My Favorite Theorem

Kevin Knudson: Welcome to my favorite theorem, the math podcast with no quiz at the end. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host, fabulous as usual, with a really good zoom background.

Evelyn Lamb: Yes, I am Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, and I'm celebrating fall with a nice zoom background that none of our listeners can see of a lovely bike trail near me with decked out in fall colors. So I hope everyone appreciates that.

KK: So judging from Instagram this weekend, you took a train trip somewhere, and it looked really cool.

EL: I did. Yeah, because I'm a freelancer and have quite a bit of schedule flexibility, I do silly things like take the Amtrak for 24 hours to go to Omaha for the weekend and then take it back. And yeah, it was, it was fun.

KK: Why Omaha, just out of curiosity?

EL: Singing, which shouldn't surprise people who know me.

KK: Sure, yeah. Well, I did none of that. I was on an NSF panel last week. That was my big,

EL: Slightly different adventure.

KK: You know, but it's important work. I mean, it really is. And and our listeners, if you happen to get asked to be on an NSF panel, you should do it. It's very interesting and important work. So anyway, now I'm back home, where I’m doing — no, it was here. It was a Zoom panel, but also that was the extent of my week last week. Now that I'm in the Dean's office, which I don't think we've actually mentioned. So I was chair of my department for six years. Now I am an interim associate dean in our college, and one of my responsibilities is that I'm in charge of the college tenure promotion committee, and that committee meets three days a week, at 8am.

EL: Oh, that's great.

KK: That is not my jam at all. And then so twice last week I had T & P first thing in the morning, followed by, you know, seven hours of NSF proposals.

EL: Yeah.

KK: But anyway, I’m glad to be back on Zoom today to welcome our guests. So we're pleased to welcome Jeremy. All Jeremy, why don't you introduce yourself?

Jeremy Alm: Thanks, Kevin and Evelyn. My name is Jeremy Alm. I am Associate Dean for programs in the College of Arts and Sciences at Lamar University in Beaumont, Texas, where it is still quite hot, even the last week of October.

EL: Yes.

JA: Before that, I was department chair at Lamar, and before that, I was department chair at a small college in rural Illinois called Illinois College.

KK: Yeah, cool.

EL: And what's your field of math?

JA: My field of math, so I wrote a dissertation in algebraic logic and universal algebra. Decided I wasn't very good at algebra, started learning combinatorics, so now I solve combinatorial problems that arise in algebraic logic.

EL: Nice. I do think it's funny, if I can interrupt for a moment. It is funny how grad school can do this to us, where you literally wrote a dissertation in algebra. And so what this means, in an objective sense, is, like of the billions of people in the world, you're probably in like the top 100th or 1,000th of 1% of people in knowledge of algebra. And yet your conclusion is “I'm not very good at algebra,” so I have had a similar conclusion that I drew about my field of math as well. So just interesting fact about higher — PhD programs in general, I think,

JA: Yeah. Well, in my case, there's some further evidence, and that is that my main dissertation result was a conditional result, and about four years after I graduated, a Hungarian graduate student proved that my condition, like my additional hypothesis, held in only trivial cases.

EL: Oh, that is a blow, but I'm glad you're using it now in combinatorics.

KK: That sounds like one of those apocryphal stories, right? Where that always gets attributed, like, I don't know, somebody's giving their dissertation defense, and somebody like Milnor, probably not, but somebody like Milnor's in the audience, and they go, “The class of examples here is empty. This is — there's nothing here, you know?”

JA: Well, fortunately, this was only discovered after I graduated.

KK: Right, right, right, right. Well, and, you know, hey, I mean, things like that happen. It's not that big a deal. So, yeah, so Jeremy and I, we go back a little bit. We actually met at an AMS department chairs workshop some number of years ago that I can't even remember anymore. We were both still chairs. I know that. But was it 2019 maybe?

JA: I think it was 2018, but one of those two years.

KK: It was in DC. Question Mark. I don't know, Baltimore? B-more? Yeah. Anyway.

JA: San Diego. Did you go to San Diego?

KK: No. It must be in Baltimore. So anyway, we've kind of kept up a virtual friendship since then. So here we are, and I thought he would — he entertains me, so I thought he would entertain our guests. So, so Jeremy, we asked you on for a purpose. What is your favorite theorem?

JA: So, my favorite theorem is that the Rado graph has certain properties.

EL: Okay, and you know, the next question.

JA: Yes, so I have to have a little bit of setup, okay? Okay, so first I want to talk about random graphs. Cool. Okay. Now imagine you've got a bunch of vertices, or I like to call them dots, because that's usually what they literally are. They're just dots, and we're going to connect two dots, or not. We call that putting an edge in and we're going to flip a coin for each potential edge to decide whether it will be present or absent in the graph. Okay? And usually we assume the coin is fair. Today, we will assume the coin is fair, although you can not assume that, you can make it whatever probability you want.

EL: Yeah.

JA: But if you assume the coin is fair, then you get the uniform distribution on the class of all graphs on some fixed number of vertices, so it's a convenient assumption that the coin be fair.

EL: Right.

JA: Now there are other random graph models. One of the ones that Erdős looked at early on was the one where you sample uniformly from the set of all graphs with a fixed number of vertices and a fixed number of edges, but then you lose independence of edges being in or not, and it's hard to prove things about that model. Even harder is the Barabási-Albert random graph model, where you start with some vertices, and then every time you add a vertex, you attach it to existing vertices, but preferentially, with preference for the the vertices with large degree.

KK: Okay.

JA: And if you do that, instead of getting a sort of binomial degree distribution, like you do with the standard model of random graphs, you get a power law.

EL: Okay, yeah, rich get richer.

JA: Yeah, yes, it's the rich get richer, right? You see this power law in, oh, like, the Facebook graph, the social network graphs, right? Most people are unpopular, and then there are some extremely popular people, but very few of them. And that was a great result in 2000 that showed how a power law degree distribution arises, and it's through preferential attachment and growth, right? But for the rest of this little talk, I just want to talk about the the coin flip model.

EL: Yeah, any graph on that number of vertices is equally likely to any other one.

JA: Correct.

KK: Right, okay.

JA: Ignoring isomorphism, right?

KK: Yeah, okay, sure.

JA: We’ve got, we've got labeled vertices so we can distinguish between two isomorphic graphs.

EL: Yeah, I guess that's kind of important.

JA: Yeah, it's very important. I don’t — I’m not sure what would happen if you worked up to isomorphism.

EL: Sounds hard.

KK: Yeah, let’s ignore that.

JA: Okay, so we're going to connect combinatorics and logic here in just a minute. So I want to briefly talk about first-order formulas. What does that mean? A first-order formula in the language of graphs is built as follows. You have one binary relation symbol that you might think of as a tilde [~]. So x ~ y means that the vertex x is adjacent to the vertex y, and then you have logical symbols — and, or, not, implies. And then you have quantification: for all x, there exists y. Okay, any sentence you can write with those symbols and variables is a first-order sentence in the language of graph theory. Okay. So for example, you could say, for all x, there exists y, x is adjacent to y, and what that says is that no vertex is isolated, right? For all x, there's some y adjacent to it. You could also say there exists x for all y, x is adjacent to y. So that would mean x is adjacent to everything, including itself — which, we're not going to allow loops today. So imagine all the things you can say with first-order formulas in the language of graphs. Well, it turns out that the first- order theory of graphs obeys what's called a zero-one law. And what that means is that any first-order sentence is either almost surely true in all finite graphs or almost surely false in all finite graphs.

EL: That is very strange.

KK: Yeah.

JA: It is.

EL: I think, I think, as a non graph theorist.

JA: Yes. So, for example, almost all finite graphs are connected.

EL: That’s funny. When you first introduced random graphs, I was I almost asked you, like, are those usually connected or not? But then I decided to just wait a moment.

JA: Yup, they are. They are usually connected. In fact, they almost surely have diameter two, which is how you prove they're connected. It turns out, and this still kind of blows my mind. One thing you cannot say with a first order sentence in the language of graphs is “this graph is connected.”

EL: Oh, yeah. Okay. I mean, I, for some reason, I don't know why, maybe it's because I feel like graphs are very tangible and, like, I should be able to understand them quickly. What I want to do right now is first of all, find a way to say that this graph is connected in this first order logic. And second of all, find some proposition that 50% of graphs are going to have, and 50% are not, just to — I don't know why. I don't dislike you, but I want to prove you wrong somehow.

JA: Well, it wouldn't be me you're proving wrong. Whoever proved this theorem, and I actually don't know off the my head, who proved it. So yeah, so we have this zero-one law. So to give an example of a statement that's almost surely false, “this graph is complete,” right? You can say that with a first order sentence, right? For all x, for all y, x not equal to y, implies x adjacent to y.

KK: Sure.

JA: Obviously that's true of few graphs, right?

EL: Yeah.

JA: So that's a good example of the zero part of it. It's almost surely false. Okay, so what do we mean by almost surely true anyway? Well, what we mean is that, if you look — so take the set of all graphs on n vertices and calculate the fraction that satisfy the property, then let n go into infinity. What's the limit? And it turns out, not only does the limit exist, it's always zero or one.

KK: Right.

JA: Okay, that's not true in general. In fact, maybe the simplest example of an intermediate property is for finite groups, the probability of being abelian is, think it's roughly a third or something. So that's a property, you can say that with a first order sentence, x times y equals y times x, and its probability, asymptotically, is intermediate between zero and one. Okay, so this is special to have this zero-one property. Okay, so now I want everyone to imagine we're going to, you know, n is going to infinity, right? And we're getting more and more graphs and the number of edges is, you know, the distribution of number of edges is converging to the normal distribution, and all this nice statistical stuff is going on. Well, what if we sort of take it to the limit and say, okay, n reaches infinity. N is countable. What happens? Well, it turns out that you get, as n goes to infinity, you get more and more of these graphs. But then something changes. When you actually go to the countable random graph, there’s one, and it's called the Rado graph. And what I mean by there's one is that there exists this graph called the Rado graph. And if you actually generate via this coin flip random process, a countably infinite,random graph, with probability one, you get the Rado graph.

KK: Okay, okay, up to isomorphism, right? Or whatever, yeah, or no? Actually, there's just the one, okay.

EL: Yeah, what does the Rado graph mean?

JA: Okay, so there, there are two ways — well, actually, there are a bunch of ways to approach it. I'm only going to talk about two. One is that it's the almost sure result of the countably infinite random process.

KK: Okay.

EL: So, so we're thinking like, let's just say we've got a vertex for every whole number and we want to and then with probability 1/2, we connect, you know, n to m for all n not equal to m. That is that what we mean?

JA: Yes.

EL: Okay.

JA: So that doesn't sound like a definition, though, right, right,

EL: Yeah, because I feel like I could make two different things. You know, in one of them there's a vertex between 2 and 3, and in the other one there isn’t. But somehow this is the same thing?

JA: Oh, well, yes. I mean, you could get different isomorphic copies of the autograph, right? But with right the probability of you getting a graph that is not isomorphic to the one I get is zero.

EL: Yeah. Wild!

JA: Yes. And more cool stuff. Going back to that zero-one law, if you have a first-order formula, it has asymptotic probability 1 if and only if it's true in the Rado graph.

EL: Wait. Can you say that again?

JA: Yes, okay, take a first-order formula in the language of graphs like this graph is complete. That formula is true for almost all finite graphs, if and only if it holds in the Rado graph.

EL: Okay, okay, thank you. So the just takes me an extra time through.

JA: So the Rado graph, in some sense, tells us what all the true first-order statements are in the language of graph theory.

KK: Is it easier to prove these things in the Rado graph? I mean, it's not complete. I get that, but I mean, you know…

JA: I don't think so.

KK: Okay, just kind of a fun fact.

EL: But it might be differently hard and sometimes that's helpful.

KK: Sure.

JA: Yes, so, yeah, I don't, I don't actually know that.

EL: Okay, so, but I feel like I'm marginally, or, you know, provisionally okay with the Rado graph, so yeah.

JA: Here’s something that will make you okay-er with it. Okay. The result of this random process actually has a simple construction that is not random at all.

EL: Okay, great.

JA: Here’s what we do. Our set of vertices is the set of primes that are equivalent to 1 mod 4.

EL: Okay.

JA: Okay, and here's how we determine whether to put an edge in. So you've got two vertices labeled p and q. Because p and 1 are equivalent to 1 mod 4, by quadratic reciprocity, they’re either both quadratic residues modulo each other, or neither. Okay, so you put the edge in if they are quadratic residues modulo each other, and you don't if they aren’t.

EL: Okay.

JA: And that gives you something isomorphic to the Rado graph.

KK: Oh, okay, no way.

JA: I know! It's just ridiculous, right? It's ridiculous. I guess. I mean, the primes are sort of pseudorandom.

KK: Yeah.

JA: You know, I to my very limited understanding, this is essentially how Green and Tao proved that the primes contain arbitrarily long arithmetic progressions.

EL: Yeah, right.

JA: Like, if they were random, then that would have to be true. And they are random enough, right? Even though they're in some sense, not random at all, they’re pseudorandom.

EL: They’re like, yeah, they are, by definition, not random in the least, right? But they act like they are to, like, any way of looking at them, it’s so wild.

JA: Yes, fascinating. Okay, so that that is the Rado graph, yeah. And my theorem is that the Rado graph is, well, in logic, we call it omega-categorical, which means up to isomorphism there's only one countably infinite model of the first-order of graphs, right?

KK: Yeah, so…

JA: Go ahead.

KK: This seems to intersect perfectly for you, right? I mean logic and combinatorics, right? I mean, this is, this is like, if you were going to define something to be like an expert in for you, this is it, right?

JA: Yeah. I mean, I really probably should have done a degree in computer science instead of math, because I like combinatorics, yeah. I like doing machine computations. In a computer science department, I would be, oh, the heavy theory guy, whereas in mathematics, I'm that guy who cheats with a computer.

EL: Hey, we can all get along. We don't need to have factions here. So how does one even begin to go about proving something like this, that your quadratic reciprocity graph construction thing that you just told us is isomorphic to any other random construction I can come up with?

JA: I’m glad you asked that. Okay, so here's the idea of what you do. So it turns out that another way to, sort of obliquely define the Rado graph is the following way. I’m going to define a property and any graph, any countably infinite graph with that property, is isomorphic to the Rado graph. Okay. So given two disjoint subsets of vertices, R and S, there exists a vertex v that is adjacent to every vertex in R and no vertex in S.

EL: So any two disjoint sets of vertices?

JA: Correct. Okay, so you name any set of vertices and then Kevin names a disjoint set of vertices. I have to find a vertex v that is adjacent to all of Evelyn's vertices and none of Kevin's vertices. And if I can always do that, then my graph is isomorphic to the Rado graph.

EL: Okay. We're both skeptical, but okay.

KK: This feels like a weird graph, but okay, all right,

EL: It’s weird property to try to…

KK: It’s because you've got an infinite collection of vertices, right? If everything were finite, this would be bad news, like this would be hard to do.

JA: Right, right. It would be impossible.

KK: But I guess, because you have an infinite number vertices — yeah, right, yeah — for every disjoint pair, you couldn't do it. But because, okay, yeah.

JA: I mean, it's weird.

KK: Yeah, okay.

JA: Another way of thinking of it is that the Rado graph contains every finite graph as an induced subgraph. Okay, anything you can dream up you can find in the Rado graph.

EL: Okay, that sounds like a property of a random graph for sure.

JA: So to go back to the proof about the the construction with the primes, we just have to show that given any two disjoint set of primes, there is some other prime that is a quadratic residue modulo every prime in the first set and no prime in the second set. And if I recall correctly, it's Dirichlet’s theorem on prime progressions, right? Something, you know, wave your hands and it all works.

KK: It’s fine, year.

EL: Call a number theorist or something.

JA: Right. I am not a number theorist.

KK: Right.

JA: I have used some number theory in my work, but I'm definitely not a number theorist. Don't ask me hard number theory questions.

KK: All right. Well, okay, I guess I could kind of see now how one might fall in love with the Rado graph, right? I mean, where'd you first come across this?

JA: I don't remember it. It was definitely not in grad school. It was later just learning about stuff. Oh, actually, I think I was asked to — yeah, I think this is what happened. I had to referee this paper that was a bit of a stretch for me, and I had to look some stuff up. And I encountered this as I was trying to figure out what was going on in this paper that I probably should have not actually refereed, but it looked interesting.

KK: Sure.

JA: Bit of a stretch. These days I would say no to that.

KK: Well, as a journal editor, let me say to all of our listeners, I edit a journal, please accept referee requests. Please? Nobody wants to referee papers anymore. You might learn something!

EL: You might learn something favorite, graph or theorem.

KK: That’s right, that's right, yeah, it's a service to the community and it's good for your brain. So this would be really nice.

EL: Yeah, so I always like to ask if this was kind of a love at first sight theorem or construction. I'm not quite sure you know what part of it would be, love at first sight, but yeah, how did you feel? Did your your love for it grow? Or develop over time?

JA: Oh, it was definitely love at first sight because I just couldn't believe the result. And it actually turns out that the Rado graph is a special case of a more general phenomenon that we don't need to get into. But this notion of a limiting structure that encapsulates all of the properties of the finite structures, that happens in other spaces too.

KK: Cool. All right. Part two, the pairing. So what pairs well with, with the Rado graph?

JA: Okay, well, the Rado graph is something taken to the extreme, right, right. Okay, so my pairing is called Huntsman cheese.

EL: Huntsman cheese. Okay, I'm a little scared.

JA: Okay, so this is something my my parents bring me sometimes when they visit from Wisconsin. It's a big wheel of age cheddar, except inside — it's a pretty tall stack — inside are two layers of blue cheese. Okay, so when you cut into it, it looks like a layer cake.

EL: Yeah!

JA: All right. It's really pretty and it's intense.

EL: Yeah, you’ve got two different kinds of intense cheese flavor, right?

JA: Yes, it's delicious, though.

EL: Yeah, no, I was a little worried there might be, like, organ meats involved or something. In some way, I'm sort of a typical American in that I'm a little not into the offal. So just wasn't sure what these huntsmen were doing with the cheese.

JA: No, I don't know why it's called that.

KK: Yeah.

EL: Oh, that's that sounds good, although maybe hard to like, just a large amount of it would be intimidating.

JA: Oh, yes. I mean, it's very rich, so you don't want to eat very much, yeah?

EL: Good party cheese. Like, get a lot of people together to help you go through it.

JA: Yes, but half of them will refuse to try it.

EL: Yeah, cool. Well, then, I mean, that's great too, because then, once you've got your party, you can start to make graphs of who was already friends and who didn't know each other when they came. Then you can start to do other graph theory. You can find some Ramsey kind of theorem examples, or Ramsey theory kind of stuff. And so you could just like, take this towards Rado graphs and towards Ramsey and other, whatever your your graph theorist heart desires.

KK: I’ve got to try this now. I mean, I was born in Wisconsin. I haven't been back in many, many years. I do not know the Huntsman's cheese, but we’ll have to find some of this.

EL: Put in a special order of the cheese monger.

KK: Yeah, right, yeah, the Florida cheese monger. Actually, we do have a local liquor store that that also does cheese, and they like all these weird — I say weird. I shouldn't say weird, unusual imported cheeses from from England and, you know, the really stinky ones and all of that. I'll have to go there. Maybe they have this Huntsman's cheese.

EL: Yeah.

JA: We like to give our guests a chance to plug anything that they're working on. Or where can we? Can we find you online somewhere? I actually do know some things about Jeremy that, again, our listeners can't see, but he's got this collection of instruments hanging on the wall.

EL: I noticed that. Yeah, do you? So it seems like you've got a variety of stringed instruments behind you. So do you record or, like, publish what you play?

JA: I do. I play several instruments poorly.

KK: I play one poorly.

JA: But yeah, I really like the songwriting process. It's sort of, you know, it scratches an itch that mathematics doesn't necessarily.

EL: Well, mathematicians are creative people, but this is using your creativity in a different direction.

JA: Yes, now that I'm in full time administration, I'm often too tired in the evening to think about mathematical research, but I can strum a ukulele. So it gives me a sort of outlet for creativity that was sort of missing for a while when I went into administration.

EL: Yeah.

KK: So you, you can be found, are you on Spotify?

JA: Yes, yes, right. The band name is the Unbegotten Brothers. And actually, a new single came out just, like, four days ago or something.

EL: Oh, congratulations.

JA: So if you want to hear yet another 12 bar blues, check it out.

KK: Yeah, Jeremy and I have an unpublished 12 bar blues too, that we had a third person lined up to do the singing, and that person never, and we won't out that person, but never followed through with the recording of the vocals.

JA: So we will just, we will shame them in private, not in public.

KK: And I only play rhythm guitar, and again, not especially well, but well enough for 12 bar blues, right? So.

EL: Yeah, it's about enjoying the music-making process.

KK: That’s right.

JA: Yes. I mean, I write it for myself, not for anyone else.

KK: Sure.

JA: But the thing I really want to plug for everyone is an open problem called the chromatic number of the plane.

EL: Ah, I’ve written about this.

JA: Ah, good. There was some shocking, at least shocking to me, progress about four years ago.

EL: Maybe might even be a little longer i could find the date on that article, because I yeah, maybe 20. Yeah, I'm not going to hazard a guess. Time kind of gets weird for me before 2020. Or my memory.

JA: I think it was pre pandemic, though, so yeah. I was absolutely shocked by that result, because I was convinced that the correct answer was four. At least four in Zermelo Fraenkel set theory with choice. I had convinced myself that the answer to the question depended on which axioms of set theory you adopt. So I was shocked when somebody came up with a finite graph with chromatic number five.

EL: Yeah.

JA: I was just like, oh, I couldn't believe it.

EL: Yeah, yeah. Well, that is — yeah, especially if you were really convinced of this and yeah, that it would you, you would require looking at that kind of set theoretic aspect of it in order to eventually prove it, I assume, then, yeah, you got your socks knocked off.

JA: Yes, yeah, I had drawn this analogy. There's an object in mathematical logic that's pretty important, called a non-principal, ultrafilter. Yep, and you can't really construct it. You have to appeal to Zorn's lemma, so you you can't write down. I mean, there are literally no examples.

KK: Right.

JA: But they exist, right? And I kind of thought that there was a four coloring, but we would never be able to describe it.

EL: Right.

JA: And maybe, maybe in different versions of set theory, the answer would be different. In fact, the thing I was originally going to talk about as my favorite theorem is, just very briefly, there exists an infinite, a graph with continuum many vertices, such that in Zermelo Fraenkel set theory with choice, ZFC, the chromatic number is 2. And in ZF plus countable choice, plus the axiom that all subsets of the reels are Lebesgue measurable, the chromatic number is uncountable.

EL: Yeah, I'm glad you didn’t. We would have kicked you off! Yeah, that is wild. So we did maybe skip a little bit for our listeners. So what is the question of the chromatic number of the plane?

JA: Ah. So imagine you are coloring all the points of the plane individually. Okay? And what we're trying to do is not have any two points that are exactly unit distance apart the same color. And it turns out that you need at least four colors. That's an exercise you could assign to undergraduates. Seven suffices. You just, there's a nice little pattern with

EL: Hexagons?

JA: Pentagons? Must be hexagons.

KK: That sounds right.

JA: I haven't thought about this in a little while. But then until 2019, or so, that was all we knew. I mean, we had some conditional results. Like someone showed that if the color classes are all measurable, then you need at least five colors.

EL: Okay.

JA: And then somebody else showed that if you use, like, even nicer sets than measurable sets — I can't remember what it was. Basically like, if you're using sort of rectilinear shapes or something like that, then you need six colors.

KK: Okay.

JA: But no unconditional results, until fairly recently.

EL: So yeah, if you're you know, needing a problem to either put you to sleep or keep you up at night, depending, that's a good one to just kind of try to try to roll around in your head.

KK: Right. Cool. All right. Well, this has been great fun, Jeremy. thanks for joining us.

EL: Yeah, thanks for joining us.

JA: Thank you for having me.

KK: Yeah. It was great. I learned some stuff today. All right, all right. Take care, man.

JA: Okay. Bye.

[outro]

On this episode, we enjoyed talking with Jeremy Alm, a math professor and associate dean at Lamar University, about the Rado graph. Here are some links you might find interesting after you listen.
Alm's website and his band the Unbegotten Brothers
The Rado graph on Wikipedia and the Visual Math Youtube Channel
Omega-categorical theory on Wikipedia
Evelyn's 2018 article about recent progress on the chromatic number of the plane

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, and I am joined, as always, by our other host. Will you introduce yourself?

Kevin Knudson: Hi, I’m Kevin Knudson. Yeah, I will. I'm Kevin Knudson, professor of mathematics at the University of Florida. It's been a while.

EL: Yeah.

KK: You know, I've actually gotten a few emails from our listeners saying, hey, where, where the hell is My Favorite Theorem? And I always have to reply, you know, we're trying, but everybody's busy.

EL: Yeah, and we're busy.

KK: And we're busy. But here we are. We are still committed. We're still into this. This is — we're going to go to year eight soon enough, which is kind of mind-blowing.

EL: Yes.

KK: I had less gray and more hair in those days. So here we are.

EL: You’re as lovely as ever.

KK: As are you, Evelyn, as are you. Yeah, although I kind of missed the green hair. I wish you would start coloring your hair again.

EL: Yeah. Honestly, like many people, during COVID, I just lost my ability to put forth more than minimal effort into my appearance.

KK: Yeah.

EL: Sorry, that sounds like a smear on other people. I just, I've heard this from other people. I'm not judging based on what I see from other people. But yes, it's just, like, the bleaching, the dyeing, it just, it's more maintenance than I'm willing to put forth right now.

KK: It’s a whole thing. And as one gets older, you just go, who cares?

EL: Yes. But anyway, we are delighted today to have Robin Wilson on the show. Robin, will you introduce yourself? Tell us where you're joining us from, and a little bit about yourself?

Robin Wilson: Yeah, hi everyone. So I am joining you from Los Angeles, currently in my office at Loyola Marymount University, where I'm a professor in the math department. And so I have been a professor since about 2007 and work in topology and math education. It's great to be here today.

KK: What part of town is Loyola in? I don't think I actually know where that is.

RW: Loyola Marymount is right on the coast, near LAX. So if you've ever visited us here and flown into that airport, then you've flown right over campus, and depending on which runway you land, you can actually, like, see the fountain.

KK: Okay. I'll be flying through LAX in December. I will try to take a look.

RW: Come say hello, yeah.

EL: And I have to say, if we were broadcasting a video of this, you have one of the best backgrounds, the beautiful bookshelf, and then the chalkboard behind you has the appropriate level of mathiness. So our listeners can't enjoy that. But I will say that, you know, it looks very math professor.

KK: It does. It could actually go in that book of math chalkboards. Have you seen this? What's it called? I can't remember. Anyway.

RW: I’m so honored.

EL: I think it’s something like Please Do Not Erase, or something. [Editor’s note: Very close! It’s just Do Not Erase by Jessica Wynne.]

KK: Please Do Not Erase. That’s right, yeah.

EL: But yes, we are so happy to have you here. And Robin and I have actually worked together before a few years ago, on a publication for — at at the time, known as MSRI. It has a new name now, the SL Math Institute now, but we worked together putting together a booklet for the math and racial justice workshops that they did in summer 2021, was it? And it was such a delight to work with you, and I'm glad that lo these many years later, we can get to chat about your favorite theorem. So what have you chosen for your favorite theorem today?

RW: Okay, so the theorem that I've chosen for my favorite theorem today, this was a tough one, and so I chose one that was sort of important for me on my journey. So the Poincare-Hopf index theorem.

KK: Oh, I love this theorem.

EL: All right!

RW: So should I tell you more about theorem?

KK: Please.

EL: Please.

RW: Okay, so the theorem, so I'll state the two-dimensional version of the theorem, which is the one that I can kind of see. So you take a surface and you add a smooth vector field on the surface, and there's an extra condition that the vector field has only finitely many critical points. And so it turns out that the sum of the indices of the critical points of the vector field is equal to the Euler characteristic of the surface. And so the theorem connects these two different areas of math, sort of analytical and topology. And so it was one that I encountered kind of at the beginning of my journey as a topologist. And so it's the one that I picked to share today.

EL: I love that, and I've got to say, I love how many people on our podcast come in and say, it connects this and this. And like mathematicians, we just love these theorems that connect, you know, calculus and topology, or, you know, algebraic geometry and topology, or something like that. It's just something, mathematicians just all love these bridges here. And so I, yeah, can you talk a little bit? You said you encountered it first as a kind of starting out as a beginning graduate student. You know? Can you tell us, bring us back to that moment, tell us about its importance in your life?

RW: Sure. So I was an undergraduate student, and there was a graduate student that was a TA for one of my classes at the time, and I asked him about what type of math he studied, and he drew this picture of a trefoil knot on the board. So shout to Aaron Abrams if you're out there, Aaron. And so I asked him, you know, how could I learn more about this? And he pointed me to a class, an algebraic topology class, that was being offered the next semester. So I signed up for it, and the class was real challenging, but I really enjoyed it. And then the following semester, I got a chance to do a senior thesis, and so I ended up going back to the instructor for that class, and he said yes. So also want to shout out Charlie Pugh for saying yes. And he chose the project that I would work on, the proof of this theorem. And so I'd never, I'd never heard of Poincare. I didn't know much about the historical context of the theorem, but it was — looking back, it was pretty significant that I got a chance to try and think about how to come up with my own proof of something that Poincare had proven, who was right there at the beginning of the field.

KK: And it's a surprisingly tricky theorem to prove. I mean, I was reading this actually, last year sometime. I was trying to remember how this proof goes, because what's remarkable about it is, there's lots of things involved, right? Because no matter which vector field — and it could have, like you said, it can only have finitely many zeros or singularities, but it might have 10 million of them, or it might only have two, but that number, the Euler characteristic, of course, is an invariant of the surface. So no matter how you wiggle this vector field, if you create something, some weird swirl on one side, well, some sort of opposite swirl has to happen somewhere else, effectively, to cancel it out. It’s really remarkable.

RW: Yeah, yeah, that's right. So I was trying to remember what we did to prove the theorem. We, you know, we really were wrestling with a lot of the tools that we used in the course, like, a lot of the details about triangulations of surfaces and trying to find the right ones and paths of vector fields and things that build mature for a very young mathematician back then.

KK: Yeah, yep, yep. So. And I guess one of my favorite corollaries is what, it's the hairy ball theorem, right?

RW: Yeah, that’s right. I was thinking about this. It's got to have, like, one of the worst names in all of mathematics.

KK: Yeah.

RW: So yeah, you can't comb a hairy ball flat without a cowlick, also known as the you can't comb a coconut theorem.

KK: Okay, that’s a little better.

EL: Yeah.

KK: A little little less innuendo, right?

RW: Yeah. That was a close, a close second for my favorite theorem. It's nice that they're connected.

EL: Yeah. We were talking earlier a little bit that, you know, it is hard for people to pick favorite theorems, and, you know, what does it mean if a theorem doesn't immediately leap to mind as a favorite? I just want everyone to know that we are, we might be mathematicians, but we are very not pedantic and mathematician-esque about definition of your favorite. We are very loose, and you know, it can be your favorite of the moment or your favorite for what it meant for your mathematical development. And we’re very imprecise with the definition of favorite on this podcast. All favorites are welcome.

RW: I must say that I had not thought about this theorem for years, until I was asked to find a favorite theorem.

KK: Well, it's sort of like on Instagram, there's this account we rate dogs. Do you know this one? So anyway, basically the guy rates dogs, but the lowest possible rating is 10.

EL: 10 out of 10.

KK: 10 out of 10. Theorems are sort of that way too.

EL: I don't know. I would say, I'm not going to name any theorem names. But I would say there are some theorems that I would put below 10 out of 10. Cancel me if you must. But you know, I’m going to put that out there.

KK: This is it. So we have to start our new Instagram account, clearly.

EL: We Rate Theorems.

RW: 10 out of 10.

KK: That’s right.

EL: Yeah. So another thing we like to do on this podcast is we ask our mathematicians, as if it weren't hard enough to choose a theorem, to choose a pairing for their theorem. You know, be it art, music, food, wine, any delight in life. What have you chosen to pair with the Poincare-Hopf [index] theorem?

RW: So I think I might have actually started with the food and then went back to the theorem. But there was this example that also really like captivated me, captured my attention as a student, and that's the hot fudge flow. So it's a vector field over a surface. And so the idea is to imagine a ball of ice cream, and you do what you do with ice cream. You take the hot fudge and you drizzle it on top of the ice cream, and you try and hit the center. And then what happens to the fudge? It sort of, you want it to expand and wrap around and then come back as a source and drip out of the bottom, if this was, you know, suspended in the air. So that's the hot fudge flow. And you can compute the sum of the indices of the critical points of that vector field, and it'll match of the Euler characteristic of the sphere. So the pairing is a hot fudge sundae.

KK: Okay.

EL: Excellent.

KK: That’s exactly perfect. Yeah.

EL: Of course we have to ask. What is your number one ice cream flavor for a hot fudge sundae?

RW: I was actually hoping you wouldn't ask that I'm the most boring ice cream person. Vanilla is my favorite.

KK: Look, you can't go wrong.

RW: Yeah.

EL: I will say, it is very unfair to vanilla that it has become this word in in our our language, for something that's boring, or pedestrian, because, like, it is an incredibly complex flavor, like, if you get an actual vanilla bean, it's like, there's so much going on. And I don't, I don't know the the history of how vanilla became “boring,” but, you know it is, it is anything but boring. Justice for vanilla.

KK: And so complicated to grow, right? It only grows in very specific places.

EL: A few places. And it’s expensive. Isn’t it, like, the second or third most expensive spice after definitely saffron.

KK: Saffron, I think, is number one.

EL: Maybe something like cardamom. Cardamom is up there too, I think.

KK: It’s not cheap.

EL: No hate to vanilla.

KK: It’s not cheap, because one little pod of vanilla, one little pod at the store is like, $4 or something. You know, it's like, it's really, really absurd. But it's an orchid, right? I mean, so, I live in Florida. We can actually get orchids to grow here, but it's still not easy.

EL: Right. Do you know if the vanilla orchid can grow there?

KK: I doubt it. If it could, they would be cultivating it left and right. I actually think it's too hot here. It's not humid enough, somehow, yeah, so some orchids will work.

EL: Because I think, like Madagascar, Tahiti and maybe Mexican? Is it grown in Mexico also?

KK: I think there might be some spots in Mexico, yeah, like, maybe in southern Mexico, Oaxaca or something. But, yeah, anyway, okay, all right, this is not a vanilla podcast.

EL: Yeah, three mathematicians speak extemporaneously on vanilla cultivation. Tune in next week for the exciting conclusion.

KK: That’s right. Yeah, so Robin, we always like to give our guests a chance to plug anything they're doing. Where can we find you online, what sort of, any big projects you're working on that people might be interested in, or anything like that?

EL: Or have done recently?

RW: Yeah, so I have a really bad online presence right now. At the moment, the website could use some dusting off. But one of the projects that I'm working on that I'm excited about right now is in math education. So we've been making videos of Black mathematicians talking about their work, their educational experiences, and giving advice to young people. And so these are for K to 12 students, but also, I think they're going to be of interest to lots of folks. And so we do have a website, but the URL isn't in on my mind to pass on to you right now. Maybe I could share it with you afterwards.

EL: Yeah, we'll, we'll get that from you and put it in the show notes, so it’ll be easy for people to get.

RW: That’ll be fantastic, but thanks for letting me make that plug.

EL: Yeah, well, and I remember seeing recently, you did a talk at the Museum of mathematics, right with and was that a conversation with Ingrid Daubechies?

RW: It was so much fun. It was a conversation.

EL: Do you know if that is available in video form somewhere? I meant to look for that before we got on. But of course, I didn’t.

RW: You know, I had the same question cross my mind as I was approaching this as well. And I think it might be available, but it could be, like, for museum members.

EL: Okay.

RW: I need to check.

EL: Yeah, I remember seeing your saying your name in my inbox, and thought, well, that's cool. And you've also, do you mind talking a little bit about the Algebra Project and and Bob Moses?

RW: Sure.

EL: Because I know that's something that you've — I know I've talked with you about it before, and Bob Moses passed away around the time we were putting that book together.

KK: Yeah, it was a couple years ago.

EL: So, yeah, do you mind talking a little bit about it? I thought it was really interesting.

RW: Yeah, sure. That's something that I could talk about for a long time. So just just check me if I start going on too long. I met Bob Moses as a graduate student, and I think I was kind of wrestling with some identity issues about my interest in math, but also, you know, interest in social issues, and kind of wanted to make a difference in my community, and trying to figure out how these two things came together, and if I was doing one, did that mean that I couldn't do the other? And so I came across his book, Radical Equations. It was about math literacy and the civil rights movement, and he brought his work in the civil rights movement together with his work as a math teacher in in Boston, and it really kind of spoke to me. And so I got a chance to meet him, and ended up staying connected with him and Ben Moynihan at the Algebra Project, and so I worked with them in different ways, attending teacher professional development. We helped spearhead an effort in Los Angeles, where the Algebra Project curriculum was used in four different high schools supported by an NSF grant. We had a second effort here, where we've been running some summer programs for students through the Algebra Project. And recently I joined the board of directors, and so I’ve been involved with them since I was in my 20s, and so it was a real honor to be asked to kind of be a part of that, that part of the leadership for the project.

KK: Bob Moses really, really impressive man. And then, this idea that you know, that every you know, things are really important. You know, education is so important to advancing, you know, civil rights and things like that. I mean, Bob Moses was really spectacular. Our listeners, if they don't know much about him, should just look him up, because he was really impressive and influential, and by all accounts, a very kind man. Like I said, I've never met him, but just a really great human being.

RW: And I think what people, a lot of people don't know about him, is he was a math teacher first, he was teaching math and and the sit-in movements happened, and he got drawn into the sit-ins. And then when, when things kind of settled down, he went right back the math classroom. And so kind of think of him as one of us.

EL: Yeah. I think reading, reading radical equations a few years ago, I remember, you know, it's just like sometimes when you're a mathematician, especially if you're really involved in the academic math world you get so, you know, drawn into these very abstract questions that you feel like have nothing to do with, you know, anything resembling reality, or anything resembling social issues, and just the way that he writes about how access to good math education, like is so important for people to be prepared to, you know, have careers that they want, be able to have financial stability in their lives then, and just the, you know, the doors that it opens to have access to math at, you know, the middle school, high school level, really reminds you as a mathematician, like, oh, yeah, we are part of this society.

RW: Yeah, that's right, and we do have a really important role to play. That's one of my biggest takeaways from him that as mathematicians, we do have a really important role to play in how this whole thing turns out.

EL: Well, thank you so much for joining us. Really great to talk with you again.

RW: Thank you so much.

[outro]

On this episode of My Favorite Theorem, we had the pleasure of talking with Robin Wilson, a mathematician at Loyola Marymount University, about the Poincare-Hopf index theorem and the importance of math education. Below are some links you may enjoy after the episode.
An interview with Wilson for Meet a Mathematician
More on the Poincare-Hopf index theorem
The 2021 SLMath Workshop on Mathematics and Racial Justice and its follow-up, to be held in May 2025
Storytelling for Mathematics
The Algebra Project
The 2025 Critical Issues in Mathematics Education workshop, to be held in April 2025, focusing on mathematical literacy for citizenship

Evelyn Lamb: Hello and welcome to my favorite theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, a freelance writer in Salt Lake City, Utah, where it is gorgeous spring weather, perfect weather to be sitting in my basement talking to people on Zoom. This is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. I don't know, Evelyn, I saw the pictures on Instagram over the weekend and it looked cold in Utah. You wrote that you rode a century, right?

EL: Metric century.

KK: Okay. Metric.

EL: Just in case — you know, I don’t want people to think I'm quite that hardcore. Yeah, at least at this point in the season. Yeah, I hadn't managed to ride more than about 25 miles since last fall because weather, travel, just things conspiring against me. The week before I was like, I really need to get 30 or 40 miles in on Monday. And then it was, like, 20 mile an hour winds and sleet and I was like, well, I guess I'm just going into this cold, but it was fine. It was actually gorgeous weather. It was a little chilly at the start, but better than being too hot.

KK: Well, you know, the muscle memory takes over, right? So you can do — I mean, 62 miles isn't that much more than 25, really, once you have the legs, so congratulations.

