get you inspired. I'll say a little bit later on about sort of trying to say what makes a good visualization? What makes something that's effective at showing something visually? And then towards the end, I'll shift to some new technologies, not 3D printing, virtual reality, and spherical video. But for now, here's some things that you might think to print. Let's do the polytonic solids, or let's do the Archimedean solids, and these work very well. And while you're at it, these look sort of like dice. So let's make some dice instead. I guess I should start handing things around, and it's good that there's some people on the front row, although there's some gaps, because the things will eventually get around. So I'll guess I'll just start over here. So this is the first mass-produced 120-sided die, which is about as big as you can get. And there's some, I won't spend too much time on this, but there's some interesting math, particularly in how you do the numberings. I mean, obviously opposite sides of this die have to add up to 121, but that's not much of a condition. So you want more conditions to decide how you should number this. And what we actually ended up doing is we arrange it so that the sum of the numbers around all the vertices of degree 10 are the same. So, and if you add up all the numbers around all the vertices of degree six, they're also all the same. And if you add up all the numbers around the vertices of degree four, they're also all the same. So it's sort of this very sort of balanced over the whole thing. And this seems to be a very difficult problem. It took us over a month with Bob Bosch as a operations research mathematician who set his code working for a month. And we were very lucky we think to find this. So anyway, there's all kinds of difficulties there. Talking of dice, these are some interestingly shaped dice. You'll notice they're not cubes. And you may wonder, are they fair? Were you wondering, are they fair? Yes. So there's a whole interesting question there. What properties of a die make it fair? And I guess maybe I won't tell you because I'm short on time. But, but mathematic, well, you can come and ask me later or you can figure it out yourself. Mathematically, these are indeed fair. Of course, no real world physical object is gonna be fair, but anyway, okay. So that's the six-sided die. Here's the 12-sided die, which is also a little skewed. It's less obvious that it's skewed because people don't know what Dedeke would look like. But it is. If you look at it up close, you'll see that they're not pentagons. Okay, back to 3D printing. So here are some other sort of reasonably obvious things that you might wanna do. Let's make some surfaces. So I made these for Swinburne University of Technology as a university in Melbourne, Australia. They had a bunch of money left over at the end of the financial year and they needed to spend it on something. So, quadric surfaces for your multivariable calculus tassels. And you'll notice that there are these much smaller ones right next to them. So here are the much smaller ones which are much, much cheaper and just as effective. So I actually used these in my multivariable calculus three classes and it really helps the students to, I mean, you can draw pictures on the board but actually getting a three-dimensional sense of what the shape is is difficult. Particularly, I mean, if you just give them the equation and say, what does this look like? They find this very difficult and this is a good way to sort of give them the answer in a sort of tactile way. And you can add all this extra information, you know, contour lines and so on and so on. Talking of sphere, so this is sort of often a different direction. This is a self-referential sphere. You can see I've highlighted in red here, S-P-H-E-R-E. So this is a sphere tiled with the word sphere. There are 20 copies of the sphere. So it's dihedral group of order 20 is the symmetry of this. This is a similar sort of idea except it doesn't have any symmetry but it's still self-describing. Maybe you can see here there's B, U, N, and Y. So this is a bunny tiled with the word bunny. 72 copies of the word bunny. So this is sort of a computer science thing here. You wanna have a sort of square grid over some arbitrary shape but you want it to be not too distorted. So there's some curvature things going on and how do you come up with this shape? So I'm a topologist and this is the standard. You know, they call it a joke but usually people don't laugh. It's not that funny but a topologist can't tell the difference between a coffee mug and a donut but. So if I was, so this is made in ceramic. You can print in all kinds of different materials. Unfortunately you cannot yet print in a gradient that sort of shifts between ceramic and actual donut because really when you get around here it should just be a donut. This was with Keenan Crane. He does computer science at CMU and this is something called conformal Wilmore flow. I mean there's an interesting question here. Where do you get these shapes? Right, yes you can, this shape's easy. This shape's relatively easy and what do you do here to make a nice smooth thing to go from one to the other? Maybe a simpler topological thing, this is another knot. This has, well, this is a trefoil. I guess maybe that's not super obvious unless you're in knot theory but for some people it will be. This is an interesting shape of a trefoil. Let me hand this one around because it rolls. So this is a knot which has no tritangent planes. So there's no plane that you can hold up to the knot so that it's tangent in three places. If it were tangent in three places then if it was sitting on the table it would be stable because it would have a triangle and this one doesn't