 Henry Seigerman is a geometric topologist who is a professor at Oklahoma State. And he's become one of the great proponents of mathematical art, and he has a book to prove it. He came out this year visualizing mathematics with 3D printing. He's given some wonderful presentations at PCMI in past years, and as you all know, he's been leading a 3D printing workshop in this virtual reality, no, non-Euclidean virtual reality demo on the past few evenings, and he will do one more 3D printing workshop this evening. Thank you very much, Henry, for coming to talk to us. All right, thanks, Rafe, and thanks all for coming. So I'm going to give, so this is a talk sort of in two parts, and I should say, I mean, in terms of the level, this does occasionally go into some sort of higher level mathematics with geometric topology, but it will pass very quickly, and there's lots of different things and lots of pictures, so stick around. So the first part of this talk is about the question of a specific question of how do you visualize a particular kind of mathematical thing. In particular, it's how do you visualize topological objects? How do you tell a computer to make a model of something topological, and of course the problem is that topology doesn't have precise geometry, it's floppy, but you need very precise geometry to give to the 3D printer. And then the second half of the talk will sort of devolve into here are some interesting projects and the sort of mathematics that went into it. Okay, so let's get started sort of how do you 3D print things? So there's a very common problem that comes up whenever you're trying to print something mathematical, in this case, this is supposed to be a knot, and the problem is that the knot is one-dimensional, but in order to print anything, it has to be three-dimensional, it has to contain volume, otherwise there's nothing there to print. And so there's a perennial problem, the first problem you always come up with, how do I thicken something up in order to make it have thickness so that it can be 3D printed? So the easy, I don't know how I'm going to keep this on my ear, I'll do what I can. So the easy thing to do here is to just put a pipe around this, have some radius and then you produce a tube around this curve, and then you can 3D print it, and you can line up your photograph of your 3D print with your virtual version of the image and wait until people are suitably impressed because it's not easy to do this. There'll be more opportunities to be impressed. Here's another example, so this is a graph in space. It's a little hard to see from just this what the graph is, if I thicken it up, maybe you can see a little bit more easily, and here's the metal version. I'm always, so whenever I give this talk, I'm always sort of curious to see if people know what this structure is, what is this an illustration of? This is less commonly known to mathematicians, but maybe more commonly known to chemists. Any guesses of what this is? Ice, that's very close, it's not ice. Salt, it's actually the molecular structure of diamond. So you've got your four carbon atoms and they have their connections out to their four neighbors. Okay, so great, moving from one dimensional objects to two dimensional objects, here's a hyperbolic paraboloid. And you might think, okay, well, you should just thicken in exactly the same way. You just thicken up sort of moving in the normal direction to make a thicken surface. I actually prefer to do something like this instead, so to put holes in it. And so there's maybe three reasons for this. First of all, it's cheaper, you're using less material and it'll print quicker. Second, it sort of lets you see through, right? There's a sort of transparency to this, it's not just a blank, flat surface you can't see through. And third, it lets you show something about the surface. In this case, it's showing the slices going in the XZ plane or the YZ plane, it's showing the contours that you see. And there's a 3D printed version. So the Swinburne Institute of Technology in Melbourne, Australia had a whole bunch of money left over at the end of the financial year. And so they had me design these quadric surfaces, these large quadric surfaces for use in their multivariable calculus classes. You'll see next to them, there's these much smaller versions, which are equally useful for teaching, but much cheaper. So I actually use these models of quadric surfaces when I'm teaching multivariable calculus. And so here's this hyperbolic paraboloid. If you give a student Z equals X squared minus Y squared and ask them to draw the graph of this, this is a difficult problem. And so this really helps sort of seal, visualize how do the contours, as you slice through different Z values, give you the surface. Okay, so far I've been talking mostly about algebraically defined objects. There's some equation that determines the curve or the surface. And so as I said before at the start, this is a problem when you've got a topological object. For example, the knot at the very start, you have to choose some geometry. So how do you choose that? Well, okay, maybe you don't have to choose the geometry. You can print something that's flexible, and then you don't have to worry about what the geometry is going to be. The 3D print somehow has the topological choice inside of it. But ignoring this possibility, what do you do? And