EL: Yes, thank you. Well, we are delighted today to be joined by Kate Stange, who is in the Mountain Time Zone, something that I always feel thrilled about because I'm constantly converting time zones when I'm talking to people, and finally, someone I didn't have to do that for! So Kate, other than being in the Mountain Time Zone, what would you like to tell us about yourself?

Kate Stange: Oh, geez. Um, well, I'm also a cyclist. And so I'm jealous hearing about your rides.

EL: Wonderful!

KS: Here in Colorado we have we have this ride called the The Buff Classic. And so it has a 100 mile option where they close Boulder Canyon so that you can bike up the canyon without any cars.

EL: Oh, wow.

KS: Then you bike along the peak to peak highway. It's just wonderful.

EL: Yeah.

KK: Yeah. That sounds great.

EL: So you're at CU Boulder?

KS: Yes. And it's run from the campus. It starts right outside the math department.

EL: Oh, perfect. Yeah, just drop your stuff in your office and hop on and ride it?

KS: Yeah.

EL: Yeah, great. Well, we are thrilled to have you here today. And I guess we can just dive right in. What you're on what do you like to tell us about?

KS: My favorite theorem, at least for today, is the bijection between quadratic forms and ideal classes.

KK: That’s a lot of words.

EL: Yeah, and I'm so excited to hear about this, because I am honestly a little nervous about both quadratic forms and ideal classes, and a little embarrassed about being nervous about quadratic forms, not so much with ideal classes. So yeah, can you tell us a little bit about what that all means?

KS: Yeah, sure. So quadratic forms is probably what sort of comes first in the story, at least sort of the way that the mathematics tells it, and also probably the historical way. And so a quadratic form is just a polynomial with an x2, a y2 and an xy. So it's like 3x2 + 7xy − y2. So that's quadratic form. And, as number theorists, one of the things that we're most interested in studying is what are the integer solutions to polynomial equations? And so first you start with linear equations. And there's a wonderful story to do with Euclidean algorithm and stuff there. And then you move on to quadratic. And really, these are sort of some of the first equations that you would start studying next, I guess. And so they go back to the classical days of number theory, Gauss and Euler and everybody. And, yeah, so they come in, what happens is that they come together in families. So different quadratic forms, you can actually just do a change of variables. And it'll look different, but it won't really be too different, particularly if you're interested in what numbers it can represent when you put in integers. So say I take x2 + y2, which is the simplest one, if I put in various different integers to that I get various different integers out. And then if I do a change of variables on that, just a little change of variables — like maybe I change x to x + y, but I leave y alone — the formula will look different after I do that change of variables, but as I put in all integers and look at all the stuff I get out, those two sets, the in and out, they're going to look the same. And so we kind of want to mod out, we want to ignore that difference. So I'm really thinking of equivalence classes of quadratic forms. So that's the first object.

EL: And that change of variables is kind of the only equivalence class thing that happens with them?

KS: Yeah. Yeah. Because they could really behave differently between the different classes.

KK: And you only allow a linear change of variables, right?

KS: Yes, exactly. Yes. Thank you.

EL: Yeah. Okay. So now, ideal classes.

KS: Now ideal classes. So this is an interesting one, to describe where it comes from, I think. So there's sort of — if you think about the history of math, I would say there are sort of two versions, there are sort of two histories of math at the same time. There's one, which is sort of the human history, which is fascinating and human and quirky. And then there's sort of the way that the math would like to unfold to human understanding the way that as any human coming to it, they might discover the pieces of the mathematics. And I don't know too much about the details of the human history. But in terms of how you might discover this, if you're just looking at the integers, you are interested in how they behave, you discover things like prime numbers, you've got addition, you've got multiplication, you've got powers, you might ask how these things interact. And at some point, maybe when complex numbers are discovered, you think about whether there are possibly other collections, or other number systems, other collections of numbers in which you could do the same kind of thing. And so one of the first examples of this would be what's called the Gaussian integers, which is where you take complex numbers, I don't know whether I should dive into complex numbers, but you take complex numbers with integer coefficients.

EL: So that means things like 1 + 2i or something. So the i and the 1 both have integers in front of them.

KK: Right.

KS: Yeah, exactly. And so this is a collection of integers, kind of, right? And we ask things like, okay, are there prime numbers? And so it turns out that there are in that system, there are Gaussian prime, so, like, 1 + i is a prime number. And so you kind of start to develop this whole theory that you have for the integers. But what you find is that in some of these systems, you lose unique factorization. So we love unique factorization in the integers, right? Every integer, up to reordering the factors and maybe putting on a minus sign, you have always a unique factorization into prime numbers. And in the Gaussian integers, that's true. But in some of these other systems, you lose that. And so what people tried to do is to try to fix it. And it turns out, the way to fix it is to add in what were I think, originally called ideal numbers. They were thought of as numbers that should be in the system that weren't in the system. And what they actually were were collections of numbers. They were sets of numbers instead of individual numbers. And the idea here is that, say, you were to take — like in the integers, if you took the number two, you could replace that idea, that idea of two-ness with just the collection of even integers. And so that's an ideal now. Instead of a number, it’s an ideal, and it's really carrying the same information. But now it's a subset. And so by moving from individual elements of the ring, of the collection of numbers, you move to subsets of them. Now you have more things, and so now you can recover unique factorization in that world. So those are ideals.

EL: Yeah. And so the Gaussian integers do have unique factorization.

KS: They do. Yeah.

EL: So this — actually, I kind of forgot, but recently, this came up in something I was writing where I wanted the example to be the Gaussian integers so bad because it doesn't have any square roots in it. But then it didn't work because it isn't true for that. I was trying to show how unique factorization could fail, but I didn't want to have to use square roots. But as far as I know, you can't do that. So then I fixed it by putting a square root of negative five in there and hoping that people would be patient with me about it. But yeah.

KK: So that's the example of one where you don't get unique factorization, right? So you take the integers and you join the square root of minus five. That's one example.

KS: That’s one of them. Yeah.

EL: And then it's like two and three are no longer primes.

KS: So if you multiply (1+ √ −5) × (1− √ −5)

KK: You get six. Yeah.

KS: You get six, which is also two times three. And those are two different prime factorizations of six.

KK: Right.

EL: Yeah, but it's so fun that you can do that, and I like your way of putting it where regardless of how these ideas actually formed, you can as a human, looking at some of these basic pieces like primes and then or primes and integers and square roots and things, you can kind of come up with this, like, what happens if I do this? And create this new thing where this this property that I know I always assumed — like unique factorization, when you're growing up, you know, when you take math classes in school and stuff, it just seems like so basic, like, how could you even prove that there's unique factorization? Because how else could you factor anything?

KS: Yeah, exactly.

EL: It feels so basic.

KS: Yeah. And this is what happened, I think, historically, too, is that people didn't expect it to fail. And so they were running into problems and it took a while to figure out that that's what was going wrong.

KK: Wasn’t this part of, was it Kummer who had a reported proof of Fermat's Last Theorem, and he just assumed unique factorization?

KS: That’s what I've heard, although I never trust my knowledge of history. Yeah.

KK: It’s probably true.

EL: Well, and there are a lot of good stories. And they may or may not be true sometimes. But yeah, okay. So we've got these, these two things.

KK: Yep.

EL: The quadratic forms and the ideal classes. So yeah, I guess either historically or mathematically, what is this connection? And how do you know that these two things are going to be related?

KS: Yeah, so they seem like different things. So I think quadratic forms were studied earlier. And at some point, people noticed that quadratic forms had an interesting property, which is that sometimes you could multiply them together and get another quadratic form, which is kind of hard to explain. But like, if you actually wrote out (x2 + y2) × (z2 + w2) and you multiplied that all out, you'd have a big jumble. But then you could factor it out. So it looked like, again, a square with some stuff inside, z’s and w's and whatever inside the brackets, plus a square. And so this meant that sometimes if you picked your forms correctly, and they had this sort of relationship, then if you looked at the values they represented, the numbers that can come out, when you're putting integers in, you would take that set of things the first one represents and the set of things the second one represents, and then you’d look at what the third one represents, and it would represent all of the products of those things. So there was this definite relationship, but the way I'm describing it to now is a little awkward, because it's a lot of algebra. But this is, I think, what was noticed first, somehow. And again, I might be mixing the human story with how math tends to want to unfold. I don't know exactly the history. But anyway, so you notice that there's this relationship. And that's kind of reminiscent of an operation, like a multiplication law. And what happens is that, in fact, that's coming from the fact that these ideal classes, each one of them — sorry, my mistake — so it's from the fact that each of these equivalence classes of binary quadratic forms, each one of them is associated to an ideal. And the ideals as the sort of generalization of the idea of number, they can be multiplied together to get new ones. And so on the ideal side, it makes sense that there's an operation because you're already living in a number ring where you've gotten an operation. But on the quadratic forms side, it's a surprise. And so that's one of things I like about this theorem is that you see some structure and you want to understand why. And the reason to understand why is just to change your perspective and realize these objects can be viewed as a different kind of object where that behavior is completely natural. Yeah, so that's one thing that I like about it.

EL: And does this theorem have a name or an attribution that you know?

KS: Oh, it's such a classical theorem that no, I don't know.

KK: Right. It's just the air you breathe, right? So what's the actual explicit bijection? So you've taken a quadratic form. What's the corresponding ideal?

KS: Well, actually, the other way is a little bit easier to figure it out.

KK: Yeah, let's go that way.

KS: So let's take the Gaussian integers, okay. And in the Gaussian integers, you've got — for your ideal, so think of it as a subset of the Gaussian integers. But because it's an ideal, it has the property that it has the same shape as the Gaussian integers. I actually usually like to draw a picture. So I'm going to try to draw a picture just out loud. So if you think of the Gaussian integers in the complex plane, they fill out a grid, right? It's all the integer coordinates in that plane. So that's a grid. And if you want to see what the ideals are, they’re subsets that are square grids as well, but fit inside that grid that we started with, maybe rotated or scaled out.

EL: Okay.

KS: But they're square again.

EL: Okay.

KS: And so, what you can do is with this example, specifically, you can take the norm of each of these elements in the Gaussian integers. So the norm of a complex number, usually I think of it as the length from the origin. But I don't want to do the square root part. So if I have a Gaussian integer x + iy, I'm going to take x2 + y2, and that's the norm.

KK: Okay.

KS: All right. And so if I take the whole Gaussian integers, which is itself an ideal, that's one of the subsets that is valid as an ideal, then if I take all of the values, all the norms of all those elements, that's all the values of x2 + y2. So from my collection of integers, I take all of the values and that's actually a quadratic form.

KK: Okay.

KS: Okay?

KK: Okay.

KS: And so you can do this with the other ideals as well. So for each one, you look at the norms of all of its elements, and that is a quadratic form and the values of that quadratic form?

KK: Right. So the Gaussian integers are Euclidean, right? So it's PID, right?

KS: It is. It’s a principal ideal domain.

KK: So everything's generated by one element, basically every ideal?

KS: That’s right.

KK: So that makes your life a little simpler, I suppose.

KS: Yeah. So the ideals, in that case, really, they're not so different than the numbers themselves. This is one of those ones where you don't have to go to ideals. But by going to it, you think about instead of just, say, 1+i the number, you think about all the multiples of 1+i and you take all of those, and you take their norms.

EL: Okay. And I told you, when we were emailing earlier, that you'd have to hold my hand a little bit on this. So yeah, sorry, if this is a too simple question or something. But like, what is the quadratic form like the x2 + whatever xy +whatever y2 that you get from the the Gaussian integers that you just said?

KS: Right. So if we take the Gaussian integers, if I take x+iy as a Gaussian integer, its norm is x2 + y2. That’s the form right there.

EL: Okay. Yeah. All right.

KS: And then if I were to take a subset, like all the multiples of 1+i, I'm not plugging in all x's and y's. I'm plugging in only multiples of 1+i, so you end up with a slightly different form popping out.

EL: Yeah, so I guess it's kind of like x+x then.

KK: 2x2 squared basically, right?

KS: Yeah. Yeah. You could have Yeah, various things in various different situations, but yeah.

EL: Okay, thank you. Yeah. And so, yeah, can you talk a little bit about how you encountered this theorem? If it was something that like you really loved to start out with? Or if your appreciation has grown as you have continued as a mathematician?

KS: Yeah, well, it's one of these things, so I think everybody has things that they're attracted to mathematically, they all have a mathematical personality. And there's some sort of particular kinds of things that attract you. And for me, one of the things is the sort of projection theorems that tell you that a particular structure, if you look at it a different way, has a whole different personality. And it's actually the same thing, but it has just become totally different. So I really love those things. And I've always loved number theory, because it has such simple questions. But then when you dig into them, you always run into such fascinating, complex structure hidden. And so this is one of those things that if you have that kind of personality thing you, just keep bumping into. And so for me, and all of the research that I've done and things I've been interested in, I keep coming back to this theorem and bumping into it in different places. It shows up when you study complex multiplication of elliptic curves, it shows up when you study continued fractions, it shows up all over the place. And it just seems so fundamental. And it's sort of like maybe one of the most fundamental examples of this special kind of theorem that I really enjoy.

EL: Okay.

KK: Cool.

EL: All right, well, then the next portion of the podcast is the pairing. So yeah, as you know, we like to ask our guests to pair their theorem with something that helps you appreciate the theorem even more. What have you chosen for that?

KS: So, when I think about this theorem, I just it's a treat. So I think the only thing that comes to mind really over and over again is just chocolate. I love chocolate. And that's what you should enjoy this theorem with because maybe you should just be happy enjoying it.

KK: I mean, chocolate pairs with everything.

KS: That’s true. It's a bit of a cop out.

KK: No, no, that’s okay. So our most recent favorite chocolate is Trader Joe's has this stuff. And it's got, I don’t even know what’s in it, pretzels and something else crunched up in these like bark of chocolate. And it’s a dark chocolate I really recommend it. So you must have a Trader Joe's in Boulder, right?

EL: Are you a dark, milk, or white chocolate person?

KS: Oh, definitely dark. Yeah, I have a dark chocolate problem, actually.

EL: Yeah, the Trader Joe's. For me the dark chocolate peanut butter cups are are always purchased when I go to Trader Joe’s.

KK: Dark chocolate feels healthier, right? It's got more antioxidants and a little less sugar. So you're like this is fine, less milk. Okay. All right. It's actually it's a fruit, right?

EL: It’s a bean. You’re having a black bean pate right there.

KK: That’s right.

EL: Yeah, well, Salt Lake is actually a hub of craft chocolate. We have some really wonderful chocolate makers here, like single origin, super fancy kind of stuff. So if either of you are here, we'll have to pick up some and enjoy together. And yeah, along with quadratic forms and ideal classes.

KS: Sounds wonderful.

EL: Yeah. So something I meant to talk about this earlier in the episode, but you mentioned that you'd like to illustrate things, and that is how we first met is, through mathematical illustration. So I don't know, maybe it's a failure of imagination on my part, but I always, I'm always fascinated by like, number theorists who are really into illustration as well, because I think of, like, geometry, as you know, it shapes it as the more naturally illustrate-y parts of math. But would you talk a little bit about it, you know, illustrating number theory? And if if you've done anything related to this particular theorem, or if there's something else you want to talk about with your mathematical illustration?

KS: Oh, yeah, that's a that's a great idea. Yeah. So there's actually building up gradually a wonderful community of people who are interested in illustrating mathematics. And so that's maybe one of the things that you could add a link for is the website for the community.

EL: Definitely.

KS: Yeah. And so I've always found that the way I think about mathematics is very visual. I mean, I think as human beings, we have access to this whole facility for visual thinking, because we're embedded in this three-dimensional world that we're living in. And another way that we think about mathematics, I think often, is we're using another one of our natural facilities, which is our sort of social understanding facility, where we imagine characters interacting with each other and having motivations and stuff like that. But for me, it was always a very visual thing. And so even though it wasn't taught in that way, in my mind, somehow these things were always very visual things. And so I've always been really attracted to situations where you can see some hidden geometry in number theory. And with this particular theorem, there is a little bit of nice hidden geometry. I mean, the first hint of this is that when I talked about ideals in the Gaussian integers, I visualized them as a lattice.

EL: Yeah.

KS: And in all of these number rings, you can do this, you can you can think about lattices. And you're really talking about lattices, and lattices have things like shape. And you know, there's lengths and angles and stuff like that to talk about. And so one of the really cool things that you can do is you can think about, for example, with the Gaussian integers or with some other ring of interest that you can put in the plane like this, into the complex plane, then you can ask this question, it's a natural question that people ask: how can I study the collection of objects instead of the individual objects themselves? So if you want to study the collection of lattices, say, two-dimensional lattices in the plane, then one way to do it would be okay, how do I decide on a lattice? Well, I have one vector that's generating it, and then another vector that's generating it. So let's put the first one, let's sort of ignore issues of scaling and rotation, let's put the first one down pointing from like zero to one. And then the other one is somewhere, but now you don't have any choice anymore. No more freedom. And so you can think of the plane itself as a sort of moduli space, as a parameter space for the collection of lattices. And this space has a lot of beautiful properties. So you might as well order your vectors so that we're just talking about the upper half plane. So the first vector is from zero to one, and the other one is an angle less than 180 degrees from that. And so when you start looking at the geometry of this plane, and you want to talk about well, okay, I want to look at lattices, but maybe I don't care so much about what basis I'm using for the lattice, you start to divide the plane up in certain ways. And what you discover is that the natural way to talk about this plane is using hyperbolic geometry, actually. And so all of a sudden, you're doing hyperbolic geometry. And I find myself doing hyperbolic geometry sometimes when I'm doing number theory, because when I want to look at these these objects and stuff, that's just the natural world in which they live. I mean, the mathematics kind of tells you what you have to do you. You know?

EL: Yeah.

KS: And so those are moments that I really enjoy, because you're doing something that you think is just some algebra, but all of a sudden, it turns out it's geometry.

EL: Very cool. Yeah. So we will include a link to your website, which I know has some of the cool illustrations that you've done available there. And to the illustrating math, there's an online seminar that meets monthly that is really nice to go to, if you can. And, yeah, it's a lot of fun. And I, yeah, just so many different fields of math represented with that in ways that I never would have guessed.

KS: That’s true. That's one of the nice things about that community is that there's people from all different areas that you wouldn't normally interact with, because usually you have a pretty narrow research circle, if you're doing research in mathematics. But there, you're talking to everybody. And it has a much more creative feel for that reason, you get surprised by people's ideas, because they come from just a little bit farther from your home base, you know?

EL: Yeah. And I think it also kind of pushes people to really think about how they're explaining things, where you have a shorthand when you're working with someone who is in, or talking to someone who's in such a close field, and since you don't necessarily have that same common background, people, I think, it seems like are very thoughtful about how they describe things and what they assume that you already know.

KS: Yeah, exactly. It's just good to get out of your little corner.

EL: Yeah.

KK: All right. Well, this has been great. I definitely learned something today. I did not know this connection between ideal classes and quadratic forms.

KS: Oh, I thought of one more, one book I'd like to plug.

KK: Okay. Please do.

EL: Great. Yes.

KS: Yeah, so Martin Weissman has written a book called — I'm going to get the title slightly wrong. It's An Illustrated Theory of Numbers, maybe? Oh, you have it. Oh, I got it right.

EL: Yeah. An Illustrated Theory of Numbers. It's been holding up my laptop, after I read it, I will say.

KS: Yeah, and so you were asking about illustrating number theory, and this is just a beautiful book. It's completely accessible. I used it when I was teaching an introduction in number theory for undergraduates. But it doesn't require any particular background because it starts from, you know, we’ve got the integers, we’ve got addition, we’ve got multiplication, let's do some stuff. And and he really looks hard for ways to turn theorems which are usually completely algebraic into something visual, and they're just lovely.

EL: Yeah, and really amazing illustrations, and full color, like everywhere, which I know is more expensive to make books, and that's why books so often have the color in the middle, like in a little section and not the whole thing. But I do think this is just much more pleasurable to read because it it is does use that aspect, too. And it's not as stark as every page being black and white.

KS: Yeah, it's so inviting. It's a wonderful book.

EL: Yeah. Great recommendation. Thank you so much for joining us. I really enjoyed talking with you.

KS: Yeah, me too. Thank you so much for having me on.

[outro]

For this episode, we were excited to talk to Kate Stange from the University of Colorado, Boulder about the bijection between quadratic forms and ideal classes. Below are some links you might find interesting as you listen.
Stange's website
The Illustrating Mathematics website and seminar, which meets monthly on the second Friday
An Illustrated Theory of Numbers by Martin Weissman
The Buff Classic bike ride in Boulder

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 I am joined, as always, by my fabulous co-host.

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, trying to remember how to do this. It's been a minute since we've recorded one of these. We kind of went dormant for the winter.

KK: Yeah, a little bit, a little bit. Yeah. But Punxsutawney Phil told us — I don’t, what did he say? Let's pretend he said six more weeks of winter.

EL: I think he usually does. I don’t know.

KK: I mean, objectively, there are always six more weeks of winter. Like, the calendar says so, right?

EL: Yeah.

KK: Anyway, yeah.

EL: And, you know, he probably is pretty good at seeing shadows if he's a prey animal because he'd be used to seeing, like, a bird coming overhead.

KK: That’s an interesting question.

EL: Do birds eat groundhogs?

KK: That’s what I was going to wonder. I mean, like, eagles, maybe, but groundhogs are pretty large, right? I mean,

EL: Yeah. What eats groundhogs?

KK: Well, that's something to investigate later.

EL: Yeah.

KK: So it is Pi Day, right?

EL: It is! Well…

KK: We’re actually, we're recording this on Pi Day. When our listeners hear this, it won't be, but we're recording.

EL: And, I always have to put in a plug for my calendar.

KK: That’s right.

EL: The AMS math page-a-day calendar on which Pi Day does not occur on this day.

KK: That’s right.

EL: There are other Pi days on this calendar, none of which is this day, my little joke here. So you can find that in the AMS bookstore.

KK: Right. Are you Team Pi or Team Tau?

EL: I’m Team whichever one works for the calculation that you’re doing. It’s not that big a deal.

KK: That’s right. That's right. Okay. All right. Enough of us, enough of our useless banter, although we did discuss what's the ratio of banter to actual talk, right, that there's, there's like a perfect ratio. But we are pleased today to welcome Karen Saxe. Karen, why don't you introduce yourself and let us know all about you?

Karen Saxe: Hi, there, everybody. So first of all, happy Pi Day. If listeners know who I am, I was a professor at Macalester College for about for over 25 years. And then about seven years ago came to work at the American Mathematical Society, where I am very happy to be the director of the Government Relations Office. So I work in DC with Congress and federal agencies. And could quite a bit about this. I'm also happy to be here because it's Women's History Month. And it will be appropriate that it is Pi Day when you hear what my favorite theorem is.

KK: Okay, good to know. So, I'm curious to know more about this government relations business. So I mean, I know that the AMS does a lot of work on Capitol Hill, but maybe some of our listeners don’t. Can you explain a little more about what your office does?

KS: Yeah, so we do a lot of things. So first of all, we communicate — I sort of view the work of our office as going two ways. One is to communicate to Congress why mathematics is important to almost everything they make decisions about, you know, our national security, health care, you know, modeling epidemics, thinking, like you’re in Florida, thinking about how to model severe weather and things they care about, and then why they should fund fundamental research in mathematics and all sciences. And then also you know, how they make decisions about education. So we tell Congress, we give them advice and feedback on our view about what they should do in those realms. And then on the sort of flip side, I tell the AMS community, the whole math community about what Congress is doing and what's happening at the agencies like the NSF, and Department of Defense and Department of Energy, that that they might care about things, things that would affect their lives. So that’s sort of it in a nutshell. I spend a lot of time on the hill. I just came this morning, I went to a briefing put on by the National Science Board, which is the presidentially-appointed board that oversees the NSF. And they put out a congressionally mandated report every few years on the state of, it's called the indicators report. I'm sure I found it more interesting than everybody else, but it's pretty fascinating. You know, it covers everything from publications around the world, like which countries are are putting out the most science publications, what the collaborator network looks like around the world, and that to sort of US demographic information about education, you know, who's getting undergraduate degrees? Who's getting two year degrees? Who's getting PhDs, that that sort of thing. It covers a lot, actually. Pretty interesting.

KK: Yeah, yeah. All that in like two hours, right, and then it's over.

KS: Yeah, all that in two hours. And then they give you the big report that you can. And I've got them sitting in front of me. But given that this is a podcast, showing things doesn't work.

KK: Well, we do it all the time.

KS: Here’s one of the reports I picked up this morning. Actually, one really, so they're, you know, they're one thing. And you might end up cutting this, but one thing that's sort of fascinating to me is they always list barriers for getting into STEM degrees. And you know, there are things listed, like college accessibility, things that — and even going back. So like, you know, school kids who say they don't have science teachers in their schools, they don't have math teachers, but they've added to this list. “I can't support my family on a graduate student stipend.” So this is something.

EL: Yeah.

KK: That’s real.

KS: And we are, we've endorsed a bill in Congress that would look that would help to improve the financial stability, I guess, you would say, or the ability to be a grad student or a postdoc. So it's looking at stipends, it's looking at benefits, you know, leave time, all that sort of stuff, making it a job that you can choose to take when you're 23, and have a family to support and could make a hell of a lot more money doing something else with a math undergraduate degree.

EL: Yeah, and not see it as something where it's like, you're kind of putting off real life for a little longer, which I think maybe in the past was more of the model, like, oh, yeah, you'll have a real career later. But you know, in your mid-20s, you'll just keep being a student and not have kids or, you know, things, you know, not have parents to support or things like that.

KS: Exactly.

KK: Yeah. Okay. That's, that's good to know. Thank you for all that hard work you do, Karen. So but this is a math podcast.

KS: Right.

KK: So what’s your favorite theorem?

KS: Okay, so first, I'm going to tell you about the three theorems that I didn't choose.

KK: Cool.

EL: Great.

KS: So — I'm sure everybody goes through this — and thinking about my research, it would probably have to be the Riesz-Thorin interpolation theorem, which basically tells you that if you've got a bounded linear operator on two Lp spaces, then it's bounded on every Lp space in between those two values of p, so I used that all the time when I did research on that sort of thing. Then, but I was primarily a teacher of undergraduates, and kind of my two favorite theorems to teach are always Liouville’s theorem and, and then the uncountability of the real numbers.

EL: Yeah.

KS: And Liouville, they’re the one that says, you know, that there's a bounded — if you have a bounded entire function function, it's got to be constant. And the result is so stunning, and it gives a great proof of the fundamental theorem of algebra, that every non-constant polynomial has a root. So I always love teaching that. And then of course, like, Cantor’s diagonal argument about the real numbers, nothing beats that proof in terms of like, cool proof, in my opinion.

EL: Yeah. All-time great.

KS: Yeah, all-time great, right. And I think it's been mentioned on your podcast before. But what I picked was this theorem that says that if you have a given fixed perimeter, then the circle maximizes the two-dimensional shape you can make, so the isoperimetric theorem.

EL: Nice! And as you said, very appropriate for Pi Day.

KS: Yeah, which, I hadn’t even thought about that, which is sort of also embarrassing. But until we started acknowledging Pi Day, I hadn't thought about that. So another way to say it, or the way you might see it in a textbook, is if you have a perimeter P and an area A, then P2−4πA is greater than or equal to 0, with equality if and only if you have a circle. So this theorem has a very long, fascinating history. Lots of great applications. And for all those reasons, I love it. I love history.

KK: Yeah.

KS: I love math.

KK: Yeah. Do you have a favorite proof of this theorem?

KS: I do, actually. Yeah. Well, I didn't know you'd ask that. So there are a lot of proofs. And the one that I like, and this comes from being an analyst probably, is in the early 1900s. Hurwitz gave a proof using Fourier series. I love that proof. And proofs are quite old, going back thousands of years to the Greeks. And then in 1995, Peter Lax actually gave a new short calculus-based proof. But I like the Fourier series proof, just because I like Fourier series.

EL: Yeah, that's a topic that I wish I understood better. Somehow I kind of missed really, ever feeling like I've really got my teeth into Fourier series. Maybe that's a little embarrassing to admit on a math podcast.

KK: I don’t know. I took that one PDEs class as an undergrad and, like, that's where you see it, you know, doing the — whichever, the wave or the heat equation, whichever one it is — maybe both? I don't know. And then that’s it, that shows you how much I remember, too.

KS: Yeah. Good. So you're not gonna dare ask me to give you that proof or anything?

EL: Yeah, generally, a proof like that on audio is not the ideal medium.

KS: It doesn’t work.

EL: Actually, you brought up these ancient proofs. So yeah. Yeah, I guess how long has humanity known this fact, do you think, or do you know?

KS: So it's considered that the Greeks knew the proof. And then it was proved around 200 BCE. It even features in Virgil's version of the tale of Dido, Queen Dido.

EL: Oh, that’s right.

KS: So yeah, I think that was around 50 or 100 BCE, after the Greeks knew the theorem. So can I say what that story is?

EL: Yeah.

KK: Yeah, please.

KS: So she apparently fled her home after her brother had killed her husband. Okay, so we're already in an interesting phase. She somehow ended up on the north coast of Africa after that, and she was bargaining to get some land. And they told her, oddly, that that somehow she could get as much land as she could enclose with an oxhide.

KK: Okay.

KS: And so she took this oxide and cut it into very thin strips, and then enclosed an area, that was the largest she could conceive of, with the given per perimeter.

KK: Okay.

KS: So there's that. So it appeared, like, 2000 years ago, or more, and then you sort of we sort of jumped into the early 1800s when Steiner gave geometric proofs. But what's kind of fascinating is his proofs all assumed that a solution existed. And I haven't looked at these proofs, at least not in a long time. But then later in that century, Weierstrass is credited with giving a proof that, well, first, he proves that a solution does in fact exist. And he did use the calculus of variations to get this proof. So that's, that's sort of the story of the, of the theorem.

EL: Yeah, this actually — you know, we say the Greeks knew this, but I kind of wonder if this is one of those things that humans would kind of intuitively know, even if they're not in a framework where they have language about proving mathematical theorems, even if that's not an aspect of, of their culture, but it seems like you're trying to get into the mentality of like, what is really intuitive or innate about mathematics for humans? And I wonder if that, you know, we kind of would understand, well, if I took a square or something, I could sort of bow it out a little bit, and get a little more area with the same string.

KS: Actually, I mean, one reason I love this theorem is you can give string to kids, and I used to do this, like in elementary schools, and tell them make the biggest shape. And you have to tell them what closed is, no, you have to describe that the string has to come back to where it started. And they all come up with a circle. And this is, you know, second, third grade kids. So it is really intuitive. Yeah. So what it's meant by the Greeks knew this theorem is not 100 percent clear.

KK: Because they didn’t even use pi, right?

KS: And then actually, Evelyn to what you just said, you know, there's something that's quite interesting to me, which is that, you know, if you think about, you know, shapes of constant width, you know what I'm talking about?

EL: Yeah.

KS: So, if you take the fixed perimeter, there's an infinite number of these, the circle’s the largest one and those Reuleaux, I think that's how you say his name, those triangles are the ones of smallest area.

EL: Okay.

KS: And you were just kind of alluding to that, like take a triangle and go puff out the sides, or something.

KK: And you can push in.

KS: Yeah. Right. And you can do it for any regular polygon.

EL: Yeah. Well, British money has a couple of these that are I think heptagons, Reuleaux heptagons? Are they all called Reuleaux? Or just the triangles? I don't know.

KS: No, but you’re right about that, they do. And so it's kind of funny, I saw something that was talking about these points, like, what possessed them to make those points? And if you have a machine that has a hole size, and you know, it could fit a circle, it has a diameter, right, but it can also obviously fit one of these other shapes. Yeah. So that works. And I think you're right. It's a heptagon, heptagonal version of those.

EL: Yeah. The first time I went to the UK, this was, I think, the most exciting things on my trip to me, was these coins. Like, who thought to make these? And I actually, I remember, I wrote a blog post about it and discovered that it was a little hard to figure out if I had the rights to use a picture because all the images of these coins are like, technically property of the Crown.

KS: That’s funny.

EL: Abolish the monarchy, man.

KK: Her Majesty relented in the end?

EL: Yeah, so strange. I was like, well, I'm not gonna beg the queen for the right to post this on my math blog. So I don't remember what happened with that. Hopefully, I'm not opening myself to takedown.

KS: I think you’re probably okay.

EL: Hopefully the statute of limitations has run out on that. Anyway.

KK: I recently came across, I was going through an old notebook, and I found — I don't know why I tucked it in there — from the late 90s. I had one of these 10 Deutsche Mark notes that had Carl Gauss on it.

EL: Oh, nice.

KK: And so I put it on Instagram. And I'm now starting to worry. Wait a minute. Will the German government come after me? Although it's not really legal tender anymore.

EL: Yeah, the pre-2000, whenever they went to the Euro, government.

KK: It was pre-Euro. Yeah, I think I’m safe too.

EL: But anyway, getting back to the math, Karen. So, has this been a favorite of yours for a long time? I guess to me, this is one that I don't think the first time I saw it, I would have been super impressed by it. So what was your experience? What's your history with this theorem?

KS: Right. So like, why did I decide I liked it? Because yeah, it's sort of like, okay, I mean, it's appealing, because everybody can understand it, it’s very intuitive. It's got this, the proof has this interesting history. But why I like it is because you probably know that I'm pretty engaged with congressional redistricting. And when they do measures of compactness of districts, this is the theorem that kind of motivates all their measures.

KK: The Polsby-Popper metric, right?

KS: Yes, exactly. And so you take the Polsby-Popper measure, which was come up in 1991. So like, different states, should I say something about redistricting?

KK: Sure, yeah.

KS: So yeah, I mean, just like the very brief thing is every 10 years, we have to do the census. This is mandated in our Constitution, for the purposes of reapportionment of the House of Representative seats to the state so then after the census is done the seats, which we now have 435 of them, they're doled out to the states. And how that's done is a whole nother you know, interesting math problem, more interesting, probably. But then once the states get their number of seats, like how many in Florida?

KK: We’re up to 27? [Editor’s note: It’s actually 28.]

KS: So let's pretend there's 27 for a minute.

KK: I think that’s right. [Ron Howard voice: It wasn’t.]

KS: Okay. Then, you know, the Florida Legislature, probably, I don't know who does it in Florida, but somebody.

KK: Let’s not talk about that.

KS: Yeah, let’s not talk about that. Whoever’s in charge has to carve up the state geographically into 27 districts, one for each representative, and how they do that geographic carving up is extremely complicated. And to answer the question, “Has this been gerrymandered?” there are certain measures of what's called compactness, and this is like a whole nother thing I could talk for hours on. And compactness sort of measures the lack of convexity, sort of, so like, are there long skinny arms going out? And this is where obviously, like a podcast is, is not the best. But in any case, you know, are there long skinny arms going out, or does the thing look like a circle? So the Polsby-Popper measure tells you how close to a circle, or a disk because it's filled in, but in any case, your district is. Well, that's kind of weird, because if you think about tiling any state with circles, it’s just not going to happen.

EL: Right.

KS: Yeah. So just to sort of fetishize circles is bizarre. But I guess, like, what are your other options? Well, there are lots of other options. But the Polsby-Popper is the most common. There's a handful of states that require specific compactness measures in their process, and many other states that require compactness, but they don't specify the actual measure. In any case, the Polsby-Popper is the most common. And the other common measure is called the Reock measure, and that also fetishizes circles. It's a similar type thing. So with the Polsby-Popper, it's kind of interesting, because they they first published it in a law journal in 1991, in this context for redistricting, but it has actually been mentioned, as far back as the late ‘20s. And can I read you a funny a funny opening line?

EL: Yeah, sure.

KS: So the it first appeared, as far as I know, in a 1927 paper in the Journal of Paleontology. Okay. And how's this for the start of a paper? In quotes: “How round is a rock? This is a question that the geologist is often forced to ask himself.” Okay.

EL: Nice.

KS: So that's a great opening sentence. And then it kind of carries on: “when he wishes to consider the amount of erosion that a stone has received.” And then the paper is actually about measuring the roundness of grains of sand.

EL: Oh, cool.

KS: So there's a lot to say here that the paper is filled with hilarious hand drawings, you know, but also, of course, that geologists seem to be male is another observation.

EL: Yeah, well, and the grammar rules of the time.

KS: Yeah, exactly. But even just this past January, I ran into a paper that was published, and uses this to measure the aggressiveness. It's in, like, a cancer journal. I can't remember which one. And I wrote it down, but of course, what do you know, I can't see it. Anyways — oh, Cancer Medicine is the name of the journal — and it used the Polsby-Popper measure to measure aggressiveness of tumor growth. So you know, it has a life.

EL: That's so so interesting. When you said Journal of Paleontology, I was just like, how is that going to come up in paleontology? But what do you say? Yeah, how round is a rock? It's like, yeah, you do need to measure that. I actually, just the other day watched this interesting video about sand grains and like, certain beaches, or, and certain dunes have different acoustical properties. Due to, like, if they've got a lot of the same sized sand grains and if they pack really well, or if they don't, sometimes there can be the squeaking effect, like when you walk on it, or in a dune, like when there's wind, there can be these like deep, deep resonances, like almost a thunder sound that happens.

KS: Oh, that is interesting.

EL: And this this video went and looked under the microscope at the sand on these different beaches, and kind of showed how some of them packed together better or worse, and some of them are more uniform. So they might secretly be using that metric.

KK: They might.

KS: That’s fascinating. I mean, I heard I've heard that squeaky sound on beaches.

EL: I never have I'm not a huge beach person. So I guess, yeah, but I'm curious about going to one of these beaches someday now.

KS: Yeah. And when you said that I was thinking of the packing, like how they pack, but that would have to do with their shape, and their size. Well, I don't know.

KK: So this is a sphere packing question now. And it's yes.

EL: Or a “how sphere-y is your sphere”-packing question.

KS: How spherey is your sphere?

EL: Not quite as catchy.

KK: Right. So the other part of this podcast is we like to ask our guests to pair their theorem with something, so what pairs well with the isoperimetric inequality?

KS: So naturally, you know, a mathematician would ask, are there analogs in higher dimensions? Right? And then back to how spherey is your sphere, so I play tennis quite a bit. So I'm going to pair it with tennis.

EL: Excellent.

KS: The shape of the ball abides by the theorem.

KK: Yes. Right.

KS: And works for so many reasons.

EL: Yeah. Well, and you are not the the first My Favorite Theorem guest to pick tennis, actually.

KK: That’s right. Yeah.

EL: Yeah, we've had Dr. Curto.

KK: Carina.

EL: Yeah. Carina Curto, paired paired hers with tennis. It was it was about linear algebra. That's right. Yeah. Yeah. Hers was about how this thing kind of goes back and forth. When you're doing this thing in linear algebra. So you picked different aspects of tennis to pair with your theorem.

KK: Yep. Do you play much do you, you play, you play a lot?

KS: I play — it’s embarrassing to put on a very well listened-to podcast — that I do play a lot, because I don't know how good I am.

KK: That doesn’t matter.

KS: But I play a couple times a week.

KK: I used to play quite a bit. So as a teenager, certainly. And then in my 30s I played a lot. I played a little league tennis. This is when I lived in Mississippi. And actually, my team won the state championship two years running at our level.

KS: Oh, wow.

KK: But I'm not any good. This was like, you know, I'm like a 3.5. Like, you know, just a very intermediate sort of player.

KS: Yeah, that's what I am.

KK: Yeah, my shoulder won't take it anymore.

KS: I still, I feel lucky. Because physically, I can do it. Right now. I'm in a 40+ league, and that's good. But next season, whatever you call it, or next season, I guess, I'm in an 18+ League, and I've done this before. It means the other players are allowed to be as young as 18. It’s a little humbling, even if we can serve, you know, we have the technical skills, like they’re, you know, like the shots you use in the 40s, like, lobbing is not a good strategy in 18+ because they can run.

KK: Back when I was in my 30s and played, I played a lot of singles still, and I could still do it. But when I would come up against the 20-year-olds, it'd be a lot harder. But then I also learned, I used to play a lot of doubles with with these guys in their 70s. And they destroyed me every time. They were just —because they knew where to be. They had such skill and good instincts for where the ball was going to be. It was humbling in that way.

KS: Yeah, it's it's fun. And I prefer playing doubles these days. It's just more fun and different strategy.

KK: Yeah, and less court to cover. That helps.

KS: Less court to cover. And it’s more social. It's a lot of fun.

KK: Yeah, so you haven't succumbed to pickleball, have you?