have any which means that it will just roll. As far as I know, I don't know that there are actually any other known examples of configurations of knots that don't have any tritangent planes. Similar sort of idea but a little bigger. So this is joint with Lee Braswell who's in the theater department at Oklahoma State and he is in sort of circus rigging and so let me just show you this video. Let's see, do I have sound? So this is somebody who actually knows what they're doing inside of this. So there's this class of circus apparatus which is, let's see, can I get that to play again? Which is you get inside of a big metal thing and you roll around. And usually they sort of go in a straight line or maybe they kind of zigzag back and forth and the question is can you do something more interesting? Can you set up the shape of the apparatus so that at certain points, if you lean one way or another, there's a choice. So you can see, you can just roll in a straight line but there's a point, there we go, where you can sort of change tracks and do something else and there's an interesting story here which I went into of how do you design this kind of thing and how do you make it and how do you make it safe? That's another part of the story. One of the nice things about 3D printing is a medium. I think it's important to use a medium well to do things in that medium that you can easily do in other media. So you can, it's in the real world so you can add real world stuff to it. So for example, here's soap film, showing minimal surfaces. There's a minimal surface on the trefoil, the rolling trefoil that I showed you and this is an example of a ciphered surface. So in topology, a ciphered surface is a surface whose boundary is a given knot or link. And so you can illustrate this with soap film. As they get more complicated, the surfaces get harder to actually make out of soap film but then you can go back to just 3D printing them. So there's some fancy sort of number theory in here to how do you parametrize these objects to get these shapes. The words fractional, automorphic form came up. I don't know what they are, asks all. He may be able to tell you. Going back down to sort of simpler surfaces, this is, well, huh, how many topologists are there in the room and can you tell me what the surface is? It's really difficult because it's not the standard form of the surface. I mean, it can be that complicated. Like, it's not got much genus. I'll put you out of your misery. This is a Mobius strip with a hole. So you can see that it's non-orientable by going around this loop and it turns over and then you sort of have to count holes and so. So one way to think about this is if you take an ordinary Mobius strip, you know, a strip of paper with a half twist and you glue it together and you look at the boundary curve, it's an unknotted loop in space. So you should be able to sort of pull it tight to make it a circle and the surface should flow along and this is what happens, or this is one answer. So this is the boundary of the ordinary Mobius strip and then there's another puncture in it around the outside. If you put two Mobius strips together, you get a Klein bottle. So there's a Klein bottle. Okay, so on to, away from topology, on to sort of different kinds of things. So this was a project with Marco Marla, who is a, he's an artist who makes mobiles. So he's usually making huge metal things that go in lobbies of hotels or fancy businesses. And so we got to sort of 3D printing these and there's some sort of interesting things here. So this is a very lopsided mobile. All of the weight is on one side. And actually, if you sort of turn this upside down, it's the same thing as a stack of dominoes and how far can you push the dominoes out before they fall over? So there's some math here and it was a lot easier to design this on computer because you know where the centroid of the objects is. You don't have to sort of actually do any work, like picking up the metal thing and deciding where to weld on the hook so that it balances and so on. These are some other mobiles we made which are rather more balanced. There's a binary tree, a ternary tree and a quaternary tree. If you look at the underside of the ternary tree, you get a Sapinski triangle. And sort of going on to the theme of fractals. So let me show you this sequence of curves. So this is a sequence of curves making, well, this is a sequence of curves that approximate a space filling curve. This is the one called the Ter Dragon curve and you often see this kind of thing done as an animation and so the idea here is to extrude this through space instead of time and so you make this sort of curvy fractal-ish sculpture. So this was one that we had made printed and made big. This was for an exhibition at the Simon Sensor at Stony Brook. Oh, let me pass these around. I've got some of these. Let's go over this side this time. And right, so maybe this is the one that maybe people will be most familiar with. This is based on the Hilbert curve and this is the Sapinski-Araher curve and this is the Dragon curve. One of the interesting things about putting this stuff out there in the world, what am I trying to do? I'm trying to say, let me illustrate the construction of this space filling curve as well as I can. But then people see this and they see something else in it and of course any time you put something out, people can interpret it in a different way. Possibly the referee of your paper can interpret it a different way as well. But anyway, so somebody saw this and said, oh, it looks like a skyscraper going for a walk. It kind of ripples off and anyway. Talking about people getting strange interpretations of things, so what is this? So this is a sculpture called More Fun Than a