so there's maybe three or four different strategies for how do you actually get some shape? How do you generate geometry? So, and I'll show some examples of these different strategies. So the manual strategy, whatever you've got available, whatever design software you have, you use that somehow with the user interface to build something. And you're not worrying too much about the precise geometry, you're just building something. I mean, it's the same sort of thing if you're drawing, if you want a picture of a two dimensional picture, let alone a three dimensional object. If you want to draw some diagram on the board, that's sort of the manual strategy. But it's the same sort of thing, maybe you've got, instead of just trying to draw your not freehand, maybe you generate in some two dimensional computer programs some parametric or implicit description of the object that you're building. So, both 3D and 2D. So okay, so first, just the manual strategy. Second, you have some parametric or implicit description of the shape. Or third, you have some iterative method that converges to some shape that's trying to optimize some value. So I'll show some examples of these. These are three trefoil knots that were generated using these three different strategies. So here's the first one. So this is cubic trefoil knot pendant by somebody called Vertigo Polka. So I put this in the manual category because this is a simple enough shape that you could just build it inside of your 3D software. But this has a lot of mathematical interest nonetheless. So one of the things that knot theorists sometimes think about is stick number of a knot. So how many straight line segments do you need in order to generate a version of that knot? And this is a little bit more restrictive still. Straight line segments, but this time the straight lines have to be along the x, y, or z axes. And I believe this is the minimal number of sticks you need in order to make this trefoil. OK, so that's an example of a manually generated trefoil. Here are some parametrically generated trefoils. I'll go into sort of similar parametrizations, but there's some trigonometric things you can do. You can imagine to generate these sort of looping curves. And then I'm going to show you a couple of examples of different trefoils that are generated by iterative methods. So this trefoil is from a program called Knotplot by Robert Schallian. And what's really going on underneath this shape here is that there are a bunch of points that are connected by sticks. And I believe these sticks are fixed length, but the points you sort of imagine they have electric charges on them. And so they're trying to get away from their neighbors. And so what happens here is you start with some shape of this. I don't actually know what shape it starts with. And then it sort of tries to relax down to minimize the amount of energy that it's got and try and get the electrons to be as far apart from each other as possible. Here's a different strategy for making a shape, in this case the trefoil. This is a minimal rope length trefoil by Jason Cantorella and various collaborators at University of Georgia. So again, where did this shape come from? So I'll show you another picture that sort of made me easier to understand, which is this one. So the difference between these two is just that one has half the radius of the tube of the other. So this is a trefoil knot that's been pulled tight as tightly as possible. So again, you start with some shape of the trefoil and then you gradually, using some computer simulation, you reduce the length of the rope as much as you can without letting it crash into itself. And then this is just with half the radius that lets you see more easily what the shape is. And so there's a subfield of, well, this kind of geometric topology where people try and prove the minimal rope length versions of various knots and links. And it's very difficult to prove precise results about, this is simulation, so they guess that the optimal is close to this, but there's no proof of these sorts of things. Okay, and I will be remiss if I didn't mention Laura Talman is at James Madison University. These are results from, I think this was a summer research program she had with a small class. And so one of the ideas of the class was, look at these papers on various ideas and knot theory and then make a visualization, a three-dependent visualization, of the idea from that paper. So this again is the cubic stick number. This is the stick number. If you don't require that things are on the XYZ axes, you can do this in six, one, two, three, four, five, six. And then there's various other versions of truffles here. I think this may be the minimal rope length truffle. Okay, so which of these things is best and which one should you be using to visualize something to make an effective visualization? And so really this is an aesthetic choice, but I think it's worth remembering that as mathematicians, most of what we do is in some sense driven by aesthetic choices. The problems that we work on, we think that they're beautiful, other people are interested in them, these are also aesthetic choices. And so it's not that different from what I'm doing here in this side of my life. And the sort of criteria you can bring to say, which of these different ways is a good way to draw a picture of a truffle? I think you can sort of