KS: I played once, on my 60th birthday. Because no one would play tennis with me. And I got invited to a pickleball thing. And I was like, Okay, we're gonna do it. And, you know, it was fun, but I haven't really. It’s a challenge in Minnesota playing pickleball because it's so windy and the balls are so light, and it’s like whiffle ball.

KK: That’s what they are, basically.

KS: The ball kind of blows around all over the place. So yeah, I haven't I succumbed to doing that. In DC I'm lucky to have enough people to play tennis with. There's a lot of them.

KK: Cool. All right.

EL: Yeah. Great pairing.

KK: Yeah, yeah. So we also give our guests a chance to plug anything they're working on. You sort of already did that. I mean, you're doing all the work. Anything else you want to pitch?

KS: I mean, back to what I do, one reason I love this new job is I get to go in and and make connections to any Congressperson. You know, they have their own interests motivated by their own history, their own life, their own constituents. And this can be — there are obvious things we think about, like people, congressional members who are interested in their electric grid, or ocean modeling for the Hawaii delegation. But it's fun. And it's a fun challenge to think of things. So there's one newish member who was a truck driver before he was elected to Congress. And, we went in and their office was like, we can't make a connection to math. And we started talking about logistics, you know, truck routing. And it was great. It turned into a great conversation where they hadn't really thought about that. So this is what I really love about my job, trying to connect math to anything they’ve got. What they’re interested in, I'm gonna I'm gonna try to connect math, and there are very few issues that that can't be connected.

EL: Yeah, well I actually have a question, something that our listeners might be interested in is like if a mathematician is listening to this, and wonders, how can I get more connected to what's happening? How can I understand what math and science, you know, representatives do on the hill? Is there a newsletter or a website or something that you have that they could look at? And, you know, maybe find ways to get more involved? Or at least more informed?

KS: Yeah, definitely. So first of all, I used to write a blog, but I don't do that anymore for the AMS. The AMS Government Relations page — so my office is the Office of Government Relations. And I believe if you search, AMS government relations, you'll get to my webpage, you know, the one that I call mine, and you'll see a lot of different things there. There are ways to get engaged. We offer felt three fellowships. Two are for graduate students, one is for a person with a PhD in mathematics to come and to come here physically and do things. One is a boot camp for graduate students, a three-day graduate boot camp to come learn about legislative policy. And then the the biggest one is a year long fellowship and working in Congress. I do hill visits with people. And you know, I'm pretty willing to bring almost any mathematician to the hill, and that can be virtual these days. So we have volunteer members through our committee work who fly in and do these hill visits. We did this last Wednesday, we had about 25 AFS, volunteers fly in, and that was a fantastic day. But I can do them virtually. I've done them with big groups of grad students from departments, and people can email me if they want. And I think you guys have my email.

EL: Yeah. Thanks.

KS: So those are the big ways. And then for AMS members who are a little more advanced in their careers, you can volunteer for AMS committees. And there's the Committee on Science Policy, which really focuses on this one. And then I'm also in charge of the Human Rights Committee for the AMS, which can be of interest to a lot of people.

KK: Sure.

EL: For sure.

KK: Lots going on there.

KS: Yeah, lots going on.

KK: Well, Karen, this is terrific. Thanks so much for taking time out of your day, and thanks for joining us.

KS: Thank you.

[outro]

In this episode, we enjoyed talking with Karen Saxe about her work as the director of the American Mathematical Society's Office of Government Relations and her favorite theorem, the isoperimetric theorem. Below are a few links you might find relevant as you listen:
Saxe's website and the homepage of the AMS Office of Government Relations

A survey of the history of the isoperimetric problem by Richard Tapia 
The 1995 proof by Peter Lax
Evelyn's blog post about 50 pence coins and other British objects of constant width
The Polsby-Popper test to measure gerrymandering
A public lecture by mathematician Moon Duchin about mathematics and redistricting
The 1927 Journal of Paleontology article that first uses the Polsby-Popper metric (though not with that name)
An Atomic Frontier video about squeaky sand
Our episode with fellow tennis-enjoyer Carina Curto

The 10 Deutsche Mark note

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right. Yeah, I was I was trying to think about what to say. And I was like, well, the most exciting thing in my life right now is that our city is starting a pilot program of food waste, like a specific food waste bin.

KK: Okay.

EL: But then I realized I also did an 80 mile bike ride last Saturday, and that's the first time I've biked that far. And that might be slightly more exciting than compost.

KK: Are you working up the centuries? Are you are you heading for?

EL: We’ll see. I felt pretty fine after 80. I also don't feel like I wanted to do 20 more miles. So we'll see. Someday, maybe

KK: The last century I did was, wow, it was 2003. It was 20 years ago. This is one called the six gap century in Georgia. And it goes over six mountain passes in the mountains of North Georgia, one of which has, like, a 15% grade, which is quite steep. It took me about eight hours. And then I hung up my bike and didn't ride it for like three months.

EL: Well, I mean, it would probably take me at least eight hours to do a flat century.

KK: Yeah, but back in my youth I could do a flat century in about five, but not anymore. Not anymore. So let's keep this banter going because I — so Ben Orlin, a former guest on our podcast, I saw Math with Bad Drawings today had various golden ratios. One of which was the golden ratio of hot fudge to ice cream in a hot fudge sundae, which he argues is one to one, but that's way too much fudge.

EL: That is so much fudge!

KK: But the podcast, like, substance to banter golden ratio, he claims is like two to one. So like a third of this should just be like us, you know, just shooting it.

EL: Saying nothing.

KK: Yeah.

EL: Well, I must admit, that's why I listen to fewer podcasts that maybe I would want to because I have a low banter tolerance. Which brings us to our guest today. Yeah, so we are very happy to welcome Corrine Yap today. Would you like to tell us a little bit about yourself?

Corrine Yap: Yes. So I am currently a visiting assistant professor at the Georgia Institute of Technology, Georgia Tech, in the math department. I'm also a postdoc affiliated with the Algorithms and Randomness Center. But I just got my PhD in the spring from Rutgers University.

KK: Congratulations!

CY: Thank you! I do a lot of, like, probabilistic combinatorics, and stuff around that. So that's sort of my main research work. I also do some performing and some playwriting as well. I actually just got back from a performance yesterday Worcester Polytechnic Institute in Massachusetts.

EL: Oh, wow.

CY: So very busy this time.

EL: Yeah. That’s actually one of the reasons that I've been wanting to invite you for a while. And I was like, well, I should wait until I've seen one of her shows. And then it just has not aligned to work out. Because I know you've done them at the Joint Meetings and things, and the times that I have been there and you have been there, it’s just not been a good time. So it's like, well, I'm not going to put this off forever. So even though I have not yet seen one of your shows, I'm very glad that that we could invite you and have you here and yeah, well, can you talk a little bit about the kinds of theater that you do, or kinds of — I don't know if it's mostly theater or more, like other? I don't know, speaking performances?

CY: Yeah. So it really started when I was a lot younger. And also in college, I primarily studied both mathematics and theatre, with no sort of vision as to what that would turn into in terms of a job or career or anything. I just really enjoyed doing both of them. And as an undergraduate I thought I was mainly interested in acting, but I started studying playwriting while at Sarah Lawrence College in Westchester, New York. And I started writing this play, which is the play that I continue to perform. It's called Uniform Convergence. And it's a one-woman play that's about math. It tells the story of Sofia Kovalevskaya, who is a historical Russian mathematician. She was born in 1850. And it tells a little bit about her life and how she faced a lot of obstacles to be successful as one of the first few women in academia. But it also has a portion that is sort of inspired by my experiences being Asian American, and also being a woman pursuing mathematics. And the setting is that of a real analysis classroom, a lecture where the character Professor….

EL: Hence, uniform convergence.

CY: Yeah, and she is lecturing to her students. So at one point, they do reach the point of the class where they do uniform convergence as a topic. So, you know, in the past, I did a lot more — like, in college, I did, you know, the auditioning for plays and being involved in rehearsals, and all this sort of stuff. But since going to graduate school, and now having an actual job, this one play is sort of the main way that I keep my ties to doing theater and the theater world.

KK: Very cool.

EL: Yeah. Well, that's cool. I didn't realize that Kovalevskaya was the subject of this. I actually just read Alice Munro's short story, Too Much Happiness, which is based on her life. And actually was not my favorite short story in the collection that it’s in, but it, you know, she is such a compelling figure and another woman who was interested in math, you know, at a time when it was a lot harder for a woman to have an academic career in any field, and was interested in literature. Wrote, I think both memoirs and fiction?

CY: She also wrote a play.

EL: Oh, wow.

CY: Yeah. But it wasn't about math. But yeah, she was very much also in both of these worlds in, you know, sort of a more artistic, creative mindset as well as a mathematical one.

EL: Yeah. Fascinating person. So yeah, that's really interesting. And hopefully someday I'll get to see it.

CY: Yeah, I'm still performing. I didn't think I necessarily would be. But it's been since 2017. I've been performing it at different college campuses, and sometimes at conferences at different parts of the country. And I still get invited places. So as long as that keeps happening, I'll keep going.

EL: Yeah, when I was still in academia and doing a postdoc, I did, you know, I'd started doing writing. And sometimes I would get invited to do both like a research seminar talk and a public engagement kind of talk. And so that that might be in your future as well.

CY: Yeah, maybe.

KK: Yeah. Broader impacts.

EL: Wearing both hats on one trip.

CY: Yeah, I actually, I forgot I am doing that. I think this is the first time I'm doing it. At Duke in October, when one day I'll be giving a seminar talk, and then the next day, I'll be performing the play.

EL: Yeah, cool. Well, we invited you on here to talk about your plays, but also to talk about your favorite theorem. So what have you chosen?

CY: Yeah, so I've chosen Mantel’s theorem as my favorite theorem. So this is a theorem that is in the area called extremal combinatorics. And I'll explain what that means. But the statement of the theorem is pretty straightforward. It says that if you have a graph, which I’m a combinatorialist, so for me graphs mean, collections of vertices with edges connecting pairs of vertices. If you have a graph on N vertices, then the maximum number of edges you can have without forming any triangles — so just three edges and three vertices connected to each other — the maximum number of edges you can have with no triangles is N squared over four with appropriate floor.

KK: Yeah, sure.

CY: And this seems like, okay, this is this is just a statement, maximum number of edges. What's so cool about that? You actually, we actually also know where the N squared over four comes from. It’s, the extremal example is the complete bipartite graph on parts of size N over two. So what that means is, you split your vertices up into two sets, each of size half the total universe. And all of your edges go between the two parts. So from one part to the other, not inside the vertices of the parts. So complete means you have all the possible edges crossing between the parts, and then bipartite because you have the two parts of the vertices, and that has N squared over four edges. And it has no triangles in it.

KK: Not even any cycles.

CY: Yes. Yeah, no odd cycles. Yeah.

KK: Okay, all right.

CY: Yeah. So, one reason I really liked this is because when I first learned it, I didn't really think much of it, I learned it in an undergraduate class in combinatorics. And there are, like, three, maybe four proofs that we learned that were all pretty short and straightforward. One of the most basic proofs is just via induction on the number of vertices, and there's nothing, there's no really heavy machinery that's needed at all. And I didn't think much of it. And I didn't have any context as to like, why do we care about this sort of thing. But every year, I learn more and more things that make me appreciate this theory, more and more, because it really was the foundation for this whole field that we call extremal combinatorics, which is really centered on these questions of, like, what are the maxima and minima of certain things that we want to count when we put certain constraints on the problem? So this is an example we want the maximum number of edges. And our constraint is we have no triangles. And you can, there are a lot of different directions you can go with this sort of theorem. One of the most sort of classical foundational ones is just to replace triangle with a different type of graph. Like you could say, Okay, if I want the maximum number of edges with no cycle of length four, or cycle of length 10, right, what can I say? Or if I want the maximum number of edges with no complete graph of size five, where complete means you know, the vertices, you have every possible edge between every pair of vertices. And this type of problem, sort of replacing triangle with other things. It's called a Turán type problem, because there's Turán's theorem that generalizes mantle's theorem to complete graphs of higher orders. And we basically know the answer of what the extremal number is, and the extremal constructions for almost every graph, except for when you consider a bipartite graph as your, instead of triangles.

EL: As the thing you're trying to avoid?

CY: Exactly. And there's a reason for this, there's a theorem where it basically fails, or it's trivial in the case that your forbidden graph is bipartite. And so there's been a lot of study, it's still a very active area of research. And what people are doing is sort of taking different flavors of this Turán type problem that sort of started with Mantel’s theorem. And my first paper in graduate school was on a topological version of this theorem, where we were looking at these higher-dimensional structures called hyper-graphs, which you can think of as a higher dimensional version of a graph, and looking at a more geometric or topological viewpoint on these hyper-graphs by making them into simplicial, abstract simplicial complexes. So we don't have to go into the details of that. But I found it, you know, when I did that project, I found it very cool that that we could take this seemingly purely combinatorial, graph theoretic statement about just counting edges, and somehow turn it into something that requires a little bit more of a geometric or topological point of view, which is not something I had spent much time with before. And so that's sort of one direction at the beginning of my grad school career, where I felt like I had suddenly a much greater appreciation for this theorem. And on the other end, where I am now, it's also connecting very heavily to the research direction that I'm currently pursuing, which is in statistical physics, which is for me an entirely unexpected application of this sort of thing. But if you think about it, this sort of characterization of the extremal structure saying, okay, we can achieve the maximum with a complete bipartite graph, you can view this as sort of a ground state, if you will, if you want to think of the vertices as like particles in some sort of distribution, and you can take a probabilistic point of view on these sorts of counting problems. For example, it turns out that the triangle-free graphs and the bipartite graphs, if you think of these two collections, triangle-free graphs and bipartite graphs on N vertices, they're very closely related to one another. In fact, almost all triangle-free graphs are bipartite. This is a theorem by Erdős, Kleitman, and Rothschild. So you can sort of ask how far does that behavior persist if you add more constraints to your problem? And you can think about it as in a probabilistic sense of thinking, well, what if I have a probability distribution on my triangle-free graphs? And I have a probability distribution on my bipartite graphs? How are those distributions related to one another? And what is the counting statement, say, in terms of the probability distributions when we when we consider a randomness point of view on these things. And the sort of magical thing is that when you go to a probabilistic point of view, there are very natural ways that you can put it into a statistical physics context, where in statistical physics, you are thinking inherently about probability distributions on certain particles, on particles in space, or different configurations of particles in space, where there's maybe a physics motivation underlying the distribution you define. But ultimately, you can distill it down into something that is, that is simply triangle-free graphs, or different discrete structures. So one thing that I'm really interested in right now is just exploring more of this somewhat mysterious, but somewhat really amazing connection between questions that arise in graph theory and combinatorics that, you know, for a long time, we have just thought of in that context, in the graph theoretic context, and how, looking at them from a more statistical physics perspective, can help us gain new insight into how to tackle these problems.

KK: Yeah, and hopefully, it'll go in the other direction. I mean, I think we have this idea that because we learn calculus, and we think about physics being based on calculus, but inherently, right, the universe has to be kind of discrete, so you can't divide stuff forever. So I mean, it sort of makes sense that the underlying business, when you get down to it, might have to involve some kind of graph theory questions.

EL: Yeah, that is remarkable that there's this connection. So this is maybe a naive question about what you're talking about doing. Like probability distributions on graphs, are you saying things like, the likelihood that that two vertices have an edge between them? Or are we talking about some other kind of probability distribution?

CY: Yeah, so there, I purposely didn't include too many details, just because there are a lot of a lot of actually interesting and all valid ways that you can think about imposing probability, you know, into this world into these problems, these extremal combinatorics problems. So one flavor is what you said, we can think of what's called the random graph model. The most common one is the Erdős–Rényi random graph model, where you simply have your N vertices, and for each pair of vertices, you flip a coin, and it can be a P-biased coin, independently, to decide whether you put an edge there. And you can analyze what happens in that graph. What are the likely properties that this graph might have, if you, for example, change P. And what's really cool about studying this model is that there are, for a lot of graph properties, you can find these thresholds with respect to P. And this is like a huge, very active area of research right now, there have been a lot of really cool things proven just this year, in the past few years with regards to a lot of open questions here. But you can sort of if you let your P, your probability that you're adding an edge, be a function of N, the number of vertices, and you imagine N going to infinity, then you can actually sort of chart what happens if you're trying to count, let's say, the number of triangles in your graph, the expected number of triangles in your graph, or other properties. And you can see how changing P changes the value of the thing that you're trying to count. And for a lot of things, they exhibit these thresholds where the probability of finding a particular structure is close to zero. And then past a certain threshold, it jumps up to something close to one, and it happens with high probability. And this is also mimicking something in the statistical physics world where we have things like phase transitions.

KK: Right.

CY: If you think of in physics, just like water’s liquid-gas sort of phase transitions. Where we're also interested in studying what happens to certain properties of your statistical physics distribution when you change the temperature of your different parameters of your model. Can you find the sort of phase transition where the behavior changes quite drastically?

KK: Yeah.

CY: And then so GNP is, is one of these ways you can sort of input probability into — you know, take a sort of probabilistic perspective on these problems. Another is simply something a little bit more physics motivated, is by just imposing a uniform, or nonuniform, or a weighted distribution on the things that you're trying to count. For example, if you want to study triangle-free graphs, you could consider the uniform distribution on all triangle-free graphs on N vertices. And then think about the uniform distribution on bipartite graphs and ask, like, are these distributions close in total variation distance? And you can conclude things about that based on what you know about how close are triangle-free graphs and bipartite graphs to one another? Well, what does that say then about the distance between the nniform distributions that you impose on each set? And that sort of thing characterizes the different sort of distributions that come from the perspective of statistical physics. There are things called, like, the Ising model, and the Potts model and the hardcore model that were defined by physicists. And it turns out that they are simply weighted distributions on things like graph colorings, and independent sets of graphs. And so you can study them in these two different contexts, in the context of the hardcore model from the physics world, or in the context of a distribution on independent sets from the graph theory world.

KK: The hardcore model, I love that name. That’s good.

CY: Yeah.

KK: Well, we’ve gotten pretty far away from triangle free graphs can have at most N squared over four edges. So you mentioned that there were like three or four proofs of this. Do you have a favorite?

CY: I have to say my favorite is the very straightforward induction proof.

KK: Okay.

CY: And the reason I like this is because it's a proof that I've done with high school students at a summer math program I teach at called MathILy-Er. And I do it as an hour-long inquiry-based activity, where I simply pose to them this question. I let N be something like six or five, something that they could start drawing examples for them, then say, how many edges can you have before you start having to find triangles? And they often come up with the extremal construction, the complete bipartite construction first. And then I asked them how can we prove that this is actually true that this is the maximum. And they've learned induction at this point, when I do this activity. And so it's a nice lesson in induction, because it requires strong induction. And everybody wants to do weak induction, first of all, and they always want to what I call induct up instead of induct down. They always want to start with an extremal example with N vertices and try and build something with N+1 vertices. And it doesn't work.

KK: Right.

CY: And I always have to remind them, you have to start with something that has N+1 vertices, and remove something and see what happens. Yeah, yeah.

KK: And the base case of one vertex is super easy, right?

CY: And then there's also some argument about how many base cases we need and whether we need one or two or three, or where do we start? And so I think it's just a really nice exercise and practice. And it's simple, but I get to give a little, tiny spiel at the end, not nearly as much as I have said here in this podcast so far. But a tiny hint as to like, you know, what's cool about this theorem, and what more could you do? And some of the students have been interested enough to try and generalize to complete graphs or higher orders, you know, a complete graph on four vertices and try and mimic the same proof. And yeah, I think it's a really nice activity.

KK: Cool.

EL: Yeah. So a complete graph on four vertices includes a complete graph on three vertices so therefore you're trying to avoid something more, so like some of these ones that have triangles could still not have the four. Sorry I'm thinking out loud here because I have very little graph theory intuition. So okay, just like which direction are we going, and how many of these are we avoiding?

KK: You and I are probably the same, Evelyn. Like, we probably took one undergrad graph theory course and then yeah, and then then became topologists.

EL: Right. It’s kind of like it came up in, my introduction to proof class, but never a specific class dealing with graph theory things. Although the times that I've taught in high school programs or stuff, it is the kind of thing that can be quite accessible because the idea of drawing a graph, it's not hard to explain to anybody.

CY: Yeah, to answer your question. So there are actually lots of triangles in the extremal example for the complete graph on four vertices.

EL: Okay.

CY: Just to give you a sense of how it generalizes, the extremal example is the complete tripartite graph where you take three parts now sides and over three, and you have all the edges between the parts, so it looks like a giant triangle.

EL: Yeah. This kind of makes me want to go think about graphs a little bit. Yeah.

KK: Well, so the other part of this podcast is we ask our guests to pair their theorem with something. So what do you think pairs well, with with this theorem?

CY: So yeah, this this question was actually harder for me.

KK: It’s harder for everybody!

CY: I thought of something right away. And then I thought, no, I can't say that. I have to say something cool. And my pairing has to be something neat that makes me seem like a cool person. But I just couldn't think of anything better. So bear with me.

KK: Okay.

CY: My pairing is tofu. Okay. And here's why.

EL: Oh, tofu is great!

CY: Yeah. Okay, great. Great. So I thought of this because I think tofu is also somewhat of an underrated ingredient. But it is also so versatile, and you can use it in so many ways. So I grew up eating a lot of tofu because I grew up in a Filipino-Chinese household. And it was just sort of a staple of the things we were eating. But then I realized that not everybody knows or appreciates tofu. The first time I met someone who had never heard of tofu before, it just sort of shocked me, but then I realized it's not a common thing everywhere. But it's used in so many ways. And so I have been vegan since 2015. And also, every year I gain more and more appreciation of tofu as an ingredient. Like, you can use it in stir fries. There's now cheese that's made of tofu, you can make eggs using tofu. You can make a pie using tofu. There are so many ways you can use tofu. And there are so many more vegan options at restaurants and grocery stores and everywhere. So I feel like you know, for anyone who hasn't had tofu before, I would recommend at least giving it a shot.

EL: Yeah, yeah. And yeah, I mean, I grew up in a household that did not eat tofu much, my parents don't eat too much. But yeah, I'm not vegetarian or vegan, but like eat a lot — we have recently been enjoying this vegan Korean cookbook, I mean, it's called Vegan Korean. You might have seen it. [Editor’s note: It’s actually called The Korean Vegan.]

CY: Yeah. I have that!

EL: And just checked out this vegan Chinese cookbook that of course, it's like I think multiple sections are tofu because it's like the tofu tofu part and the tofu skin part, and all of this stuff.

KK: And, you know, do you use silken or what.

EL: But yeah, we’re a high-tofu household now.

CY: Nice. Yeah, there are so many different levels of tofu that you can have.

EL: Yeah, so many different textures, like, the Korean soft tofu is different from like the soft tofu in the cardboard package. Yeah, and so I finally found a Korean grocery store in Salt Lake that I could get to and got, like, the real stuff and oh man, great. That soft tofu soup, so good. And I can actually eat the kind I make because when I get it at a Korean restaurant, it's way too spicy. So I cut — in that cookbook, I think I cut at minimum, sorry, maximum spiciness is, like, a third of what the recipes start with, sometimes a sixth and see if I can work up.

KK: The correct answer level and there was you know, N squared over for, the floor.

EL: Yeah. I am impressed by the spice tolerance of Koreans.

KK: Asian cuisine in general, we once years ago, I was director of the University Honors Program, there was this place in town. It was an Asian place, and they have various stir fries. And you could ask for your spice level from zero up to no refunds, right? And so we had a student worker who was from Bangladesh, we went to lunch there one day, and he got the “no refunds.” And we were like, how is it? And he just went, eh. Like, it's just not very hot. And it's just an interesting cultural thing. Because, you know, I grew up in the Midwest, my mother's family was German, you know, we ate a lot of fried potatoes and sausage, like no flavor, you know? And it's just all what you get used to. Right?

CY: Yeah.

KK: All right. Well, this is we like to give our guests a chance to plug anything. Where can people find you online? Or you've talked about your plays, so that's good.

CY: Yeah. I mean, you can find my website. I recently updated it. And now it's got an all-purple background, which I'm very happy with. And it's corrineyap.com. That's Corrine with two R’s and one N, in case you forget. And, yeah, I continue to perform my plays. So if you're ever interested in bringing me out somewhere to perform, I am always happy to consider doing that. And I've done it at a lot of math departments. I did it at some conferences, but I don't have any conference performances coming up. So mainly like seminars and colloquia slots, things like that. So Evelyn, if you have any universities around you in Salt Lake City who might be interested in hosting a performance, you can let me know.

EL: Yeah. I'll make you vegan Korean food.

CY: Oh, amazing. But yeah, I mean, I just do this for fun. So it's not something that I'm trying to, I'm not trying to schedule, you know, 100 performances on my show this year. I just do it whenever someone is interested in having me there, but I'm always open to new inquiries. So yeah, that's, I guess, the one thing that I'll plug.

EL: Okay, great. Well, it was lovely to have you I'm so glad we finally got to at least meet online.

CY: Yeah, you as well. Thank you. Thank you so much for inviting me. This is a lot of fun.

KK: This was great.

[outro]

On this episode, we enjoyed talking with mathematician and playwright-performer Corrine Yap about Mantel's theorem in graph theory. Below are some related links you may find interesting.
Yap's website
MathILy-Er, a summer math program for high schoolers
Wikipedia on Turán's theorem, the generalization of Mantel's theorem
The Korean Vegan

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and I'm joined as always by your other and let's be honest, better, host.

Evelyn Lamb: I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. And tomorrow is my 40th birthday. So everything I do today is the last time I do it in my 30s. So, like, having my last mug of tea in my 30s, taking out the compost for the last time in my 30s, going for a bike ride for the last time in my 30s. So I'm, I'm kind of enjoying that.

KK: Well, congratulations. Let's not talk about how long ago I passed that landmark. I will say there's a switch that goes off when you turn 40. So riding your bike will be more difficult tomorrow, I assure you.

EL: Well I’d better get one in then.

KK: Any big plans?

EL: I’m actually going to the Janelle Monae concert. She's in town on my birthday. I'm sure that's a causal relationship there.

KK: It must be.

EL: So yeah, I'm excited about that.

KK: Okay, so fun fact, my Janelle Monae number is, is two. So I have a half brother. Very long story. I have a half brother, who also has a brother by — his mother has two children with — my dad was one of them. And then another man was the other one. So this other one, his name is Rico. He was a backup dancer for Janelle Monae.

EL: Wow. So yeah, brush with celebrity there.

KK: I mean, of course I've never met Janelle Monae, but you can — actually if you look him up, so there's a style of dance, sort of Memphis Jook, it’s called. Dr. Rico. He's something else. Amazing dancer.

EL: Wow. Interesting life.

KK: That's right. That's right. So anyway, enough about us. We have guests on this show. So today we're pleased to welcome Allison Henrich. Allison, introduce yourself, please.

Allison Henrich: Hi. Yes. I'm Allison Henrich. Happy birthday. I'm so excited for you.

EL: Yes, you get to be on my last My Favorite Theorem of my thirties!

AH: Yes, awesome! I feel so special. So I'm Allison Henrich. I'm a professor at Seattle University, and I'm also currently the editor of MAA Focus, which is the news magazine of the Mathematical Association of America.

KK: I have one on my desk.

AH: Woo-hoo! Is it one of mine?

EL: Yeah, and when we were chatting before we started recording, you made the mistake of mentioning that you've done some improv comedy. Is that something you do regularly?

AH: So I wasn't an improv artist. This is such a cool event. This science grad student at the University of Washington started this type of improv comedy where they have two scientists give short five-minute talks. And then this improv comedy troupe does a performance that's loosely based on things that they heard in the science talk. And so I gave a talk about some basic knot theory ideas, and it was so funny. I wish everyone could have the experience of an improv comedy troupe doing a whole set about your like research or your job. Yeah, it was so amazing.

EL: Cool. But also, it sounds a little stressful. A little bit. Yeah.

AH: Yeah. You want to not be too boring. And you gotta, like — it's really interesting. The other speaker tried to work in things for them to make jokes about, and they totally didn't take the bait. And they found like more interesting things to make jokes about, but I definitely tried to work in some things that would help them riff off of my talk, and it worked pretty well. Like just referring to knots with quirky names and making jokes about knot theorists and whatnot.

KK: Sure. What-knot. Hahaha.

AH: There are a lot of good knotty puns.

KK: Sure. Okay, so this podcast does have a theme. And the question is, what's your favorite theorem?

AH: Yes! This is a hard question.

KK: Of course.

AH: I’ve decided to tell you about my second favorite theorem. Should I admit that?

KK: Sure.

EL: I’m sorry, that’s a different podcast, My Second Favorite Theorem. It has two slightly worse hosts.

AH: It’s the cheap knockoff.

KK: No, it's the sequel, once we get rid of this one, we're gonna move on.

AH: Just, we're all out of mathematicians, we’ve got to go through them again. So my, let's call it my favorite theorem.

KK: Sure.

AH: My favorite theorem is the region crossing change theorem. So I have to tell you a bunch of stuff before I can explain what this theorem is.

KK: Sure. But it must be about knots.

AH: It is about knots. So, you know, knots we represent, typically, with two-dimensional pictures called knot diagrams, where you have ways of representing when a strand is going over and when a strand is going under at a crossing. And so every type of knot that there is has infinitely many diagrams you can draw of it. But no matter how you draw a diagram of whatever your favorite knot is, it can always be unknotted if you're allowed to do a special kind of move called a crossing change. So if you have your favorite knot diagram, and you're allowed to switch the over and under strands on whichever crossings you want, you can always turn that knot diagram into the diagram of an unknot, which is like a trivial knot that'll fall apart if you unravel it a little bit.

EL: Basically just a circle, right?

AH: Yeah, a circle. I mean, all knots are circles, so I have trouble. Like, a geometric circle.

EL: A boring circle.

AH: Yeah, a boring circle.

EL: And so this theorem, does it come with like, a number of how many of these crossing changes?

AH: Ah, so this is not my favorite theorem. This is a theorem that's going to help us understand my favorite theorem.

EL: Okay.

AH: So this theorem has a really interesting proof that Colin Adams calls “proof by roller coaster.” So the the theorem that says you can unknot any not diagram by changing crossings, you can accomplish unknotting using a certain algorithm where you choose a starting point to travel around a knot, and you decide that every time you encounter a crossing for the first time, you're going to go over it. So the fact is that you're kind of like always traveling downwards. And then when you get to the very end, you take a little elevator back up to where you started. So this will always create an unknot. So it's not that surprising that this is true, that if you're allowed to change whatever crossings you want, you can unknot things. What is surprising is my favorite theorem, which is that region crossing changes can unknot any knot diagram. So let me tell you what a region crossing change is. So you have your knot diagram in the plane. A lot of us kind of imagine that this plane is on a big sphere. So can we picture not diagram on a ball? Is that okay?

KK: Sure. Make it a big enough ball, and it looks like a knot diagram.

AH: Exactly. Yup. So we've got a knot diagram on a ball, and the knot diagram basically separates the surface of the ball into different regions, right? So this amazing theorem uses this operation called a region crossing change. And what a region crossing change is, is you choose a region in the diagram, and you change every crossing along the boundary of that region. So in my head, I'm picturing kind of like a triangular region in the diagram. And if I do a region crossing change on that region, I'm going to change all three crossings that are kind of around that region. So this is the amazing result: every not diagram can be unknotted by region crossing changes. So you no longer, seemingly, have control over individual crossings, you can only change groups of crossings at a time.

KK: Okay.

EL: But you can still do it.

AH: Yes, you can still do it.

KK: Right. That seems less likely. The other one, you told us and I thought, Well, yeah, I can kind of see, before we even saw the proof, I could sort of imagine, well, yeah, you just lift them up basically.

AH: Exactly. You lift it up, and then if it gets stuck, you know, change that crossing. But you can only change groups of crossings with the region crossing change. But amazingly, it's still an unknotting operation. So that just blew my mind when I heard that.

KK: Okay, so now I have questions. So, more than one, right? You can't expect to be able to just do one of these, right?

AH: Right. I mean, so if you have a region that just has one crossing on it, it's like a super boring region, because it's just a little loop.

KK: Yep.

AH: And that's actually called a reducible crossing.

KK: Sure.

AH: If you just have a little loop, it doesn't matter which way, which is going over and which is under.

KK: No, but I guess I meant, so you know, you've got one region, right?

AH: Yeah.

KK: So there might be multiple regions, you might have to change many of these, right?

AH: Yes, yes.

KK: What if two are adjacent, then you do one flip on one and one flip on the other, then you're undoing some of the flips from the other.

AH: Exactly.

KK: Is that why it works, maybe?

AH: That is why it works. So it’s a really cool proof. It's actually a proof by induction, which is so cool, that you can have like a proof on knot diagrams that's a proof by induction. But it's by induction on the number of reducible crossings. So the number of these crossings that you could sort of flip out of the diagram. They're not really necessary for the knotedness of the knot. But the base case is the most interesting part of the proof, where you have a knot diagram that has no reducible crossings. So no extraneous little loops or flips on it. But it's very constructive, and it uses things like checkerboard colorings, and it uses splices, or smoothings, which is where you take a crossing and you turn it into — like, you basically get rid of the crossing by cutting it and reattaching ends so that it's just — I’ve got this picture in my head, how do I say it? What's the best way to say that? So you have a crossing, and you want to get rid of it by cutting it and reattaching ends so that there's no crossing anymore. Does that make sense?

KK: Well, it’s sort of like a braid, right? I mean, so you imagine sort of a braid cross, you just clip the string above and below and then you just reattach, then you don't have it, right? Is that what you’re doing?

AH: Okay, yeah, what you just said totally makes sense because I could see your fingers.

KK: This would be a better video podcast, I suppose.

AH: I know. Yeah, at least for topology, or geometry. But the proof basically creates a checkerboard coloring that tells you how to find a collection of regions where you can basically control which crossing you're going to change. So I can change just one crossing, by carefully selecting a group of regions where exactly one, or exactly three of the regions involved in that crossing are going to get changed, but every other crossing in the diagram is next to either zero, two, or four regions that are being changed. So if it gets changed, it'll get changed back and look like it like it started.

KK: Right. Okay. All right.

EL: So I have not thought about knot theory, probably since we talked with, like, Laura Taalman on this podcast years ago. It's not something I think about a whole lot. And so I was not expecting this induction to be on the number of reducible crossings because they're so silly, you can just undo it, and then your diagram doesn't even have it anymore. So, yeah, why not the number of crossings or the number of regions or something?

KK: Yeah.

AH: So the reason reducible crossings are annoying for region crossing changes is because at a reducible crossing — you know, at any crossing, if you zero in on it, it looks like there are four different regions involved in the crossing, but with a reducible crossing, two of those four regions are actually the same region.

KK: Right.

AH: So it can look locally like you're changing two regions, so that you know, the crossing shouldn't flip. But you're really changing one, so the crossing does flip. So that's why reducible crossings are the annoying thing that you need to carefully control.

KK: Okay. All right.

AH: Yeah. And so once you get into the inductive step, you basically want to take a reducible crossing, change it so that you have two pieces, one has one fewer reducible crossings, and you know how to deal with that. And then one is a totally reduced diagram of a knot.

EL: Yeah. But the base case is the hard part, it sounds like.

AH: Yes, yes, yes. Totally.

EL: Interesting.

AH: Yeah. So yeah, so one of the reasons I love this is because I love unknotting. In general, I find unknotting questions really interesting. And I highly recommend everyone go listen to Laura Taalman's My Favorite Theorem podcast because she talks a lot about unknotting problems. But also the woman who proved this result is named Ayaka Shimizu. She’s a Japanese mathematician, probably my age, maybe a little bit younger, maybe she's about to have her 40th birthday or something, I don't know. But she is one of the coolest mathematicians I've ever met. She's definitely the cutest mathematician, and her talks are so cute that you're like, oh my gosh, I'm watching such a cute talk! And then you realize, oh my God, this result that she just proved is really amazing! So she's just super, super cool. I love her so much, and I think it's amazing that she proved this result that, you know, the Japanese math community wondered about for a long time, but no one came up with a proof before her. And she must have, maybe she was even a grad student at the time, or she was definitely a very young mathematician when she proved this result. So I love it.

KK: So here's a question: why would you want to allow such operations? I mean, because physically, changing the crossing, I mean, that would be great when your shoes are knotted, right? Like, you could just go Oh, snap, that's unknotted. Right. Is there a practical reason? And by practical, it could be including things like, it doesn't change the knot invariants or something? Or I don't know, it must if you get to the uknot, but I mean, is it… or is it just fun?

AH: Well, so the other thing — yeah, it's just fun. The other thing you need to know about me is that I study games that you can play on knot diagrams.

KK: Okay.

AH: And this result enabled this Lights Out-type game, they actually have a website. You can search for this game called Region Select. It's a really fun solitaire game that's a lot like Lights Out if you've heard of that game. And basically, the fact that the region crossing change is an unknotting operation basically means that any lights out game that you can think of or any region select game you can think of is playable, so you can have a knot diagram. Basically, at each crossing, instead of a crossing, you have a light. So it looks a lot like a graph, actually. You have a light and the lights are, some of them are on and some of them are off, and you need to select regions to try and turn all of them on or turn all of them off. And it's a really fun solitaire game that comes from this.

KK: Okay.

AH: But I’ve actually use the region crossing change to invent one of the many games that I've studied. It's called the region unknotting game. And basically, I'm super interested in these types of two-player games, where you start with a knot diagram, or maybe the shadow of a knot diagram. And you have two players doing something to the diagram, and one player wants to create the unknot and the other player wants to create something knotted. And so we have many games of this variety we've invented. One is the knotting-unknotting game. There's the region unknotting game. I’m about to publish a paper with some students called the arc unknotting game. And there are more. I could go on and on listing games.

EL: Kind of like you know that you can always unknot these things, but it's like, can you unknot it faster than someone can knot it? Is that sort of what's hard about playing this game?

AH: It doesn't have to be faster, necessarily. So the game, these games always are of the form, each player is going to move and they're going to go back and forth until everything is completely determined. And then at the very end, you see whether you have a knot or an unknot. And so you could be playing the long game, like, Oh, I'm just gonna wait it out playing on these little crossings over here to force the other player to play in this region of the knot diagram first, so that I can, you know, have the last move and turn it into a knot at the very end. So yeah, they're combinatorial games, topological combinatorial games, which is cool, because then you, then there is a player who has a winning strategy. And so your goal is to figure out which player is it? And what is a strategy that will always allow them to win?

EL: You said that the proof is constructive. So does that mean that given a knot diagram, you — someone who knew the proof — could actually say, okay, I can, you know, look at this knot diagram and do some sort of wizardry on it and say, Okay, the second player is definitely going to have a way to win this game. Or first.

AH: Yes, it can help. But of course, when you're playing two-player games, the other player can always thwart it. Like, let's say, I have to change these three regions in order to make this unknotted. Well, the other player knows that too. And so they're going to make it so that I can't change one of those regions. But actually, the the constructive way that the proof goes for region unknotting, the region unknotting operation, like basically, there's a complementary set of regions that will have the same effect. You can either do all the moves on this set of regions, or you can do all the moves on this other set of regions, and it will have the same effect on the diagram. So that does help inform game strategy, although we haven't looked at the types of diagrams that are terribly difficult to see how to unknot, because those are already hard enough to figure out strategies for.

KK: Right, right, right. Maybe it's like NIM, right? Like, if when you're playing, and if you have a huge numbers or piles of toothpicks, or whatever, you just kind of play randomly for a while, right?

AH: Yeah.

KK: And then when it gets small enough to where you can kind of analyze it, then you start to do it.

AH: Yes. Actually, I was giving a talk on not games at the Canada-USA math camp. And John Conway was in the audience. And he and all the students who were obsessively playing with him got really excited about calculating numbers for these topological combinatorial games. It's kind of a funny story. He said, he doesn't usually come to talks that speakers give at the math camp, or he didn't. And he said, normally he would leave before the speaker started speaking, because he was afraid to make speakers nervous. Like he didn't want them to be too nervous with him in the audience. But he was so intent on thinking about some problem that he was thinking about with a student there, that he just accidentally ended up in the room until it was like too late to leave. And so he told me afterwards about this dilemma he had, like, would it be worse for him to stay? Or worse for him to get up in the middle of my talk?

EL: Oh yeah, I’m glad he stayed. It would feel like a snub.