Hypercube of Monkeys. This is joint work with my brother. So this is the, oh and I should mention, like you've probably figured it out already. I'm gonna show you a whole bunch of stuff and I might say some words that don't make sense to you or me, but it's fine because I'll just move on to something else. So with that warning in mind, so this is the stereographic projection of a sculpture in the threesphere, which has one monkey in each of the eight cubes of a hypercube readily projected onto the threesphere and these monkeys are symmetric in the sense that if you apply any elements of the eight element quaternion group to that structure in S3, it's invariant. So there we go. You can do the same thing with the 24 cell and you get more fun than the 24 cell of monkeys or with the 120 cell or you can do it in virtual reality. This was joint work with Vi Hart and some other people and let me show you, if you go to monkeys.hypernom.com then you can check this out and it looks something like this. So you can fly around on your keyboard. So this is the eight monkeys and you can see every few seconds what's gonna happen. This monkey's gonna turn into this monkey. This monkey's gonna turn into this monkey. So this is sort of a smoothed out version of one of those symmetries, which is moving this thing around. And then if I press some other buttons we get the 24 cell version and oh yeah, you can fit two 24 cells together because the 24 cell is self-dual. You can fit another 24 monkeys in the gaps left between the first 24 and there's the full 120 monkeys. And again, these all have sort of different symmetries which are subgroups of, finite subgroups of the symmetries of the three sphere. Okay, enough monkeys. So, right, so what makes a good visualization? And maybe I could get the lights down. Is it, does somebody have access to the light switch that I should have said beforehand? For this one? So let me play with some light switches. What I'm gonna do is I'm gonna show you maybe one of my most effective, do I have the light switches? We'll never get, ah, perfect. Okay, thank you. I hope they can come on again. I guess we'll find out. So, okay, so this is supposed to be what I think is a good visualization. So I've got here a sphere. It's made out of plastic. It's got these grid lines on it. And if I arrange the light in precisely the right way, then this grid of squares is illustrated to actually be this flat Euclidean grid of squares. Okay, so we can try and get the lights back on again. We have the standing next to the light switches. Uh-oh, okay, it's never coming back. Thank you. Okay, so there's a better picture of the same thing. So, well, I can't resist to tell you what this is illustrating, of course. So this is illustrating stereographic projection. Ah, the lights are not quite the same as they were before. Oh, well. Ah, yeah. Look. Sure, let's go with that. So light rays come from the north pole of the sphere and they go down and they inside the sphere and they hit somewhere and they carry on down and they hit the table. And this is the map of stereographic projection which goes from the sphere to the plane. And so, I mean, so what's sort of effective about this? Well, so there's sort of a wow factor, right? You're not expecting or you don't know what's gonna happen and then there's something that's surprising. And of course, you wanna draw people in to ask questions about what's going on. So there's a surprise there and it's, you know, the mathematics is completely accurate, of course. Like, the light rays really are doing what this projection is doing and it gets you into the core of the concept without needing to go too far into the equations and the formulas and so on. I mean, of course, you can write down the formula for this and it's not that complicated but you don't need to know what it is to understand what this map is doing. So I guess one other thing I should say about this, I guess, and it's using the medium, right? So here's a 3D printer, here's a 3D printer, just a piece of plastic and a light, a flashlight. There's no sort of cheating going on. If this were a computer animation, you're like, well, yeah, sure. I mean, it could be anything. This is really real. It's what light rays do. And I guess the last thing to say about this is this was not an easy photograph to take. So you may have, as I was sort of lining this up, a tiny, tiny movement in where the flashlight is changes the shadow quite a lot. So I'm there holding the flashlight. I'm also taking the photograph. How did this happen? So really what's going on is that this flashlight is attached to a vertical beam and there's a crossbow and a couple of clamp stands either side. It's all arranged so that it's in exactly the right place. Oh, that's perfect. Thank you, lights person. So the hand is purely decorative. It's just trying not to move the light out of the way at all. It's not necessary at all. But it makes people believe that it might not be a computer render, which is a problem with these. You can get your graphics to be too good and then people say, oh, that's a nice computer render. So there are some other patterns you can do. You can do any pattern you want, really. Sabetta showed this last week in her talk, getting different hyperbolic projections out of this hemisphere model of the threesphere. We did an exhibition a summer or two ago in Edinburgh. This is a globe which is stereography projected onto the wall. The 3D printed globe is reversed so that the shadow is the right way around and there's this grid. This is about four meters by four meters. And right, these walls sort of interactive. You could move the light around in some way and see how it changed the projection. And this was the highlight of the show. So this