Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and here is your other host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to rehydrate myself after taking a long bike ride yesterday in the Utah desert in July, basically.

KK: That’s not cool. Okay, so here's my here's my story. So the last 10 days of June, my wife and I were out in Vancouver visiting our son. And it was lovely. It was, you know, 65 degrees every day, and we took a side trip to Banff. And which, if you've never been, I cannot recommend enough expect ACULA really beautiful. We had a wonderful time. We took the redeye Wednesday night back from Seattle to Orlando. Thursday morning. I had a sore throat for a couple of days.

EL: Uh oh.

KK: Yeah. I took a COVID test and it was negative. Okay, but I still don't feel great. Thursday I didn’t feel good. Friday morning I'm feeling worse. Take another COVID test. Guess what?

EL: It got you?

KK: It got me. I had a good run. It was two and a half years. But anyway, so

EL: This is a podcast from quarantine, although it's exactly the same as all our other podcasts because we’re always on Zoom anyway.

KK: Yeah, so anyway, here I am. So if I sound a little froggy, that’s why. I'm feeling a lot better and so is Ellen, but yeah, it's been a rough few days in the Knudson house. And it's 100 degrees here and miserable.

EL: Right? Yeah. A little less conducive to fun.

KK: Yeah, yeah. But enough of my petty problem problems, which — look, you know, everybody, if you're not vaccinated, get vaccinated, right? I'm of an age where I can be double boosted. And you know, I just, I got a bad cold. That's it. So get your shots, people. There, enough political — anyway. It shouldn't be political, but somehow it is. So today, we are pleased to welcome Philip Ording to the show. Phillip, why don't you introduce yourself?

Philip Ording: Hi, thank you, Kevin. Thanks, Evelyn, for having me on the show. Yeah, I am a mathematician and a writer and I teach at Sarah Lawrence College in Westchester, New York state. It's not as hot here right now. It was over the weekend.

EL: You’ve got your moments up there, I’m sure. It can be very oppressive.

KK: Yep.

PO: And if you hear some some child background noise, that's because COVID got the summer camp up here. My son came back from upstate Catskills camp, because they had to shut it down after a week.

KK: That’s a bummer.

EL: Oh, man. Yeah, that's rough. Well, we have invited you on the show to talk about your favorite theorem. But first, I wanted to sort of digress to a theorem that you have written about quite extensively.

KK: A lot!

EL: That apparently is not your favorite theorem. But I wanted to invite you on here because of this amazing book 99 Variations on a Proof that came out a few years ago, and I read it, you know, last year, or something, and I kept thinking, “Oh, I should invite him on here.” And it's 99 proofs of a theorem — maybe we might not even call it a theorem, a statement.

PO: It’s generous to call it a theorem.

EL: Yeah, that about the roots of a cubic polynomial, one particular cubic polynomial, and you just talk about it, you have 99 different ways to prove that the roots of this polynomial are 1 or 4. Sorry, a little minor spoiler for this book. I think you’ll still be able to enjoy it. So yeah, can you talk about that? Like the how you had the idea to write this book and kind of maybe tell us about some of the styles of proof or styles of presentation that you've included in here?

PO: Yeah, sure. Thanks for bringing that up. And the, the book, yeah, it’s not my favorite theorem. I chose it almost at random. And the book is really about everything around it. So I was interested in whether or not you could fill a book by thinking about the expressive material of mathematics outside of the content, or almost parallel to the content. I had a friend in grad school who said that he — he said this, I think, over drinks, but with gravitas — that he thought that the thing that mathematics had over other subjects was that it has so much content; you know, if you make one statement in mathematics, it's the kind of thing that not only is very condensed, and is probably the result of a long, long track to study, but it's also something you can return to a lot. So I was interested whether or not even a very humble equation and solution, something that anybody who has been exposed to math would recognize as mathematics, would be able to support that kind of an investigation of something — mathematicians don't talk about style that much. I think philosophers maybe are starting to talk about it more recently — and just carry it through the things that I like about math, the things I don't like, the history, and some of the folklore as well. So the titles are kind of the style for each of those chapters. And they range from things like “Psychedelic” to “Medieval” to “A proof that's found in a book.” And everything in between. So there are proofs from school, from graduate school, from college. There are person different languages. There are proofs that are linguistic, I guess, you could say, that draw attention to the particular notations, or the sound, that the proof reads as. Yeah. And it was a lot of fun. It kind of was a project that once it started, it took over and had a life of its own, which was probably what what got me to the very end of it, even though it was a long project.