AH: Yes, he did stay. And we had a nice conversation about it afterwards, which was amazing. Because if you don't know about John Conway, he was, like, the king of knots and games and all of these things that I care about. So it was very cool.

KK: Yeah, that is cool. All right. So part two.

EH: Yes.

KK: What does this theorem pair with?

AH: This is it was such an obvious answer to me. With this paired with, because are you familiar with Nancy Scherich and her math and dance work?

EL: No, I don't think so.

AH: Nancy Scherich is a knot theorist. Actually, she works with braids. And she's also an amazing dancer, aerial acrobatics person. Acrobaticist? Acrobat. Aerial acrobat. And when she was a grad student, she won the Dance Your Ph. D competition, representing cool things about braids with dance. And since then she has recorded a number of other videos demonstrating mathematical ideas. And my husband is a musician, and he makes the music for her videos.

KK: Okay.

EL: Ao she just had a video that came out within the last month that is showing the proof of Alexander's theorem, which is a theorem about braiding. And the music that my husband composed for her dance piece, my husband's name is James Whetzel, was just beautiful, and just beautifully went with this performance that shows how the theorem works. And so I would recommend James Whetzel’s music.

KK: Unbiased, of course.

AH: I’m totally biased, and he has a new, actually, so I always forget if it's under Whetzel or James Whetzel because he has two different music personas. Right, so he has a new EP under Whetzel, W H E T Z E L, and the title track is “I want to go about my day,” and I think “I want to go about my day” would pair very well. Oddly enough, he also has songs called “Reidemeister Moves” and “This Is what Topology Sounds Like” and “Mama Proves a Theorem.” So he has some various songs with mathematical titles.

EL: So interesting that he came up with those and you also have done with things with these. What a weird coincidence.

AH: I know. How strange, isn't it? Yeah. So, but anyway, I think that everyone should go check out Nancy Scherich. I mean, you could probably just go to YouTube. Scherich is S C H, E R I C H. And check out Alexander's theorem. It's so beautiful. She does pole dancing to it.

KK: Okay, cool.

AH: Because it's about how you can turn any projection of a knot into a projection that always revolves in the same direction around a pole. So it works really well with that medium.

EL: That is so neat. So we have some things to watch and listen to after we're finished with this episode.

AH: Yeah.

KK: So we would like to give our guests a chance to plug anything that they're working on, or where we can find you on the intertubes.

AH: Yes, this is very timely because I'm trying to get out the word about an interesting event that I'm cohosting at the Joint Math Meetings.

KK: Okay.

AH: So, last Joint Math Meetings, a bunch of folks associated with Center Minorities in the Mathematical Sciences put on a storytelling event at the Joint Meetings. And it was so amazing and lovely. And they're doing it again, this Joint Meetings. But my friend Aaron Wootton and I were so inspired by this that we decided to also host a storytelling event at the upcoming Joint Meetings. And the theme is, people will be telling stories about some professional rejection that they experienced that was pretty crushing that ended up turning into something even better. So Aaron and I realized we both have stories like this where we didn't get something, we were totally feeling awful about it. And then it ended up being like a way bigger, more awesome thing. So we have a number of speakers lined up, but we need more. And so we have a web form that I created a bit.ly URL for. If you're interested in in telling a story no more than five minutes in length of the Joint Meetings, go to bit.ly/JMM2024STORY, all uppercase. Well, the bit that l y is lowercase, uppercase, JMM2024STORY, and we'd love to have people submit requests to speak, and I really hope that we have a good turnout for the event itself. It's going to be on Friday afternoon at the Joint Meetings. So mark your calendars. And let's see, the session is called Inspiring Stories: How an Academic Rejection Led to Something Amazing.

KK: Okay. In San Francisco, here we go.

AH: Yes. Can I add one more thing?

EL: Of course, you also, you also host a podcast, right?

AH: Yeah.

EL: I don't know if that's the thing you wanted to plug, but you should plug it too, and whatever you were about to say.

KK: You can plug as many things as you want.

AH: Okay, I have a lot. Okay. So we're just finishing up a book that's a handbook for math majors, called Navigating the Math Major: Charting Your Course. And it's going to be published through MAA Press by the AMS, and that should be coming out by MathFest of next year. So be on the lookout for that, especially if you're at a university that has one of these one- or two-credit freshmen seminars for math majors, or, like, an intro to the math major course. But also, it'll be good for just advisors and mentors to recommend to students and for students who might just be starting out in their college career, and they need some advice about, you know, what communities they should try and be a part of, how to apply for an REU, what kind of weird jobs are available for people. And this is what Evelyn was talking about earlier, because she did a wonderful interview for us about science writing, and that career path for math majors. So I definitely want to plug that. And regarding the podcast, it's a collaboration that I do with my friend who's an artist, Esther Loopstra. The podcast is called Flow into Authenticity. But what it's about is, if you're stuck in your life, it could be professionally or it could be personally, how can you use creativity and intuition to get unstuck? And we're actually writing a book on this called Think Like an Artist, Create Like a Mathematician, that's going to be published by 619 Wreath, which is Candice Price and Miloš Savić’s new publishing company. So that might be coming out in 2024, as well.

EL: Yeah. Well, and that sounds like something all of us can probably use it at some point.

KK: Sure.

EL: We always feel a little stuck.

AH: Yes. It's designed — we’re kind of aiming it professional stuckness in the book, but it's really broadly applicable. So very excited about that. And Esther Loopstra is amazing. She's a fine artist. She used to be an illustrator. You know, she used to work for, like, American Greetings and Target and all these places doing illustration. But now she's a fine artist and creative coach, and just super insightful about how we can use creativity to get unstuck.

EL: Cool.

AH: So, yeah, so the podcast is flow into authenticity. And the book is Think Like an Artist, Create Like a Mathematician.

KK: Cool. All right. Well, we'll try to link to everything that we can find links to.

AH: Okay, thank you.

KK: All right. Well, Allison, this has been terrific. Thanks so much for joining us.

AH: Thanks. Thanks for having me. This is such a fabulous opportunity, and I really appreciate getting to talk to you about today.

KK: Sure.

EL: Yeah. It was a lot of fun.

[outro]

On this episode, we talked with our delightful guest Allison Henrich, a mathematician at Seattle University, about the region crossing change theorem in knot theory. Here are some links to things we mentioned that might be interesting for you.
Henrich's website
MAA Focus magazine
Ayaka Shimizu's paper about the region crossing change theorem
Region Select, a game you can play where you try to unknot a knot using region crossings
An article Henrich coauthored about the region unknotting game
Nancy Scherich's YouTube channel, where she shares videos of her dances about math
James Whetzel's song I Want to Go About My Day on Bandcamp
The signup form for the mathematics storytelling event Henrich is cohosting at the Joint Mathematics Meetings in January 2024
Flow into Authenticity, the podcast she cohosts with artist Esther Loopstra

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm your host Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. It's Friday. Hooray!

EL: Yeah, yeah.

KK: Long Weekend. Yeah.

EL: It’s the start of a new month. Everything — anything is possible.

KK: Right.

EL: Including a great conversation with our guest.

KK: Yeah. I think it will be good. It's been an okay day so far.

EL: Great.

KK: The hurricane notwithstanding.

EL: Yeah.

KK: But yeah, that went by. But yeah, Hurricane Idalia really did some serious damage. And it’s, yeah, it's rough.

EL: Yeah, and there was recently the tropical storm on the other side of the country that actually kind of affected our weather, and today, I am hoping that the gale of wind outside my window isn't too much, too hear-able on the audio.

KK: I don't hear it, so it must be okay. Yeah.

EL: Great. Well, anyway, we are here today to talk with Tom Edgar about his favorite theorem. So Tom, would you like to introduce yourself?

Tom Edgar: Yeah, sure. Hi. Thanks for having me. It's fun to be here. I love your podcast, as you both know, but now everybody knows I love your podcast. I'm Tom Edgar. I'm a professor of mathematics at a small, comprehensive university in Tacoma, Washington called Pacific Lutheran University, just south of Seattle, about 35 minutes, maybe. Depending on traffic, like an hour and a half. I'm also currently the editor of Math Horizons, which is the undergraduate-level periodical from the Mathematics Association of America. And spend a lot of my time on those two things right there and just getting ready to go back to teaching here starting next week.

KK: Oh, you guys start after Labor Day. Okay, good for you.

EL: Oh, yeah. That is nice. Yes. And I think we've worked together a little bit on various Math Horizons things.

TE: Yeah, both of you have. So I mean, Kevin's on my editorial board, and he's written a couple of things. And then, Evelyn, I met you I think it in person at ICERM back forever ago. And I remember you were nice enough to do a piece about your awesome calendar, which I still have. I actually have a second copy now because I just have two now.

EL: Excellent. Yeah. Well, I would recommend getting one for every room.

TE: It doesn't hurt: one for the office, one at home.

EL: I’m not biased at all.

TE: No, one for your for your classrooms for your students. It's a great idea.

KK: Right. And it's universal. It's not year-specific. So reminder to all of our listeners, go to the AMS bookstore where they seem to be having a sale all the time, right?

EL: Yeah. Can’t afford not to! That's right. Anyway, Tom, now that you've so kindly plugged my calendar for me, what is your favorite theorem?

TE: And just that wasn't planned either. Right? That was just, you know, it's a nice thing that you've done. It's really cool. Yeah, so my favorite theorem is a hard thing. Because I've been listening your podcast for a number of years, and I was like, hey, if I ever get a chance, I wonder what I would talk about. And I had one that I was going to talk about, but I I've changed recently. There have been some projects that I've done in the past few years that kind of have changed my viewpoint. And so the theorem that I want to talk about is a pretty elementary theorem, in some sense. Most mathematicians will have seen it, a lot of, any math-adjacent people will have seen it. And it's the formula for the sum of the first N positive integers. So if you were to add up, say one plus two plus three plus four plus five, right, you can do this addition problem. My son, who's eight, can do this addition problem. But is there a quick way to get to the answer? And so the result is that if you add up one plus two plus three plus four plus five, you can actually get that in sort of fewer computations by multiplying five by six and dividing by two. And so the general formula is, if you were to add up the first N positive integers, pick your favorite number to stop at, N, then the theorem says that that sum should be N times N plus one divided by two. So the number that you stop at, multiplied by the next number, and then take half of that. So I really love this theorem for a variety of reasons.

KK: So there’s the apocryphal, probably apocryphal, story about Gauss, right?

TE: Yeah, for sure. So I definitely enjoy this aspect of it because most people think, oh, there is this story. So the story is, I'm not even going to tell the story because I've read — Brian Hayes has an article where he tries to get to the bottom of this actual story and where it came from, but the general idea is that, you know, some teacher of Gauss gave this as an exercise, to find this sum and expecting it to take a long time and Gauss produces the answer almost instantaneously. I like talking about this because a number of people have changed that story over the years. And so it gets more dramatic, or things like that, or a lot of people think that this is Gauss’s sum formula, that Gauss was the very first person to come up with this, like in the 1800s, like, nobody knew that, you know, this was it. But this has certainly been known — you know, one of my favorite proofs is the picture proof where you imagine the sum of the first N integers is sort of almost like a staircase diagram, one box at the top, two boxes below that, three boxes below that, and so on. And you take two copies of this staircase diagram, rotate one 180 degrees, and stick them together, and you have an N by N +1 rectangle. And Martin Gardner attributes this to the ancient Greeks, right? So presumably, people been drawing this in sands, and all sorts of things, for as long as people been thinking about counting, right?

EL: I must admit, I do — like, that story always bugs me because people, I don't know, people will use it as evidence of like this amazing genius. And I'm sorry, if this is, I don't know if I sound like I’m bragging or something. But like, I figured this out when I was in school, and I'm not a Gauss, by any stretch.

KK: Don’t sell yourself short.

EL: And it's like, you sit around playing with numbers a little bit, then, you know, you can figure this out, it's figure-out-able, which I think is good for people to know, rather than think, Oh, you have to be, you know, some native genius to be able to figure something like that out.

TE: Yeah, for sure. And, and I think, like, I don't know if you've read Brian Hayes’s article on it or not.

EL: I think so.

TE: Yeah. He brings up the point that maybe the reason people like it is because it's sort of, like, the student having this victory over the the mean classroom teacher. And somehow we just love this idea, not necessarily the genius myth, but this idea that like, oh, the the student won, or something like this. But yeah, but it's fun to talk about too. And just that always opens up the conversation with people about all the misattribution that we have in mathematics, right? Theorems named for people that maybe don't even have anything to do with that theorem, for one reason or another.

KK: So let's talk proofs. So you mentioned the one that Martin Gardner did with the picture. Okay. What's your favorite proof? Do you have one?

TE: Yeah. I mean, that one's pretty amazing, if you ask me. You know, I mean, another reason I like this is that this is sort of, if not the, it's probably the standard first induction proof that any undergraduate sees, right? So you learn about induction, and then you prove this formula by induction. I dislike that proof in one sense, and I love that proof in the other, right? So it's nice from learning induction. On the other hand, it's like, man, it's induction. I didn't get anything out of that. Whereas that picture proof from the ancient Greeks, right, just tells you exactly what what to do, right?

EL: Yeah. And I'm trying to remember is there a book or something called, like Proofs without Words or something like that? And it's a great proof without words, because it doesn't take a whole lot of scaffolding to show this picture and the numbers and to see exactly what's going on.

TE: For sure. Yeah, yeah. So Roger Nelson has three compendia now, like Proofs without Words, right? So this is three books, maybe almost a total of 600 pages of diagram proofs. And that one is in the first edition. And it's definitely — I mean, there's a couple iconic proofs without words, and I would put it as one of the top four iconic proofs without words. There's the Pythagorean theorem with a couple, and a couple of other ones that go along with it. But that's that. But my favorite proof actually — well, so, back in, like 2019, right at the end of 2019. Right, the beginning 2020 Before the before, sort of all the craziness, a mathematician named Enrique Treviño, who's a professor at Lake Forest College in Chicago, he was posting some things on Twitter about different proofs of this theorem and I knew a couple and I sent it to him, he's like, Hey, we should write these all up. So we got together and wrote these all up. And so we have a compendium that's online of 35 proofs so far, of the of the fact. And we finished that just before — I think it was end of January 2020, we sort of finished it. We've been working on it here and there ever since. But one that came out of there that's my favorite — and it's hard to describe, so I'll see what I can do — but it's also a picture proof. But instead of taking two triangular diagrams, so two staircase diagrams, you take eight staircase diagrams. The same kind of picture, instead of two and you just glue them together and you get a rectangle, you take eight. So again, the visual here should be sort of a right triangular stack of squares, N squares on the bottom, one square on the top, and then it's right oriented. And when you put eight of these together, you get a perfect square, except there's this one missing cell in the middle. And so it tells you that eight times this, this number, which these are called the triangular numbers, because they fit into these triangular arrays. So eight times the Nth triangular number is basically the Nth odd square. So (2N+1) squared, except missing one, missing one cell, so minus one. And this proof to me, it's much more complicated, in some sense. Like, why don't you just use the real picture proof, the easy one with two? But this one indicates that there are a lot of other things going on. So you can use this proof essentially, to prove that odd squares are congruent to one mod eight and these kinds of things right here. I mean, it sort of falls right out of that. And then this was key to Gauss’s — what's it called? — three triangle theorem, which says that every positive integer can be written as the sum of three triangular numbers. And so this fact plays a role. This visual proof plays a role there.

KK: Okay.

EL: Oh, nice.

KK: Very cool.

EL: Yeah, I'll have to draw that out later. I'm not quite sure I believe you, but I'll take your word for it for now.

TE: You’re going to have to draw it out, for sure. I was like, Oh, should I? Kevin asked my favorite. I wasn't going to necessarily going to talk about that one, but for some reason, I liked that one because it opened my eyes to a lot of other things going on in math as well. So it just has a connection, you know, thinking about what are called figurate numbers. So these are numbers that can be arranged in certain geometric patterns. So the triangular numbers, the squares, these are familiar ones to us, but there are just so many cool mathematical ideas that somehow I never picked up as an undergraduate or a graduate student about these, like Euler’s pentagonal number theorem, or Fermat’s polygonal number theorem, just amazing facts out there that I just never would have come across.

EL: Yeah, well, I guess that one is kind of an overpowered proof for that particular formula. But like you said, yeah, it kind of opens the door to a few different things, a sledge hammer for a mosquito.

KK: I like that.

TE: Those are some of my favorites of the ones that that Enrique and I compiled. One of the ones that sort of blew my mind that we came across was this idea that you can use Euler’s polyhedral formula for planar graphs, right? So the the planar graph version, you can use this and it proves the sum of the integers formula if you just find the right graph, and that's like a sledgehammer!

KK: Oh, nice.

TE: But it’s a beautiful, really powerful theorem for topologists. I think both of you somehow are topologists or topology-adjacent. Am I wrong about Evelyn? Not you?

EL: Yeah. Oh, yeah. Why not?

KK: No, it's true, Evelyn.

TE: The fact that you can use you know, this Euler’s polyhedral theorem, which I know has been featured on your podcast before, and maybe even recently, you know, to me was really powerful, like, oh, you're using something really strong. But it's also a way that you can introduce people to a cool idea with this relatively simple fact, elementary fact that they might be encountering as early undergraduate-level mathematicians, or even earlier than that.

KK: Very cool. All right, so I know visual proofs are kind of your thing. So have you animated this one? I know you like to animate these things. I see them on Twitter occasionally.

TE: Yeah, so I spend my time animating. For the past year and a half, two years this, this arose out of the pandemic, right, we all went online, and some of us were teaching online and kind of upset with maybe some of the digital content that we could produce. And so I spent some time trying to figure out how to how to do some animations. But yeah, so this one I animated, I animated 12 of them, so a dozen of the proofs from Enrique and I, that we compiled I animated a dozen of them last year. This was part of, I submitted as part of Three Blue One Brown, Grant Sanderson, runs this summer of math exposition stuff. So I submitted that last year as my video, the idea being that you should think deeply about simple things because you can encounter a lot of things along the way. And this is not my quote, this is a quote from Ken — the person who started the Ross program, and I'm forgetting the Ross program, I'm forgetting the founder. His last name is Ross but I can't necessarily remember the first name. Okay. So yeah, so I have animated some of them. And I believe I've animated, I think I've animated Euler’s polyhedral theorem, Pick’s theorem, the classic visual proof, there's combinatorial proofs. So there's like, a double counting proof. And then there's one that uses bijective proof. So just some really cool ones out there to see and explore.

KK: On YouTube? They’re on YouTube, right?

TE: Yeah, that’s on YouTube. Yeah. Mathematics Visual Proofs is the name of the YouTube channel at this point. Who knows? It changes if you have to change it, right?

KK: Well, we'll link to it. We'll find it.

TE: Okay. I appreciate that. Thank you. All right. Cool.

EL: Yeah. And so you said maybe this isn't the theorem you would have picked, if we had asked you, you know, three years ago or something. So how, how did this theorem get — Was it this project with Enrique that got you interested in it?

TE: Yeah, I mean, I've always loved the theorem, but sort of seeing all of all of the available proofs and the ways that it could open me up to things. It’s given me well, a couple of things. So when you teach a discrete math course, you can essentially teach the entire discrete math course using this theorem. You can talk about so many different discrete mathematical ideas using this and so it can be fun that way. So I've done that in a discrete math class and really enjoyed that experience with students as they see the connections being made. It's maybe a little more fun to talk about than some of the the others, I mean, the other theorem that I probably would have talked about is called Kummer’s theorem. And that one is fun to talk about, but it requires a little bit more knowledge, or a little bit more technical detail sometimes. So I like the accessibility in this one. I like that I get to speak with people — whenever I get to talk about this, I speak with people about the fact that mathematicians are looking for other proofs sometimes, right? I think mathematicians know this, we know this, that you're not always just looking for one proof. Some people say you're looking for the best proof, the so called proof “from the book.” I don't know if I agree with that. I just like the idea that we're looking for other proofs, other ways to try to understand these things to give that broad picture. And somewhere along the way, before or after, I came across this quote, It's my absolute favorite quote from a, from a mathematician, maybe ever, it's from Bill Thurston, who was a Fields medalist in the late 20th century and passed away only about a roughly a decade ago, maybe. He says, what did he say, “we're not trying to meet some abstract production quota of definitions, theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics.” And I just, I wish I had said that. If I could have said that, I think I could die happy, like that was my quote. But I like the idea that we're not just — mathematicians aren't just sitting in the room trying to pump through more results, that we are actually interested in understanding and communicating and trying to get those ideas out.

EL: Yeah. And that, you know, what insight can we get by looking at this problem in a different way even if we already know the answer?

TE: Exactly. I think a lot of people just don't think that way about mathematics. People who are not, who haven't been around mathematics long enough, think that it's just one and done, right? You do this problem, and you move on to the next.

KK: Right, right. Or that we're just sitting around, like, doing arithmetic with really big numbers, right?

TE: Yeah, that's kind of what — that’s actually what that's what this is. This is arithmetic with really big numbers!

KK: That’s right. But clever arithmetic! They think we would just sit there and add it all up. It's like, why would I do that? I don't want to work that hard.

TE: Yeah. Yeah. I'm kidding. That's good.

KK: All right. The other thing we like to do on this podcast is ask our guests what it pairs with. What pairs well, with this formula?

TE: Yeah, so this is the greatest part about your podcast, not that there not other good things about your podcast, right? I think you two are great together. And it's fun, you know, but I think the idea of this and I was — this is the challenging part with with the other theorem I was thinking about. I was like, wow, what would I pair it with? I don't know. Presumably, I would come up with something. But this one was fairly easy for me. When I was younger, a movie came out, and over time, I guess it's become somehow I read online, that it's one of the greatest comedies of all time. I'm not sure if I agree with that. But I watched this movie a lot. And this movie is called Groundhog Day.

EL: Oh yeah!

TE: Have you seen Groundhog Day?

KK: Many times!

TE: Exactly.

EL: My thing about Groundhog Day is like watching it once is like watching it several times. Right. And then if you watch it more than once you've just like really increased your your volume of Groundhog Day.

TE: Right. So you you have no idea, exactly, you have no idea how many times you've seen this movie, you're sure you've seen this movie 30 times, but maybe you've only seen it twice. Right? But for people who haven't seen the movie, the premise is Bill Murray is a weatherman from Pittsburgh, Pennsylvania, and he's tasked with covering Groundhog Day and Punxsutawney Phil and he doesn't want to go there and essentially ends up in sort of a time loop where every morning he wakes up and it's exactly the same day and he's the only person who thinks he's reliving the day and everyone else is treating the day as the same. And so he does various things to try to, I guess the idea was to sort of “get it right,” sort of be the best possible person. But from my perspective, this is exactly — what would a mathematician do if they ended up in the Groundhog Day situation? Well, which is every single day I would just find a new proof of the sum of the integers formula and I would maybe never be bored. Maybe I'd never get it right and get out of the time loop. But I liked this idea because essentially in the movie, he learns a lot about himself, he learns a lot about the people around him. And this is sort of what happened with me working with Enrique and learning a lot of the things that come along with this theorem. You learn a lot of stuff and like, oh, this is stuff I didn't know, and it's led me to a lot of other things that I didn't know and connected me with other people. And so it's kind of like that movie, I guess. So, you know, sit down and watch that movie and figure out a couple of new proofs of the sum of the integers formula.

KK: And remind yourself of the genius of Sonny and Cher.

EL: Yes.

TE: A song that you probably probably can't listen to ever again, without automatically thinking about the movie.

KK: No, probably not.

EL: Yeah.

KK: No, that's a great pairing. I like that.

EL: Yeah, that's a nice one. I think. So I think in the movie, one of the things he does is he becomes this great piano player, right? Because he has so many times through the day. And you know, he goes, at some point, I think goes to his lesson and is like, oh, yeah, I've never played piano before and just busts out something. I always thought, like, oh, that would be — what would I have the dedication to do something like that if I got this time?

KK: What else do you have to do?

TE: Well, it's a great, that's what's so cool about the movie is, like, really, if you put yourself in that situation, you could do whatever you want. Right. I think that was what was so good about it in the end, he learned to play the piano, he learned to be a good person, I guess as well. But you know, like, you just learn a lot of things. He

KK: He learned to do ice sculpture!

EL: Yeah, that’s right.

TE: Yeah. The end scene, like, the last day when he does everything right, it’s just it really puts it, it brings it together so nicely. Like, oh, he saves that person's life and builds his ice sculpture and he's really filled himself out, right? I mean, there are some dark parts of the movie as well, but it ends nice. I can see why people might say it's the greatest comedy of all time.

EL: It’s up there, for sure, I think.

TE: And from the mathematics — there's this one scene, like from mathematics point of view, mathematicians, they famously love their coffee. And there's this one scene when he's kind of at one of his low points, and he's just eating all of the foods at the diner and he grabs this thing at coffee, and he just drinks it straight like that. I'm like, oh, okay, I could see a mathematician doing this in Groundhog Day.

EL: Yeah.

KK: All right. Well, this has been great. We always like to give our guests a chance to plug anything they want. So you've plugged the YouTube you've, you've plugged a little well, we plugged it for you.

TE: Yeah. Thank you. Oh, yeah. Plug the YouTube I appreciate.

EL: And Math Horizons, which I'm still involved with for one more year. And then there'll be someone taking over there. Yes. Yeah. It's been a long time. I don't know if I have anything else to plug otherwise, I appreciate you all having me on. It's fun to come and talk about these things. I guess I could plug — No, I don't know, for mathematicians interested about this favorite proof that I mentioned of the sum of the integers formula, this somehow told me that there's a connection between, there's sort of three famous proofs that you see as an undergraduate math major, would be the sum of the integers formula for induction, the fact that the square root of two is irrational. And then maybe the arithmetic mean, geometric mean inequality, you might learn as a first inequality type proof in a in a real analysis course or something. But somehow, there's a visual proof for all of these and the visual proof is somehow the same. So I think that possibly those theorems are somehow the same, in some realm. And so I spent a little time trying to prove one of those theorems using different techniques. So I recently had an article if people want to check in Math Magazine about the arithmetic mean, geometric mean inequality, where you prove it using moments of mass and centers of mass. And I was inspired to do this because David Treeby proved the sum of integers formula using moments of mass and centers of mass.

KK: This one? [Kevin holds up Math Magazine.] It happens to be sitting on my desk.

TE: That’s a different one.

KK: That’s not you?

TE: I didn't — I didn't know that — No, that is me, and I wasn't going to plug them both. But that's where I use the centers of mass to prove that the square root of two is irrational.

KK: Okay, that's what it is.

TE: So somehow this proof allowed me to connect those things together. And so it's been fun to play around with ideas that I that I don't know. So if you're interested in how balance plays a role in pure mathematical ideas, I would check those out. So that's one thing I can plug.

EL: Yeah, we’ll link to those. Those sounds really interesting.

TE: Thank you.

KK: All right. Well, Tom, thanks so much. It's been terrific.

TE: Yeah, thank you both. I know it's hard work, the work that you all do, but I think the community needs it and we appreciate it and it's great for my drives to work.

KK: Okay, thanks.

EL: Well thank you.

[outro]

On this episode of the podcast, we chatted with Tom Edgar of Pacific Lutheran University about the formula for the sum of integers between 1 and n. Here are some links you may enjoy:
His website and Twitter profile
Math Horizons
His collection, with Enrique Treviño, of proofs of the sum formula
His YouTube channel, Mathematical Visual Proofs, including his video on the 8-triangle proof of the sum formula
His article about proving the square root of two is irrational using centers of mass
His article about using centers of mass to prove the arithmetic-geometric mean inequality

Also, Brian Hayes’s article about Gauss: https://www.americanscientist.org/article/gausss-day-of-reckoning

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

Evelyn Lamb: Hi, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we sadly are past our beautiful, not too hot spring and fully into summer. So we enjoyed it while it lasted. I didn't have to turn on any air conditioning until after the start of July.

KK: I think we started air conditioning in March.

EL: Slightly different.

KK: Little different vibe down here in Florida, but that's where we are. So anyway, it's summertime here, which means that there are tumbleweeds rolling through my department and I'm answering a few emails a day and trying to work, trying to do math. And boy, sometimes it's hard, you know, but sometimes it isn't. So. Anyway, so today, though, we are — this is great — we are very pleased to welcome Tatiana Toto, who will introduce herself and let us know what she's all about.

Tatiana Toro: Thank you very much for the invitation. I'm very glad to be here. And in fact, I'm very glad to see Evelyn's cloud that I had heard about in other podcasts. So I'm Tatiana Toro. I'm a mathematician at the University of Washington, where I have been a faculty member since 1996. And currently I am the director of the Simon's Lab for Mathematical Sciences Institute, formerly known as MSRI. And I'm in Berkeley, California, and summer hasn't arrived yet.

KK: It never will.

EL: Yeah, that’ll be November, right?

KK: I had actually forgotten that the name of MSRI had changed to the Simon's business. That’ll take some getting used to. I think I mentioned before we started talking, I spent a semester there, way back in 2006, and my son came with me, and my wife did too, and he was seven at the time. And now he's an adult living in Vancouver. It's weird how things change. I love that building, though. And the panoramic view you have the bay, and you can watch the fog roll in through the gate at tea time. It’s just a really wonderful place. So congratulations. How long have you been director? Has it been a year yet?

TT: It’s almost a year, a year August first.

KK: Yeah. That's fantastic. What a terrific position. And I'm glad that you're willing to take it on. Do you split your time between Berkeley and Seattle? Or are you mostly in Berkeley these days?

TT: I am mostly in Berkeley. My students are still in Seattle, so I see them mostly on Zoom. But once in a while on a Friday, in Seattle.

KK: Oh, so you go there. You don't fly them down?

TT: Some of them have come, actually one of them this year to the summer school.

KK: All right. So what is this podcast about? Favorite theorems. And you told us yours ahead of time, but we'll let you share. What is your favorite theorem?

TT: Okay, so my favorite theorem is the Pythagorean theorem, and I know that everybody's gonna say what on earth are you talking about?

EL: No, I really, really love this choice. And, you know, I've said this on many other iterations of this podcast, but I love that, you know, we'll get things that span the gamut from Pythagoras theorem, or the infinitude of primes, or something like that, all the way up to something that you, you know, you need to have been researching for 20 years in some very ultra-specific field to even understand, and so, you know, it just like shows how math connects with us in different ways at different times in our lives, and how we can appreciate some maybe things that seem very simple about math, even when we have had math careers for for many years. So yeah, tell us about how did you end up settling on the Pythagoras theorem?

TT: So, actually, it has played a very important role in my career. Like, when I describe it to my students, when I'm teaching a graduate class and I talk about the some of the theorems I'll describe in a minute, I tell them, you know, one of the key ideas in my thesis was the Pythagorean theorem. So let me explain. It appears in many other results in this area of geometric analysis. So for example — let me give you two examples. So what was my thesis about? You have a surface, a blob in space, and you're trying to — two dimensions in R3 — and you're trying to understand if you can find a parameterization, which means a good way to describe it in terms of the plane. So can you deform the plane in a nice way so that it covers the surface? And a nice way means that distances are not changed too much. So I had some specific conditions for this surface, and the answer, the key, is in the situation I was looking at, yes, you could do it. And when you go and deeply look at what makes this possible, it is the Pythagorean theorem because the basic point is that if you can control how distances are distorted, you can control how the whole shape is mapped from the plane. And at the time, it looked like a curiosity. You know, I graduated many years ago. At the time, a few years earlier, Peter Jones had solved the analyst’s traveling salesman problem, which I'm gonna — just in general terms, let's imagine you have a lot of points in a square, and you're trying to understand whether you can pass a curve of finite length to all of these points. You're going to tell me, “If they’re finite, of course you can.” But you want to do it in an efficient way, in a way that doesn't depend on the number of points. And so he had found the condition that told you if this condition is satisfied, then yes. And there's not an algorithm, that doesn't exist yet, that tells you what's the best curve, but there's a curve, and he tells you that the length is no more than something. And what's behind that is the fact that if you have a straight triangle that has sides, A and B, and the other one is B, A squared plus B squared equals C squared. And it really is understanding that. And there's another important thing, the fact that the square root also plays an important role in these, but really, really, if you ask me, “What are the tools you need in this area?” I'll tell you how the square root behaves in the Pythagorean theorem, and then a couple of good ideas and you're able to reconstruct the whole thing.

KK: I’m now curious about this traveling salesman problem. So there's no algorithm though?

TT: No, there's no algorithm. I used the word analyst’s traveling salesman problem because the analyst wants to know whether you can pass a curve of finite length. Maybe you can say you're not ambitious enough. You don't want the shortest possible curve. To build the shortest curve, there’s no algorithm. And the construction of Peter Jones builds a curve, but it's not necessarily the best one.

KK: Sure. Yeah.

TT: It doesn't tell you it tells you the length is no more than D. But it's not. Yeah, no.

EL: Yeah. I'm trying to remember if, like, I think there probably are some algorithms or some, like results that say like, you can get within a certain percentage of something. But yeah, the algorithm for the actual fastest path doesn't exist yet. Which is, you know, it's one of those things, it's like, huh, that's kind of surprising that we don't have a way to do that yet. Just means that there's still work to be done. Still jobs out there for mathematicians.

KK: Well, because the combinatorial on the graph theory one is, is NP complete, right? I mean, yeah. So that that are NP-hard, or whatever. NP-something. I've never been clear about the differences. But is this one known to be that too?

TT: I believe.

KK: Okay. All right.

TT: But you can construct — you know, so this was what was interesting about the problem, the result of Peter Jones, is that — the result of Peter Jones, and I have to say, I was very ignorant of that result, which had just happened a few years prior to my thesis. I have to remind the young audience that at the time, there was no internet the same way, and there was no arXiv, and you know, there was no Zoom. And then Peter Jones had a couple of postdocs at Yale, Stephen Semmes and Guy David, who started working on this. And the truth is, may I tell story about my thesis?

EL: Yeah.

KK: Please do.

TT: So my thesis came out of misunderstanding. I went to my advisor, and I showed that these surfaces that I was looking at, which were some that he had looked at, that there was this property about distances over the surfaces, like if an ant traveled on the surface between two points, you know, taking the shortest path, it was comparable to the Euclidean distance. And so I went to my advisor, Leon Simon, and I told him, you know, I've been able to do this about these surfaces. And then he told me, oh, then I guess they have about they admitted bilipschitz parameterization, which is this good description. So okay, so I went to the library, and I looked through every possible book that I could find, and I couldn't find that. So I went back two weeks later and asked if he’d mind giving me a reference for this results, and he said, oh, I don't have a reference. That must be true.

KK: It must be true.

TT: And that became my thesis problem. And then, oh, there were many iterations of attempts. And I could do specific cases, but I could not do the general case. And on May of my fourth year, finally, somebody gives a colloquium where he talks about good parameterizations. And he talks about things like what I was thinking. I was thrilled. I mean, I thought, oh, I'm going go read everything this guy has written and my answer will be there. And then I told my advisor afterwards, I think I'm going to go read this guy's work. And this guy was Stephen Semmes, and he comes from harmonic analysis. And my advisor says, no, stop reading, I don't want you reading anymore. You just prove that theorem and that’s it. I don't want you reading. But one good thing, you know, harmonic analysts use squares, rather than balls. That's the most useful comment my advisor had.

EL: Huh!

TT: And what's interesting is that Stephen Semmes was talking about a broader class of surfaces than mine. And for those, he was asking, “Do bilipschitz parameterizations exist?” And for those the answer still is not known. And if I had gone and read everything that he had written, I mean, he was the big shot, I was the student, I might not have gotten my result. And I remember when I told Stephen at some point in the fall, oh, you know, I proved this, his first question, his first reaction, was, “I don't believe you.” And he said, “How did you do this?” And I said, “Using the Pythagorean theorem.” And so that's why the Pythagorean Theorem really is very dear to my heart.

EL: Yeah. So I imagine that you saw the Pythagorean Theorem many years before you were in grad school. Do you remember, did it make a big impact on you when you saw it in school for the first time? I don't know what what year that would have been, elementary or middle school or whatever it was?

TT: So I remember, I think I remember when I saw it because I remember the book. I had a beautiful — I went through the French system. I'm Colombian, but I went through the French system, and in the French system at the time, they tracked us very early on. And so we had these beautiful math book that, you know, I still remember how it smelled, and it was in there. But I remember the book, not especially the theorem. I never thought much about it until I got to graduate school. I used it other times.

EL: Right. I mean, I think maybe the beauty of that kind of thing isn't necessarily what you're looking at, when you're a kid and first seeing math. You’re more like, okay, how can I use this to do the problems on the homework or something like that? So you were tracked into math pretty early on? You knew very early on that you were interested in math?

TT: Yeah.

KK: It’s nice they let you just do math. I think in the US what happens, I think, is students who are good at math are told they should be engineers. As if they're kind of the same thing, and they're not.

TT: But that, you see, now, you feel free to remove this if you want. That's what the boys were told. The girls — since math was roughly like philosophy, and I come from a South American country, it was okay.

KK: That’s fascinating. Okay, interesting.

EL: Yeah. Well, I mean, there's a lot of different, you know, philosophies about whether tracking that early, you know, kind of deciding on what direction you want to go that early, is good or not. You know, it works for some people and not others, definitely.

TT: Absolutely. I think it worked for me very well. And it didn't work on any of my classmates who were in the same class. I mean, I thought everybody loved it the same way I did and had as much fun. And then, it's interesting. Later on, I've learned that that wasn't the case. And then some of them suffered through it, you know. But to me, it was great.

KK: So this is a French system in Colombia? Okay, this is a bit — okay, let’s get there. How did that actually happen? Why were there French schools in Colombia?

TT: Well, I'll explain why there were French schools in Colombia and how I got into a French school. So there's something that's called a cooperation agreement between France and developing countries, where they have schools. The primary reason to have them is so the kids of their diplomats can continue their studies, but then they also offer them to the general population at a very reasonable price. They are private schools, but they are not as expensive. They're a fraction, or they used to be a fraction, of what the other private schools were. And at the time, so Colombia for a long time was what was called a Sacred Heart country. And so the ties with the Catholic Church were very strong. And so in terms of education for the girls, it was most girls went to nun school. But I am not Catholic, and therefore I couldn't, that was not an option for me. And so we needed a coed school. I mean, my parents wanted a coed school. The girls schools were all nuns. They wanted a coed school, and we needed an affordable coed school, and public schools were not good, and still unfortunately are not good. That's how I landed in the French school.

KK: Fascinating.

EL: Wow. Okay. Yeah.

KK: Our listeners are learning all kinds of stuff, right?

EL: Yeah, yeah, we've wandered a little away. But luckily, we know, thanks to the Pythagorean theorem, that we can walk back in a certain amount of time. So yeah, the other things that we like to do on this podcast is have you pair your theorem with, you know, some food, beverage, sport, you know, whatever, delight in life you would like.

TT: So I actually will pair it with walking. So I'm going to give myself the title of urban hiker. I do walk long distances around town and in cities on a regular basis. I mean, I walk about two hours a day, at least. And so I pair it with that, because most often when I walk, I'm actually doing exactly the opposite of the Pythagorean theorem. I want to go the longest possible way, not the shortest possible way. But once in a while, I take the diagonal. And now that I'm living here in Berkeley, there's a beautiful diagonal that I take. And so I think about that here often.

EL: Yeah. Do you like the hills?

KK: Yeah, I was about to say, do you actually hike all the way up to the building there? Because that is quite a hike.

TT: Not when I'm coming to work. But sometimes on weekends I do. You know, I want to crease and it depends. It depends what I'm doing while I walk. I use walking as a way — if I am listening to a book, then I can go up the hill. But if I want to talk on the phone, I need to go down the hill, because the reception here is terrible! I know exactly at what point on the hill, you lose AT&T.

KK: That’s true. Yeah, like I said, I was there some time ago and cell phones weren't quite as good as they are now. But yeah, my reception was terrible at the institute.

TT: Well, your cell phone might have improved, but the reception hasn’t.

EL: Yeah. I love this pairing I love walking and biking as like, ways to, you know, see the city on a human scale instead of when you're in a car or something and you just almost teleport from point A to point B, you don't like see the — you kind of don't get the same environment around you, that kind of effect. So I like that. Even though walking is also a great time to sort of, like, let your mind wander and not think about what's around you, listen to your book, or talk on the phone with someone, or think about proving that next theorem, or anything like that. So it's kind of that, it has both of those things.

KK: You live in two great cities for walking.

TT: Ye.s. With respect to, you know, seeing things differently, one thing I find amazing is that depending on what side of the street you walk, you see things differently.

