Kevin Knudson: Welcome to My Favorite Theorem, math podcast and so much more. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined today by your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer, usually based in Salt Lake City, but today coming from the Institute for Computational and Experimental Research in Mathematics at Brown University in Providence, Rhode Island, where I am in the studio with our guest, Edmund Harriss.

KK: Yeah. this is great. I’m excited for this, this new format where we're, there's only two feeds to keep up with instead of three.

EL: Yeah, he even had a headphone splitter available at a moment's notice.

KK: Oh, wow.

EL: So yeah, this is—we’re really professional today.

KK: That’s right.

EL: So yeah, Edmund, will you tell us a little bit about yourself?

Edmund Harriss: I was going to say I'm the consummate unprofessional. But I'm a mathematician at the University of Arkansas. And as Evelyn was saying, I'm currently at ICERM for the semester working on illustrating mathematics, which is an amazing program that's sort of—both a delightful group of people and a lot of very interesting work trying to get these ideas from mathematics out of our heads, and into things that people can put their hands on, people can see, whether they be research mathematicians or other audiences.

EL: Yeah. I figured before we actually got to your theorem, maybe you could say a little bit about what the exact—or some of the mathematical illustration that you yourself do.

EH: So, yeah, well, one of the big pieces of illustration I've done will come up with a theorem,

EL: Great.

EH: But I consider myself a mathematician and artist. And a part of the artistic aspect, the medium—well, both the medium but more than that, the content, is mathematics. And so thinking about mathematical ideas as something that can be communicated within artwork. And one of the main tools I've used for that is CNC machines. So these are basically robots that control a router, and they can move around, and you can tell it the path to move on and carve anything you like. So even controlling the machine is an incredibly geometric operation with lots of exciting mathematics to it. When I first came across—so one of the sorts of machine you can have is called a five-axis machine. That's where you control both the position, but also the direction that you're cutting in. So you could change the angle as its as its cutting. And so that really brings in a huge amount of mathematics. And so when I first saw one of these machines, I did the typical mathematician thing, and sort of said, “Well, I understand some aspects of how this works really well. How hard can the stuff I don't understand be?” It took me several years to work out just how hard some of the other problems were. So I've written software that can control these machines and turn—in fact, even turn a hand-drawn path into a something the machine can cut. And so to bring it back to the question, which was about illustrating mathematics: One of the nice things about that idea is it takes a sort of hand-drawn path—which is something that's familiar to everyone, especially people in architecture or art, who are often wanting to use these machines, but not sure how—and the mathematics comes from the notion that we take that hand-drawn path, and we make a representation of that on the computer. And so you've got a really interesting function, they're going from the hand drawn path through to the the computer representation, you can then potentially manipulate it on the computer before then passing it again back to the machine. And so now the output of the machine is something in the real world. The initial hand-drawn path was in the real world, and we sort of saw this process of mathematics in the middle.

Amongst other things, I think this is a really sort of interesting view on a mathematical model. you have something in the real world, you pull it into an abstract realm, and then you take that back into the world and see what it can tell you. In this case, it's particularly nice because you get a sense of really what's happening. You can control things, both in the abstract and in the world. And I think, you know, to me that really speaks to the power of thinking and abstraction of mathematics. Of course, also controlling these machines allows you to make mathematical models and objects. And so a lot of my my work is sort of creating mathematical models through that, but I think the process is a more interesting, in many ways, mathematical idea, illustration of mathematics, that the objects that come out

KK: Okay, pop quiz. What's the configuration space of this machine? Do you know what it is?

EH: Well, it depends on which machine.

KK: The one you were describing, where you can where you can have the angles changing. That must affect the topology of the configuration space.

EH: So it’s R3 crossed with a torus.

KK: Okay.

EH: And so even though you're changing the angle of the bit, you really need to think about a torus. It's really also a subset of a torus because you can't reach all angles.

KK: Sure, right.

EH: But it is a torus and not a sphere.

KK: Yeah. Okay.