EL: Yeah, a long time to be thinking about one cubic equation. I was flipping through today, and I did you know, towards the back, you have a mondegreen, which is one of those kind of misheard lyrics sort of things. “Their omelet: eggs, beer, eel” is the the first line of it. So you know, “their omelet” instead of “theorem: Let”

KK: Yeah. Yeah.

EL: And I read through it. This is one, you just have to concentrate so hard to read it and try to figure out the math version of it, but yeah, so you got, yeah, so many different ways to roll over this equation. So yeah, I hope people will check that out. It's a lot of fun.

PO: That was a that particular proof was a lot of fun to work on. I had some students helping me in the summer, and we just turned over the language of one of the simplest proofs in the book from a mathematical point of view. I think it comes from a kind of sleight of hand that could easily be misunderstood if somebody wasn't paying attention. And I remember when I was in college, I had a friend who said that she liked math, and she'd taken some courses but gave up after a calculus course in which she couldn't understand what the professor was saying. And all she remembers is this professor would get very excited and say “Knees the baby, knees the baby.” And she didn't have any idea what that meant, but she knew it was important. And so I tried to think of things that sounded very similar.

KK: Yeah.

PO: And I think it's an experience that everybody has, at some point, you're sitting in a talk and you kind of are reading the person's emotions as much as you're reading, you're listening, to that particular details of the techniques that are used, and there's often things that are lost in that channel. So it was fun to make fun of that phenomenon that I think most people who have studied mathematics at a certain level have experienced.

KK: What’s knees the baby? I can't figure it out.

PO: I still don't know. If anyone figures it out.

EL: Listener submissions.

PO: Multivariable calculus

KK: Okay, well, anyway, that’s, okay. We’ll try to figure it out offline.

EL: We’ll try not try to be thinking about that the whole time we’re recording this.

KK: That’s right. Okay, so you've told us what isn't your favorite theorem. You do have an actual favorite theorem. Why don’t you tell us about it?

PO: I do. And I love this question, because it’s, to me, it's very appealing. It's also very challenging. It's not the first time, actually, somebody asked me for my favorite theorem. The first time it wasn't for a podcast, it was for bathroom. I had a friend, some family friends, that had remodeled their apartment and they thought that this bathroom they had designed, it was like black paint or wallpaper inside. And they thought it would be fun to have their mathematician friend make a theorem or some kind of statement of gravitas in the bathroom. Or maybe they just thought it would go well with the marble sink or something. I'm not sure. But I thought about it for a long time. And I thought, okay, you know, is this going to be like something that is, I think, the most important piece of mathematics? Or is it going to be something really personally meaningful? Or maybe, like, I was in a bathroom at a bar once downtown and some, I think it was a grad student at NYU maybe, had done, like, the de Rham cohomology sequences, and I thought that looked cool. Maybe it should just be graffiti. But yeah, I sort of never got around to it, because I felt like I didn't really attach that much meaning to particular theorems. But anyway, what I came up with is something that's instead of a theorem, it's an idea. So it's called the Erlangen program.

KK: Okay.

PO: And it's credited to Felix Klein, German mathematician, 19th century. And a program — yes, so it's kind of a project or an assessment of the state of mathematics at the time, but also a direction forward. So to say what it is, it's, I reread the Erlangen program, which is a lecture that he prepared, actually. It’s named after the university that he was going to be teaching at, a professor. And actually, there are no theorems in it. The thing that's maybe closest is a statement that says that if you want to learn about geometry, you can find everything that you want to know by studying the motions of geometric objects in that space. So what does that mean, to give some example — Have you heard of this, by the way? I don’t —

EL: I definitely knew the name. I could not have told you. The first thing of what it actually was.

KK: I think it's actually been a remarkably influential idea for the last 150 years, right? I mean, I think it's driven a lot of of what happened in the 20th century.

PO: I think so. My background is in geometry and topology. So I might, you know, I might be biased.

KK: Yeah, me too.