KK: Absolutely. All right, this has been terrific.

EL: Gotta be some metaphor in here.

KK: I’m sure, I’m sure. Yeah. So it's always nice to get another perspective on the Pythagorean theorem. So we didn't even — it's one of those things that everyone knows so much, we didn't even tell them what it was. I think it was embedded in there somewhere.

EL: Yeah.

KK: But the idea that it is still vital, like still important in modern research mathematics, you know, is a really interesting thing to know about. We all just sort of take it for granted. Right?

EL: Yeah, this theorem that has been known by humans for millennia. And, you know, still is important.

TT: One of the things that these ways of building parameterizations, so they developed into a whole field, and then they moved to other areas. So there are some recent results by Naber and Valtorta trying to look at a singular set of minimizing surfaces, varifolds, you know, that minimize some sort of energy. And they have been able to give a very good description of the singular set by using these type of parameterizations. And they're all basically, the basis is always the Pythagorean theorem. It's really, that's how distances change.

KK: That’s right. It’s completely fundamental.

EL: Thank you so much. This was really fun.

TT: Thanks for the invitation.

[outro]

In this episode, we were happy to talk with Tatiana Toro, mathematician at the University of Washington and director of the Simons Laufer Math Foundation (formerly known as MSRI), about the Pythagorean theorem. Here are some links that you may find interesting.
Toro's website and the SLMath website
Our episodes with Henry Fowler and Fawn Nguyen, who also love the Pythagorean theorem
The analyst's traveling salesman problem on Wikipedia
Naber and Valtorta's work on singular sets of minimizing varifolds

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. Or perhaps today we should say the maths podcast with no quiz at the end. My name is Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. It's Juneteenth.

EL: It is, yeah.

KK: And I'm all alone this week. My wife's out of town. And yesterday was Father's Day and I installed cabinets in the laundry room. This is how I spend my Father's Day, something we've been talking about doing since we bought the house.

EL: That’s a dad thing to do.

KK: 14 years later, I finally installed some cabinets in the laundry room. So it looks like you had a good time in France, judging from your Instagram feed.

EL: Yes, yeah. And I'm freshly back, so I'm in that phase of jetlag where, like, you get up really early. And so it's 9am and I already went for a bike ride and did some baking and had a relaxing breakfast. At this point, I'm always like, “Why don't I do this all the time?” But eventually my natural circadian night owl rhythms will catch up with me. I'm enjoying enjoying my brief, brief morning person phase.

KK: Yeah. Never been one, won’t ever be one as far as I can.

EL: Yeah. Just keep moving west, and then you’ll be a morning person for as long as you can keep jetlag going.

KK: That’s right. That's right. Yeah.

EL: So yeah. Today we are very happy to have Sarah Hart on the show. Sarah, would you like to introduce yourself? And tell us a little bit about, you know, what you're all about?

Sarah Hart: Ah, yes. So my name is Sarah Hart. I'm a mathematician based in in London in the United Kingdom. I'm a professor of mathematics, but my true passion is finding the links and seeing them between mathematics and other subjects, whether that's music or art or literature. And so I think there's fascinating observations to be made there, you know, the symmetries and patterns that we love as mathematicians are in all other creative subjects. And it's fun to spot them and spot the mathematics that's hiding in all of our favorite things.

EL: Yeah. And of course, just a couple of months ago, you published a book about this. So will you tell us about it?

KK: Yeah,

SH: So this book, it's called Once Upon a Prime: The Wondrous Connections between Mathematics and Literature. And in the book, I explore everything from the hidden structures that are underneath various forms of poetry, to the ways that authors have used mathematical ideas in their writing to structure novels and other pieces of fiction and the ways that authors have used mathematical imagery and metaphor to enrich their writing, authors as diverse as you know, George Eliot, Leo Tolstoy, Marcel Proust, Kurt Vonnegut, you name it. And then I also look in the third section of the book at how mathematics itself and mathematicians are portrayed in fiction, because I think that's very, very interesting and shows us the ways in which those things at the time the books are written, how is the mathematics perceived? How has it made its way into popular culture? And how mathematicians are perceived as well, that tells us something fascinating, I think, about the place of mathematics in our culture.

EL: Yeah, definitely.

KK: We’re always portrayed as either mentally ill. Or just, like, absurd geniuses, you know, when really, you know, we're all pretty normal — most of us are pretty normal people, right?

SH: Yeah. Well, we are, as everybody, there's a range. There's a range of ways to be human. And there's a range of ways to be a mathematician. But yeah, we're not all tragic geniuses, or kind of amoral beings of pure logic, or any of those things that you find in books. So yeah, and there are some sympathetic portrayals of mathematicians out there, and I know I talk about some of those, but yeah, it's very interesting how these these tropes, these stereotypes can creep in.

EL: I must confess I'm about three quarters of the way through, I haven't quite finished that last section. But the first few sections that I've read, I've definitely — I keep adding books to my “Want to Read list,” so it’s a little dangerous.

SH: Oh yeah, it should have a little warning, the book, saying “You will need a bigger bookcase.” Unfortunately, you know, you will want to go and read all of these books. And yeah, “Sorry, not sorry,” I think is the phrase.

EL: Yes, definitely. I downloaded — so I don't need a bigger bookshelf because I put this one on my ereader — but I downloaded The Luminaries, which sounds like a really interesting book and excited to get to that, you know, in the neverending list of books that I'd like to read.

KK: Right, we were talking about talking about our tsundoku business before [tsundoku is a Japanese word for accumulating books but not reading them]. So I actually I did, with a friend in the lit department, or in the language department, we taught a course on math and literature a few years ago.

SH: That’s fantastic.

KK: It was. It was so much fun. It's the best teaching experience I've ever had. But I was glad to read your book because we missed so much. Right? I mean, of course, we only had 15 weeks, you know, we and we talked about Woolf, like To the Lighthouse is kind of an interesting one. And yeah, I did finish the book. So sorry, Evelyn, I won. But no, it's it's actually, you know, it is spectacularly well written and, and I'm glad you're having success with it. Because it's — again, I like this idea, that you're sort of humanizing mathematicians and mathematics and showing people how it's everywhere. Isn't that part of your job? Aren’t you the Gresham professor, is that correct?

SH: Yes, I’m the Gresham professor of geometry. So Gresham College is this really unique institution, actually. It was founded in 1597 in the will of Sir Thomas Gresham, who was a financier at the Court of Queen Elizabeth I in Tudor times. And in his will, he left provision for this college to be founded that would have seven professors, and their whole job was to give free lectures, at the time to the people of London. Of course now it's all livestreamed and it goes out and is available all over the internet. And anyone could go and it was just, you know, if you wanted to learn these subjects — and he thought there were seven most important subjects at the time that he said, I still say, geometry and mathematics more broadly, very important — but it was geometry, music, astronomy, law, rhetoric, physic, which is the old word for medicine, and I perhaps I’ve forgotten one. But yeah, these subjects, and so still today, this is what Gresham College does, free public lectures to anyone who wants to come. Now, you used to have to give them once in Latin and once in English. Now, you do not have to do it, thank goodness.

KK: Yeah. Who would come?

SH: I don’t know. Yeah, if I had to suddenly give my lectures in Latin, that might be slightly more of a challenge. My role there is to communicate mathematical ideas to anyone who wants to listen, so a general audience. And some of them will have mathematical training, but many will not. And they they're just kind of interested people who find things in general interesting, and mathematics is part of that. I love that idea, that mathematics is part of what a culturally interesting person might want to know about. And that is something that perhaps used to be more so than it is today. And I really would like mathematics to somehow be rehabilitated into what the cultural conversation involves, rather than it seems to be perhaps in a little bit, sometimes it's pigeon holed or put to one side, you have to be a geek to like mathematics. You have to be unusual. And it's really not true. It's not the case.

EL: Yeah. Wow, that sounds like a dream job. I’m writing that down and putting it on my dream board? It's yeah.

KK: I seem to remember, so I read the review of your book, I think by Jordan Ellenberg, who's also been on.

SH: Yes.

KK: It mentioned that the first person who held your chair invented long division. Is that right?

SH: It’s true.

KK: That's what used to get you a university job, is you invent long division.

SH: Yeah. So that's, you know, what a lineage to be part of. I really feel honored and humbled to be in that role. And, actually, I'm the first woman to do this job in its 400 and whatever year history which, yeah, okay, you could say, yes, we might be a bit late with that one. But I feel it's a real privilege to do it.

EL: Yeah. Well, that's wonderful. So we have invited you on this show to tell us what your favorite theorem is. So have at it.

SH: Okay, so, my favorite theorem, I guess it's could be called a collection of theorems really, but the properties of the cycloid. So the cycloid is, it’s my favorite curve. And it's my favorite curve that probably unless you're a mathematician, you may not have heard of it. So people have heard of ellipses and circles and parabolas. And they've heard of shapes like triangles and things, but cycloids, people tend not to have heard of. And for me that's a surprise because they're so lovely. And the history of the study of the cycle of which, you know, we can we can talk about, is so fascinating and fun, and so many of the most famous mathematicians that people have heard of, like Isaac Newton, and Leibniz, and Mersenne, and Descartes and Galileo, and Pascal and Fermat, all of those people worked on the cycloid and were fascinated by it. And so there are these beautiful properties that it has, which we can bundle up into a theorem. And that would be my favorite thereom.

EL: That’s great. And yeah, in case anyone listening to this doesn't know about the cycloid, it’s a cool curve. And it's actually, you know, it's a curve that a lot of people haven't seen as such, but it's one that does kind of arise sort of in everyday life, kind of. So yeah, do you want to describe what a cycloid is?

SH: You can make a cycloid quite easily. It’s a fairly natural idea, I would say. Imagine a wheel rolling along the road. And now somewhere on the rim of the wheel, you paint the put a little blob of paint, or something like that, or if it's in the dark, you can put a little light. And then and then as the wheel rolls along, that blob of paint or little light will be following a particular path, as the wheel rolls.

EL: Going up and down.

SH: Kind of up and down. And eventually, sometimes it'll touch where the ground is. And then we'll go up and down again. And what you get is a series of arches, they look like arches. And that's what the cycloid is, normally you take one arch and call that the cycloid.

KK: Right.

SH: So this is quite a natural idea, what kind of shape will that be? And what is this arch shape? And the first thing you can say is, yeah, is it something I already know about? So early on in the study of this curve, which is first written down as a question, what is this shape? About 1500. Marin Mersenne, who is famous for Mersenne primes, among other things, so he thought maybe it's half an ellipse. And that's not too bad an approximation, but it isn't quite that. And so that's sort of question one. Is it something we already know? And it wasn’t. So then, people like Galileo started to ask, well, what do we what do we like to know about shapes and curves? So there are two questions really, at the time, they were called the quadrature question and the rectification. So quadrature is what's the what's the area? So if you make this arch, what's the area underneath this arch, between the arch and the road, I guess. That's question one. And the other one is the rectification: what's the length? So how long is this arch in terms of the circle that makes that makes the arch, the cycloid. And Galileo didn't know how to calculate either of those things. But he actually made, he physically made a cycloid. So he got a piece of sheet metal, and he rolled a circle along it, and he got the path. And then he cut it out and he weighed, he weighed the bit of metal that he had.

EL: Oh wow!

SH: To find an estimate for the area. Okay? So this is a real hands on thing.

EL: Yeah, that’s commitment.

SH: Because he did not know. So he physically made it and weighed it. And he got an answer that was around about three times the area of the of the circle that makes it, roughly speaking, and he said, Okay, if we all think, what’s a number that's roughly three, that's to do with circles, right? And so he wondered, could it be pi times the area of the circle? It isn't. It isn't pi times the area of the circle! Galileo never managed to work out exactly what it was. But this guy Roberval, Gilles de Roberval, did manage to work out what the area is. He didn't tell anyone how he'd done it because at this time in history, there were all these priority disputes, who sorted this thing first, who has done what first? People would sometimes go to the length of writing their solutions in code. So Thomas Hooke, who was another Gresham professor, when he worked out what we call Hooke’s law now, he wrote Hooke’s law down as an anagram in Latin, before he told anyone else. And then if anyone else came up with it, he could say, look, here's my anagram that I did earlier to prove that I thought of it first. So there were all these weird and wonderful things that people did at that time to establish priority. But Roberval, he had this incentive for not telling that he knew the area under a cycloid. And the incentive was this — it was not a good idea for them to do this — the job he had at the time, Roberval, was renewed every three years. And to get the job every three years, there were some questions that were set. And if you could answer those questions the best out of all the people who tried to do it, you could get that job for the next three years. But the person setting the question was the incumbent professor. So if you're the incumbent professor, you need to set questions that only you know the answer to, and then you get to keep your job. So for a few years, Roberval could say, you know, what's the area under this cycloid, and no one else knew. So he worked it out. And his proof was quite nice, but it wasn't published until 30 or 40 years after his death. But it actually — and this is the first lovely thing about the cycloid — the area, if you have a circle that's making this cycloid by rolling along road, the area underneath one of these arches is exactly not pi times, exactly three times the area of the generating circle. So a lovely whole number, simple relationship between the arts.

EL: What are the odds? It’s almost miraculous.

SH: Fantastic. So here's another equally miraculous thing that kind of adds to the first one. Then people try to work out what's the length of this cycloid? And the person who managed to solve that was, in fact, Christopher Wren. So he's well known as an architect, and he designed St. Paul’s, the wonderful dome of St. Paul's in London, and many other churches in London. But he was also a mathematician among many other things. So he solved the rectification problem, what's the length, and if the circle that makes this, the cycloid has diameter d. So we know that the circumference of that circle, the length around the circle would be pi times d. Well, another beautiful whole number relationship, the length of the cycloid arch is exactly four times the diameter. A beautiful whole number relationship. It's fantastic. So you've got these two lovely properties of the cycloid. And people were fascinated by it. So it had this nickname, the Helen of geometry, as in Helen, you know, face that launched a thousand ships.

KK: Right.

SH: It was a very beautiful curve with beautiful properties. But there's another reason why it was called the Helen of geometry. And it was because, like Helen of myth, it started lots of squabbling. So I mentioned Roberval, who had proved the area formula for the cycloid. Someone else came along a few years later, and found out this this result, and Roberval immediately accused him of plagiarism. And this guy was like, No, I didn't do that. But they argued about it. I think it was Torricelli. And and When Torricelli died a few years later, team Roberval said he's died of shame because of being a plagiarist. He may have died of shame. But he also happened to have typhoid at the same moment. So you know.

KK: Sure.

EL: Shame-induced typhoid?

SH: But you know, so that was one squabble, but then Fermat and Descartes had an argument because they both proved something about the tangents to the cycloid. And they hated the way each other done this. So I think it was Fermat did have a particular method. Descartes said that this method was ridiculous gibberish. So you know, he's not mincing his words, he’s not saying “I prefer my method” but “Fermat is speaking gibberish nonsense.” So they argued. But, you know, this beautiful curve has other exciting properties. And this is where it goes for me from, “Okay, nice whole number relationships, cute.” But then one of the things that we all love in mathematics is where something you've studied over here, reappears in a completely different context. And this is what happens with the cycloid. So it comes up to in connection with trying to make a better clock. So there's this mathematician, Christiaan Huygens, who is trying to make a better clock. And he comes up with a pendulum clock. And so pendulum clocks improved timekeeping dramatically. Before the pendulum clock came along, basically, it was a sundial or nothing, really. There were no good mechanical clocks. And the ones that existed would lose about 15 minutes a day or something of time. The pendulum clock comes along. And so you can do kind of the mathematics of a swinging pendulum, and if you make a little approximation, so the approximation that you make is that for a small angle, theta, the sine of theta is approximately theta. So you can make that approximation. And it's pretty good for small angles. And if you do that, then when you work out what the forces are acting on the pendulum, you find that, roughly speaking, it'll take the same time to do its swing wherever you release it from. So it has this kind of constant period, basically. And that's why pendulum clocks are useful for telling for time. But they're not perfect, because we had to use an approximation to get to that point. So Christiaan Huygens is wondering, is there actually a curve that I can make, that will really genuinely have this constant period property, that wherever I release a particle from on this curve, it will reach the bottom in the same time?

KK: Right.

SH: Because that's what the pendulum almost does, but doesn't quite do. And so he said — and this problem is known as the tautochrone problem, because it's “the same time” in Greek. And it turns out, guess what, the cycloid solves the tautochone problem. It's precisely — so we have an arch, you've got to turn the arch upside down. So now you can roll, your particle can roll down. And wherever you release a particle from on the cycloid, it will reach the bottom in exactly the same time.

KK: Remarkable.

SH: I mean, assuming you know, it's smooth, no friction or whatever. It's just rolling down under gravity. And I mean, it's not even clear that such a curve could exist, right? It's quite a thing to ask. And yet, the cycloid has this property, and it's fantastic. So that's an amazing thing. And few years later — so Huygens worked this out. A few years later, a different problem was posed. It's kind of a related question, or it's something to do with particles anyway. And the question here is called the brachistochrone problem. And it was proposed by Johann Bernoulli, one of the Bernoulli brothers. And he posed this kind of publicly in a journal saying, Okay, if you now have two points A and B, A is above B, and you want to have a curve such that when a particle rolls down that curve from A to B, it will reach point B in the quickest time, so what might that be? Is it sort of a parabola, maybe a straight line, what's it going to be like? And this problem was posed to the mathematicians of Europe as a challenge, and quite a few big names enter this competition to see if they could do this. So Leibniz was one, Gottfried Leibniz, Bernoulli himself solved it, his older brother solved it, and then they got this anonymous entry. And it was so beautifully done, and elegantly produced, the solution to this, that, even though it was anonymous, when Bernoulli he saw it, he said this famous phrase, “I recognize the lion by his claw.”

KK: Right.

SH: And it was Isaac Newton, who had solved this problem. And guess what? It's the cycloid again. The cycloid solves this problem as well. So you've got this amazing curve, which is a natural idea. It's got these lovely whole number relationships about its length and its area, and then it suddenly also can solve these totally different questions about particles rolling down in the quickest time or constant time. And so that is why I love the cycloid so much. Everybody’s worked on it. It's got this amazing history, it's really beautiful.

KK: This sounds like a good public lecture.

EL: Yeah.

SH: I just get really.

EL: The cornucopia of the cycloid.

KK: Yeah, so question, the original area calculation that Roberval did, did he use calculus? Or was this a geometric argument?

SH: So he used something that isn't quite calculus yet, Cavalieri’s principle. If you're comparing areas, if you have got two shapes where if you slice through, the length of those slices is the same at every point, then the areas are the same. So he used that principle, which you can extend to volumes as well. And he kind of did a particular, so he managed to do this. And he had the curve that you make for the cycloid, he made it up from three different pieces. And he did this sort of slicing argument to compare it to with things he already knew, one of which was the sine curve, although I don't think he noticed it was a sine curve at the time, but we can now see that. So now, you would make that argument with calculus. But it's the same basic idea. You're slicing something very finely.

KK: Right. You could almost imagine Archimedes figuring this out.

SH: Yeah. Yeah, exactly.

EL: Yeah. So I mean, you've made a very compelling case that this is a very cool curve that has all these properties, So like, why is this your favorite? Or I know it's hard to pick a true favorite. But yeah, can you talk a little bit about, like, how you encountered it and what makes it so appealing to you?

SH: Well, there's at least two things. There might be three. One is, I love the simplicity of the results about the area and the length, that they are just lovely, simple relationships there comparing to the circle that makes this this curve, which itself is easy to think about what it is. So it's not contrived at all. It arises fairly naturally from just thinking about wheels rolling along roads. You get this curve, and then these relationships are very simple. The second reason I love it so much is because of this unexpected appearance of the cycloid in this totally different context from from how you imagined it. When it's generated by just, you know, a wheel, but then a curve that has these other properties, that’s very surprising. There are other things we could talk about to do with it. involutes, and other kinds of things where it crops up, but that for me, it encapsulates why it's such an exciting thing. And it's like when you first encounter pi or something, or you see the e to the i pi plus one equals zero, it gives you that same kind of feeling, that thing's from over here, and this other constants from over there, you know, that they're linked together seems really surprising. But the final thing, I suppose this kind of links in again with what we were saying about mathematics and literature, is how the cycloid has caught people's imagination over time. And it's both of mathematicians, but outside. And there are several books that mention cycloids. So Moby Dick is one. That's got a lovely little passage about cycloids. But also, Gulliver’s Travels mentions cycloids, Tristram Shandy by Laurence Sterne, this amazing, crazy 18th century book talks about cycloids. And those are just three that are really classic books. It was in the air at the time, and perhaps we don't necessarily — like, a modern and modern person may not have heard of cycloids. But certainly if you were educated in the 18th, 19th century, you may well have heard about cycloids. And that, to me, is very interesting too.

EL: Yeah, do write a little bit about this in your book that Moby Dick part, I have gotten to that part. And apparently, did you say that Melville apparently had some amazing math teacher in high school. And so, you know, kind of was able to really capture his imagination about math and then bring that into literature later, which is just kind of a cool thing to think about as math teachers, people who teach math. It's like, yeah, even if your your students don't end up in math or something, they might, you know, hopefully bring some of what you teach them that direction.

SH: Yeah, absolutely. I mean, it’s the value of having a great inspirational teacher. Just look at with Melville. So he had a teacher. He went to a school called the Albany Academy, and he was good at school in some areas, mathematics was something he was particularly good at. And he actually won a prize for being the first best at ciphering, was what it was for. Cipher, the old word for calculation.

KK: Right.

SH: His prize was a book of poetry, which I liked, because for me, that's absolutely a natural prize, but it wouldn't necessarily be thought so. But his teacher was a man called Joseph Henry. And Joseph Henry was no ordinary schoolteacher. He was a very good scientist in his own right, he went on to become the first secretary of the Smithsonian. So you know, pretty impressive. But physicists will know the name Henry, because the Henry is the scientific unit of inductance. And that's for Joseph, that is Herman Melville's maths teacher at school. So he was by all accounts an exceptionally good teacher, to the extent that some of his classes were actually, members of the public were allowed to come in and attend as public lectures. So there's a record that says, a request of his that he wants to have additional books for the more advanced students to entertain them beyond the normal curriculum. And so I don't know, and we can't know for sure, how Herman Melville learned about cycloids. But I could very easily imagine that a lesson on Friday afternoon, let's just talk about this fascinating curve because it's really interesting. And Melville did have a love, then, of mathematics, which just comes out in his writing. You can just see it, the way he chooses metaphors and imagery, they're often mathematical. And you can just see it's, it's not thinking “I must include some mathematics.” It's just the sheer pleasure of it. The delights, the joy of mathematics just comes out in his writing, which is wonderful to see.

EL: Well that was such a cool story that I read in there. And I have loved to revisit this. I don't think I've actually thought about cycloids since I taught calculus, right, which, it's been quite a while since I taught calculus. It is a fun, it’s a very common example in calculus books now. You'll kind of go through and solve some of these, these things. And I think, when you do parametric curves, maybe?

SH: Yes.

EL: So yeah, lots of fun, but I don't think I had really appreciated it as this whole whole thing before. So the other thing we like to do on this podcast is ask our guests to pair their theorem, or their bouquet of cycloid facts, with something else in life. So what have you chosen for your pairing?

SH: Well, so I've chosen Moby Dick.

EL: Okay.

SH: Because, I mean, he does talk about cycloids in the book. It’s not just because of that, but with the cycloids is this lovely passage where Ishmael, who is, you know, traveling as a deckhand on a whaling ship with Captain Ahab, who perhaps is not entirely sane, and we discovered that through the book, but there are many — Ishmael sort of has these wonderful meditations, he's just thinking about things. And some of them are mathematical, and some of them aren't. But there's one particular point where he is cleaning the the try pots. A try pot is something you had on a whaling ship, where there's great cauldron like pots where they kind of render the whale blubber down, and then you have to clean them. And so he says, you know, this is a place for wonderful mathematical meditation. And he and he talks about, as his soapstone is circling around the inside of the try pot, he says, I was struck by the fact that in mathematics, the cycloid is the curve where you can you can release something and it falls to the bottom in a constant time. And so he's just sort of drops in, the cycloid, just mentions it while he's daydreaming about something else. But Moby Dick, it's full of mathematical ideas. And it’s, you know, they are interested in numbers, to the extent that Ishmael keeps, he has the data or information about whales, measurements and statistics about whales, he has them tattooed onto his body, because as he says, you know, I didn't have a pen to hand, kind of thing, there was no other way to record. So he just has them tattooed on his body. Ahab is doing calculations on his ivory leg, you know, there are all these discussions about number. But there are lovely pieces of imagery around the infinite series of ripples in waves in the sea. There's a metaphor about loyalty where Ahab says to the cabin, boy, you are loyal as the circumference to the center, you know, the circumference always stays the same distance from the center. And it's just lovely little pieces of mathematical imagery throughout, and throughout all Melville's work. So I thought, yes, Moby Dick would be a very good pairing.

KK: Yeah. And so you actually have a paper about this in the Journal of Humanistic Mathematics, right?

SH: Yeah.

KK: Ahab’s arithmetic?

SH: Yes. And that itself is a little bit of a reference to a discussion that happens in Moby Dick, which is where two of them were talking about a book called Daboll’s Arithmetic, which was the kind of classic text in American schools, I think, at the time, which had all these rules about how to do calculations. And you could do mysterious things with with this book because, you know, if perhaps the mathematics hadn't been taught by a teacher like Joseph Henry, perhaps you learnt you've learned these rules off by heart, you don't quite understand them. And so they talk in the book about cabbalistic contrivances of producing these things. And at one point, someone says, “I have heard devils can be raised with Daboll’s arithmetic.” So, you know, this is the other side of mathematics, where people sort of hold it in or but also, perhaps, they have some suspicions around what do all these symbols mean? And it's very interesting, if you look at that book, Daboll’s Arithmetic, it isn't like a mathematics book would now be. So when he talks about how to find the areas of circles, for instance, pi is not mentioned at all. He says, you square the radius, and you multiply it by 22/7, or if you want a more accurate thing, you could multiply it by what's that other approximation right? 355/113? But he doesn't say “because these are approximations to pi,” it's just like, you can do this or you can do that.

EL: Here’s a number.

SH: Tust where's that come from? So that's a very interesting thing. And so there are mathematics books discussed or mentioned in Moby Dick as well. And if you know a little bit about them, so Euclid, of course, is mentioned a little bit. Yeah. So the book is full of mathematics. And I really wanted to think about in the article I wrote, why — how did Herman Melville know all this stuff? Why, you know, where does it come from? Because, you know, he's not a mathematician. And this is why, you know, nowadays, we're sort of taught to believe, or somehow we come to believe, quite often, that you're either a mathematics personal, or you're not. And if you're not, then you don't know any and you don't care. But this is absolutely not the case for one of our greatest writers, Herman Melville. And so you know, yeah, where did that come from? And, you know, it was just lovely to, to find out a little bit more about what he knew and how he knew it, and where it all came from.

KK: Very cool.

EL: Yeah, that's great. I have confessed to you already, but I will confess to our listeners that I have not read Moby Dick, but it is on my list that I hope to get to this year. It's a little daunting.

KK: You’d better get cracking.

EL: I know. I’ve only got six months.

KK: I read it at bedtime. That's when I tend to read, and so I read it, you know, maybe 10 or 12 years ago, and it took me quite a while. Yeah, yeah. It's pretty dense too.

SH: It is. I mean, I didn't read it till I was older. Because, you know, you hear this is the “great American novel,” and you should, everyone should have read this book, and then you feel bad that you haven't read it, and then you feel annoyed that you feel bad that you haven't read it. So there's all these barriers that you put up for yourself. And, you know, I'm so glad that I did eventually read it, because I loved it. It's so rich. And there is, you know, many many, many layers of interpretation and depth in the writing, but it is a great book. So yeah, I hope you will enjoy it when you read it.

EL: Yeah.

SH: You know that there are books that we all — I haven't yet read, I don't know if I will ever read, maybe one day, Finnegans Wake. I do mention it in the book because James Joyce, I talk about Ulysses a little bit and Dubliners in the book. But Finnegan’s Wake for me, I tried and I didn't quite quite get there. All I can say is in the middle of Finnegans Wake, there is a picture which could have come straight out of Euclid’s Elements. It's got equilateral triangles, two circles intersecting. But yeah, that for me, maybe one day, maybe I'll have a sabbatical one day and that will be what I do in that sabbatical.

EL: There just, there is so much. There's so many good books published now, you can't you can't read them because you’ve got to read last year's good books. But I mean, it's just you — Yeah, anything you read is great. And you’re never going to get to all of it. Enjoy what you read.

SH: Exactly. Amnesty of all our unread books. It's fine. We forgive ourselves.

EL: Yeah. Thank you so much. for joining us. This has been a lot of fun. You know, we do like to give our guests a chance to plug things but we've already talked about your book quite a bit. Is there anything else that you'd like to to mention about what you're working on or other things that you've published that you'd like us to share?

SH: Oh, no, I think I'm alright. So coming up. I mean, not for US listeners, but I've got an event coming up in a couple of weeks is going to be really fun because we're going to watch a classic B movie from the 1950s, which is this film about giant ants terrorizing the New Mexico desert. It’s called Them! with an exclamation point.

KK: Yeah, I've seen the posters.

SH: Yeah, right. Yeah. Which is super fun. But that's about, yeah, something has happened. Who knows? But there are giant ants. They have a lot of fun with it. But we're going to watch the film at the Barbican Centre in London. And then we're going to talk about, yeah, what does mathematics tell us about what life is like? Could giant ants exist, could giant spiders exist? Or giants, or tiny people like Lilliputians. And so that's a kind of fun thing that's coming up. But yeah, you've already, if you look at my book, you will already have a reading list that’s like 100 new books that are gonna be fun, fun to read and explore. So yeah, there's plenty to go on.

EL: Great.

KK: Thanks so much, Sarah, this has been great fun.

SH: Thank you for having me. Yeah. I’ve loved it.

EL: Bye.

SH: Bye.

[outro]

In this episode, we were delighted to talk with Sarah Hart, the Gresham Professor of Geometry at the University of London, about the serendipitous cycloid. Below are some links you might enjoy as you listen.
Hart's website and Twitter profile
Her book Once Upon a Prime and its review in the New York Times
Hart's article Ahab's Arithmetic about mathematics in Moby-Dick
The Wikipedia entry for the cycloid, which has links to many of the people we discussed

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 I am joined today as always by my fabulous co-host.

Evelyn Lamb: Well, thank you. I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City. And anyone who's on this Zoom, which is only us and our guest, can see that I am bragging with my Zoom background right now. We just got back from a trip to southern Utah, and I took possibly the best picture I've ever taken in my life. And 95% of the credit goes to the clouds because they just — above these red rock hoodoos outside of Bryce Canyon, I turned around and looked at it while we were hiking, and I was like, Oh, my gosh, I have to capture this.

KK: It is quite the picture.

EL: My little iPhone managed.

KK: Yeah. Well, they're pretty good now. Yeah. Anyway, so yeah, I'm getting ready to — I have three trips in the next three weeks. So lots and lots of travel, and I'm gonna make sure I mask up and hopefully I don't come home with COVID, but we'll see.

EL: Yes.

KK: Anyway. So today, we are pleased to welcome Matthew Kahle. Matt, why don’t you introduce yourself?

Matthew Kahle: Hi, everyone. Thanks for having me, Kevin and Evelyn. I'm a mathematician here at The Ohio State University in Columbus, Ohio. I've been here for 11 or 12 years now, and before that, I spent a good part of my life in the western United States. So those clouds look familiar to me, Evelyn. I miss the Colorado sky sometimes.

EL: Yeah, just amazing here.

KK: You did your degree in Seattle, right?

MK: I did. I did my PhD at the University of Washington.

KK: Yep. Yep.

EL: Great. And what is your general research field?

MK: I work a little bit between fields. My main interests are topology, combinatorics, and also probability and statistical physics. And I think I usually feel most comfortable, or maybe I should say most excited mathematically, when there's sort of more than one thing going on, or when it's in the intersection of more than one field.

KK: Yeah, lots of randomness in your work. He’s got this very cool stuff with random topology. And I remember, some paper you had few years ago, I remember really sort of blew my mind, where you had some, you're just computing homology of these random simplicial complexes, and, like, some four- or five-complex had torsion of order, you know, I don't know, 10 to the 12th, or some crazy torsion coefficient. Yeah.

MK: Yeah. So we were really surprised by this too, and we still don't really have any way to prove it, or really understand it very deeply. Kevin was mentioning some experimental work I did with some collaborators a few years ago. But yeah, that is the gist of a lot of what I think about, is random topology, which I sometimes try to sum up as the study of random shapes. And one of the original motivations for this was as sort of a null hypothesis for topological data analysis, that if you want to do statistical methods — if you want to use topological and geometric methods, and statistics and data science, you need a probabilistic foundation. But one of the things we've discovered over the last 15 years or so is that these random shapes are interesting for their own sake as well. And sometimes they have very interesting, even bizarre, properties, where we don't even know how to construct shapes that have these properties at all, but they're they are there. And we know they exist, because of the probabilistic method. Yeah.

EL: So let me be the very naive person who asks, like, how do you, I guess, come up with — like, what do you randomize about shapes? Or you know, if I think about, I don't know, randomly drawing from from some sort of, I don't know, bucket of properties, is it that or is it… Just what is random? What quantity or quality is being randomized?

MK: Right. So a lot of the random shapes or spaces that I've studied have have been on the combinatorial or discrete side. So for example, there are lots of different types of random simplicial complex that people have studied by now. And typically, you have just some probability distribution, some way of making a random simplicial complex on n vertices. And n can be anything, but then the yoga of the subject is that typically n goes to infinity. And then we're interested in sort of the asymptotic properties as your random shape grows. So one of the early motivations, or early inspirations, for the subject of random, simplicial complexes was random graph theory. So you can create random networks various ways, and people have been studying that for for a bit longer, probably at least 60 years or so now, with new models and new interesting ideas coming along all the time. For example, there was originally the Erdős–Rényi model of random graph where the edges all have equal probability, and they're all independent. This is a beautiful model mathematically, and it's been studied extensively. We really know lots and lots about that model of random graph now, although surprisingly, people can continue to discover new things about it as well. But in today's world, some people have studied other models of random graphs that they say may have made better model real world networks, for example, social networks, or what we see in epidemiology, and so on. The Erdős–Rényi model is something that's tractable, and that we can prove deep math theorems about, but it might not be the best model for real world networks. But, you know, I think of the random simplicial complexes that I study sometimes as just higher-dimensional versions of random graphs.

EL: Okay.

MK: So as well as as well as vertices and edges, we can have higher-dimensional cells in there, and and that starts to sort of enrich the space. It's not just one-dimensional now, it could be two-dimensional, or it could be any dimension.

EL: So you might not know. You've got some large number n, and you might not know what dimension this random — you’re, like ,attaching with edges with some sort of probability between any two things. And so you might not know what dimension your simplicial complex is going to be until after you randomly assign all of these edges and faces and, you know, and n- whatever the word is for that, n-things. [Editor’s note: it’s n-simplex.]

MK: Yeah, absolutely. That's right. It could be that the dimension of the random simplicial complex is itself a random variable. And you know, that we don't ahead of time even know what the dimension of it is.

EL: Cool!

KK: So there's lots to do here. This is why Matt has lots of students and lots of lots of good projects to work on. But anyway, we invited you on not just to talk about this really interesting mathematics, but to find out what your favorite theorem is. So what is it?

MK: Okay, so I've been thinking about this. Well, I have to admit, I think I asked myself this just knowing of your podcast in case I ever got invited on. And then I've been thinking about it since you invited me. I would say my favorite math theorem, probably the one I've thought about the most, the one maybe that affects me the most, is Euler’s polyhedral formula, which is V−E+F=2. Right? So let's just start out saying, well, you know, what do we mean by this? I think my understanding of the history of it is that it was something that as far as we know, the Greeks didn't observe even though they were interested in convex polyhedra. And sometimes people consider the classification of the perfect Platonic solids is one of the peak contributions of Euclid’s Elements. But we don't know that they recognized this pattern that Euler noticed thousands of years later. If you take any convex polyhedron, a cube or an icosahedron, or a pyramid, a bi-pyramid, any kind of three dimensional polyhedral shape that you can imagine that's convex, V, the vertices is the sort of number of corners of the shape and E is the number of edges. And then F is the faces. It always is the case that V−E+F=2. So Euler noticed this. And it's not clear if he gave a rigorous proof or not. I don't even know if he felt like anything needed to be proved, maybe it was obvious to him. And nowadays, we have many, many beautiful proofs of this fact. But one of the things that strikes me about it is that, it’s sort of in hindsight, is that this is just sort of the tip of a very big iceberg. There's a much more general fact that we are just kind of getting our first glimpses of, and nowadays, we would think of this as not just a phenomenon about convex polyhedron, 3-dimensional space, that it’s just a general phenomenon in algebraic topology, or you can say =more generally, in homological algebra. It's just sort of a feature of nature somehow.

KK: Right, right.

EL: I think something that I really enjoy about this fact is you can present it at first as a theorem or as a fact. But then this fact kind of leads you to this new definition that you can observe about all sorts of different shapes, you know, this number that is the vertices minus the edges plus the faces, hopefully, I got it in the right order, yes. Then you can assign that, you know, you can say, like, what does, you know, if you've got a torus, like a polyhedral torus, or, you know, a higher-genus object or a higher-dimensional thing, you can sort of use this, and so it's like a fact becomes a definition or a new thing to observe.

MK: That’s right. Are you saying, for example, you know, we have the Euler characteristic is an invariant of a space?

EL: Right.

MK: And that might, if you're introduced to a new topological space, that might be one of the first things you might like to know about it. And so yeah, it becomes its own invariant. It’s a way of telling some different spaces apart, for example.

EL: Yeah. So do you have a favorite proof of this favorite theorem?

MK: I do. I present it and the graduate combinatorics and graph theory course when I teach this course. So already, we're looking at a little bit more general formulation than what Euler looked at. We don't just have a convex polyhedron in 3-dimensional space, what we have is a connected planar graph. So we have some kind of network with nodes and connections between them, and it's one that you can draw on the plane without any of the edges or connections crossing. And in this case, the faces now are just going to be the connected components, or the regions, in the complement of the graph that then comes with an embedding into the plane. And then V is the number of vertices of the graph, and E is the number of edges. So V−E+F=2 in this case, so for for just any connected planar graph, this might seem totally unrelated, but it's actually a more general version than what we just saw with convex polyhedra because you could take any convex polyhedron and unwrap it, or stereographically project it into the plane and get a planar graph. But planar graphs could have lots of other features. So when I present this in class, I tend to give three or four different proofs of it. There's a beautiful proof that I've heard attributed to John Conway, where he says something about, like, letting in the ocean or something. So your graph is connected, but there may be some cycles in it. And anytime you have cycles, the Jordan curve theorem tells us there's an inside and outside. So John Conway wants to let the ocean in. The ocean is the sea, is the outside of the graph, let it in until it touches. So what he's saying is if there's any cycle, delete one edge from it, and so what this does is it reduces the number of edges by one because you deleted an edge, but it also reduces the number of faces by one because that two regions that were inside and outside of that cycle are now the same region, so V−E+F has stayed the same.

KK: Right.

MK: And then eventually, you've just got a tree. There's no more cycles left, but your graph is still connected, so it must be a tree. And we know that every finite tree with at least two vertices has a leaf, has a vertex of degree one. And again, you can prune away that, and then you've reduced the number of vertices by one and the number of edges by one. And V−E+F is again not changed. So at the very, very end, we're just left with a single vertex in the plane. There's one vertex and there's one region, which is everything except that vertex. So at the very end, V−E+F=2. But through all those steps, we know that it never changed. So it must have been V−E+F, it must have been 2 at the very beginning. So I love that proof.

EL: Yeah.

MK: There's another proof that I think I like even better, which is that you consider the dual graph and a spanning tree. You pick a spanning tree on the original graph and a spanning tree and a dual graph at the same time.

EL: So the dual graph being where you replace, you swap vertices and faces.

KK: Yes. For every face there’s a vertex and you join two when the two faces share an edge.