is a zoetrope. How many people know what a zoetrope is? So some but not everybody. So a zoetrope is sort of the precursor to film. So maybe you've seen this but don't know what the word is. So you have the sort of cylinder here and the slits cut in the cylinder and it's spinning around. And you look through the slits to the other side where you see a frame of an animation and that sort of loops around. So this is doing the same thing except that there's no slits and there's no cylinder. The disc here is spinning around. I think it's once every two seconds or something like that. So it's pretty fast. And there's a strobe going off precisely when the 3D prints are in the right position so that it looks like it's animated. And this is doing some sort of rotation of the hypercube in the three sphere. So I should hasten to add 3D prints don't do this. They're rigid, they're made out of plastic. You can't do this but this is an animation. So here is an animation of something which is real. So it's not an animation at all, it's a movie. So these are these three gears which are rotating around and through each other. There's another shot. So this was sort of an engineering challenge. I'm not gonna tell you the whole story of how this came to be but I'm gonna tell you one story of where the question came from. So maybe once you start seeing these things you can't help but notice them. There's graphic design elements of three gears that are meshing with each other. So this was in Manchester, there was supposed to be three different transportation systems that were gonna come together and everything was gonna move very smoothly. But of course if you look at this, if this one's turning this way that means this one has to be turning this way which means this one cannot turn. It doesn't work at all, nothing moves. And over here the teacher, the students and the parents come together and nothing is achieved. So the question is how can you make a system of three gears which are symmetrically arranged with respect to each other and yet they do move. And so this was one solution. There's a few other solutions. This is another solution with three gears except they're sort of linear. And the one that's on the screen is has got four sort of linear things that move through each other. And we've just more recently done one with five straight lines that go through each other in some complicated way. So other sort of moving things. Moving things seem to be a really good effective thing because again it's something that 3D printing can do that you believe more than if we were a computer animation. So this is illustrating that's a torus, right? And the geometry on a torus is supposed to be flat and Euclidean and this one really does so you can open it out onto the flat plane and see that it really is flat Euclidean geometry. Oh yeah, this one's fun. So this is a model of the hyperbolic plane made out of flexible well hinged triangles. So you put more triangles, more angle around each vertex than you should have and the thing starts getting wrinkly. So the real sort of interesting mathematical question here is how did I print this? So you need to give a file to the printer which says here are these triangles though in this shape and then I'm gonna print this thing. Oh, I should mention this was printed in place. I did not, so the hinges came out of the printer like that, I did not print 300 tiny triangles and then piece them together somehow. So I had to arrange these in the printer in a way which was a valid configuration and how do you come up with that configuration? So we used a sort of iterative method when we run this animation. So we started with the triangles laid out in the Poincare disk model, which means that they're not crashing into each other and the commonatorics are correct but the lengths are not correct. As you go further out into the Poincare disk model further towards the boundary, the edges get smaller. So that's not correct. I want all the edges to be the same. And so we put springs on everything and shake and eventually it sort of converges and you get this shape and then another bit of programming puts those triangles, puts the models with hinges and so on on the positions of those triangles and then that's how we made it. And there's an interesting question here which is still open. So if you take equal out all triangles and you fit seven around each vertex and you sort of keep building outwards, can you keep going forever or putting them into three-dimensional space or must they crash somehow and what's the largest number? So there's no proof that you can go forever, sorry, there's no proof that you cannot keep going forever but if the triangles have any volume at all then there's a sort of rates at which things grow argument that says that you have to fail at some point but it's still open. First and I think I thought about this a little bit. Is there a limit theoretically to how far out you can go? And there's a puzzle which is made out of a combinatorial version of the hopf vibration on the 120 cell in the three-sphere and these are some of the strands of them which you can take apart and put together again. I recently got interested in expanding objects so this is a somewhat portable dodecahedron. Whoops, let's play that again. So these are called auxetic mechanisms so you pull on it and it sort of gets bigger in all directions. This is a bigger one which is, so that's based on a diamond structure so the molecular structure of carbon atoms in a diamond. Let's just see that one again and this one is based on cubes and if you're interested you can go to mathmechs.com and we've actually made this into a product you can get a little set of these for 20 bucks or something like that. Other sort of expanding