EH: So if you think about how to get from one position of the machine to another, you really want to—if you think about moving on a sphere, it's going to give you a very odd movement for the machine, whereas moving along a torus gives the natural movement.

KK: Sure, right. All right. So, what's your favorite theorem?

EH: So my favorite theorem is the Gauss-Bonnet.

KK: All the way with Gauss-Bonnet!

EL: Yes. Great theorem. Yeah.

EH: And I think in many ways, because it speaks to what I was saying earlier about the question: as we move to abstraction, that starts to tell us things about the real world. And so the Gauss-Bonnet theorem comes at this sort of period where mathematics is becoming a lot more abstract. And it's thinking about how space works, how we can work with things. You're not just thinking about mathematics as abstracted from the world, but as sort of abstraction in its own right. On the artist side, a bit later you have discussion of concrete art, which is the idea that abstract art starts with reality and then strips things away until you get some sort of form, whereas concrete art starts from nothing and tries to build form up. And I think there's a huge, nice intersection with mathematics. And in the 19th century, you've got that distinction where people were starting to think about objects in their own right. And as that happens, suddenly this great insight, which is something that can really be used practically—you can think about the gospel a theorem, and it's something that tells you about the world. So I guess I should now say what it is.

EL: Yeah, that would be great. Actually, I guess it must have been almost two years ago at this point, we had another guest who did choose the Gauss-Bonnet theorem, but in case someone has not religiously listened to every single episode—

KK: Right, this was some time ago.

EL: Yeah, we should definitely say it again.

EH: So the gospel out there links the sort of behavior of a surface to what happens when you walk around paths on that surface. So the simplest example is this: I start off, I’m on a sphere, and I start at the North Pole and I walk to the equator. At the equator, I turn 90 degrees, I walk a quarter of the way around the Earth, I turn 90 degrees again, and I walk back to the North Pole. And if I turn a final 90 degrees, I’m now back where I started facing in the same direction that I started. But if I look at how much I turned, I didn't go through 360 degrees. So normally if we go around a loop on a nice flat sheet, if you come back to a started pointing in the same direction, you've turned through 360 degrees. So in this path that I took on sphere, I turned through 270 degrees, I turned through too little. And that tells me something about the surface that I'm walking on. So even if I knew nothing about the surface other than this particular loop, I would then know that the surface inside must be mostly positively curved, like a sphere.

And similarly if I did the same trick, but instead of doing it on the sphere, I took a piece of lettuce and started walking around the edge of a piece of lettuce, in fact, I’d find that when I got back to where I started, I’d turned a couple of hundred times round, instead of just once, or less than once, as in the case the sphere. And so in that case, you've got too much turning. And that tells you that the surface inside is made up of a lot of saddles. It's a very negatively curved surface. And one of the motivations of creating this theorem for Gauss, I believe—I always find it dangerous to talk about history of mathematics in public because you never know what the apocryphal stories are—one of the questions Gauss was interested in was not whether or not the earth was a sphere. Well, actually, whether or not the earth was a sphere. So not whether or not it was round, or topologically a ball, but whether it was geometrically really a perfect sphere. And now we can go up into space and have a look back at the earth, and so we can sort of do a three-dimensional version of that, regard the earth as a three dimensional sphere, but Gauss was stuck on the surface of the earth. So he really had this sort of two dimensional picture. And what you can do is create different triangles and ask, for those triangles, what’s the average amount of curvature? So I look at that turning, I look at the total area, the size of the triangle, and ask does that average amount of curvature change as I draw triangles in different places around the earth? And at least to Gauss’s measurements—again, in the potentially apocryphal story I heard—the earth appeared to be a perfect sphere up to the level of measurement, they were able to do then. I think now, we know that the earth is an oblate spheroid, in other words, going between the poles is a slightly shorter distance than across the equator.

KK: Right.

EH: I believe that it was only a couple of years ago that we managed to make spheres that were more perfect than the Earth. So it was sort of, yeah, the Earth is one of the most perfect spheres that anyone has experience of, but it's not quite a perfect sphere when your measurements are fine enough.