have similar ideas to the mathematical ideas. So here are my proposals for good ways to make choices for how you make some topological model or some geometric model of a topological object. So the first thing is you should make as few choices as possible. You don't wanna be making extraneous choices if you don't have to. You would like there to be a single best one. And the second is that you should be as faithful as possible to the object that you're trying to represent. If it has certain symmetries, your model should have those symmetries and so on. And really these, I mean, this is another way of getting to I want a canonical geometric structure associated to my topological object. And so in some sense, this is in the same thread of a lot of the past few decades in geometry and topology, you know, taking a topological object and then finding canonical geometric structures. And once you have such a canonical geometric structure, there's a very clean path from the mathematics, the mathematical ideas over to a physical object that illustrates it. And so, well, you go via a computer model. If you have a canonical structure, a canonical mathematical structure, then you should hopefully be able to easily convert that from a mathematical idea into an accurate computer model because you can code it up. Well, a canonical geometric structure you hope is simple enough that you can write code that generates it. And once you have a computer model, then 3D printing will convert that over into a physical model in a very accurate way. Okay, so let's look at some examples of different ways of making the same model and why I think our way is better. So this is a popular thing for people to 3D print or this sort of theme of these Mobius ladders. So it's a Mobius strip, but it's got sort of rungs and holes in between those rungs. And then you somehow have them twisting around so that they interpenetrate each other. And so there's various examples of these and this is, well, my version with one of my collaborators Saul Schreimer is at the University of Warwick. And so this is our version of essentially the same object. Hopefully I've got it on the table somewhere. Here it is. So it's actually in two pieces, it jangles because they're not the same object. So I'll start handing things around. Rafe, you're on the front row so you're gonna get lots of them first. So okay, so why do I like ours better than all of these other ones? And so there's sort of, you know, I want there to be a canonical geometric structure why I want as canonical as possible. And these things are somehow very blobby, right? They're sort of kind of mushy shapes. And I don't know, it's what are the angles here? They seem to be changing in some strange way. Over here, what do we have? We have always right angles between the rungs and the poles of the lattice. And why do we have this? It's because we have a parametrization. So this is actually parametrized, these mobius strips are parametrized in the unit sphere in four-dimensional space. And then, well, here's the parametrization. There's some cosines and signs. And the trefoil knots, the parametrized trefoil knots that I mentioned earlier, there's also parametrized in similar sorts of ways. So you've got a function of two variables. Theta is the direction that's going around the mobius, oh sorry, no theta is going across the mobius strip and tau is going around the mobius strip. There's actually a third parameter here because we need a thickness for the mobius strip but I suppressed that. And so, well, so why are rungs and are ladder poles at right angles to each other? Well, what you can do is you can take the derivatives in the theta direction and the tau direction. You can show that they are perpendicular to each other in four-dimensional space and then we use something called stereographic projection to go from four-dimensional space to three-dimensional space and then this is the result. And stereographic projection is conformal, it preserves angles and so we still have our right angles. So that's where that comes from. Just talking about stereographic projection, this is sort of illustrations of stereographic projection from the sphere in three-dimensional space to two-dimensional space. And you get the, these are actually, so stereographic projection is one of the few kinds of projection from one space to another that you can literally do with rays of light. And so there's a very nice sort of illustration of that. So here's another illustration using the same sorts of ideas. There's these various models of the hyperbolic plane. So maybe the most familiar model of the hyperbolic plane is the Poincare disc model. So if you've seen the Escher pictures with angels and demons and so on, where the angels and demons get smaller and smaller as you go out towards the boundary of the disc, it's a similar sort of thing to this. Here I've got triangles, which is going further out to the disc. And this is the shadow of this sort of bowl-shaped object here. And then there are other models of the hyperbolic plane that you can get out of this bowl by casting shadows in a different way. So if I raise the light very far up, then the rays come in down parallel to each other and you get the Klein model. And if instead you put the light on the equator of the bowl and turn it upside down, then the shadow that casts on the wall is the upper