MK: That’s right, exactly. So one thing that's tricky about that is that now the dual of even just a nice planar graph might be a multi-graph. So just imagine a triangle in the plane. The dual graph has two vertices, one inside the triangle and one outside, but there's three edges connecting those two vertices, because there's three edges in the original graph. And the edges in the dual graph have to correspond to edges in the original graph, and that's important. They cross them transversely. So then you choose a spanning tree on each one, and you and you realize that — you count the number of edges in each and you somehow — now I'm getting a little stuck remembering the proof, but the punch line is in the original graph, I guess the number of edges is V−1. And then the dual graph, the spanning tree, the number of edges is F−1. And these have to be in correspondence. So you just immediately just write down V−1=, sorry, no, I don't remember exactly how that the end of that proof goes. But there was something about it I liked. It seemed like the other proofs, you're kind of doing induction on either the number of vertices or the number of edges or the number of faces, and that you have to make some arbitrary choices. And this proof by duality doesn't use any induction and doesn't require any choices. It just kind of comes for free. And you sort of immediately see where the 2 comes from, because there's a V−1 on one side and an F−1 on the other side, so the 2 just sort of pops out immediately from the proof. There’s — I think it’s Eppstein? — some mathematician collects proofs of Euler’s polyhedra formula on his website, and he has at least 10 or 20 different proofs. And when you read them all, some of them start to remind you of each other, and who knows what counts as the same proof or different proofs.

KK: Sure.

MK: But there are some neat contributions in there. One of them he attributes to Bill Thurston in the middle of some very influential notes that Thurston had in differential geometry. And he's talking, he's giving his own proof, I think, that the Euler characteristic of a differential manifold really is an invariant of the manifold, for a smooth manifold, let’s say. You could triangulate it, and then the Euler formula, the Euler characteristic, you could just say is the alternating sum of the faces of every dimension. But why doesn't that depend on which triangulation you pick? And Thurston gave a really beautiful kind of almost physical argument with, like, moving charges around. I like to show the class this one also. At that point, we leave — I don't know how to make that proof work for planar graphs, but it works beautifully for polyhedra, for convex polyhedra, like what Euler first noticed. And apparently, it works also for higher dimensional manifolds, too, although I've never gone through that proof carefully.

KK: Yeah. Right. Well, the proof that you said might be attributed to Conway is sort of the one that I always knew, and I never heard it attributed to him, but that's good. It's sort of nice. You can explain that one to just about anybody right? You just sort of imagine plugging away an edge and a face at the same time basically, yeah.

EL: Yeah. A proof that proof that is of something that is so visual, but you can really understand over a podcast, is a special proof. Because I do think that it doesn't take a whole lot of you know, imagination, to be able to follow this audially.

KK: Audially, is that a new word?

EL: There is a real word that is embedded in that word. Aurally, that’s the real word I was trying to say.

KK: Yeah, so is this sort of a love at first sight theorem? I think I first learned this theorem in the context of graph theory.

MK: I think for me, too.

KK: And then I became a topologist kind of later. And then of course, now I think of it as, oh, it's the alternating sum of the Betti numbers, but that those two quantities are equal is an interesting theorem in its own right.

MK: Right. Yeah. So I was trying to think about this. When did I learn about this theorem? And I think I first learned it in graph theory also. But then I know now that it's much more general, and I don't even know if I ever remember anyone telling me that specifically in a class or reading it in a particular book or paper. I think this to me, maybe part of what I like about the Euler formula is that I feel like my understanding of it has just deepened over time, and that there’s kind of a series of small revelations. At some point, I started thinking of it as the alternating sum of the Betti numbers, and things like that. And since I like the combinatorial side of topology, and have simplicial complexes or cell complexes, also the alternating sum of the number of faces of each dimension. But then even in the last couple of years, my understanding has continued to develop because now I think, you know, well, you could just have a chain complex, and all you know is the dimensions of the vector spaces, but it makes sense to ask what's the homology of the chain complex, so they're the Betti numbers again, and again, the alternating sum of the Betty numbers now is the alternating sum of the dimensions of the vector spaces of your chain complex. But I think I probably first saw, you know, the graph theory version of it, maybe in an undergrad or a first graduate combinatorics course.

KK: All right. So the other thing on this podcast is we like to ask our guest to pair their theorem with something. So what have you chosen to pair Euler’s formula with?

MK: Well, you know, I've been stumped by this. But you know, thanks for the warning that I am going to get asked this question. So I had a little time to think about it, and I'm not totally stumped on the spot. But the thing that keeps coming to mind the most when I ask myself that question is some of Bach's music. Johann Sebastian Bach is really known for his four-part harmonies and for counterpoint, and it feels a little bit like this: You're listening to a beautiful piece of — it could be anything, you know: a fugue on an organ, or four-part harmonies that were written for choral music or something like that. And when you listen to it, you can listen to a recording of it two or three times and each time pick out a different voice to follow along. And there are just these independent melodies, harmonies that he's somehow weaving together. You can also just relax and just let the whole thing wash over you. And honestly, that's most often how I listen to music. But it's completely fascinating to just hone in on one particular thread. And so I think a lot of people feel like Bach's music has maybe a mathematical feeling to it, or that it's mathematically perfect or precise. So you could say that Bach, pairs with mathematics already, but the reason I want to try to connect it with the Euler formula that I like as my favorite theorem is that there are these sort of different layers. And just the same way you can kind of listen for one voice, and then tune your ear and listen to a different voice and emphasize that, I feel like this is one of these areas, of one of these kinds of mathematical phenomena, that’s just sitting there in, you know, platonic space, or wherever it lives. And you can look at it. So if you look at it from the topological point of view, it's the alternating sum of the Betti numbers, the number of holes in each dimension. But if you look at it through a combinatorial lens, then it's the alternating sum of the number of faces of each dimension. Or you can just step back and it's just its own thing. It's just an invariant of the space, the Euler characteristic, and these just happen to be different ways to compute it. But it has that feeling to me that you can look at it different ways. But you're really always looking at the same thing. Just we're putting on different glasses or looking at it through different lenses, and so it reminds me of that sort of, I don't know, counterpoint and music or something.

EL: Yeah. Oh, I love this pairing! I'm also, I play viola and I sing, and when you get to a point when you’ve learned a piece that you've learned it enough that you don't have to be just concentrating on, like, am I singing the right note at the right time, but you can actually start hearing like, oh, I didn't originally hear that the parallel that the bass and the soprano line has right here, or the way we come in and then the altos come in and something like that. I've been singing a lot of, you know, things that have these fugal sections in them, which is — I haven't actually sung much Bach recently, but similar things — and I just love that pairing and how seeing the same thing, or singing the same music over and over again, you hear something different every time. You know, just a little easter egg that you didn't pick up the first 20 times you practiced this piece, and then now you hear and you say, oh, next time I really want to make sure that I, you know, do that crescendo with the tenors just perfectly or something. I love that.

KK: Yeah, so I see the edge of a keyboard there in your Zoom, Matt. Do you play?

MK: A little bit. I mean, nothing to write home about, but it's something I enjoy. I took it back up during the pandemic as a hobby. And I've been practicing a little bit. This over here, I have a little portable keyboard, and then I have an electric piano out in the living room. But I've been practicing music with one of my friends. We get together, like, once a week and and just play some cover songs. And I like what you're saying, Evelyn, about hearing different things. Even, you know, we'll be playing some song by REM or somebody that I've known, I don't know, it seems like my whole life, it’s very familiar. But once we start to play it, once we start to sing it, then I hear all kinds of different things in it that just listening to the same recording that I've listened to before all of a sudden, I'm like, wait, Mike Mills is actually doing some really interesting harmonizing and this track, and not only is he harmonizing, like singing different notes, than what Michael Stipe is singing, he’s actually singing different words. He's saying something in that song I never even noticed he was saying. So anyway, music and mathematics, I think that's probably another big thing that they have in common, is that, you know, a little bit can go a long way, and even just entry-level, you can already start to appreciate the beauty of it, but that it's sort of almost inexhaustible how deep it goes and that you can always, there's always more to learn. There's there's always more to notice.

EL: Yeah, with Bach specifically, you know, as a viola student, I think I started playing the Bach cello suites, an octave up on the viola, I was probably 10 or 11 years old? And it's like, I will still play those same suites that I started learning when I was in fifth grade. And it's like, it always has something to teach me. It's something that I can always get something more out of.

KK: Yeah. I think we can all agree that that Mike Mills is REM’s secret weapon. I took up the guitar about 10 years ago, so I'm terrible, and I play alone. But it's still something that I enjoy to do. It's certainly, it's a good way to exercise your — what was it Leibniz said? That music is the pleasure the brain derives from counting without knowing that it's counting?

EL: Oh, yeah. That’s a good little quote, to file away for us math-musician-type people.

KK: That’s right. All right, well, so we always like to give our guests a chance to plug anything. Where can we find you on the interwebs?

MK: Yeah, I don't have anything particular to plug, but you can find, you know, all my mathematical work on my professional webpage, matthewkahle.org. There's links to all my papers and everything there. And, you know, if my friend and I get our REM cover band off the off the ground, we’ll keep you posted.

EL: All of our Columbus area listeners can find you.

KK: I can play rhythm guitar on some of the tracks if you need somebody.

MK: All right. We'll have to all get together if you come out and visit in Columbus.

KK: Well this has been great fun.

EL: Thanks so much for joining us.

MK: Thanks for having me today.

[outro]

On this episode, we were delighted to talk with Matthew Kahle of the Ohio State University about Euler's polyhedral formula, also known as V−E+F=2. Here are some links you might find useful as you listen to the episode.
Kahle's website
His paper about torsion in homology groups of random simplicial complexes
The Erdős–Rényi model of random graphs
Euclid's Elements, book 13, is devoted to the classification of Platonic solids. Also found herestarting on page 438.
The Jordan curve theorem has made a previous appearance on the podcast in our episode with Susan D'Agostino.
David Eppstein's website with 21 different proofs of Euler's formula. Thurston's proof is here.

Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knutson, professor of mathematics at the University of Florida. And today I am flying solo while I am at Texas Christian University in Fort Worth, where I'm serving as the Green Honors Chair for the week. And I've been given some talks and meeting the fine folks here at TCU. And today, I have the pleasure to talk with some of their students. And they're going to tell us about their favorite theorems and what they pair well with. And we're just going to jump right in. So my first guest is Aaryan. Can you introduce yourself?

Aaryan Dehade: My name is Aaryan. I'm a sophomore computer science major at TCU, and I'm from India. And I've chosen to go with the fundamental theorem of calculus.

KK: Okay. The fundamental calculus.

AD: Yeah.

KK: Okay, so now there are two parts of the fundamental theorem of calculus. Do you have a favorite part?

AD: So that's what I like about it. Honestly, I can't choose a favorite part because one of the parts is very important, and the other is interesting.

KK: Yes.

AD: So the first part, basically, it tells us the relationship between the integral and derivative. The second part tells us — like, basically, you can use that second part supply and to get solutions for questions in calculus. Does that makes sense?

KK: Sure.

AD: So what I like about this theorem is that it's not like other theorems where it has two parts. So it's kind of interesting how everything in calculus is based off of these two theorems. And if you didn't know what the relationship between the derivative and the integral was, probably you wouldn't be able to do anything with mathematics with it.

KK: Sure, maybe. Yeah. So you said one part was useful, and one part was interesting. Which part do you think is interesting?

AD: I feel like the first part is interesting, because it's almost intuitive. Like, you know that should happen, like the relationship between integrals and derivatives should be, like, one is an inverse of the other. And that is intuitive. So it's almost given. And it's interesting that we have to say that.

KK: Really, you think that’s intuitive? I mean, I’m not — I don’t know, when I first learned the fundamental theorem, I thought it was kind of shocking that somehow this this thing, this integral, which is sort of defined as in terms of these Riemann sums, somehow that went, you know, if you let x be the upper limit there, and you differentiate that function, you get your original function back. That’s intuitive? That’s amazing.

AD: Yeah, basically what it is, it's just an area of a rectangle. So I just thought of it as just decreasing the width of the rectangles, and then you get smaller and smaller rectangles, so you get the area. And then if you take the function at that point…that’s what I think.

KK: You’re cleverer than I am. I was just kind of dim, I guess, and I didn't think it was so intuitive. I mean, I saw the proof and believed it. But then, yeah, then the second part is how you actually evaluate integrals.

AD: Yeah, so that's what you use to calculate the area between two points.

KK: Yeah. Although I guess the problem is, right. So the theorem says that, you know, if you want to find the, the integral, the value of this definite integral, all you have to do — and our listeners can't see me doing the scare quotes — “all” you have to do is find an antiderivative of the function. Right?

AD: Right.

KK: Yeah. And then you spend all of Calc II learning how to find antiderivatives.

AD: Yeah.

KK: And even then, if I hand you an arbitrary function, you can't even do it, right? Like that's the sort of disappointing part of that theorem, is that most functions, you can't find a closed form antiderivative for. And so what do you do? But you’re a computer science major. You know what you do, right? You do it numerically, right?

AD: Yeah.

KK: Okay. Very cool. So you've known this theorem for quite some time, I guess.

AD: Yeah, I've done it since high school.

KK: So you love the theorem.

AD: I really, yeah, I do. Because when I was in high school, I used to sit at my dining table and study because I wanted to have some snacks at the same time.

KK: Sure. As we all do.

AD: I would just spend hours just doing sums on integrals, or basically just integrals. That was difficult at that point.

KK: Sure.

AD: And yeah, it was interesting, because I got used to that at some point. And then it got easier. And I just started liking the satisfaction of being able to do this. That was fun.

KK: Cool. All right. So on this podcast, we also like to ask our guests to pair their theorem or something. So what pairs well with the fundamental theorem?

AD: So as I said, I used to sit at the dining table and have snacks. And there's this really, really popular biscuit in India called Parle-G. And I used to have that with tea while doing my sums. So that was the highlight of it, that's why I used to look forward to studying, just for those biscuits.

KK: Okay, so I assume there's an Indian market in town somewhere, right? Can you get these?

AD: Yeah, I do have them in my dorm right now. Yeah. I have them every day.

KK: All right. So what are these called again?

AD: Parle-G.

KK: Parle-G. Okay. So I'll have to go to the Indian market when I get back home and see if I can find these because I am always on the lookout for a good new biscuit.

AD: Yeah, they’re amazing. Okay, so you should you should know, our very first episode of this podcast, aur guest, who was Amie Wilkinson, who is on the faculty at University of Chicago, chose the fundamental theorem as her favorite theorem. So you're in very good company, because she's a phenomenal mathematician. And okay, thanks.

AD: Awesome. Thanks so much.

KK: All right, up next, we have Toan. So why don't you tell us about yourself and what your favorite theorem is?

Duc Toan Nguyen: Okay. My name is Duc Toan Nguyen. People usually call me Toan. I’m an international student from Vietnam, and I'm a sophomore majoring in math and computer science

KK: Okay, great. So, favorite theorem. What’ve you got?

DTN: Yeah. So as my peer Aaryan, he chose the fundamental theorem of calculus, right?

KK: Yes.

DTN: But I want to bring another fundamental theorem in analysis, which is the mean value theorem.

KK: Oh, okay. So I have a theory, okay. I call the mean value theorem, the real fundamental theorem of calculus.

DTN: Yeah, me too!

KK: So why do you like it so much?

DTN: Yeah, I think I have the same idea with you of why it is called the real fundamental theorem. I think, because to prove the fundamental theorem of calculus, you need the mean value theorem.

KK: You absolutely do.

DTN: Also for analysis, the most popular and common tool in calculus, which is a derivative test, also has the mean value theorem behind it.

KK: That's right. Yeah, that's right.

DTN: So when I first so I first approached the mean value theorem when I was in high school. I took the Math Olympiad in Vietnam. So I had to prepare for that, and there is a section about that.

KK: Okay.

DTN: So it's called the Lagrange Theorem, it was kind of very fancy. Yeah. And it usually applies to — so you know, in the exam, we had some of the problems related to the continuous version, and f(a) minus f(b), something like that. Most of time, we used the mean value theorem. So yeah, it was kind of cool at the time, but I really enjoyed that until last semester, I took real analysis. So I could see the whole process was using the mean value theorem. That's why it can be taught in one lecture or one unit. Even today, this semester, I’m taking multivariate analysis. And it's also a very fundamental thing in proof, everything from differentiability on. It also even has a mean value theorem in it higher-dimensional space.

KK: So maybe we should remind our listeners what the mean value theorem actually says,

DTN: Oh, okay. So, let f be a function defined on an interval [a,b], so that f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). So the theorem say that there exists a point c between a and b that is not is not inclusive so that f(b)−f(a) is equal to f’(c)(b−a). So I think the mean value theorem, the name comes from the quantity f(b)−f(a) divided by (b−a).

KK: Yeah. Right. So the average rate of change over the interval is equal somewhere to the instantaneous rate.

DTN: Yeah.

KK: Yeah, that's right. That's how you prove the fundamental theorem, too, because it's just a telescoping sum when you write it out correctly. And the mean value theorem sort of pushes everything away, and then you're done.

DTN: Yeah, that's also my favorite part. Because it can tell you the relationship between the integral of functions and their derivatives.

KK: Right, right. So what do you want to pair with your theorem, what pairs with the mean value theorem?

DTN: Yeah, I want to pair with something really weird. Which is a phone with FaceTime. Okay, so I'm here. I study. I'm far from my home. My home is in Vietnam, which is on the other side of the Earth.

KK: Almost exactly opposite, right?

DTN: Actually it takes 20 hours from this time to my country time. So it's actually like, opposite, it's more. So yeah, and the FaceTime, why? Because through FaceTime, I can see what people in my home are doing and they also [can see me]. So it's kind of a bridge or relationship that connects what I'm doing here and what my family is doing there. And, you know, my family is always wants the best for me and hopes everything is good for me here. And me too. So that's a very meaningful thing for me.

KK: Yeah. That's great. And I'm glad that technology exists. When I was in college, the internet didn't exist. So you know, I had to call people on the phone and phone calls to Vietnam, I imagine, would be — I can't imagine what that would cost. It was expensive enough to call my girlfriend who lived four hours away.

DTN: It’s kind of more romantic. And you can give them a love letter.

KK: We wrote letters too. All right. Well, Toan, thanks so much. That was great.

DTN: Yeah. Thank you.

KK: Up next, we have Maiyu Diaz. You can introduce yourself.

Maiyu Diaz: My name is Maiyu Diaz. I'm a second-year graduate student here at the department of mathematics at TCU.

KK: Cool, and you have the best shirt. You win the shirt contest today.

MD: Oh, thank you.

KK: I’m actually kind of wishing you would like give me that shirt. [Editor’s note: How dare you say that and not send us a picture?!] All right, so you’re a second-year grad student here in the math department. Okay, great. And so yeah, so what your favorite theorem?

MD: So my favorite theorem is — I don't see it really formally presented, but Stirling's formula where n factorial can be approximated by n^n e^−n times square root of 2πn.

KK: Yes.

MD: That is my favorite theorem.

KK: Yeah. So that’s a really interesting approximation for factorials. So okay, where did you come across this?

MD: I first came across this, I want to say, when I was first studying the factorial back in grade school, and it was just more of like, looking up on a Wikipedia page, which was something that I really relied on when it came to writing essays for my English classes. And when I found out that it could also be used as a resource for mathematics as well, I thought, oh, factorial, let's learn interesting things about this. And that's where I found an example about Stirling's formula, under that. It wasn’t until much later that I was able to understand how that was derived, and there were a variety of proofs for proving Stirling's formula. One of them relies on probability distributions, which are looking at how the factorial works. And then there's another way of finagling a little bit with the integral formula for n factorial that comes from the gamma function, and that's another way of deriving Stirling's formula.

KK: Do you have a favorite proof of your favorite theorem?

MD: I do, but it's a very uncommon proof.

KK: Okay.

MD: The proof relies on a contour integral involving the derivative of the Riemann zeta function.

KK: Oh! I don't know if I know this proof, but what contour do you use?

MD: So you're going to go ahead and do a contour that is on the half strip, so you're just going to fix a number like σ and since the derivative of the Riemann zeta function, you just want to pick a σ just a little bit to the right of where it converges. So real part greater than one, and that's going to be a line integral from σ − i infinity to σ + i infinity, okay? And what you want to do is that you kind of want to start pushing that back a little bit so you can start picking up the residues of the derivative of the zeta function. You’re going to go ahead and possibly the non trivial zeros you could possibly hit, though. You’re definitely going to hit the pole at s=1. That's where the n log n minus n term comes from, when you look at the log of n factorial.

KK: Right, okay, sure.

MD: You keep pushing that more, and then you're going to be picking up the rest of the terms from the Stirling’s formula as well. It's pretty interesting.

KK: Yeah. All right. I haven't seen this proof. That's very cool. Yeah, okay. Cool. All right. So what do you think pairs well with Stirling's formula?

MD: I am going to say chicken tikka masala.

KK: Chicken Tikka Masala. Okay. I do like chicken tikka masala. What in particular makes you want to link those two things together?

MD: So Stirling's formula, I would say is a little bit spicy. And I underestimated it at first.

KK: Right.

MD: Because that was just something I would not have been — I just, if you’d asked me if the formula was intuitive, it’s just like, absolutely not. Where's where does this e term come from? Where does the square root of two pi come from?

KK: Sure.

MD: It’s not until you start familiarizing with the proof more. It's like, okay, at this point, it's not that I'm used to that, I’ve just seen this too many times.

kk: Sure. Sure. Yeah.

MD: But I want to say it’s spicy because of a paper, a PhD thesis, that was published in 2014 by Matthew Lamoureux, whose advisor is Keith Conrad, who wrote a paper on Stirling's formula, just devoting a series of notes on it, you can see all the things he has compiled and I like reading through his notes. And this PhD thesis, he was looking at a generalization of the factorial, which is the factorial for number fields. So instead of looking at the derivative of the zeta function, what it was looking at was a modification of the derivative of the Dedekind zeta function because they had a function defined over number fields. And that's the same technique as well. You want to keep pushing it along the line, be able to pick up non-zeros, possible poles, and it just more or less the same outline as well, and spicy because number one, there is a ton of information coming about the factorial just from the location of zeros and poles of these zeta functions. I was like, Whoa, this is some pretty advanced stuff. I don’t want to be messing with this. But then it's also really delicious because one, I think chicken tikka masala is very delicious.

KK: It is. Agreed.

MD: But also delicious in the context of these formulas, because the approximation formula for factorials is reliant on the poles and zeros of these zeta functions. So it's like, whatever I want to know about on the left side over here, the approximation for the factorials I'm looking at, all I’ve got to know is the information about the zeros and poles of these zeta functions.

KK: Right.

MD: That’s the spicy and delicious part.

KK: Very cool. All right. That's a good pairing. I like that. Cool. Thanks, Maiyu.

MD: Thank you.

KK: Up next, we have Hope Sage. Why don’t you introduce yourself?

Hope Sage: Hey, I'm Hope. I'm a junior physics major at TCU.

KK: Cool.

HS: My favorite theorem is Bell's theorem.

KK: Okay, I don't think I know this theorem.

HS: Okay. It’s kind of Physics-y. It's from quantum mechanics. And John Bell wrote it as a response to the EPR [Einstein-Poldosky-Rosen] paper, which is kind of a famous paper in quantum physics, where it talks about how quantum physics is probably incomplete, and there's probably some hidden variable that's underlying it. And Bell uses this inequality to calculate the probabilities based on what you would expect classically. So if the quantum particles are not entangled, then you would get this expected probability. And then he shows that experimental results kind of conflict with that. And so the underlying assumption is that from that, local realism isn't a thing. So then the universe is like, super wacky.

KK: I think we knew that, right?

HS: Yeah.

KK: So okay, so Alright, so my quantum mechanics is — well, calling it rusty would be, like, an insult to rusty things. So the probability of what? The state that a particle is in?

HS: Yeah, so whenever you have two entangled particles, one might be spin up, one might be spin down, okay? And you can run an experiment. They have a beam splitter and two detectors, and you measure the different states. It's kind of a traditional example. And the probabilities depend on the angle that everything is situated at. So you have nine, and then you were to take all those probabilities, he basically proves this inequality that just shows that mathematically, it can't be possible for there to be hidden variables, which means that things are paired and they would have to be, like, communicating at faster than the speed of light, which doesn't happen. So then there are all these theories about what could theoretically be the underlying nature of the universe from that okay, in different interpretations.

KK: Okay. So which is your favorite interpretation of what might be going wrong here? Or right, whatever the right word is.

HS: There’s this interpretation kind of extended from this called the many worlds interpretation of quantum mechanics. I don't know if I’d necessarily say it is the most likely to be correct. But I think it's the most fun one. It's also fun because you can read cool science fiction books about it. Like, every action you take, there's a different universe, different paths. I think it's kind of fun.

KK: Sure, right. So so right, so like, right now, what we're doing, we could, like, split into any number of paths. And there are all these weird different outcomes that could happen, depending on whether or not some quantum state is what it is or not.

HS: Yeah, basically. One of my favorite books is Dark Matter. It's by Blake Crouch. And it's about, like, every action that you take, there’s a different universe. And then there's infinitely many possibilities based on every single decision you make, which is kind of interesting, because every decision that you make does create a different next possible decision.

KK: Sure.

HS: But yeah.

KK: Okay. Well, all right, so our minds are getting blown. All right, what pairs well with this theorem?

Well, sometimes it's called “spooky action at a distance” and so I was going to go with Halloween candy. Because spooky.

KK: Yeah, yeah. All right. Okay, so what what's your favorite Halloween candy though?

HS: Probably Reese’s.

KK: What? Okay, I have strong Reese’s opinions. So which, like the full size or the miniatures or what?

HS: Okay, a Reese's Peanut Butter Cup, but the dark chocolate version.

KK: Okay. All right. I can respect that. I am team miniature. I think that's the correct ratio of chocolate to peanut butter. But the dark, I get it, I understand. Okay. All right. That was great. Thanks, Hope. Jonah, why don’t you introduce yourself?

Jonah Morgan: All right. Well, I'm Jonah Morgan. I'm a freshman engineering major here at TCU, and my favorite theorem is Gödel’s incompleteness theorem.

KK: Gödel’s incompleteness theorems. So that's more than one theorem. All right, so let's remind our listeners what what at least one of them is.

JM: Sure. So the first one: The first of Gödel’s incompleteness theorems is effectively any — I'll call it interesting, okay — any sufficiently interesting or complex set of axioms, it fundamentally has theorems or statements that cannot be proven, but which are true.

KK: Cannot be proven inside the system, right?

JM: Yeah, cannot be proven inside the system. Right.

KK: And okay, you hedged around, but I think “sufficiently complicated” just means, like, you can do arithmetic.

JM: Yeah, you can add numbers. Because if you just can't do anything, then well, you can’t say anything.

KK: Okay. Yeah. So that's the first one. What's the second one? I think I don't even remember the second one.

JM: So the second one was, I think — I'll make some background. I just think background’s kind of fun to know.

KK: Sure.

JM: His first theorem kind of says, Okay, well, if you have a set of axioms, you can have effectively a statement that says this statement cannot be proven by the axioms. So if you have that statement, then well, if that statement is true, then it's a true statement within the system that cannot be proven by the axioms. But if it's false, then it is a statement which cannot be proven. Or it is a false statement, which cannot be proven. I'm a little rusty on that aspect of it. But effectively, the idea is, so there's this weird statement that you can have in any system, which makes it he says incomplete, where incomplete is the word for it. But then, so mathematicians are like, “Okay, well, we want to prove things. And you gave one example, it's a bit of a weird example. I don't want to prove that a statement is unprovable.” So then he has a second theorem that says, well, also there are true statements which are unprovable which we cannot prove or unprovable.

KK: Yes.

JM: And so that's the second theorem. And that's like, okay, so you can spend your life working on a theorem or working on a problem, and then you can't even know whether or not you can know the answer to this problem within within the set of axioms that you're working with.

KK: Right. So it's hopeless, in some sense. Yeah. You can't fix this issue.

JM: No. It’s — some people say math is broken. But really, it just incomplete. There are certain things —

KK: Yeah, I mean, I think it caused a crisis amongst certain elements of the mathematical community, but I'm with you, I just sort of view it as, well, okay, so there are unprovable statements. It doesn't mean the bridges that we build are going to fall down,

JM: Right. Everything we have proven still stands. You can still prove a lot of things.

KK: All right, still lots more to prove. Where did you come across this?

JM: I think some time in grade school, middle school, high school, I got really into just watching videos about mathematics. And at first they were just little conjectures and little fun things and then Gödel’s Incompleteness Theorem stuck out because it's like, I was diving into the world of math for the first time. And there are so many things that you can prove, and even these, like, crazy things that I never thought were provable, or like things that are so complex. Fermat’s last theorem took 300 years to be proven.

KK: Right.

JM: And I don't understand any part of that proof, to be honest with you.

KK: Same.

JM: But we did it. And it took a long time, but sort of my idea after seeing all of this was that, well, anything can be proven if you have a sufficiently — or maybe we're not smart enough to find the proof, but everything should be provable. Mathematics, it’s a language, you should be able to explain things in that language. And then Gödel’s incompleteness theorem says no. And also, there are things that you just — it kind of changes your perspective on that.

KK: Right.

JM: And so then you get questions like, well, you know, the Riemann zeta hypothesis, one of the most famous unsolved hypotheses. And a lot of people see this and they're like, Oh, well, does this mean that this this million dollar problem, one of the millennium problems, could just be unsolvable? And we can't even know that it's unsolvable? And I think that was the first thing I looked up when I heard about this theorem. And something that was even more interesting to me was that if the Riemann zeta hypothesis is false, it’s provably false. Because it is equivalent to saying that there exists some number on the real part one half line, yeah. So you can write an algorithm, and given infinite time, you will find sure if there is one, you will find it. So it is provably false, which also means that proving that the Riemann zeta hypothesis cannot be proven means that it must be true.

KK: Okay. Yeah.

JM: So I can't give you the formal explanation on that. I don't know what it is. But the general idea is that you can prove that something is true by proving that you cannot prove it.

KK: Yeah, that's a little mind-twisting. But I can see why this would appeal to you, as you’re coming into your own intellectual being sort of state and moving out of being a kid. Yeah, that's really great. Okay, so. So what do you think pairs well with the incompleteness theorems?

JM: So, this isn't sponsored, but GrubHub, or Uber Eats or whatever.

KK: Okay.

JM: Because, occasionally, I order food there. It's easy, it's convenient. Most of the time, you get what you ordered. And sometimes your driver takes a nugget and you don’t — and, you know, that's gone. And sometimes he just doesn't show up at the door, and you're left wondering, where's the stuff that I paid my money for? And I think the feeling is similar there that you can you can pay for something, like you can spend your time working on this theorem, and it just unprovable, and you'll never know where it went or where it goes. And also you can pay for your food and just have no idea where.

KK: Yeah, well, that's a good pairing right there.

JM: Thank you.

KK: Thanks a lot, Jonah.

JM: Thank you.

KK: All right, up next, we have Anna Long. Anna?

Anna Long: Yeah, so my name is Anna. I'm a senior math and French double major. Actually.

KK: Nice!

AL: I’ll stick with English for you.

KK: Je ne parle pas bien le français.

AL: Tres bien!

KK: So I can say I don't speak the language really well in several languages.

AL: Well, that’s all you really need.

KK: That’s right. It was only that and “toilette,” you know, yeah.

AL: Yeah, so my favorite theorem I picked is the invertible matrix theorem from linear algebra.

KK: Okay.

AL: And it's really a pretty big theorem. It's 24 equivalent statements for a square matrix, that'll call A. So just a few of my favorite little statements in there. So obviously, we have that the matrix is invertible, that the columns then form a linearly independent set. So there's always a solution. And then that, for it to equal zero, it's only the trivial solution, that your vector is zero. That it has n pivot positions, or that it has full rank. And then that the linear map is both one-to-one and onto, and that the determinant is nonzero and that zero is not an eigenvalue.

KK: Right. And that's only, like, six of the equivalent conditions. So yeah, if you open up a linear algebra book, there will usually be at some point, some page where they list all these, I had forgotten there were 24. I can probably — so I'm teaching our senior-level analysis course this year. And part of it, we do some stuff with operator theory, and they remind, in our text that we wrote ourselves, they use of 11 of the equivalent conditions, but not all of them. So yeah, I can't imagine. I don't even think I know what some of the other ones are. I'm sure I would if you told me.

AL: A lot of them are, like, jumbled together. So, like, n pivot positions and full rank I've seen defined as two different things, but they’re really the same. But essentially, because of the theorem, they're all the same.

KK: Yeah. So yeah. So why do you love this theorem so much?

AL: I just love it because it's so useful. I’ve had two linear algebra classes now, and it just makes life so much easier trying to prove that any various things in class. So yeah, it's just great being able to find the easiest statement and prove that one, and then you just know all of them are true.

KK: Right. Yeah, I do like those things when, like, 50 things are equivalent. that's really nice. Yeah. Okay. So I guess you learned this in your linear algebra course. You had a second linear algebra course? What’s the second one?

AL: I’m in applied linear algebra right now.

KK: Okay. All right. So you're doing, like, singular values and things like that?

AL: Yes, we are.

KK: Okay. All right. So that's super useful stuff. Linear algebra, of course, I think is one of those things that we don't teach enough of. And basically, any problem in math comes down to either making some estimate, like analysis, or some linear algebra problem, it seems to me. so yeah, the more you learn, the more you know, then the better off you'll be. So you're a senior. What’s next for you?

AL: Graduate school.

KK: In what? In French?

AL: No. In math. Yeah. So I haven't decided on a school yet, but I'm looking at PhD programs in applied or computational math.

KK: Excellent. That’s great. Well, good luck to you. So what pairs well with this theorem?

AL: Yeah, so I picked chicken tortilla soup, mostly because it's my favorite soup. But you kind of have all sorts of different things piled in there. You've got your spices, you've got your chicken, you got your, you know, pieces of tortilla, you've got maybe chives or your different little vegetables in there. So you have a whole bunch of things that may or may not look very similar or different to each other. But you get one scoop of it, and you have the whole thing. So it's kind of like a 24-for-one deal.

KK: It’s like 24 equivalent soups in one.

AL: Right!

KK: Okay. Very cool. Yeah. So are you from Texas?

AL: I’m from Oklahoma.

KK: Okay. So this region, yeah, that sort of makes sense. That’s a popular sort of soup. Yeah. Okay. All right. Excellent. Well, thanks so much, Anna, that was great.

AL: Thank you.

KK: Up next, we have Matthew Bolding. Matthew, welcome!

Matthew Bolding: Hey there. Thank you. Yeah. So my name is Matthew. I am a senior dual degree student for mathematics and computer science, and I'm from around here as well.

KK: Okay. Cool. All right. So what's your favorite theorem?

MB: My favorite theorem today, or really, I guess all time, is the four color theorem.

KK: The four color theorem. Oh boy. Okay.

MB: Are you familiar?

KK: Oh, yeah, I am. So there's a lot to be said about this theorem. So yeah, tell us what it is, and then we'll unpack it.

MB: Sure. So I suppose in the most dry mathematical language, the four color theorem states that the chromatic number for a graph, for a simple planar graph specifically, is no more than four. And I guess there are some things to unpack there.

KK: Yes. Right.

MB: So a planar graph is a graph that can be constructed, drawn, if you will, such that no two edges cross one another. And a simple graph is one that does not have any self loops. And I guess the other part of it is, what's a chromatic number, right? And so a chromatic number is essentially the smallest k for which the graph is k-colorable. And then that also takes us down the rabbit hole with what is k-colorable? So a k-coloring of a graph is an assignment of at most k colors to the vertices of the graph in such a way so that no two adjacent vertices are the same color.

KK: Right? Okay. So most people might know this in terms of maps.

MB: Yes, that is correct. That actually what got me interested in it. It seems so deceivingly simple, I guess, you know, four colors, all you need are really at most four colors. You could do it in three or two, depending on on the map or graph.

KK: Sure.

MB: But, you know, you can ask people, what do you think? How many colors you think you might need to color this graph under these conditions and constraints? Oh, six, seven? No, you only need four.

KK: Right.

MB: And although I'm no cartographer, you know, I've never really colored a map maybe since pre-K, I think it is just so interesting, and sort of out of the blue, that you only need four colors. And with that, I also think it is really interesting that the proof that the chromatic number is no more than four hasn't been proved by humans, by hand.

KK: Right.

MB: And we've had to rely on computers to facilitate that proof. And being a computer science, or within the computer science field to study, I think that's really, really interesting.

KK: I thought that was part of part of your motivation here. I mean, so although, yeah, recently there, some people announced a proof by hand of the four color theorem. And it's wrong.

MB: Really?

KK: So I mean, anybody who's tried to prove this thing just by hand has come up short. And, you know, so the question of the maps, right? So, for a map, it's like, you know, you don't want two states that share a border to be colored the same, you know, and you can draw examples where you need four, but it's interesting that you can always do it with four, but then it does turn into a graph or a question, because how do you how do you create a graph out of this? Well, you stick a vertex for each state, and you join them if they share a border, and now you've converted it to a graph theory question. So that's how they come along.

MB: Exactly. And actually, I found this, or was presented this theorem in graph theory last spring. And I mean, I really — of course, with a computer science background, I mean, I just, you know, jumped in headfirst. I thought it was the coolest class. You know, it maybe doesn't have the same rigor as, like, really analysis or something.

KK: Oh no, it’s hard.

MB: Well, don't get me wrong. There are some difficult concepts. You know, I guess, with a computer science mindset, you know, I mean, of course, not every topic, you know, had roots in computer science, but maybe with Huffman encoding, or shortest path algorithms, you know, it was just so interesting and fascinating.

KK: Yeah, graph theory stuff is vital in computer science. I mean, it's everywhere, and having good algorithms for that is really important. You know, decision trees and all these kinds of things that you need to know. Cool. All right. So, yeah, but it's true that the first proof was given in, what, 1976?

MB: Around there.

KK: And yeah, and basically, it reduces to some couple hundred special cases that you just check. And then you get a computer to check it. And yeah, so for mathematicians, that's unsatisfying, right? We would just like a nice clean, wordy proof that works instead of relying on computer code. But I mean, I'm okay with it personally.

MB: I am too, but according to the Wikipedia page for this theorem, there are still many doubters. I guess we just have to prove it by hands.

KK: I guess. So that's how it goes. All right. So what pairs well with the four-color theorem?

MB: I really struggled trying to think about something that went along with this, but I landed on something that you could actually, you know, show the four color theorem with, and that would be Skittles.

KK: Okay.

MB: You know, you could lay them out all flat, you know, make it a planar collection of Skittles, basically. you could arrange the Skittles in such a way that no adjacent Skittles share the same color. Of course, there are more than four colors in a Skittle pack.

KK: Right.

MB: I would think it's been a while since I've had Skittles.

KK: Yeah, too sweet. Although, so the five color theorem is really not so hard to prove. Apparently, I've been told. I think I may have even read the five color theorem proof. It's not so bad. But four is tricky.

MB: I looked back at my graph theory notes, and we worked from a chromatic number no greater than six to five. And then we sort of just had a blank, you know, statement. Well, you can prove that the chromatic number is no more than four. But right, the ones for five and six aren't, too, too — I mean, compared to having to do it on a computer.

KK: Right. Right. Compared to people, you know, not necessarily believing the computer proof, right? Yeah. Yeah. We’re convinced about five, so we’re good. All right. Well, Matthew, that was great. Thank you.

MB: Thank you.

KK: All right, up next we have Brandon Isensee. Brandon?

Brandon Isensee: I'm Brandon Isensee. I'm a math major at TCU. I'm a senior. I'll be graduating this semester.

KK: What’s next?

BI: Grad school at Rice University.

KK: In math?

BI: Yeah. Computational and applied mathematics.

KK: Okay, that’s That’s great. It’s a terrific university. You're going to have a great time there. Well, I mean, grad school is what it is. It's fun and work and all those things. But it will really a great experience. All right. So, what’s your favorite theorem?

BI: So my favorite theorem is called Sharkovskii's theorem.

KK: Sharkovskii's theorem?

BI: Yes.

KK: Okay.

BI: Have you heard of it?

KK: I have. But let's tell our listeners.

BI: So kind of the theoretical way of saying it is that the theorem relates to discrete equations that are in, it’s one-dimensional discrete equation, so you only have one variable, and if a certain period exists — if a certain periodic orbit exists — that implies the existence of other periodic orbits. And if there's a three-cycle in particular, that implies the existence of all the other cycles

KK: Yes.

BI: And so that's more of the theoretical way of saying it. But if we put this in more concrete terms, if you have an equation that models a population over time, and it's discrete time, so it's years 0, 1, 2, 3, etc. So this equation tells you the population values for each year, right? And say there's a population growth parameter within this equation that you can vary, so we'll call it K. And this parameter tells you how fast the population is growing. And so say your growth parameter is two. And for this growth parameter, no matter which population value you choose, your population ends up oscillating between three different values across time.