KK: So what's the actual statement of Gauss-Bonnet?

EH: So, the statement is that the holonomy, which is a fancy word for the amount of turning you do as you go around a path on the surface, is equal to—now I’m forgetting the precise details—so that turning is closely related to the integral of the Gaussian curvature as you go over the whole surface.

KK: Right.

EH: So it's relating going around that boundary—which is a single integral because you're just moving around a path—to the double integral, which is the going over every point in the surface. And the Gaussian curvature is the notion of whether you're like a sphere, whether you're flat, or whether you're like a saddle at each individual point.

KK: And the Euler characteristic pops up in here somewhere if I remember right.

EH: Yeah. So the version I was giving was assuming that you’re bounding a disk in the surface, and you can do a more powerful version that allows you to do a loop around something that contains a donut.

EL: Yeah, and it relates the topology of a surface, which seems like this very abstract thing, to geometry, which always seems more tangible.

EH: Yeah. Yeah, the notion that the total amount of curvature doesn't change as you shift things topologically.

EL: Right.

EH: Even though you can push it about locally.

KK: Yeah. So if you're if you're pushing it in somewhere, it has to be pooching out somewhere else. Right? That's essentially what's going on, I guess. Right?

EH: Yeah. You know, another thing that's really nice about the the Gauss-Bonnet theorem, it links back to the Euler characteristic and that early topological work, and sort of pulls the topology in this lovely way back into geometric questions, as Evelyn said. And then the Euler characteristic has echoes back to Descartes. So you're seeing this sort of long development of the mathematics that's coming out. It’s not something that came from nowhere. It was slowly developed by insight after insight, of lots of different thinking on the nature of surfaces and polyhedra and objects like that.

EL: Yeah. And so where did you first encounter this theorem?

EH: So this is rather a confession, because—when I was a undergraduate, I absolutely hated my differential equations course. And I swore that I would never do any mathematics involved in differential equations. And I had a very wise PhD advisor who said, “Okay, I'm not going to argue with you on this, but I predict that at some point, you will give me a phone call and say you were wrong. And I don't know when that will be. But that's my prediction.”

KK: Okay.

EH: It did take several years. And so yes, many years later, I'd learned a lot of geometry, and I wanted to get better control over the geometry. So I sort of got into doing differential geometry not through the normal route—which is you sort of push on through calculus—but through first understanding the geometry and then wanting to really control—specifically thinking about surfaces that were neither the geometry of the sphere, the plane, or the hyperbolic plane. Those are three geometries that you can look at without these tools. But when you want to have surfaces that have saddles somewhere and positive curvature—I mean, this relates back to the CNC because you're needing to understand paths on surfaces there in order to take our tool and produce surfaces.

And so I realized that the answers to all my questions lay within differential equations, and actually differential equations were geometric, so I was foolish to dislike them. And I did call up my advisor and say, “Your prediction has come true. I'm calling you to say I was wrong.”

EL: Yeah.

EH: So basically, I came to it from looking at geometry and trying to understand paths on surfaces and realizing from from there that there was this lovely toolkit that I had neglected. And one of the real gems of this toolkit was this theorem. And I think it's a real shame that it's not something that's talked about more. I’ve said this is a bit like the Sistine Chapel of mathematics. You know, most people have heard of the Sistine chapel.

KK: Sure.

EH: Quite a lot of people can tell you something that's actually in it.

EL: Right.

EH: And slowly, only a few people have really seen it. And certainly a very few people have studied it and really looked and can tell you all the details. But in mathematics, we tend to keep everything hidden until people are ready to hear the details. And so I think this is a theorem that you can really play with and see in the world. I mean, it's not a—there are some models and things you can build that are not great for podcasts, but it's something you can really see in the world. You can put it put items related to this theorem into the hands of people who are, you know, eight or nine years old, and they can understand it and do something with it and and see how what happens because all you have to do is give people strips of paper and ask them to start connecting them together, just controlling how the angles work at the corners.