half-plane model of hyperbolic space. Okay. So we actually used some of these ideas. It was an exhibition in Edinburgh that Saul and I and various other people put on. This was earlier this summer. So we had 1,500 or so visitors. They were a three-week period. So this is this sort of grid projecting onto the wall. This was a globe being stereotypically projected onto the wall. You could rotate the globe around to any orientation you wanted and put your favorite country in the middle of the stereographic projection. We had various tiling. So this is the 532 tiling of the sphere and then the stereographic projection of that. You can see the conformality. We had this room of different tiling. So this was the 632 tiling and the 732 tiling. So thinking about different ideas about curvature. And then we had a Zoetrope, a 3D printed Zoetrope. Okay, this thing isn't gonna load, but that will load. So a Zoetrope is one of these things where you have a sort of disk that's rotating around and the Victorian Zoetropes, the way they worked is they had a cylinder with slits in it. And as the cylinder rotated around, you would see through the slit only for a very short amount of time. And this was sort of the first kind of animation things that people saw before film. So our version uses 3D printed models and strobe lights to spin the disk and flash the right way. And this is a stereographic projection of a cube that's being projected down into three-dimensional space. And there's a four-dimensional rotation that the animation is showing. Okay, so another example or a whole other class of examples of comparing different ways to turn topology into geometry. So this is an object called a ciphered surface. So a ciphered surface is a surface, this sort of orange surface, whose boundary is a given knot or link. A link is just a collection of knots together in the same space. In this case, the link consists of three loops. There's a yellow loop that goes around here and a blue loop and a red loop here. And these are images from Cypherview by Jack van Waik. And so he starts with this, well, this is a parametric way of drawing this surface, but it's not a very pretty one, but it is parametric. And so what is this? So if you imagine sort of rotating around this cylinder, there's a braid going on. So you've got three strands, a red strand, a yellow strand and a blue strand. And then as we rotate around here, the red strand is doing nothing and the yellow strand and the blue strand are switching over and you carry on and then you produce this surface that goes through them. And then he follows an iterative process, the same sort of thing as for the truffle knots that I showed before. They kind of smooth this out. It's reducing some sort of energy and making something that looks like a nicer surface. So in comparison, this is our, again, this is joint work with Saul Schleimer. This is our version of the same surface. And I won't go into too much detail about where this comes from. I'll just sort of mention, well, so we have a parametrization. There's a sort of implicit description of what this surface is that's easier to understand. So this comes via these Milner fibers. So okay, so what's going on? So really this thing lives in four-dimensional space again, but I'm going to think of four-dimensional space as having two complex dimensions, which are given by W and Z. And it turns out if you work out what are the solutions to this equation here? W cubed plus Z cubed? Well, so the first thing to notice is that if you have a solution to this and you scale W and Z by the same number, you have another solution. And so this sort of has a cone-like structure as it's sort of projecting into the center. So let's chop it off along this sphere. In this case, the sphere in four-dimensional space and then stereographically project down to three-dimensional space and get something. And it turns out where you actually get are three circles. It's the same three circles as in here, at least topologically. And so this thing is some equation for a circle, for three circles. It turns out, okay, so this thing here is the equation of a surface. It's the argument of the thing from before. So you take the argument of a complex number, you get some answer, some real number, and so just something to think about here. So if you take a complex number and you take the argument, this is gonna work unless you're taking the argument of zero, because zero doesn't have an argument as a complex number. So you're gonna get a surface that has boundary, it stops making sense on this equation here. So it has a surface that has boundary on these three loops and you get this cyclic surface. Okay, I won't go into where it comes from, where this parameterization comes from. There's some scary things involving fractional automorphic forms. I'll just show a few sort of pictures of where this tiling of triangles comes from. There's a parameterization from the upper half plane into the sphere and four dimensional space, which then gets so definitely projected down to this. And then this pattern comes from a bunch of conformal transformations to this and then to the triangle. But let's compare the two. So maybe you can tell that they're really the same surface, sort of tricky. So you can see there's this red loop around here and that's the same as this circular loop here. I've tried to line these up in