KK: Right.

BI: So that's the long term behavior. So as time goes on, your population oscillates between, say, four individuals, five individuals, six individuals, and it keeps repeating. So it's 4, 5, 6, 4, 5, 6, 4, 5, 6. So if that's the case, then that implies that there is a four-cycle. So maybe it's 8, 9, 10, 11, there's a five-cycle, there's a six-cycle, there’s a 1-million-cycle, all the other cycles exist.

KK: Right.

BI: And so it's quite interesting, because when you look at particular examples of equations, like the discrete logistic map, and you look at where the three-cycle exists, you only see that three-cycle. So you may be wondering why? Why am I only seeing that three-cycle and not the four-cycle or the five-cycle? That's because Sharkovskii's theorem tells you that these cycles exist, but doesn't tell you whether or not they're stable. And so when you see the three cycle, it's stable, because that's where your populations are oscillating between. But when it's unstable, well those unstable cycles repel the population values away from them, and it ends up settling at that three cycle. So it's almost like there's an infinite number of fixed points, if I'm understanding this correctly. It’s like there's an infinite number of fixed points where you see the three-cycle, but all of them are unstable, except for the three-cycle, because that's what you're seeing on the graph.

KK: Right. Yeah. So sometimes this theorem is stated as “period three implies chaos.” Right?

BI: Right. So there, so there was actually like a difference between those two. So Sharkovskii's theorem is the one that's stronger, because that tells you exactly which periods imply the existence of other periods.

KK: Yes, right.

BI: And so period three, that relates to the three-period, right, the three-period implies everything, but Sharkovskii's theorem tells you exactly which periods imply the existence of others.

KK: Yeah. And if I remember right, he puts some weird order on the natural numbers.

BI: Yes. So the order the order is, if you're doing this in rows, the first row is your odd numbers, so 3,5,7…. Your second row is your odd numbers times two. And then the next row is your odd numbers times two to the second power.

KK: Sure. Right.

BI: So it's odd numbers times powers of twos. It’s very interesting. It took me some time to understand the order, but now I get it.

KK: And then, like, one is at the end or something, right?

BI: Yeah. Your two-cycles and such, you know, your periods of twos are all the way at the bottom.

KK: And then this is very important for discrete dynamics, right? It's just kind of the whole story. That's very cool. It's very, very cool. All right, so what do you think pairs well, with Sharkovskii's theorem?

BI: So maybe it's very a superficial connection.

KK: That doesn’t matter.

BI: There's a story with Alan Turing. He's the mathematician — for those that don't know, he solved the Enigma code during World War Two, the Germany Enigma code, and because of that, I forget the exact estimates, but it's like at least a million lives were saved because of that in, like, two years.

KK: It was vitally important.

BI: That shortened the war by two years. And so there's a story with Alan Turing where he rode his bicycle, and after a certain number of revolutions of the bike wheels, the bike chain would fall off. But instead of him just fixing it, he would just count the number of revolutions as he's riding the bike, and right before the bike chain would fall, he’d get off the bike and he would just put the bike chain back on.

KK: I wonder if that’s true.

BI: I suppose so.

KK: It’s a good story either way. Yeah. Okay. That's that's a good pairing. Excellent. Thanks.

BI: Thank you.

KK: All right, and our last willing volunteer today — I think they were all willing — is Julia Goldman.

Julia Goldman: Hi. Yeah, I'm Julia. I'm in my first year of grad school here at TCU.

KK: In math?

JG: Yes.

KK: Okay. How do you like it so far?

JG: I like it so far.

KK: Math’s pretty cool.

JG: I think so.

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

JG: So my favorite theorem today is Brouwer’s fixed point theorem.

KK: Brouwer! All right, good. Finally a topology theorem. Good.

JG: So last week, I was trying to come up with a theorem talk about, and one of my professors suggested this one. And when I was going online, and reading it and learning about it, I was looking at all these proofs of it that seem fairly technical, and I’d probably want to take a topology class to really get into it.

KK: Sure, right.

JG: But the theorem itself is, I think, very understandable. And I came across so many of these cute little fun real world examples that make the theorem pretty explainable to anyone of any math background, and I really appreciated that aspect of it is approachable.

KK: Okay, so what's the theorem? Let’s remind everyone.

JG: Oh yeah. The theorem is for any continuous function of a convex compact set onto itself, there's going to be at least one fixed point.

KK: Right. One point that doesn't move.

JG: Exactly. I think that’s really fun. There’s a couple examples. Like if I had a map of Fort Worth right now, and I laid it on the floor, there would be at least one point on that map lying directly on top of the point it's supposed to represent.

KK: That’s right.

JG: That’s kind of fun.

KK: Yeah. That's a good example.

JG: Then my other favorite example is a cup of tea. You stir the cup of tea. When you're done stirring, there’s going to be one little bit of your tea that's in the same spot as when you started stirring.

KK: That’s right. Okay. Are you a tea drinker or a coffee drinker?

JG: A little bit of both. Maybe it's because I'm not a topologist, but I read that example, and I just thought, I feel like I could stir my tea enough.

KK: Sure.

JG: But Brouwer says I'm wrong.

KK: That’s right. That's right. Okay. So maybe you don't know enough topology to prove this yet? Did you find a favorite proof that sort of made sense to you?

JG: Not a favorite. I just glanced over it.

KK: Right. So the algebraic topology proof involves, like, something with the homotopy groups, or homology groups and things like that. So that's kind of weird. There are sort of analysis-type proofs. So we just talked about dynamical systems a little bit. So if you just pick any point, and you start iterating the function, right? Just keep pushing it around, then eventually it will converge to a fixed point. Well, some subsequence of it will, because you're in a compact set. So that sort of analysis thing. So that's another way to think about it. But yeah, this is a popular theorem among topologists. You know, we all really dig this theorem a lot. It's probably one of our favorite examples. It is a fan favorite. Absolutely. Yeah. Okay, good. I like the map example, too, because that's really illustrative. Okay. So what do you think pairs well with Brouwer’s fixed point theorem?

JG: Well, I wanted to pair my favorite theorem with my current favorite TV show, which I'm only a little bit embarrassed to say is a reality show called Love Island.

KK: Okay. I've heard of it. I have not watched it.

JG: I think they must have some topologists on set there because I think the show itself is kind of an example of a theorem if you stretch some definitions a little bit.

KK: Okay. Let’s hear it.

JG: So if you're unfamiliar with the show, it’s a dating show. The very first episode, all the participants are put into couples. And then there’s, like, 65 episodes of just, like, fighting and breaking up and getting into other couples, whatever. From the seasons I've seen, at the end of the show, there's always at least one couple that ends up back in their original pairing.

KK: Right.

JG: So thinking of the participants as our convex set, and all the show drama as a function, then there’s your example of the theorem.

KK: Yeah, it's funny how there are always two people who were like, you know, we were we were right all along.

JG: Exactly.

KK: Do they do it randomly? I’m sure the producers don’t actually do it randomly.

JG: I mean, they kind of mix it up. They let the girls choose the first time, or the boys.

KK: All right. Okay. Well, I’ll have to check this out.

JG: Great show. Highly recommended.

KK: Okay. All right. Excellent. Okay, well, thanks, Julia.

JG: Thank you so much.

[outro]

In another Very Special Epsiode of My Favorite Theorem, Kevin had the privilege of asking a group of nine TCU students about their favorite theorems. We loved the variety of theorems and pairings they picked! Below are some links to more information about their favorites.

Aaryan Dehade led off with the fundamental theorem of calculus.

Duc Toan Nguyen's favorite is the mean value theorem, which some would argue is the real fundamental theorem of calculus. It was also a hit with our past guests Amie Wilkinson and Aris Winger.

Maiyu Diaz shared Stirling's formula for approximating factorials.

Hope Sage chose Bell's theorem from physics, which was a response to a paper by Einstein, Podolsky, and Rosen.

Jonah Morgan shared his love for Gödel's incompleteness theorems, which also came up when we talked with math students from CSULA last year.

Anna Long chose the invertible matrix theorem, a behemoth of a theorem that gives scads of ways to show that a matrix is invertible.

Matthew Bolding highlighted the four-color theorem. 

Brandon Isensee chose Sharkovskii's theorem, which was also the favorite of past guest Kimberly Ayers.

Julia Goldman finished out the episode with a perennial MFT favorite, the Brouwer fixed point theorem. We have talked about it on past episodes, most recently with Priyam Patel. See if you agree with Julia that it is the mathematics underlying the TV show Love Island!

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Evelyn Lamb, one of your co-hosts, coming to you from snowy Salt Lake City, Utah, where I feel like I've said that the past few times we've been taping. Which is great, because we really need the water. It is beautiful today, and I am ever so grateful that the life of a freelance writer does not require me to drive in conditions like this, especially as someone who grew up in Texas where conditions like this did not exist, and so I am extremely unconfident in snow and ice. So yeah, coming to you from the opposite side of the weather spectrum is our other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. It's true. It's the opposite end of the spectrum, but hey, you know, I was putting up my Christmas tree the week before last and I was sweating. So this is my reality.

EL: Yeah.

KK: It’s hard to get in the mood, you know, you put on the Christmas music and you you get the tree out of the attic. And then I'm in, like, shorts and a t-shirt and sweating.

EL: You can sympathize with Australians, who have to deal with that every single year.

KK: That’s right. That's right. Yeah. So anyway, we're looking forward to a nice holiday. My son's going to come home after Boxing Day because he has a part time job at a bookstore in Vancouver and his boss said no one gets Boxing Day off.

EL: Yeah, that's that's a thing in some places.

KK: In the Commonwealth. I think it's a big thing. Right? So yeah, he'll be home on the 28th. So we're looking forward to that. But anyway, anyway, this will be after this will be after the holidays when people hear this anyway. So they'll go, gee, I wonder how that went?

EL: Yeah. Waiting with bated breath for updates about your son’s Boxing Day experience.

KK: That’s right. That's right.

Yes. Well, today we are very happy to have on the show Cihan Bahran, coming to us from I don't know what kind of weather. So yeah, could you introduce yourself and tell us about the local conditions?

Cihan Bahran: Yeah. Thanks for having me. I am joining you from Ankara, Turkey, which is the capital of Turkey in the middle. So it's a continental climate, I would say. But it has been rather mild. We haven't had any snow yet or really anything that close to freezing temperature. So yeah, it's chilly, but yeah, I like it.

EL: Yeah. And so what what kind of math are you interested in

CB: Right. So I am interested in representation theory, especially with functorial methods, and I am doing a postdoc here about that at this at this time. So actually, maybe I'm at a little bit of a disadvantage in that the theorem I will share is not necessarily directly from my expertise, so I'm not really, maybe on top of the literature or the methods, but I thought I would pick that because I find it really interesting.

EL: Sometimes, honestly, that could be a little better, because we are also not experts in that.

KK: That’s right. Yeah.

EL: But yeah, and you run a Twitter account, and I meant to look up the exact — is it called some theorems?

CB: Yeah, it's called some some theorems. The username is something like Cihan posts theorems [Editor’s note: It’s @CihanPostsThms] Okay, let me talk about that a bit. So I guess it goes back to maybe 2020 or something, not this account, so that was the pandemic time and for me, maybe psychologically a difficult time that I was seeking out somewhere to connect with the math world. And I found initially a Facebook page called Theorems. And it is it is still running, I guess. But I started posting there. And I had a lot of, like, some bits of knowledge about some interesting theorems that I would, like, share with my friends. And it became like, I was almost daily posting, like the group became dominated by my posts, to the point that people started asking, like, what are you really doing, et cetera. And then maybe since last year, I've been more on Twitter, and I posted some of these on my personal Twitter account. But then for some reasons, I had to make my personal account private. And at some point, I thought I might repost these things that I have had collected, because that group in Facebook was actually a private group, not everyone can see it before joining. And I thought I would post those on Twitter, and I find it, like, when it gets some responses, it's like a dopamine hit for me.

KK: Sure.

CB: And I was actually almost aggressively posting in the summer because I had all this sort of backlog. And at this point in time, I have posted most of the past stuff, and I post much less regularly. When I see something interesting, I post them to the to that account. And I suppose how that's how I am maybe known in math Twitter-verse.

EL: Yeah. And so as a person with with much knowledge and love for theorems, what is your favorite favorite zero?

CB: Okay, so I don't know if it's my favorite, but at least for this episode of My Favorite Theorem, the theorem I would like to share is the so-called — well, so there's this problem, and the theorem says that this is algorithmically undecidable. So what's the problem? The problem is called matrix mortality.

EL: Which is a really an inviting name.

KK: It’s a great name, right? It sounds like a video game or something. Yeah.

CB: Yeah. So, in the most general sense it asks, so the input is a finite list of square matrices of the same size. And the the the decision problem is whether a product of these things in some order, possibly with repetitions, could be ever zero or not. So, if an algorithm would say yes or no to each such collection. And I think at first this was shown to be undecidable for already 3 × 3 matrices in the 70s. And then there were some further developments as to because of course, if I give you one matrix, then matrix mortality becomes is this matrix nilpotent, and you can determine that by the characteristic polynomial, so that is decidable. So how few, how short can the list get and remain undecidable? I think for 3 × 3 matrices, it has been shown in 2014 or so that six 3 × 3 matrices, the problem is undecidable. So like A, B, C, D, E, F, F, that’s six 3 × 3 matrices. So that's like, what, like 54 entries of integers? These are all integer matrices, by the way.

KK: Okay. I was going to ask that.

CB: Snd then the question is, is some product ever zero or not? There can be no algorithm answering that for every possible input. And if you make the, if we allow the matrices to be a bit bigger, there is a version which says that when you make the size 15 × 15, it is undecidable for even two matrices. So just, like, two matrices of size 15, A and B, the decision problem, is ever a sequence of A's and B's equal to the zero matrix? And such an algorithm cannot exist, it's undecidable. It's rather striking.

EL: Yeah, I guess — I'm actually a little more upset about the six, 3 × 3 than the two 15 × 15’s. Because honestly, I just imagined trying to write down the entries of a 15 × 15 matrix, and I give up maybe 30% of the way through, I'll just, okay, whatever.

CB: Well, there is still a gap in knowledge. So let me talk a bit about what's known. For I think, two, 2 × 2 matrices, just two of them, it has been maybe recently shown that that is decidable. So but when the list is, when you have three or more matrices, I believe open. Also, I believe it's still open, whether if you're given, like, five, 3 × 3 or four the lowest boundary we know is six, although from from the development, you might — I would guess that it will remain undecidable for even two 3 × 3 matrices. But that's, I think, unknown at the moment.

KK: So once you show that it's undecidable for a certain, so for six, 3 × 3’s is undecidable, so that means it's undecidable for six of any size larger than 3 × 3, correct?

CB: Yeah.

KK: Because it sort of stabilizes, right? You can put those inside of the next size up by just sticking a one down on the lower corner with a block.

CB: Exactly, you can even you can even pad them by zeros, right?

KK: Sure. Sure.

CB: The mortality problem will not change once you've artificially made your 6 × 6 matrices into 10 × 10 matrices by writing zeros everywhere else.

KK: I see. So the question is for a fixed n, can you what's the minimal number k for which it's undecidable? Right?

CB: Yeah. So there are two parameters, how many matrices and the size of the matrices.

EL: But I guess there's a chance that it's three for 2 × 2 and two for everything else.

CB: Maybe. If I were to bet, I might bet that 2 × 2 is special and would be decidable always, and like the 3 × 3 introduces — but that's just a hunch. I don't really know much about how these things are done, because, like — I mean, I did look a bit to the into the two 2 × 2 matrices, and the algorithm is by computing some some eigenvalues or such, and I and 2 × 2 is so small that I would guess that is enough information somehow, but I don’t know.

KK: Now, I'm just thinking about this, right? This is sort of different from — so I would think of this in terms of the group generated by these matrices, but that's not at all what you're doing, right?

CB: Yeah, it's more like a monoid because it becomes zero. Also, I would like to, for people who know about the word problem, this this reminds people of the word problem for groups. And there is, of course, a relationship, but I would object to the argument that, “Oh, because the word problem is undecidable, that’s not so surprising.” And my objection is that we can always multiply the matrices. A specific instance is always decidable. We can just multiply them and see. For the word problem, there are even specific instances, which remain, like, is deciding whether a word is trivial can be made into a specific presentation and remain undecidable already there. It’s not like because you have maybe relations between the words, you don't know how to change your word into something. But with matrices, we can always, we can multiply like multiplications doable. But when you when we allow, is there ever zero among arbitrarily long multiplication that that is where the problem is. I think the word problem, the problem arises earlier than that.

KK: And that direct analogy with the word problem, you'd be looking for products where you get the identity, right, as opposed to zero. So I guess you don't want any of these matrices to be invertible. It's allowable, I imagine. But but if you have an invertible one, that’s not going to help.

CB: Yeah, I mean, the invertible ones, you can always — I guess, well…

KK: I don't know, though, maybe you need a permutation matrix to make some product work out correctly? I don't know. This is an interesting question. I like this question.

EL: Yeah.

KK: We’re not going to try to solve it on the spot.

CB: I’m not sure. Like, my first thought is that you can probably even, like, throw the invertible ones out. But now I'm not so sure. Maybe they might help in some way of arranging the zeros.

KK: So where did you come across this theorem? This is an interesting result.

CB: Yeah, well, undecidable problems always have fascinated me, and I guess I might have been looking at some of these, maybe it was, I don't know where I came across it. Maybe it was some survey paper of undecidable problems, or maybe a Math Overflow question. I'm not sure. But yeah, somewhere along those lines.

EL: But it's a nice one that's maybe a little more accessible to most people who have taken, you know, a few upper-level math classes than some of the undecidability things, which are just like, Okay, I need to climb this whole mountain to even understand this. You know, we all take linear algebra at some point, you know, if you're a math major or something, and so it's very concrete, you can immediately understand what it is if you’ve seen matrices.

CB: I agree. The description is rather elementary. You don't need to introduce Turing machines and halting arrays or some abstract presentations of groups and such. It's rather — the operations are ones that, as you said, any linear algebra student has seen before, but somehow the problem is already like, not even difficult, it's impossible in some sense.

EL: It is always really interesting to see, like, what are the limits, not just of our knowledge, but of what we can know about our possible knowledge.

KK: Right. Yeah. So the other thing we do on this podcast is we invite our guests to pair their theorem with something. So what pairs with this theorem that doesn't really have a name, but we'll call it the undecidability matrix theorem or something?

CB: Yeah, okay. So I'm not really a food person, so I didn't think of a food. What this — I would say that it pairs well with a decent table tennis service. Because it — there’s some, like, it’s not a killer service but decent, so you can have a decent back and forth, as we have just had, as to like, how small you can make it, how bad is it, that sort of thing. So that's what it reminds me of.

KK: Yeah. All right.

EL: Do you do you play table tennis?

CB: I like table tennis. I don't play as much as I would like to, but occasionally I do play it, and I like playing it.

KK: I’m much better on the Wii than I am in real life. The Wii table tennis is really fun.

CB: Ah, okay.

KK: In real life, not so much. I mean, I like it, but I'm not very good.

CB: I also heard people play on VR online. You can even like see a table.

KK: It’s not quite the same.

EL: I have not played since I was probably in sixth grade or something, when I think I was pretty capable of beating all of my opponents, who were my younger siblings. So, you know, at that age, you’ve kind of got just some advantage by being a little older. And so I tried to take advantage of that whenever I could, as the oldest sibling. You know, I really, we played two-on-one basketball sometimes, and I always kicked their butts at that because, you know, I was way taller. It helped a lot. I couldn't really play but like, against someone four or five years younger…

KK: My problem with all racket sports is that I played a lot of tennis when I was in high school. And so I play all racket sports like tennis. So you know, with big swings, so that doesn't work in table tennis. I's a much faster, yeah, just shorter.

CB: Much more of a wrist play than the whole arm.

KK: That’s right. Yeah. And racquetball is the same way. Like, I want the ball at my waist. I don't want to be reaching down to my ankles. I'm too tall. I can't get there. You know. So all these things were a challenge for me.

EL: Yeah, well, I do really like this pairing, because just like this theorem is sort of this meta- about, not just a specific case of matrices, but like, what we can know in general, given, you know, any set of information, your pairing was not just about the theorem, but was also about our discussion of the theorem.

CB: Yeah.

EL: So I think this was this was very elegant. I appreciate that.

CB: Thank you. I also like that we still have a gap in knowledge. I think that as, I don't know, like, teachers, we introduce — I remember being as a student, that that would really pique my interest, like, when teachers discover, you know, this is not known. And I think it offers a different landscape versus a completely furnished theory.

EL: Yeah, well, I know, when I was in college, I liked my math classes, but I didn't understand that math was still this active area of research. I was very naive about that kind of thing, and even now, you know, you've run into people who don't know what math research means. It's like, well, we know how to add and multiply numbers. We know how to do all of these things. Like, what else can there be to know? and this is something that doesn't take as much — you know, it's one of those examples that you can give.

CB: Right.

EL: You know, it has a lower, or, you know, a more basic way that you can enter this and like, understand, Oh, we're still trying to figure out this kind of thing. And so, I like that.

KK: Yeah.

CB: Also another thing I like, it's a bit upsetting that this is not decidable. I like those sorts of results. Actually, my account in Twitter has been referred to “the account that posts cursed math facts.” A lot of people say that, and that was not my intention, but it kind of fits with that.

EL: Yeah, well, that's very true because yeah, when I first saw it, I was just like, well, how can we not just, you know, just try all the ways to multiply it. At least in theory, you could do that, but not if it's arbitrarily long.

CB: Yeah.

EL: You're allowed to have as many as you like. Okay, if it was just one copy of each one, well, that's trivial. I mean, not trivial to actually do it, but it's trivial to know how to do it. But it's kind of funny that once you allow yourself multiple copies, it's just like, everything goes out the window.

CB: Yeah.

KK: All right. So you've already plugged your your popular Twitter account. We like to give our guests a chance to let us know where we might I find you online or anything else you're you're you're trying to promote or anything like that. It’s okay if you don’t.

CB: There’s my account. It’s called some theorems. So I think I can just put that in Twitter.

KK: Well for now.

CB: Yeah. I think I won’t add more to that.

KK: Okay. Well, Cihan, this has been great.

EL: Thanks so much for joining us.

CB: Yeah. Thank you for having me.

[outro]

On this episode, we were excited to talk with Cihan Bahran about the undecidability of the matrix mortality problem. Here are some related links you might enjoy:
Bahran's website and Twitter account, where he posts "cursed math facts"
The 2014 paper establishing the undecidability of the matrix mortality problem for, among other cases, six 3 × 3 matrices
The word problem in group theory

We recorded this episode before the devastating earthquake in Turkey and Syria. Our hearts go out to all who have been affected. If you would like to contribute to relief efforts, Doctors Without Bordersand Ahbap Derneği are two organizations doing work in the area.

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcasts with no quiz at the end. I'm your host Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right. I got to take an overnight Amtrak trip last weekend, my first time, so that was pretty fun. Went from Salt Lake to Sacramento and got to see lots of beautiful Nevada and California landscapes on the way.

KK: Yeah, I did an overnight Amtrak once and it was less fun. It was from Jackson, Mississippi to Chicago. And — which, I mean, it's, you know, it's all night, right? So you don't really see anything. And it's remarkable how many times have to pull over for the freight trains, right?

EL: Yeah.

KK: This is how American rail is really different from European rail. You're at the mercy of all the freight, but that's okay. Anyway, yeah.

EL: I guess, today, living on the only portion of Amtrak's corridor for which they actually own the tracks, is our guest, Juliette Bruce. At least I hope I'm correct, that that's where you're living. Otherwise, that was a weird introduction. So please tell us a little bit about yourself.

Juliette Bruce: Thank you so much for the introduction. I'm Juliette Bruce, as you said, and I am a postdoc at Brown University. So in fact, I am in the northeast along the Acela Express corridor. In fact, I've never taken that Amtrak corridor, I've only taken the very slow ones that you were talking about, but I hope to take it soon.

EL: Yes. Find yourself someplace to go between New York, DC, Boston, I guess to Boston, you don't really need the Acela. It's already pretty close.

KK: You can walk to Boston.

EL: If you're really dedicated.

JB: It’s a pretty far walk.

EL: Yes. So I guess this isn't the train cast. This is a math podcast. So, so yeah. What are your mathematical interests at Brown?

JB: Yeah, so my area of math is kind of in the intersection of algebraic geometry and commutative algebra, which is all about studying the interaction between this algebra, coming from kind of the symbolic equations we get when we write down systems of polynomial equations, and the kind of geometry we can look at when we study the zero set of those equations. So we can look at the simultaneous solutions to the system of polynomials, and that's some lovely geometric object. And alternatively, we can look at these symbols we write on our paper, and somehow, in some point in math, we learned that we can do lovely things, like finding the roots of a quadratic polynomial by graphing them on our graphing calculator pictorially, or we learn we can use symbols and write down things like the quadratic formula, and magically they give the same answer. A lot of my research is sometimes a generalization of this fact that there's two different ways to study the solutions to a system of polynomial equations.

EL: Right. I must admit, I'm pretty naive about algebraic geometry, but there is this kind of magic in it, which is — you know, like, in, what, seventh or eighth grade or something, you start learning to graph the zeros of polynomials. Maybe you might not use that exact language for it, but you start to understand that you can intersect two different polynomial equations and find these intersection points and stuff like that. And yet, this is also like cutting edge math, you know, just add a few variables, or bump up the powers of the the numbers that you're using. And suddenly, this is stuff that, you know, people are getting PhDs in. I's kind of kind of cool,

KK: Right? Or work over a finite field, whatever those are. Yeah.

JB: I mean, I always find it fascinating with just how many different areas algebraic geometry has touched in mathematics and in the world. It seems to start from such a lovely and beautiful, simple idea that we learn in, you know, middle school or high school, and just kind of grows exponentially. And it turns out, it's actually a very deep idea that maybe we don't always appreciate when we first see it. I know I certainly did not.

EL: Yeah.

KK: All right.

EL: So then what is your favorite theorem?

JB: So my favorite theorem, or the theorem I want to talk about today, I know it as Petri’s theorem. I know some people know it as the Babbage-Enriques-Noether-Petri theorem. I'm not sure exactly on the correct attribution here, so I'll stick with Petri’s thereom and apologize to Babbage, Noether, and Enriques, who maybe want the appropriate attribution here. And this is a theorem from classical algebraic geometry, which means from the 19th century, and it's about understanding the interaction between thinking about systems of solutions of polynomial equations abstractly, and how we can realize that abstract solution set concretely as solutions to an honest-to-God set of polynomial equations that we could write down and describing what those polynomials might look like.

KK: Okay.

JB: And so the statement of the theorem, I'll state the theorem, and then we'll walk through it, maybe. And you can ask questions, because I know when it’s stated, it's a little bit of a mouthful and a little scary, is that if I have a curve that is non-hyperelliptic, and I embed it via the canonical embedding, then the image of the canonical embedding is cut out by quadratics unless the curve is trigonal, meaning it admits a three-to-one map to the Riemann sphere, or it's a curve in the plane of degree five. So that's the statement of the theorem. That's a mouthful, I know, to get through.

KK: Yeah, sure. Right.

EL: That’s interesting. So, you know, as I already confessed, this is outside of maybe my, my mathematical comfort zone a little bit. And how, how should I think about these exceptions? Like how exceptional are the exceptions? Is it, like, a lot of things? Or just a couple of little things that and otherwise, everything falls under this umbrella?

JB: Yeah, so that's a fabulous question. And so I gave — there are two exceptions to this theorem, right? If a curve admits a three-to-one map to Riemann sphere, so there's a map that goes to the Riemann sphere, that kind of every preimage has three points, it kind of looks like a sheet wrapped up three times around the sphere. Or it's a very specific curve in the plane of degree five. And so these exceptions, there's an infinite number of them. But it turns out if you think about them correctly, it's kind of a small proportion, or it's not most curves that will satisfy this. So this is somehow saying, with these few exceptions aside, we can actually understand the image of what's called the canonical bundle. So maybe I should say, what is actually going on here. It's something a little deep. So kind of the starting point of algebraic geometry is that we want — I said, we want to study the solution sets of polynomial equations. Well, it turns out that that's how the field started. But pretty quickly, people realized, well, this is some geometric space, it's a set of points. And instead of looking at the solution set to a particular set of polynomial equations, we can kind of abstract this away and forget the polynomial equations together and just think about what possible sets of solutions could I have, and think about that kind of abstractly in the ether. There's no polynomial in sight, we can just say, oh, you know, this is a solution set to some system of polynomial equations. We don't know which. And it's a lovely theorem that, you know, if we're talking about curves, it turns out algebraic geometers have this very weird convention that curves would look to people like us, like a two-dimensional surface. This is because I like to work over the complex numbers. So my polynomials have solutions and the complex plane is two-dimensional. So we have this weird terminology. So abstractly, a curve, if it's smooth and has to satisfy some other conditions, just looks like a closed surface, possibly with some holes in it. So we'd have a genus g surface. So if you've seen a doughnut, or a torus, that's just an algebraic curve of genus one. And if you seen a sphere, that's just an abstract algebraic curve of genus zero. And the beauty of these is that somehow, if we take these objects, we can realize them in space, we can put them into some large, complex space, or some large projective space, and once we've done that, you can ask, well, I know there is some set of polynomials that cut the space out, we have this algebraic variety. It's a system where we know it's by definition, a solution set to some polynomials. And you could ask what polynomials actually cut it out under this realization in space. And often, there are many different realizations. So for example, you could look at the parabola, a very simple example. We can look at the parabola, x2−y=0. This gives us the normal parabola going through the origin. It’s realized in space. But we can also abstractly think about just kind of the parabola floating around, no coordinate system at all. And we could also realize that same parabola in space by just, you know, shifting it up or down the y-axis and moving it around, and the polynomials that cut it out when I start moving it around, we learn, are different, right? We learn how to do transforms, we knew somehow, like (x−1)2−y=0 gives a different solution set, but it looks the same in the plane, just moved around. So you could ask, when I put my abstract curves in space, what are the polynomials that actually cut this thing out? And so those are kind of the input to the theorem, is these abstract curves. And we put them into space. And what are they cut out by? So that's kind of the input. And the theorem is answering that question, what are they cut out by? What are they defined by?

KK: Right. So are you assuming you're starting with a plane curve? Or?

JB: No, so this curve doesn't have to be in the plane, although it's kind of just this abstract notion of a curve, so it’s somehow, just in general, a kind of smooth looking surface that's compact and has g holes, so maybe like a 2-holed torus or a 3-holed torus. It's kind of some very weird donut-looking shapes, essentially, is what the curve goes in, what is the input of this theorem?

KK: Right. And you mentioned something called the canonical embedding. So that might require a little terminology.

JB: Exactly. So what do I mean by the canonical embedding? Defining it exactly is complicated. And it's not something I would want to try to do on this podcast.

EL: Especially with audio.

JB: Especially with audio. But instead, let me just kind of give this notion. I said, you know, if we're looking at perhaps standard parabolas in the plane, there's a lot of different ways we could put it in the plane. We could put it through the origin, we could put it so like the vertex is at (1,1) or (2,1), or we could do all these things, and there isn't, doesn't seem to necessarily be a natural best choice for how we put a parabola in the plane.

EL: Right. It feels very arbitrary.

JB: It’s very arbitrary. And when we change our arbitrary choice, we change the set of polynomials that define the parabola in the plane. It turns out that when we're kind of working in a slightly more abstract setting, where instead of looking at parabolas in the plane, but we're looking at these two-dimensional surfaces, which are what algebraic geometers will think of as curves, because we're looking at the complex set of points, there’s an almost canonical way to put them into some kind of space. And that's called the canonical embedding. It kind of arises by looking at ways you can kind of differentiate on your surface. It comes from looking at what are known as differentials on your surface. And I won't say anything more than that, other than to say somehow, it's this beautiful fact that was developed by people in the 19th century that there exists such a thing that allows you to transport these abstract curves into different spaces in a way that has beautiful properties. And somehow, it's a great tool for studying curves.

EL: Not quite sure if this is the right question asked, but you know, you have this input to this theorem, and then it tells you something about like, you know, what polynomials can be your solutions? How specific is it? Like, would it output something that, you know, we would have recognized as a polynomial in seventh grade? Or does it output something that maybe has a little more technical machinery behind it?

JB: This is an absolutely fantastic question. This is a fantastic question. So, right, as you're saying, the input is I input this abstract surface abstract Riemann surface of genus g that satisfies some properties and the output of the theorem and saying if it doesn't satisfy, if it doesn't fall into these two exceptional collections, which are relatively small when it comes to lists of exceptions, then we know that the defining equations are degree two. And you might ask, well, does the proof actually give — like what are the polynomials? Can you actually write them down? And in part, the version of Petri’s theorem I know, in fact, gives you those polynomials in some sense. There are some choices that have to be made, and those choices kind of arise from some technicalities about defining exactly what is the canonical embedding. There are some choices there. But once you've made those choices, Petri’s theorem actually comes down and says we can write down an honest set of degree two polynomials in a lot of variables now. The number of variables is the number of holes on my surface, is the genus of my surface. So it's a lot of variables. But we can write down an honest set of equations that cut this out. And this is this beautiful thing that takes an extremely abstract thing, you know, this curve that's abstract sitting in our mind, and realize it in space, and it outputs a list of polynomial equations.

EL: Okay, wow.

KK: So okay, so now I'm thinking about elliptic curves in particular. So you mentioned there's just one variable. So that's a torus. Right? You can you can actually write this with one variable?

JB: Yes. Yeah. So if you're looking at elliptic curves,

KK: But I always think of elliptic curves as being, like, y2=x3 plus some change, right? That's not degree two, is it?

JB: That’s not degree two. Right. And you're calling me on a on a little technical point that I swept away in the beginning, which is that I said — when I said theorems carefully, I said that if our curve is non-hyperelliptic, right, and it happens, that elliptic curves will kind of not be in a case where the canonical embedding is, in fact, an embedding. Somehow you can talk about what that map might be, and for an elliptic curve, that map would take your elliptic curve and map it all to a single point.

KK: Yeah, that's a bummer.

JB: And that's a bummer. So sadly, elliptic curves, it doesn't quite work. So it is this interesting issue where if we're looking at abstract curves of small genus, so like elliptic curves are doughnuts of genus one, so there's one hole, or if we're looking at curves of genus two, so there's donuts with two holes, this theorem doesn't really apply, because those curves are extra special. And in fact, that's some of the beauty of things like elliptic curves. But once we're looking at more higher genuses, like genus three or four, and so on, you start to see very interesting things. So for example, if you take a genus three curve, and it satisfies this property being non-hyperelliptic, whatever that means, you can realize this curve in the projective plane, which is kind of a three-dimensional object cut out by some polynomials of degree four.

KK: So I mean, is this this love at first sight? Like, did you, you know, as a student read this in, is this in Hartshorne somewhere, or is it somewhere else and just fall in love?

JB: Yeah, so this is a great question. And it's actually as far as I'm aware of, not in Hartshorne, which is kind of a weird thing, because Hartshorne is notoriously quite comprehensive.

KK: Sure.

JB: But all the players are in Hartshorne. And in fact, the lead-up to kind of this theorem is in Hartshorne. And so the build up to this is to get to this theorem, you spend a lot of time in this fairly hard textbook, and you get to the end, and all of a sudden, they say, let's look at curves. And you think, wow, that's pretty simple. I've spent a year and a half, two years of my life learning all this complicated machinery and now you're going to tell me we're doing the simplest case. And you start looking at them and you see these beautiful things where all of a sudden, you built this machinery that lets you compute these equations in specific cases, so say small genus. And later on, you can read this amazing theorem of Petri and see that there's actually a full argument there, of how you can write down these polynomials, and it's kind of this beautiful synergy of all the things you learned coming together in one.

EL: I mean, I guess you've kind of answered this a little already, but what do you think draws you to this theorem so much that makes you love it?

JB: Yeah. So what draws me to this theorem, I think, is a number of different things. So (a) is this beautiful combination, or culmination, of learning so much, so many of these kind of complicated tools that don't seem closely related to the spirit of algebraic geometry, which is again, studying polynomial equations and their solution sets. But also, its has this amazingly surprising thing, which is that somehow if I take this abstract curve, and I take some realization of it in space and I ask for the equations that cut it out, those equations should really depend on how I put it into space. You know, if I put them into space a different way, I'll get different equations. And what this theorem is saying is that since these exceptions to this theorem don't depend on how I put it into space, those things only depend on the actual curve itself in its abstract form, that somehow, there's this beautiful thing that sometimes when you put things into space in the correct way, and you look at their defining equation, that's telling you something very, very special about not just that particular realization of your curve, but kind of the abstract, ethereal curve that lives kind of off in our imagination.

KK: So part two, we like our guests to pair their theorem with something. So what pairs well with Petri’s theorem?

JB: Yeah, so the thing that I thought of when I was thinking what pairs well with Petri’s theorem, is I was thinking about how, when I first started to see the, you know, glimpses of this theorem, it was Chapter Four of Hartshorne, so partway through this huge book that I had spent, you know, multiple years trying to get to that point. And all of a sudden, you get there and you see this beautiful vista of mathematics, these beautiful examples, that use all the tools you’ve got, and there was no easier path there. And it made me think of another one of my favorite hobbies, which is mountain climbing, or mountaineering. And how, you know, oftentimes you have to slog through these very tedious, long, hard, difficult and exhausting things, exhausting work, not always the most enjoyable — sometimes you're tired, sometimes your legs hurt, sometimes you're just kind of looking around and saying, wow, this is kind of a dusty desert, this isn't very pretty. But all of a sudden, if you put in that work, you get to this vista, and you see kind of the beauty of the world around you. And to me you get to this theorem, and you see the beauty of algebraic geometry, and kind of the essence of why you did what you did why you put in that work.

EL: Yeah, so what are some of your favorite vistas or mountains that you've gotten to climb?

JB: Yeah, that's a great question. So I have lived in California with my partner for a while and so a lot of the things I've liked to climb and do are kind of in the Sierra Nevada range or the Cascades, and not all of these ones I've actually fully summited, but I think things like looking off from Mount Shasta in Northern California has a beautiful view where you see the changing from how it's beautiful green forest around the mountain where it's snowy to this dry browner desert as you move off into kind of Northern California. There's some beautiful vistas out near Lake Tahoe and you kind of climb these peaks and get to the top and all of a sudden you can see this absolutely gorgeous lake spread out in this beautiful forest with peaks kind of circling it as a rim.

EL: That does sound amazing. As you can see, but our listeners can I've got my Zoom background is from a hike I did recently where — I must say this hike is basically beautiful the entire time. So there, there wasn't so much slogging, but you know, it was covered in aspens and the evergreens and things and just, you know, you do sometimes come around this curve, and suddenly you can see the Salt Lake valley, below where before it had just been trees, and there is something really special about that.

KK: Yeah, absolutely. I'm an East Coast kid. So Appalachians, which is still of course very beautiful, but different vibes. All right. So we'd like to give our guests a chance to plug anything they're doing where can we find you on on the intertubes?

JB: That’s a great question. I guess I would say I have a professional website. Google my name, you'll find it. Of course I’m also on Twitter for who knows how long.

EL: Limited time offer.

JB: A limited time offer, potentially, also under my name. And you know, otherwise I'm happy to respond to emails or things like that. And I'll just plug as a final thing, you can't reach me this way, but I am the president of Spectra, the association for LGBTQ+ mathematicians. So I'm also very heavily involved and happy to talk, and, you know, look at that sort of work. So if you're interested in LGBTQ+ mathematicians, I'd plug looking up Spectra and the work we've been doing there.

EL: Yeah. And did they just kind of recently, sort of — I got the feeling maybe it was a little more of an amorphous organization, and now it's sort of coalescing into something that has has a little more structure. I've tried to make an algebraic geometry analogy here and it’s just not working. But yeah, this is relatively recent, right? So are you the first president of Spectra?

JB: Yeah, so that's absolutely right. Spectra has kind of a long and amorphous history, coming from kind of a lot of grass roots activism in the ‘90s, through the 2000s. And it's existed in some form, at least with a website for, you know, a number of years. But in the last few years, we've really been trying to grow and formalize and expand our reach and the ability of support we're able to give LGBTQ+ mathematicians. Part of this includes kind of creating a formal board structure, and we did that over a number of years, going to effect this last year, and I was lucky enough to be chosen by the previous board members to be the inaugural president for this year. So I've been lucky to kind of take the reins and guide the organization through its first kind of formal year this year, although building upon all the amazing work a number of extremely dedicated and thoughtful people have done many years previously.