And depending on whether those angles add up to less than 360 degrees—well not the angles at the corner—depending on whether the turning gives you less than 360, exactly 360, or more than 360, you're going to get different shapes. And then you can start putting those shapes together, and you build out different surfaces. And so you can then explore and discover a lot of stuff in a sort of naive way You certainly don't need to understand what an integral is in order to have some experience of what the Gauss-Bonnet theorem is telling you. And so this is sort of it's that aspect, that this is something that was always there in the world. The sort of experiments, the sort of geometry you can look at, through differential geometry and things like the Gauss-Bonnet, that was available to the whole history of mathematics, but we needed to make a break from just geometry as a representation of the world to then sort of step back and look at this result that is a very practical, hands-on one.

You know, if you really want to control things, then you do need to have solid multivariate calculus. So generally, the three-semester course of calculus is often meant to finish with Gauss-Bonnet, and it's the thing that's dropped by most people at the end of the semester, because you don't quite have time for it. And there's not going to be a question on the test. But it's one of those things that you could sort of put out there and have a greater awareness of in mathematics. Just as: this is an interesting, beautiful result. I would say, you know, it's one of humanity's greatest achievements to my mind. You don't have to really be able to understand it perfectly in order to appreciate it. You certainly—as I proved you—can appreciate it without being able to state it exactly.

EL: Yeah, well, you've sold me—although, as we've learned to this podcast, I'm extremely open—susceptible to suggestion.

KK: That’s true. Evelyn's favorite theorem has changed multiple times now. That's right.

EL: Yeah. And I think you brought it back to Gauss-Bonnet. Because when when we had Jeanne Clelland earlier, who said Gauss-Bonnet, I was like, “Well, yeah, I guess the uniformization theorem is trash now”—my previous favorite theorem, but now—it had been pulled over to Cantor again, but you’ve brought it back.

KK: Excellent. All right, so that's another thing we do on this podcast is ask our guest to pair their theorem with something. So Edmund, what pairs well with Gauss-Bonnet?

EH: Well, I have to go with a walnut and pear salad.

KK: Okay.

EL: All right.

KK: I’m intrigued.

EH: Well, I think I've already mentioned lettuce.

EL: Yes.

EH: Lettuce is an incredibly interesting curved surface. Yeah. And then you've got pears, which gives you—

KK: Spheres.

EH: A nice positively curved thing. But they're not just boring spheres.

EL: Yeah.

EH: They have some nice interesting changes of curvature. And then walnuts are also something with very interesting changing curvature. They have very sharply positively curved pieces where they're sort of coming in but then they've got all these sort of wrinkly saddley parts. In fact, one of the applications of the Gauss-Bonnet theorem in nature is how do you create a surface that sort of fits onto itself and fills a lot of space—or doesn't fill that much space but gives you a very high surface area to volume ratio. So walnut is an example—or brains or coral—you see the same forms coming up. And the way many of those things grow is by basically giving more turning as you grow to your boundary.

KK: Right.

EH: And that naturally sort of forces this negatively-curved thing. So I think the salad really shows you different ways in which this surface can—the theorem can affect the behaviors of the surfaces.

EL: Yeah, well, what I want now is something completely flat to put in the salad. Do you have any suggestions?

KK: Usually you put goat cheese in such a thing, but that doesn't really work.

EL: That’s—well, parmesan. You could shave paremesan.

EH: Yeah, shavings of parmesan. Or maybe some thin-cut salami.

EL: Okay.

EH: And so even though those things would bend over—I mean, we’re now on to a different theorem of Gauss, and I don’t meant to corrupt Evelyn away—but you know, when you thinly cut the salami, it can it can bend but it doesn't actually change its curvature.

KK: Right.

EH: Your loops on that salami are going to have the same behavior that they had before. And I guess I should also say that I did create a toy that makes that paper model that I talked about easier to use. You don't have to use tape. You can hook together pieces. And so the toy is called Curvahedra.