PO: Yeah, so, I mean, to give an example, if you wanted to understand, say, points in the plane, the idea is that you can understand points just as well as anything that you might do with, say, intersections of lines, or coordinates, or quadrants, or distance, by just studying, say, rotations of the plane that fix that point. Or collections of rotations that fix a set of points. Or if you wanted to study line geometry in the plane, you could study, well, I don't know if this qualifies as a motion, but the transformation takes every point on one side of the line to the point reflected across the line. So just studying reflections in the plane that fix axes, you can really express everything you'd want to know about lines in the plane. And just to give some sense of, you know, why is this interesting, and not just a complication, so if you wanted to have — say you had two reflections, and you compose them, so you reflect across your first line, and then you reflect across the second line — and that's two operations, you can you can combine them, you're going to get something back — the result is going to be a rotation about — if there's a point where those lines meet, when lines typically do — you're going to get a rotation about that point. And when I first sort of started to get this idea, and use it, it's sort of a yoga, you get used to it after a while, of going back and forth between the world of geometric objects and the world of the structure of motions, or the group of motions. I loved it, and it was very useful, and it seemed like it joined together areas of my brain that were were divided before. So I think that's why it’s my favorite, but I could say more than that too.

EL: Yeah, so where did you first encounter it?

PO: I think it was in my senior year in undergraduate, I was given a project by my senior thesis advisor, wonderful professor and Troels Jørgensen, and he cut his teeth studying hyperbolic geometry. So one of the things that I think is really amazing — and this was maybe Klein's motivation for introducing the Erlangen program — is that if you have many different geometries, so if you've been introduced to the idea that there isn't an absolute singular geometry out there, what we call now Euclidean geometry, that used to be just geometry, and now there are non Euclidean geometries or even wilder things, like topology that we don't call a geometry, exactly, then you might want to know, how are you going to do anything in those weird spaces? And if you have the Erlangen program, it's telling you, as long as you can understand the structure of the transformations, which we think of as generalization, or restriction of congruence. So congruence is the word we usually use for those motions of the plane that that fix them, that preserve them. Okay, so what I had to do is understand something about the hyperbolic, the non-Euclidean analog of a pyramid, tetrahedron is the term. And so tetrahedra are kind of, like, the dumbest of the platonic solids, I mean, maybe it's got four sides, four corners, it's a good shape for a die if you want to have just four options, because it's so symmetric. But when you go into the this world of negatively curved space, you can study tetrahedra that are formed by points that are at infinity, meaning that you don't see, actually, a finite object in front of you, you just see these sets of lines that are going off into space. But it turns out, they bound a finite volume, which is very, very bizarre. And if you want to understand anything about them, you're kind of left scratching your head, if you're just going to be limited to the tools of Euclidean geometry, measuring things like area and volume in the traditional way. It turns out that if you take that idea of, okay, I'm not going to think about the tetrahedron as made up of lines, I'm going to think of it as constituted by rotations in that space, it turns out, you can write down those rotations, that whole set, quite easily using once you've gotten used to using some matrix algebra, so kind of higher dimensional generalization of the regular algebra on the real numbers. And that — so combining those kinds of representatives of lines, you can just go to town computing things, and you can compute intersections, just like I said, with these compositions of reflections in planes, instead of say, instead of Euclidean planes, now hyperbolic planes. So that's a long answer to where I first encountered it. And yeah, I wouldn't have known really how to approach the problem I was assigned if I hadn't had those tools.

KK: Sure. Yeah. I mean, trying to think about the actual geometry of 3d hyperbolic space is sort of weird, right? I mean, like you say, you can't see it. You can, but you can't, and you might draw it, but you have to remember the metric is different and the distances don't look — things that look finite aren’t. And, yeah, it's a very bizarre feeling to try to move into that space.

EL: I love those representations of hyperbolic space. I mean, they're stunning. And they produce some of the most interesting kinds of ornaments out there. But it is hard to know where to start when you're just looking at these dazzling representations or models. And I guess, the other thing that I was made to understand was that this Erlangen program is a little bit like a Rosetta Stone, because it's not only telling you how you can work within any given geometry, by studying its associated group of transformations, but if you know that a geometry has among its transformations a subset or a subgroup that has this kind of coherence, then that becomes sort of a sub-geometry. And you can relate them. And I think this was going back to Klein, when he was up to — they had all these great methods in projective geometry, one of the kind of early alternatives to Euclid, and they were able to use those to study and relate geometries one to another in this kind of zoo that exploded in the 19th century.