mechanism things this is a jitterbug. So Buckminster Fuller invented this jitterbug mechanism. I guess we'll go over this side. And we put gears on it so that it helps it. One of the things getting into these sort of moving things is you start becoming less of a mathematician and more of an engineer. Because it should just work, it should just be perfect but of course hinges aren't like that. So the gears are not sort of mathematically necessary to make this move the way that it does but they help sort of keep things aligned and they keep the movement sort of smooth. And here's a generalized version based on the cube octahedron instead of the octahedron and so you get square faces and triangular faces instead of just triangular faces. Okay, so this is all very well but as any educator will tell you, I mean seeing things is not as good a way to learn something as doing it yourself as making yourself. So to that end, I have a 3D printing lab in the math department at Oklahoma State and I teach a course which is essentially 3D printing for math majors. So the students learn how to use, it's sort of junior, seniors kind of project-based capstone course and they learn how to 3D print things. This is I think some homework from week four of the semester where they're printing parametric curves in space so they learn how to use the software, they learn how to use the 3D printers. You too can learn how to do these things if you come along seven to nine tonight. And then their projects are sort of illustrating mathematical things or indeed whatever else they want to do as long as it's got some interesting mathematical content. They've got these sort of pine cone geometry and Sapinski, this is a, wait, what is this? This is a menger sponge that's been cut in half along a diagonal. It has a beautiful pattern, lots of interesting things. This was the final project from one of my students this year. So this is similar to the fractal curves except instead of generating steps of a polygonal approximation to a space filling curve, this is doing Fourier series. So these are getting closer and closer to a sawtooth curve or to a square wave curve just by stepping down what are the next terms in the Fourier series. I think this is just a wonderful illustration of that. You can just hold on your hand and see how it's changing and you can see whatever this thing is, what's this thing, the little jump? Gibbs, yeah, so I mean, there it is, right? You can touch it. And it's just a wonderful quote from George Hunt who's a fantastic mathematical artist who's been doing this for many years. The more math you know, the more stuff you can make. And so absolutely, if students come in and they know more math, then there's some direction they can take in and there's some other strange things they can do, strange and wonderful things. So let's see. So let me move on to talking a little bit about virtual reality and a little bit about spherical video as well. So Sabetta showed off some of this last time, well I guess on Wednesday, so I won't go through this. This was the first, and what she was talking about was sort of this first iteration of the simulation that we were working on. So again, this is a simulation of three dimensional hyperbolic space and there's all of these cubes, actually these are truncated cubes and they're arranged with six cubes around each edge, in this case both vertically and horizontally, so this is a tiling of three dimensional hyperbolic space by these cubes. And this is the new and somewhat still under development version of this project using rather than sort of the traditional polygon pushing way of doing computer graphics. So most computer games use polygons which are sort of taken from the world and projected onto your screen to decide where to draw them. Instead this is made using something called ray marching, so this is similar to ray tracing. So what happens there is for every pixel on the screen you send a ray out into the world and you see what it hits and depending on what it hits you decide what color it is, what color it should be. And the advantage that this way of doing things has is that the map goes the other direction, so there's sort of a covariant thing going on. When you take polygons that are in the world and you project them onto your screen you have to figure out this map. When you look at pixels and you wanna know where did you guys go, you're figuring out this map and there's a difference between those two cases, not so much in simulating Euclidean space but if you're simulating, well, we're trying to get to some of the other Thurston geometries, these very strange and screwy geometries and some of them have, if you're trying to do this map it's not single valued. There's multiple things that can get to the same, from the same object that can get back to you and so that makes everything nastier. If you're going forwards then it's hopefully easier and so I'll mention at the end there's an ISEM semester that I'm one of the organizers for coming up next semester on illustrating mathematics and one of the things we're gonna try and do there is make the other eight Thurston geometries. Maybe we'll skip the Euclidean one because that's sort of been done but the other ones there's a lot to do and actually, yeah, so I think I've got time. Let's see if we can actually take a look at this. If this will work, let's hype the array. So you can go to this on your, I used to say to people, actually let's use the mobile version because it will run faster on my poor computer. These sort of simulations are pretty heavy on the graphics card and so this is one that is supposed to run on your phone. So if you go to that address, oh, I don't know how visible that is, but if you go to that address on,