KK: I was going to say, you should promote your toy. Yeah.

EH: I’m terrible at self-promotion, yes.

EL: We will help you. Yes, this is a very fun toy. I actually got to play with it for first time a few weeks ago when you did a little short thing and I think when I had seen pictures of it before I thought it was not going to be as sturdy as it is. But this is—yeah, it's called Curvahedra—look it up. It’s these quite sturdy—you know, you don't need to worry about ripping the pieces as you put them together—but you can create these things that look really intricate, and you can create positive curvature, or flat things, or negative curvature in all these different conformations. It's a very fun thing to play with.

EH: And it is a sort of physical version of exactly the Gauss-Bonnet theorem. As you hook together pieces, you're controlling what happens on a loop. And then as you put more of those loops together, you can get a variety of different surfaces, from hyperbolic planes to spheres to—of course, kids have made animals and creatures with it. So you get this sort of control. In fact, it's one of those things that, you put it into the hands of kids, and they do things that you didn't think were really possible with it because their ability to play with these ideas and be free is always so inspiring. So that's what I said, this is a theorem that you can—people can understand as something in the real world. And then you can tell the story of how this understanding of the world is linked directly back to abstract, esoteric mathematics, of the most advanced sort.

KK: Right. One of my favorite things about Curvahedra, though, is the video that you put online somewhere—I think was on Twitter—of it popping out of your suitcase, like you compressed it down into your suitcase to travel home one time?

EH: Yes, I have a model that's about to a two-foot cube. And so you can’t travel with that easily, but it can compress very small. And that same object has been in my suitcase and other things several times, and it's now sitting in my office here.

KK: That’s great fun. And also you've made similar models out of metal, correct?

EH: Yes. So the basic system—not the big one you can crush down to put into suitcases.

KK: No, certainly not.

EH: I’ve made a couple of the spheres. And we're currently working on a proposal to go outside the Honors College at the University of Arkansas. That grew out of a course—it was a design that was created from Curvahedra and other inspirations—by a course I taught with Carl Smith, who is a landscape architect in our landscape architecture school. And so there's going to be—hopefully at some point there's going to be a 12-foot tall Curvahedra-style model outside the Honors College at University of Arkansas.

KK: Very nice.

EL: Nice.

KK: Yeah, this has been great fun. Anything else we want to talk about?

EL: Yeah, well, do you want to say a website or Twitter account or anything where people can find you online?

EH: So I’m actually @Gelada on Twitter, and there is @Curvahedra, and my blog, which is very rarely updated, but has some nice stuff, is called Maxwell Demon.

EL: Yeah, and can you spell your Twitter?

EH: Yes, so Gelada is spelled G-E-L-A-D-A. They are baboons in Ethiopia, or it’s a cold beer in Brazil. I discovered that latter one after being on Twitter, and I regularly get @-ed by people in Brazil, who were not wanting to talk to me at all, but they're asking each other out for beers.

EL: Ah.

EH: And yeah, so then there's also curvahedra.com, where you can get that toy.

EL: Cool. Thanks for joining us.

KK: Yeah, thanks Edmund.

EH: Thank you.

[outro]

On today’s episode, we were pleased to talk with Edmund Harris, a mathematician and mathematical artist at the University of Arkansas, who is our second guest to sing the praises of the Gauss-Bonnet theorem. Below are some links you might find useful as you listen to the episode.  

Edmund’s Twitter account, @Gelada 

His blog, Maxwell’s Demon 

The website and Twitter account for Curvahedra, the toys he makes that help you explore the Gauss-Bonnet theorem and just have a lot of good fun with geometry 

Our episode with Jeanne Clelland, who also chose the Gauss-Bonnet theorem 

Edmund and Evelyn both attended the Illustrating Mathematics program at the Institute of Computational and Applied Mathematics (ICERM). The program website, which includes videos of some interesting talks at the intersection of math and art, is here.