EL: Yeah, and that's, you know, thinking about, I guess, when most of us go to grad school in math, you know, one of the powerful things that we do is see this relationship between the algebra, you know, group actions, and geometric objects in some way. And so, this is building from that connection, I guess, from the Erlangen program. Is that somewhat right?

PO: Exactly. Yeah. I mean, there are other connections between geometry and algebra, right? I mean, we learn with Descartes, and once we start plugging in coordinates for points, and then writing out lowly cubic equations for expressing pictures of curves. So, you know, I think that when — you know, the process of learning math is usually, even though math seems like a very strict discipline, it has its own subfields. And those subfields don't always work together in obvious ways. So we tend to teach them in by these isolated textbooks, you know, algebra, or group theory, and analysis and geometry, and so forth, or calculus. And I think Klein was very much a synthesizer. And I like this idea. As was William Thurston, who was the person setting out programs for geometry when I was a grad student, and I think we're still untying some of the things that that he that he set out.

EL: Definitely.

KK: Yep. All right. So another thing we do on this podcast is we invite our guests to pair their theorem, or in this case program, with something. What do you think pairs well with the Erlangen program?

PO: Oh, yeah. Okay. So, yeah, this is something I thought about. And along the lines of the question of your favorite theorem, I went to a kind of personal, like, trying to think about, taking this question very sincerely. Because I like the idea that there might be a connection between our personal taste and the things that we do. I mean, it's a high bar for mathematics, I guess, because we're working with abstract things. But there's a piece of sculpture that I would pair with this theorem that's in the Museum of Modern Art. It's by a sculptor, it’s a postmodern piece by Richard Serra. He’s an American sculptor. From 1967, I think. And it doesn’t — it sits on the floor, it doesn't have a pedestal. And it's not much to — you might you might step on it by accident, if you didn't, if there wasn't a cord around it or something, but it's a rectangular piece of rubber, like black vulcanized rubber. And it looks a little bit like, it has a graceful form, it rises in the middle and then descends to the floor. It looks like maybe like a cowl over a monk at vespers or something, I don't know. Or like the hills on the screen, the green screen behind Evelyn of Utah. And the name of the piece kind of says it all. It's called “To Lift.” And it's part of a series that that he made. I think he was inspired first by dancers that he was seeing, choreographers at the time, in downtown Manhattan. But the idea, he made this this kind of Erlangen program for himself that was called the verb list. And it starts out, like, to crease, to fold, to roll, to twist, to torque. And he made pieces for many of these that kind of instantiate this verb by applying it very simply to material. So it's not — a lot of them are I think were rubber, but others are lead or steel. Later on, he got into more. And yeah, so the thing about the Erlangen program, besides its connective properties between different disciplines, I really like the way that it takes things that I've thought of as more solid, geometric objects, concrete things, and then trades them with verbs, with actions or motions or transformations. So it makes geometry much more dynamic, anyway in my way previously of thinking of it as this static world that you kind of enter and measure things. Instead, it's this world where you have all of these permutations of the space that you're looking at. And you kind of play. I think of it as more playful. But that's the sense I got from, I had a chance to work as a grad student in Richard Serra’s studio. And it's very serious business. I don't want to pretend like it's kindergarten, but sometimes it has that feeling of like, this is a proposition, can we make a form that embodies this movement, or this daily kind of task or transformation? And he made videos around that same time, “Hand Catching Lead,” and they're very simple, but they just act on you in a way because you start imagining your own participation with the material world around you in this way. So that would be my pairing.

EL: Yeah, well, your description I haven't, you know, I'll look it up later, to see if I can find a picture of the sculpture. But your description, you know, it sounds like you had a sheet of rubber on the ground and just lifted it up. And it does sound like the simplest thing. I'm sure it wasn't the simplest thing to actually make something that that gives you that feeling. But yeah, I guess that there's that — when you see it, you immediately understand what the aim was. And it's funny, when you're listing that these other verbs that that he did, so many of them also have mathematical things that you can almost imagine a textbook that's telling you, like, Okay, this is what a Dehn twist is, and shows you a simple example of that. This is what something with torsion is or this, I don't remember exactly, all the words, but even “lift” has a mathematical meaning as well. And so these choreographic and artistic things, also connecting to the these mathematical ideas we have.