the same way. This yellow loop that's sort of coming towards us and away from us is this loop here. You can just see up here there's the front of the loop and then it goes around behind and so on. So okay, so let's check our aesthetic conditions. So they have the same symmetry actually as objects. So they have the same symmetry of the topological object. You can see this two-fold rotational symmetry. If you look at them above, you see a three-fold rotational symmetry. There's another sort of two-fold axis. This is where I think there's a difference. So this is looking straight onto this red loop here and straight onto this circle here. And the difference is that ours has an actual round circle. Topologists sometimes add the adjective round to circles because some of our circles are not round. But okay, so it's an actual circle whereas this is just a loop here, which is not surprising and we have this nice parameterization. We know all kinds of things about where it comes from and we built it so that in particular it has lots of nice sort of geometric properties which the iterative method can't really hope to find. It's not gonna find these things. So in that sense ours is sort of showing more about the object. On the other hand, so the problem with our method is that it only works for torus knots and links. So these are knots and links you can put on the surface of a torus whereas this iterative method can work with anything. So here's an example. This is the Borromean rings, not the same as the link that I was showing before. Borromean rings, these famous sets of rings where no two of them are linked but if you cut one of them, the other two fall apart. So somehow a symbol of unity. So this is again this image from SciFitView. So this was turned into a mesh and then a 3D print by Bathsheba Grossman, who's one of the real pioneers in 3D printed mathematical artwork. And again, getting this beautiful shape we wouldn't have been able to do because the Borromean rings are not a torus link and so our techniques don't work. We don't have enough math to do it. Whereas the iterative method doesn't care, it just gets on with it. Okay. So one last sort of case of this, what are the choices of turning topology into geometry except rather than talking about topological objects, the difficulty of making a model being that it's topological, here the problem is that it's too high dimensional. So this is the hypercube again, I guess we already saw this picture but maybe I'll go just into a little bit more detail how to go from four dimensions to three dimensions in a way which tells you lots about the objects. So what do we do? So just in a little bit more detail because I went through pretty quickly earlier on. So this is showing how to turn a cube, a three dimensional cube into a two dimensional picture of the cube by analogy with turning a hypercube, the four dimensional cube into a three dimensional picture of the hypercube. So what do we do? So this is a cube inside of a sphere and we radially project this outwards onto the boundary of the sphere and then we stereographically project from that down to the plane. And this is a very nice model of the boundary of the cube. In particular, it's nice because it's a one to one map. There's no overlaps between features on the cube down onto the plane. So I think this is the nicest sort of way to convert from polyhedra or to two dimensional pictures of those polyhedra or from four dimensional polytopes to three dimensional versions of those polytopes. So there's the hypercube. I can hand this around if I can find it. Here we go. So there's the hypercube, a 3D printed version of a hypercube and again this is radially projected onto the sphere in four dimensional space and then stereographically projected down to three dimensional space. And there's a render of the same thing. There are all of these other four dimensional polytopes. These are the four dimensional versions of the three dimensional polyhedra. And they all have sort of bit of etymology that they're named 16 cells. This one has 16 cells like the cells of a honeycomb or biological cells. They're three dimensional regions that are packed together in the same way that the two dimensional polyhedron has faces which are packed together over the surface of a sphere. So whereas you have dodecahedron 12 faces, the 16 cell has 16 cells. And these various very strange four dimensional objects. With Sol again, we made some puzzles based on the 120 cell. 120 cell is somehow the four dimensional version of the dodecahedron. So just to demonstrate one of these things, there are these various chains of dodecahedron inside of the boundary of this four dimensional polytope. And this is sort of, well, let's see. So this is a picture of some of the cells of that dodecahedron. And there's a sort of interesting way that they packed together. So I'll have this round as well. And by taking various chains of those cells, well, you can make a puzzle. So this is not a finished sculpture. This is a puzzle that comes apart into lots of different pieces like this. And then you can try and put it back together again. The mathematics behind this is the 120 cell has a sort of combinatorial version of something from algebraic topology called the Hopf vibration that more or less makes these puzzles work. Anyway, there we go. Okay, so more sort of visualization of four