PO: I love those those suggestive verbs and those little diagrams in kind of combinatorial, or cut and paste topology. Yeah, definitely.

KK: So you mentioned that you worked in the sculptor’s studio as a graduate student. How did that come about?

PO: Oh, yeah, that's right. So it was kind of one of those “only in New York” moments. I think his studio manager reached out to the tutoring email at Columbia University where I was a graduate student. And I think they were kind of stuck with the communication between the studio and the engineers that make the large-scale sculpture that that Serra's known for most widely today. So there was some kind of communication breakdown there. And they thought, you know, we're not understanding what the engineers are telling us about what is not possible, and what is possible. And it had gone, it had left the converse the bounds of what was engineering-ly, possible, like, it was actually what was formally possible. So at that point, you know, this was in the early 2000s, he was already working at a very sophisticated level, in the sense that, to make some of the large-scale forms, they were using cutting-edge architectural design and engineering tools. And once they had a design, they could send it out, and people would sort of develop it to a point where it would be stable, rigorous, and so forth, pass the test. So yeah, when I saw that, I jumped on it because I've kind of always been interested in places where mathematics might speak to the arts. And had at that point already, I think I was tutoring a professor in the architecture department. And his name is Peter Macapia. And he was, he became a very good friend, but he also kind of, once he knew I was going to go try to meet Serra, he gave me a bit of a crash course on why Serra might be interested to talk to a geometer or mathematician. And so I don't know if that's the reason that I was asked and invited to work in the studio, because I had done a little homework? It might have just been that — sometimes I would joke that I was a math therapist, a geometric therapist. I would listen to the things that they were trying to do and ask them why they talked about them the way they did. And often I think they came up with their own solutions. But certainly it was a formative experience for me. And part of the reason I wrote the variations was to see if there was a way to take seriously reversing the direction from instead of applying math to art, to see if I could borrow some of the ideas of art-making to do math.

EL: Yeah, that is such a great opportunity you had as a grad student, even, to get to do that kind of thing. And yeah, it’s, I think both of us, a lot of people love that murky boundary between math and art and the ways that, you know, we can apply the very — I guess, I think of math often as a theoretical art, we're doing a similar kind of thing. I mean, I know this isn't original, but it's the similar kinds of thing. We've got aesthetics, we've got rules of our discipline, and like, we apply it to abstract objects, and using that to apply it to concrete objects after that is very cool.

KK: Yeah.

EL: Thanks. I'm looking forward to looking up that sculpture. Yeah, after we get off the call.

KK: All right. So we've talked about your book, we always like to give our guests a chance to plug anything else, or where we might find you on the worldwide web or anything like that.

PO: Oh, sure. Yeah. The book came out in paperback last fall, so you can find it even more affordably priced, I would say, from Princeton University Press. I want to plug my friend's book. I don't know if you've had Jessica Wynne on.

KK: No.

PO: She’s a photographer that made a beautiful book called Do Not Erase.

KK: I have this book.

PO: Yeah. Okay. Yeah. So she was going to ask me to make a board and for her to photograph and I wrote — I used the maximum word count, I think — to write a little bit about my senior thesis advisor Troels. So if you want to page through that, but the book is amazing, and there are many celebrated mathematicians boards in there. So that's fun. I think your listeners would enjoy that if they haven't seen it already.

EL: Yeah. We'll put a link to that in the show notes. Yeah, thank you for joining us. I really enjoyed getting to talk to you and think more about that connection between math and art.

KK: Yep.

PO: Thank you for having me. It's been really fun to talk to you.

KK: Yeah. Thanks, Philip, it’s been great.

[outro]

In this episode, we talked with Philip Ording, a mathematician at Sarah Lawrence College, about the Erlangen program. Attached are some related resources you might enjoy.
Ording's website
The website for his book, 99 Variations on a Proof
John Baez's links related to the Erlangen program, including Klein's original paper on the topic
Royce Nelson's page about 3-dimensional hyperbolic geometry
Jessica Wynne's book Do Not Erase about mathematicians and their chalkboards