dimensional things, getting them down into three dimensions, more fun than a hypercube with monkeys. This is showing a particular kind of symmetry in four dimensional space that doesn't really exist in three dimensional space. So there are sort of these chains of monkeys that are sort of looping around each other and connected in various ways. There's actually one monkey here for each cube of the hypercube, but this doesn't have the same amount of symmetry as the hypercube because the monkey has less symmetry. So here we go. So here's a monkey inside of one cube of the hypercube, and then he's got six neighbors and the six neighboring cubes. And you can see there's this sort of, if I go from this central monkey to this one over here, there's sort of a left-handed twist. This kind of twist motion is a symmetry that's possible in four dimensional space because there are two planes that you can rotate around simultaneously. And that's not possible in three dimensional space. So you sort of get extra symmetries and you can see them using these. So there's the more fun than a hypercube of monkeys. You can do the same sort of thing with the 24 cell, showing a different kind of symmetry of the four dimensional space. And you can do the same thing with the 120 cell, the sort of 120 monkeys that are kind of twisting around each other. I would show you the interactive computer animation that goes along with this, but my wifi isn't working, so I can't. But you can look it up for yourself, monkeys.hypernome.com. Okay, so now we're well and truly moving away from topological things to here are some interesting projects. So this is a sort of mechanical question. This is, again, joint work with Saul Schleimer. This one I did load up. Did I load this up? No, I did not. Okay, so I'll just have to show you what this is. So there are three different rings here going around like this. And here it is. Here are these three different rings. You can rotate all of them together. The three of them can rotate together. You cannot rotate just one on its own. So it's sort of like gears that are meshed into each other except that they're actually linked with each other. And then there's this extra axle you can put it on that makes it easier to make it go. There we go. So if you just, oops, that's not what it's supposed to do. There we go. If you just sort of hold it vertically and let it run down, then you can see the three rings running against each other. So I'll hand this around so you can take a look. So where did this come from? Here's one story for where this came from. Once you start seeing this motif in graphic design, you can't stop seeing it. So I grew up in Manchester and this was a bus sign advertisement for, there was some sort of three different transportation systems that were gonna come together and make the city work wonderfully. But of course, nothing can move here. When this is rotating this way, this has to be rotating this way because they're meshed and then what way is this supposed to go? And this one over here, the teachers and the students and the parents come together and nothing happens. So three pairwise meshing gears are usually frozen. So there's a challenge. Can you find a triple of pairwise meshing gears that moves? And so this was, this thing is some sort of answer to this question. So how do we design this? So there's a lot of different sort of areas of mathematics that are involved here. And we started out with, well, sorry, this is another inspiration before I go on so that how do we do this? So there's a couple of similar sorts of objects. This is a sculpture called Umbilic Rolling Link by Helen Ferguson. So there are these two parts and they sort of twist into each other. So if you rotate this one around, the other one is forced to rotate I think on a third of the speed. So there's this sort of gearing into each other. If you're ever at Stony Brook University, if you go to the Simon Center for Geometry and Physics, whatever it's called at Stony Brook, there's a very large, like 20 meter tall version of, maybe 15 meter tall version of one of these kinds of sculptures. And this is another one. This is not a gear by Oscar van de Venter. So there's a three, two tour is not here and a two, three tour is not here. And as you rotate one of them, the other one has to rotate at relative speeds of two and three. So we wanted to do the same thing with three of them. So how do we do this? So the very first question was, what is the topology, right? So I wanna have three rings that are linked together and that have gears on them. There are some choices of how those should topologically be linked together. So for example, you might think, oh, we should use the Borromean rings, the most famous of the three ring links. And unfortunately that couldn't work for us. So the reason for this is that it's a result that you cannot make the Borromean rings from round circles. They have to be sort of ovals. And you want round circles because the things are supposed to rotate. So if you rotate your gears and they're ovals, then something is gonna go horribly wrong pretty soon. Whereas if they're basically circles, then hopefully things should work out. So we wanted them to be, so we said we used the three, three Taurus link. So again, it's a little hard to see, but there's a Taurus that sort of goes around through all of these and these curves.