EL: Yeah, and I think it really has been maybe a lifeline, or really as a place that, you know — young LGBTQ mathematicians have maybe sometimes felt isolated where they are, and able to be like, is there anyone else like me? And, like, of course, there are a lot of people like you. And it's been a place that people can find, and I think that's really special.

JB: Yeah, that's exactly the goal. One of the goals we have is trying to make sure people see other visible LGBTQ mathematicians and see people they might be able to aspire to and reach out to or seek advice from or support from. So that's been one of our goals of formalizing and trying to increase our presence.

EL: Well, that's great. Yeah, check out Spectra. And yeah, send Juliette an email, you know, about anything related to that.

KK: Or algebraic geometry, or mountaineering.

JB: Or mountaineering. Yeah.

KK: All right. Well, this has been great fun. Thanks for joining us, Juliette.

JB: Yeah, thank you so much for having me. It's been a pleasure. I've really enjoyed listening to your podcast prior to this. So I really appreciate the opportunity to tell you some hopefully coherent words about my favorite theorem.

EL: Yes, thank you. I just love all the different perspectives we get by talking with so many different people here.

KK: Yeah, that's the best part. All right. Take care.

[outro]

On this episode, we were happy to talk with Juliette Bruce, a mathematician at Brown University, about Petri's theorem. Here are some links you might enjoy as you listen to the episode.
Her website and Twitter profile
The canonical bundle and Petri's theorem on Wikipedia
Robin Hartshorne's (in)famous Algebraic Geometry textbook
Spectra, the association for LGBTQ+ mathematicians

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 I am joined as always by my fabulous other host, co-host? I don't know,

Evelyn Lamb: Co-host. It’s a host but going in the opposite direction.

KK: That’s right. We reverse the arrows. Haha, math joke.

EL: Yes, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. Actually just got back to Salt Lake from a wonderful trip this past week where I got to meet two new additions to my family, ages three months and three weeks. So that was, that was pretty fun, to hold one of the tiniest babies I've ever held. So yeah, very nice little fall trip to take. And now I'm back here and talking about math.

KK: Yeah, well, last Friday night, I drove two hours over to Ponte Vedra, which is sort of near Jacksonville, by myself to a concert. So this is where I am in life. So I went to see Bob Mould, who many people may or may not know, but he was — Yeah, Christopher's shaking his head yeah. He was at Hüsker Dü and then Sugar. He's been doing solo albums forever. And I've been a fan for going on 40 years, which is also weird to say. Had a great time, though. By myself, that's great. This is what one does in his 50s I suppose. Anyway, not as exciting as holding a newborn but, but still pretty good. So anyway, hey, let's talk math. So today, we are pleased to welcome Christopher Danielson to the show. Why don’t you tell us about yourself.

Christopher Danielson: Yeah, I am coming to you from St. Paul, Minnesota.

KK: Nice.

CD: Bob Mould, also a fellow Minnesotan. [Ed. note: Bob Mould is actually from upstate New York.]

KK: He went to McAllister, right. Yeah.

CD: Nice. Right up the street from where I'm standing right now. I work a day job at Desmos Classroom, which is now part of Amplify, designing — working with a number of colleagues to design math curricula. We are currently working on an Algebra I curriculum, about to wrap that up, and moving on to Geometry. And then on the side, I have many projects, some of which will come up in our work today. But I think I understand that you two are familiar with the Talking Math With Your Kids blog that grew into then a large-scale playful annual family math event at the Minnesota State Fair called Math on a Stick.

KK: Cool.

CD: And I also am Executive Director at a small nonprofit that seeks to create playful, informal math experiences for children and families in the same spirit as the work we do at Math on a Stick, but designed for a variety of other sorts of spaces. That nonprofit is called Public Math.

KK: Very cool.

EL: So I'm probably doing that thing where I generalize from a small number of examples. One of my best friends in grad school was from Minnesota, and just loved the State Fair. So I think that Minnesotans just have it this special relationship with the State Fair. And so I did — I am really interested in hearing more about how you do Math on a Stick at the Minnesota State Fair.

CD: Yeah. Should I pick that up right now? Or is there more on the agenda?

EL: Yeah, that would be great!

KK: No, go ahead.

CD: Yeah, so the Minnesota State Fair, it's the second largest state fair in the country behind only of course, Texas.

EL: Where I am from.

CD: Oh, nice. Texas lasts for a month. Ours is 12 days. 12 days of fun ending Labor Day is one of the mottos. The other is the great Minnesota get-together. The location of the fairgrounds is especially convenient for large attendance. The fairgrounds are right, sort of on the border between Minneapolis and St. Paul. And they have been, for probably the past 20 years, have been working on developing some educational and family friendly spaces, out of a perception that it is expensive to go to the fair, which is true, but then once you're in that there isn't much to do besides look at animals and buy a bunch of food.

EL: On a stick.

CD: Yeah, on a stick. So they’ve been working on that. And that led to a lovely literacy space called the alphabet forest that is about 12 years old now. And the first time I sat down there, it was their fifth year and was like the sky, the clouds parted and the angels sang, and I was like, I’ve got to figure out how to build a math version of this. And so together with some organizational support from the Minnesota Council of Teachers of Mathematics and a bunch of expertise from folks that I know through the blog work and through my work in math education, put together a pitch, and after many very boring meetings, it became a thing. So we've got about 15 to 20, different mathematical, playful, creative math activities, everything from a big table full of tiling turtles, to a set of numbered stepping stones that you just see kids jumping up and down happily counting, counting by twos, creating all sorts of fun things to do with. We have a different visiting mathematician or mathematical artist every day, each of the 12 days and they bring whatever sort of hands-on thing they're into. Sometimes that's sort of the standard stuff with, like, Mobius strips and hexaflexagons, and sometimes it is new and new and delightful, creative things that the world has never seen before. So yeah, Math on a Stick, come on out and play with us. 12 days of fun ending Labor Day, always starts on a Thursday, runs through a full week, two weekends and then ends on Monday.

EL: Yeah, that does sound like a neat thing. Sometimes I go to the farmers market here or something like that, and I just think, like, where are there opportunities to kind of create, like you said, these playful, you know, a non-classroom math experience for people?

CD: Yeah, my, one of my Public Math colleagues has a project called Math Anywhere, Molly Daley. She's in Vancouver, Washington, and also does some stuff across the river in Portland, Oregon. And farmers markets are one of the more successful spaces for her. So she'll pay for, for a booth, she has grant money, she'll pay for a booth and just set up a much smaller version of Math on a Stick stuff, as well as some other stuff that she's designed or harvested from other places, but three or four activities, and yeah, delightful times ensue. However, I had a recent experience at the Mall of America, largest shopping complex, also here in the Twin Cities. And it was really interesting, because the way that kids’ families move through the Mall of America is wildly different from how they move through the state fair. So just an invitation to a big STEM/STEAM carnival. And we brought some — one of our favorite things is called a pattern machine or punchy buttons, a nine by nine grid of punchy buttons that you can drop pictures on. And each button is clicky and on a ballpoint pen. So we bought a bunch of those. And then we also had the mega pattern machine, which is just thousands of buttons from all these machines smashed together to make a nice big floor space. But the way that kids come into Math on a Stick is that there’s, like, this long elastic band between parents and children at the fair, not in the super crowded spaces in the fair, but in the less crowded spaces. And so often kids will see those stepping stones that, by the way, start at zero, and then continue on to 23. Yeah, so they'll start on the zero, and they'll lead the way into the space, like we deliberately set up those stepping stones to that the edge of this outdoor space. And by the time kids get to 23, now they're surrounded by eggs that they can put into — little plastic eggs they can put into large egg crates, and tiling turtles and pattern machines and all sorts of fun things to do. And families will sort of follow along behind. At the mall, there’s none of that. There's none of that. Families move in really tight units. There's no, like ,a child leading the family into a space, which is just a really interesting dynamic. And having been out in Portland a couple of weeks ago with Molly when she was at one of the farmers markets, it felt very much more like the fair. A mom or a dad might be much more likely to say, okay, sweetie, you keep playing with these turtles, I'm going to hop over there and buy some apples and I’ll be back in two minutes. They kind of keep their eye on them and everything, but that that elastic band is much longer. Nobody ever says, okay, sweetie, you know, their four year old, I'm going to I'm going to just hop across to, you know, the department store over here, you keep playing with something in the hallway. So Public Math is our project where we're trying to think about how do you design for those kinds of spaces? What would have been a better design than the one we had for something like them our time at the Mall of America?

EL: Yeah. Interesting different kinds of math problems to solve. Different optimization.

KK: That’s right. Yeah. All right. So this podcast does have a name, though. So presumably, you have a favorite theorem. So you want to tell us what it is?

CD: I do! And it is — Yeah, my favorite theorem is, I'll state it simply. And then I guess we get to talk about like, why it’s my favorite and things?

KK: Yeah, sure.

EL: Yeah.

CD: It doesn't have a name. I feel like maybe, maybe it should have — maybe it has a name. Maybe you'll know a name for well,

EL: We’ll brainstorm about it.

CD: But yeah, let me state it simply, which is that the vertices in a polygon are in one to one correspondence with the sides of the polygon. So for example, the three-sided polygon has three vertices. Is there a name for this?

EL: So, yeah, well…

CD: The polygon theorem or something?

EL: I don't know. Yeah, that’s

KK: I mean, a polygon is just a cyclic graph. There must be some graph theory name or something.

EL: I kind of you know, this has a little bit of an interesting linguistic thing, right? Because we call polygons a little bit differently at different sizes, like we call it we talk about triangles not trigons, or trilaterals. When we talk about quadrilaterals, like I think I have heard quadrangle, that must be the tipping point. Then we get to pentagon, so I guess that's not lateral or angle.

KK: That’s just gon. Then it’s gon after five.

CD: But gons are angles. So you are counting —

EL: Okay, is that the Greek word for for angle, and angle is Latin?

CD: So goniometer is the is the thing that you can use to measure your range of motion. I'm gesturing, so that's great on a podcast.

KK: We do it all the time.

CD: Like in your arm or knee? Yeah. So yeah, gon is angle.

EL: Okay. Learn something new.

CD: So it's only the quadrilaterals whose sides you count. Everything else, you count the angles. And, by the way, we also have elided the fact that the vertices and the angles themselves are in one to one correspondence, right? That’s also, maybe a corollary perhaps.

EL: Yeah. Okay. So maybe I'm playing devil's advocate a little bit here. But why is it a theorem that the angles and sides are in one to one correspondence? Why is it not obvious, other than the fact that, like, I've experienced these shapes my entire life and have never experienced one that did not have this property?

CD: Yeah! So I learned that this was a theorem, and its necessity, by working with five-year-olds. So I wrote a book called Which One Doesn't Belong, which was an adaptation, both of the Sesame Street routine, but also playing on some of the routines that I had seen other people playing around with. But for me, the thing that was novel about which one doesn't belong, was that when my children were small, all the shapes books that they had an opportunity to encounter were wildly simplistic. There would be, you know, a triangle page, and then there'd be a square page, and then a rectangle page, and never a square, never a square on the rectangle page. That's confusing for kids. And all of the triangles would be equilateral and oriented on one of their sides, all the hexagons were regular, and again, sitting by their sides, or maybe if they're feeling a little wild, straight up and down balanced on a vertex. But orientation isn't a thing, like, there's all this work that we know is important to come to understand a mathematical idea that just doesn't get doesn't happen in books that get published for young children, even though if you've ever been around four or five or six year old children, they can think about complex relationships, they can think about complex ideas. But somehow we don't understand or value that when we're creating books for kids. So Which One Doesn't Belong was my way of producing, taking ideas that other people had had and condensing them down into what I thought of as a shapes book that was more worthy of children's minds.

EL: I just want to insert that it is a really fun book. I don't remember when or how I obtained a copy. But I have enjoyed going through it myself, and I probably should have asked permission, but I actually used it as an inspiration for one of the pages in this page-a-day calendar I put together a couple of years ago, where I made just one where, you know, it's a bunch of shapes that all have slightly different properties, and you know, you decide which one doesn't belong.

CD: By the way, I’ll give you a little tip before explaining again, why this theorem is important. If you ever try to design a “which one doesn't belong” set, what you want to do is think about whatever your domain is, so say it's shapes, you want to think about four properties of shapes, and then cover up the first one, and design one that has these three, but doesn't have the first one. And then cover up the next one, design one that has those three, but doesn't have this one. And by the time you're done, you'll either realize that your set of four properties is more intertwined than you had originally thought, and now you’ve got to go back and revise, or you'll have a set where you know for sure that there's at least one reason for each not to belong. But then extra, an important key to this is that you have to be open to the possibility that some kid will see a reason for a shape to not belong that wasn't the reason you'd intended. Right?

EL: Yeah.

CD: this isn't a game of “guess which of the four is right.” But it's also not a game of “guess what was in my head when I designed the set.” Instead, we want to offer up something that we know is rich, and then be open to learning from the kids. So I made this book. I was trying to shop it around to get it published, but also needed to, you know, test drive it with children. So I went on what I called my Twin Cities shapes tour. And visited, I think it was three different elementary schools per week for four or five weeks. So I got into just a ton of different situations, worked with kids, kindergarteners, through, like, fourth graders, all in classrooms, like 20 minute bits, and we just had a ball. And frequently, I would hear from kids, like, one kid would say, you know, that shape doesn't belong because it has three sides and the others have four. The opening page of the book is a triangle, and then there are three rhombuses of various types and orientations. So a kid would say that one doesn't belong, because it has the wrong number of sides, right? It has three sides, the others have four. And then somebody else would talk about some other shape. And then another kid would say that one doesn't belong, because it has three corners, and the others have four corners. And in my mind, the first, like, 12 times I heard this from children, I thought to myself, yeah, you're not listening. Some other kid just said that. Didn't say it out loud, kept it to myself. But it was after about the 12th time that I heard it that I said, “Wait a minute. Wait a minute, you heard you heard when this kid over here said said different number of sides?” And they'd be like, “Yeah, and I said different number of angles.” And so it was at that point that I realized that — they’re kindergarteners, right? They haven’t — I know that they haven't seen any good shapes books, right? So they haven't had the opportunity to consider the relationship between the number of sides and the number of angles. And in my adult mind, I had this idea that it was obvious, which is so true of mathematics, like always, right? That if there's something that we ourselves have internalized and experienced for a large number of years, even if it was hard for us to learn at the beginning, we've probably forgotten about that.

KK: Right.

EL: Yeah.

CD: So yes, that's our that's our theorem. And that's why it's important. It's the thing that you actually do have to learn, it isn't obvious when you're first exploring these mathematical objects. I imagine that's true for those who are studying combinatorics. So we were talking about graph theory earlier. Lots of results that feel obvious in retrospect, because you use them all the time, so much that they're sort of internalized, and you don't even think about them anymore. But there is some some point where that thing had to be learned.

KK: So I'm sitting here trying to think of a proof of this theorem. And of course, the dumbest one that just popped in my head is to use the Euler characteristic.

EL: Is that what the five- and six-year-olds do?

KK: I love using sledgehammers to drive nails! Okay, so all right, this is a theorem; it must have a proof. So let's, let's construct one that doesn’t require Euler characteristic.

CD: Yeah, well, I feel like I would start with a line segment that a line segment has two vertices, right? And then every time — so then now I'm going to add another line segment to get what I remember formally being a polygonal curve, right, made up of straight line segments. And when I add another line segment, now I add a segment and a vertex. So I’m always going to have an extra vertex. Until such time that I come back around.

EL: Yeah, and you add a segment and no vertices.

KK: This is exactly the Euler characteristic proof, just in reverse.

EL: Yeah, it's funny, because my mind actually, I think, basically was the dual of what you said, where I swapped out, so instead of that, I was thinking, when you start with an angle, you've got two line segments, and the vertex, and then I was actually kind of thinking, like, the number of angles you have, they each have two segments, but to connect them, you overlap the two. So you divide by two.

KK: Right, so the number of angles is the number of lines.

EL: Yeah, Little, it may be maybe slightly different, but similar sort of idea.

Yeah. Okay. So it's interesting that children see this as two different facts. Children are more literal, right? I mean, in my experience, one of my favorite stories about my son was we were at open house for eighth grade. And he walks in and his soon-to-be math teacher says, “Do you know what eight times seven is?” And he said, “Yes.” Right?

EL: Yeah.

KK: I mean, she was expecting him to say 56. But children will just give you the most literal answer that you can ever imagine. Yeah. So, okay, well, we usually ask if this is a love at first sight sort of theorem. But I don't know. Maybe that's not the right question here. Although maybe it was for you. I don't know.

CD: Well love at first noticing, right?

EL: Yeah.

CD: For me, the noticing that this thing that I had interpreted as being — these two statements that I interpreted as just being equivalent and repetitious of each other, noticing that that was a thing that required learning, and that these kids were absolutely listening to each other. And it gives me an opportunity as a teacher, right? I'm only in there for 20 minutes or so, but it gives me an opportunity to say, “Wait a minute, is that gonna always be true?” The generality is that this one had three sides and three corners? And these all have four and four. Is that always true? Can we imagine a polygon that has some different number of sides and corners?

EL: And what do kids conclude about that? Or do they have, like, ways that they reason about why they have to be the same? Or do they develop pathological shapes that don't have this property?

CD: Yeah, I haven't had time to dig into that in in depth with a group of students. I've had a lot of sort of related experiences. But yeah, I don't know. That would be super fun to to step in. Posed as an offhand question, kids absolutely will both think that it is probably, be willing to believe that it is true, and there will also be kids who will imagine that maybe there is some shape that they just haven't had a chance to meet yet that isn’t. Of course what that investigation with kindergarteners, that's going to get you into a lot of a lot of really interesting kinds of conversations, because they don't have polygon yet as a defined category of mathematical objects. So we're going to have to start to think about whether a circle is a polygon or whether curvy sides count as sides.

EL: Or if you’ve got, like, a square with a handle on it that's just a line segment, what’s that?

KK: Very cool.

CD: But yeah, that kind of, you know, monster creation, from Lakatos’s Proofs and Refutations, that kind of potential counterexample, and then dealing with whether the counterexample is really a counterexample, that kind of stuff goes on at all levels of mathematics, for sure.

EL: All right. I like this. It is not a theorem I have thought about as a theorem ever in my entire life.

KK: Right. Well, I think I see why you love it. Because it actually it's more of a meta-result than the actual theorem. The theorem itself is less important than kind of the questions that it can trigger. And to get kids thinking about things in an interesting way.

CD: But it’s definitely not a Postulate. Like if we're in Euclid, it’s not a postulate, nor an axiom.

KK: No, it isn't. It’s a theorem.

CD: And there are certainly lots of results about triangles in which we know there are three sides, and so there are also three angles, because it was a triangle. Yeah. So if you don't have it, if you get rid of it — like, we can say it's not important, but if you get rid of it, there's a lot of geometry you're not going to be able to do.

KK: Oh, okay. So right. So now instead of non-Euclidean, we might have sort of non-polygonal geometry. So we don't insist that our polygons have the equal numbers of sides and corners.

CD: Yeah, I was just imagining a world in which the theorem is an undecided result, or that we can’t count on. So anything, any place that we assume it, we've got to work around it or prove it again.

EL: Or we can only use theorems about angles.

KK: All right. So the other part of this podcast is we ask our guests to pair their favorite theorem with something. So what pairs well with this?

CD: I have two pairings.

KK: Okay, good. Good.

CD: I don’t know if that counts.

EL: Yes.

CD: Or we need a new word for a pairing.

EL: Yeah. No, that's great.

CD: Yeah. So I'm going to pair it first with a claim and then with an admonition. The admonition is related to what we've already been discussing. But the claim is, it's going to be controversial here, I imagine claim is that a diamond is a shape.

EL: Okay.

KK: A 2-d diamond or a 3-d diamond?

CD: Oh, yeah. So I'm still in plane geometry. Surely there is some corollary for 3-d geometry. But yeah, I got my start in math education teaching seventh and eighth grade. And I used to, when I was a seventh and eighth grade teacher, mid 90s, I was in a camp that is still still very active in which if a child says diamond, I say again, “No, no sweetie, rhombus, you mean rhombus.” Like we call it, we're sophisticated mathematicians, we don't use the word diamond. But again, through working with the kindergarten kids, I came to understand that they don’t — like, diamond and rhombus are absolutely not the same thing to them. So if we treat mathematics as a human construction, right, then the mathematical ideas that a five-year-old has are worth testing and exploring. And one of those ideas that they have is that orientation of the shapes matters, right?

EL: Yeah, I was wondering.

CD: A square standing on its corner is a diamond, a rhombus standing on a vertex is a diamond. But also, if you cut the top off that rhombus, you now have a pentagon. Still a diamond. It's got a vertical line of symmetry, still a diamond.

EL: Right, right.

CD: So not only is there not a correspondence, because rhombus is a thing that doesn't depend on orientation while diamond does, but also that not every diamond has to have four sides in the way that a rhombus does. They don't have to be equal sides. You can you can stretch it. So you've got short sides and long sides.

EL: Yeah, I was wondering if a kite is a diamond.

CD: Yeah, absolutely. Kites are diamonds. And so the thing that I would be very excited about would be a world in which instead of we as math teachers saying, “No, no, sweetie, that's not diamond, you mean rhombus. Diamond isn't the word we use, it doesn't really count.” That it instead be a place where we press on that in all the ways that we press on mathematical ideas and try to get at definitions. Right? So now we're going to make a whole bunch of different examples. Draw me a diamond that looks different from anybody else's diamond. And we create this category. And so I think the best understanding I have of a definition of diamond that would satisfy most kindergarteners, it’s something that has to have a vertical line of symmetry. And it has to be convex. So darts are not diamonds. And somewhere between four and probably, like, eight sides. Triangles are never diamonds. Never, never, never. But four or five.

EL: And it has to have a vertex on the bottom.

CD: Yes, a vertical line symmetry that goes through the vertex at the bottom.

EL: Oh, yeah.

KK: Yep.

CD: Okay, excellent. So that's my claim: a diamond has a shape and therefore worthy of investigation rather than of dismissal.

EL: I’ll buy that.

KK: The admonition?

CD: Sure. The admonition is stop showing children only the special case.

KK: I seem to remember a Twitter like, like you were…

CD: I started yelling at a publisher

KK: You were you were hot about this on Twitter.

CD: Yes. Okay. It’s a really interesting — I think the thing you're remembering was actually almost the reverse, which is something I alluded to earlier, the thing that there's never a square on the rectangle page. So I went to a public library, doing some research on children's books for some work that I'm doing and happened — of course, was in the shape section and happened to see this book about rectangles. Like literally its title is Rectangles. This is a book all about rectangles, it has no other purpose. And I pick it up and just, like, want there to be a witness to this — but of course, there wasn’t — of my predicting, there's not going to be a single square in this in this rectangle book. And I flip through the pages and of course there isn't. So it's just one of these small sort of regional publishers that publishes educational titles for libraries and school libraries and whatnot.

KK: Right.

CD: But I DM them on Twitter to say, hey, maybe we could liven this up a little bit. And they said, Well, no, according to state standards, you know, we're responding to state standards, blah, blah, blah.

EL: Oh no.

CD: I was like, Oh, that's really interesting. I'd love to see the standard that says that you can't say a square is a rectangle. What they came back with was a Texas standard at kindergarten that says at kindergarten, you are supposed to be studying special examples of shapes such as squares being special rectangles. And this publisher was publishing a book for four-year-olds. And so because it was a pre-K title, they couldn't put the kindergarten standard in. It wouldn’t be well-aligned.

KK: Don’t let them get ahead.

EL: Yeah, it would be too advanced to know that a square is a rectangle.

CD: And we have this idea that we can't provide, again, we can't provide complex ideas. We can't give kids interesting things to think about, or conundrums or puzzles. So yeah, admonition isn't quite that, right? My admonition is stop showing them only the special case, but also please, let's show them the special case and help them integrate the special case with the general one. But yeah, all the shapes books with the triangles that are on their bases. And yeah, you know, it's like if we were teaching kids about even numbers and the only even number we showed them was 2, end of story. It seems like maybe we need a little more.

EL: I’m kind of wondering, you know, if, like, guerrilla math person with like square stickers, like going into all the shapes books, putting them in the rectangle pages…

CD: That would be a fabulous public math project.

KK: It really would. That's good. All right. So we like to give our guests a chance to plug themselves and things they're doing. Where can we find you on the line? Where can we purchase your wares? You have excellent wares for sale.

CD: Yeah, thank you. So Talking Math With Your Kids is the blog and also the online store where tiling turtles and pentagons, hexagon puzzles for small children that have widely varying examples of hexagons, are all available there. The Twitter feed is trianglemancsd. Unfortunately, triangleman was already taken by the time I got to Twitter like 12 years ago, and so I had to tack my initials CSD Christopher Scott Danielson.

KK: But not by They Might Be Giants. So who took triangleman?

CD: Yeah, I don't know, some guy who never uses it. I think he lives in Florida. Never tweets.

KK: Sure.

CD: And yeah, by all rights, it should have been turned over to me long ago. But yes, the Twitter handle is in honor of both They Might Be Giants and my love of shapes and geometry. Okay. So that's the Twitter feed. Yeah, and public-math.org for some of the projects, we're up to over there, but you can get to it all through the through the Twitter.

KK: Okay.

EL: Yeah, thanks.

KK: Excellent. Thanks for joining us and for making us think about the fact that it's a theorem. That's, yeah, that's useful.

CD: Truly a pleasure. Thanks for having me on.

[outro]

On this episode, we had the pleasure of talking with Christopher Danielson, who works for Desmos and is involved with several programs to help kids have rich, creative mathematical experiences. Here are a few links you might find useful after you listen.
Danielson's Twitter account
Talking Math With Your Kids
Math on a Stick
Public Math
Math Anywhere
Evelyn's Page-a-Day math calendar, which takes inspiration for August 8's page from Danielson's book Which One Doesn't Belong?

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. My name is Evelyn Lamb. I'm a freelance math and science writer in beautiful Salt Lake City, Utah, where fall is just gorgeous and everyone who's on this recording, which means no one listening to it, gets to see this cute zoom background I have from this fall hike I did recently with this mountain goat, like, posing for me in the back. It kind of looks like a bodybuilder, honestly, like really beefy. But yeah, super cute mountain goat. So yeah, that really helpful for everyone at home. Here is our other host.

Kevin Knudson: I’m Kevin Knudson, a professor of mathematics at the University of Florida on the internet, they would call him an absolute unit, right?

EL: Definitely. At least they would have five years ago. Who knows these days?

KK: Shows how out of touch I am. That's right. Yeah, here we are. Yeah, it's actually lovely in Gainesville. Like I've got short sleeves on, but it's like 75 and sunny and just everything you want it to be.

EL: Perfect.

KK: And tomorrow, tomorrow's homecoming at the university, which means that it's closed — this is bizarre — for a parade. But as it happens, tomorrow is also my birthday.

EL: Wow!

KK: So I get the day off, and it’s unclear what I'm going to do yet.

EL: Well, just having a day off to lie in bed as long as he want, you know, drink your your coffee at a leisurely pace.

KK: Absolutely.

EL: It’ll be great. Yeah, and we are recording this shortly after Hurricane Ian. And so you're here, so you made it through okay. I actually don't know my Florida geography well enough to remember where Gainesville is.

KK: Gainesville is north central. And weirdly, this cold front sort of pushed just south of town right before. It was about 65 degrees for three or four days, which is freakish. The hurricane, of course, took its very destructive path entering around Fort Myers, went across over Orlando, then to the Atlantic side. We got about a half inch of rain. It was — I mean, we were expecting, like, 10 inches, and then that that weird path happened. Of course, a lot of a lot of our students, you know, their homes have just been devastated. It's a rough time, but you know, the governor and the President are at least putting aside their differences temporarily and making some good progress. Well, we'll see. It's gonna be a long rebuild down there.

EL: Yeah.

KK: And it's a beautiful part of the state, and I feel bad for everyone down there.

EL: Definitely.

KK: But yeah, it's not the first time, you know?

EL: Yeah. Well, yeah, I hope it it continues to progress on the cleanup and everything. And today, shifting gears entirely, we are very excited to have Kimberly Ayers on the show. Welcome, Kimberly, would you like to tell us a little bit about yourself?

Kimberly Ayers: Hi, thank you. Yeah, I'm super excited to be here. And happy early birthday, Kevin.

KK: Thanks.

KA: So I am an assistant professor in the math department at California State University San Marcos, which is about half an hour north of San Diego, so for those of you who are less familiar with California geography, and my research is in dynamical systems and ergodic theory.

KK: Cool.

EL: Nice. And I said “shifting gears” because I know you're also a biking enthusiast like I am.

KA: I am. Yes. I love to get out on my bike. And California weather is — San Diego weather, it’s hard to be outside. So I’m a big bike fan.

EL: Yeah, I — the other day, someone on a local social media thread was posting like, you know, “We shouldn't have good bike infrastructure in Salt Lake because, you know, we're not San Diego, so there's so little time that you can bike here.” And I was like, well, first of all, that's just not true. But you do live that dream of the, like, San Diego biking weather all year.

KA: Yeah, it’s — I can't complain about it.

KK: Sure. Well, it doesn't really snow that much in Salt Lake right. I mean, it hits the mountains. But yeah, I mean, so when I was a postdoc in Chicago, I cycled a lot, but come November I was finished, right?

EL: Yeah, because the roads just never get all the way clear, but here it's dry enough that they do get cleared. And so — you know, I am not an especially hardy person. But, you know, if you’ve got some layers on and the ice is off the road, it's actually doable.

KK: It’s not a problem.

EL: I discovered. I mean, this was a pandemic discovery because I grew up in Texas, and I would just put my bike away in, like, November here, but decided I mental health-wise that I really needed that during especially the height of that covid winter — our first covid winter. Anyway, lovely to have, I guess three people who enjoy biking on this show, but we are not here to talk about biking. We are here to talk about Kimberly's favorite theorem. So yeah, what is that?

KA: So my favorite theorem is a theorem called Sharkovskii's theorem, which is a pretty famous theorem in dynamics. To back up a little bit, when I talk about dynamics, right now I'm talking about discrete dynamical systems, where the idea is if you have a function that has the same domain and codomain, you can think about compositions of that function with itself, right? You can take successive compositions over and over again. And so as a dynamicist, I'm interested in looking at these sequences, like if I start with the point x in my domain, and then I apply f, so I get f(x), and then apply f again, get f(f(x)), and then f(f(f(x))), and so on and so forth, right? This is a sequence. And so we can ask questions about sequences, right? We can ask, like, do they converge? Or maybe if they don't converge, do they have a convergent subsequence? Do they ever repeat themselves, and that repeating themselves is actually what Sharkovskii’s theorem is all about. So Sharkovskii's theorem is about continuous functions on the real line. So it's important to say that there is at the moment, no, like higher dimensional-analog to Sharkovskii’s theorem. This only applies to one-dimensional functions. But Sharkovskii’s theorem says, okay, so we have to take a really weird ordering on the natural numbers. So we're going to start with all of the odd numbers except for 1. So starting at three, we'll take all the odd numbers in a row, so 3, 5, 7, 9, 11, so on and so forth. And then, once you're “done” with all the odd numbers, yes, then you'll consider 2×3, 2×5, 2×7. And again, all of those, right, and then 22×3, 22×5, taking higher powers of to multiplied by 3, 5, 7,.… And then again, once you're “done” with all of those, the only numbers that you haven't included are the powers of 2, so then you take all of the powers of 2 in descending order, until you get back to 1. So it's not a well-ordering, right? You have to kind of wrap your mind around this fact of like, once you're “done” with the odd numbers, then…

EL: Yeah, right.

KA: But it is a total ordering.

EL: Yeah, so you can take any two numbers, you can tell, like, which one is before the other one.

KA: Exactly.

KK: Right.

EL: But you can't say this one is 17th. Well, you can actually say which one is 17th in the series, but, like, a number that isn't an odd number, you can’t say what position it has.

KA: Exactly, exactly. If I were to count, like, the natural numbers in their usual ordering, you know that eventually, you're going to get— like, if you asked me are you eventually going to hit 571? Yes, I will. Right. But if I were to try to do this with the with Sharkovskii ordering, that’s not going to happen, right? So it's kind of weird that there's a minimal and a maximal element in the Sharkovskii ordering. So I haven't told you the punchline yet.

EL: Yeah, we’re just wrapping our head around.

KA: But yeah, exactly, right. So we start by taking this really strange ordering on the natural numbers. And then what Sharkovskii’s theorem says is if you have a period — so I guess I didn't quite define which way which the ordering goes, but let's say that 3 is the maximal element and 1 is the minimal element. So Sharkovskii’s theorem says that if you have a period N orbit for some discrete mapping on the real numbers, then for every M that's less than N, you also have a periodic orbit of that period. So, for instance, if you have a period 2 orbit, you have to have a period 1 orbit, which is what we just call a fixed point, right? If you have period 64, then you also have a period 32, 16, 8, 4, 2, 1, etcetera, and probably most excitingly, is that if you have a period 3 orbit, then you are guaranteed to have periodic orbits of any other period.

KK: Right.

KA: So sometimes people talk about Sharkovskii’s theorem, and what they say is period 3 implies chaos.

EL: Yeah.

KA: Now, I haven't told you what chaos is. And I kind of joke, actually, that chaos in the dynamics community is sort of a bit of a chaotic concept in and of itself. Because there is no really one universally accepted definition of chaos. There's several different types of chaos. There's what we call like Devaney chaos or Li–Yorke chaos. But this definition of chaos, which I believe is Li–Yorke, says that in order to have what we call chaos, you need periodic orbits of all periods. So there's that period 3 gives you that condition, you also need an orbit that is going to be dense in your space. So an orbit that kind of fills up your entire space. And then you need this other thing, which is probably the most famous aspect of chaos theory, which is the sensitive dependence on initial conditions, otherwise sometimes termed the butterfly effect, which basically says that if you have two points, no matter how close together, your initial points are, if you apply f enough times, their sequences, their orbits eventually grow some distance apart from each other, no matter how close together, they start. So there's essentially like no room for error if you have a chaotic system.

KK: Right, right.

KA: Sothat's why once you have period 3, you're guaranteed at least one of the requirements for a chaotic system, right? So my students asked me the other day, they were like, what's your favorite number? And I was like, Oh, that's a really hard question to answer.

KK: Three!

KA: But I think it has to be three because of this, like, you see a period 3 orbit, you kind of automatically get excited, because those are, like, pretty rare. So that is, yeah, I guess I have to say that three is my favorite number

EL: So we get a favorite number for free on this episode.

KA: A favorite number and a favorite theorem!

EL: Okay, so I want — I'm dragging a little bit this morning. And so having a little trouble with, you know, putting these things together. So what I mean, so we have this weird order on the natural numbers. Like sure, I'll let you do that. Can't really stop you. So what kind of — what is our f? What? What kinds of dynamical systems are we talking about here?

KA: Yeah, so let's talk about a couple of famous examples. So oftentimes, I'm talking, I've been talking about functions on the real line, but I guess really dynamicists really like studying functions on compact sets. Because if you have compactness, you're guaranteed things have convergent subsequences. So so we get what we call this limit behavior. So a lot of dynamicists study functions on, like, the closed unit interval. And so some examples are what's called the logistic map, which is a quadratic function. So it's some parameter R times x times 1−x. So it looks like an upside down parabola, right, it intersects the x-axis at 0 and 1. And then in order to make sure that you map back to things in between the 0 and 1 interval, we're going to require that R be between 0 and 4, right?

KK: Right.

KA: If R is bigger than 4, then the top of that parabola bumps up above 1, and, and there are actually cool things that you can study with that as well. But I'll talk about that another day.

KK: Oh, the bifurcation diagrams. That's what you want to talk about. Right?

KA: Yeah, once you vary that, so that's the cool thing, is like as you vary this parameter R, if you start at 0, and then kind of think about what happens is you increase R a little bit, you see these periodic orbits appear in exactly the order that Sharkovskii tells you, they're going to happen. So you start with a fixed point, so that's your period 1 orbit, right? And then I believe it as once R gets above 3, that's when that period 2 orbit shows up, right? And then if you bump up R a little bit more, then you see a period 4 orbit appear, and then a period 8, and we call this a period doubling cascade. And then, of course, as you can imagine, there's only finite room for R, right, we only consider up to 4, but there's infinitely many kinds of periodic orbits that have to appear, so they have to start coming at you faster and faster and faster. And then once R gets above — I want to say it's roughly 3.87, and it's only a number that we've sort of gotten that numerically, unfortunately — then you enter what we call the chaotic regime, which means that period 3 orbit shows up. And then we know we have out there somewhere periodic orbits of all periods. So people might be familiar with this bifurcation diagram, which shows kind of like, where the periodic orbits appear. And it starts off, like on the left hand side, looking pretty simple and smooth. But as you go to the right hand side, it gets more, like, fractally looking. And that's because, once again, those periodic orbits have to show up more and more and more quickly, right, as R increases in value. It’s very cool.

EL: You are just blowing my mind, that there isn't just, like, what I thought was going to happen here was like, we would only get these powers of 2 ones. And then you're going to tell me a different dynamical system later that would have — but like, because it makes sense, it somehow makes sense to be like, at 3, we get this, at, you know, 3 1/2, we get this, at 3 3/4, we get that and, like, the fact that, that like, we some at some point do shift to this period 3 place?

KA: Right!

EL: And we can't even know what the number is? I’m sorry!

KK: And these functions are so simple, right? It's just a quadratic. It's a tent map, basically. Right?

KA: I know! It blows my mind every time I think about it. Yeah, so okay, so you're seeing the beauty in this theorem now. Like, just how cool it is. And I guess I'm gonna, you know, shift gears, pun completely intended there, once again, because not only is the statement of the theorem really cool, but the proof of the theorem is really incredibly beautiful. And so I'm going to do my best to explain it without being able to draw anything. But let's talk specifically about the period 3 implies periodic orbits of every other period, that kind of statement right there. So let's suppose that you have a period 3 orbit, so you've got three points on the real line, call them a, b, and c, such that f(a)=b, f(b)=c, f(c)=a, so they go in this cycle, right? You can label — so that's going to partition, at least between those, into two different intervals, right? So let's say that you have a, b, and c arranged from left to right. And I'm going to, for purposes, it's going to be a little bit weird. But there's a reason I'm going to call I1, the interval between b and c, and I2 the interval between a and b. So what you see happening, then, is that because of continuity, that interval I1, when you when you think of what it maps to under f, it has to cover that entire interval of I1 and I2, right? And also I2 maps to something that at least contains I1. Right? And so you've got this, I'm gesturing a lot with my hands, which I realize nobody can see.

KK: We do this all the time.

EL: Just to slow it down a little.

KA: Yeah.

EL: So the I2 has to cover I1, it’s kind of obvious-ish if you've drawn this like I have.

KA: Right, because you think about where the endpoints go.

EL: So yeah. Can we slow down on this I1 thing?

KA: Yes. So remember, b goes to c and c goes back around to a. v EL: C goes to a. Okay.

KA: Right? So if you think about what I1 does, is it kind of flips upside down, and then maybe kind of stretches and comes back around, right? And so once again, I'm assuming that f is a continuous function, so you're guaranteed fixed points when you map an interval to itself, right if you have a continuous function. And so since I kind of have this structure, what I'm going to do is I'm going to draw a graph, a directed graph, where my two vertices are going to represent I1 and I2. And then the way that you can kind of visualize this is that I1 has a loop that maps, because you think about I1 kind of covers itself. Right? There's an edge that goes from I1 to I2, because we said that I1 covers I2, and then there's also an edge that goes from I2 to I1. Okay?

KK: So you’ve got a directed loop in this graph.

EL: Yeah, a lollipop.

KA: Exactly. And now the really cool thing is you can create closed paths of any length in that graph, right, just by starting at I2, going up to I1, and going around, I1 as many times as you need and then coming back to I2. Right? And that — traversing along that graph is essentially what tells you about what the periodic orbits are doing. There's some analysis here, when you think about intervals mapping to themselves and guaranteeing fixed points. And so we say a periodic orbit sort of follows this loop if it kind of travels between these intervals, and then comes back to itself.

EL: Yeah, so you've got like some other theorem kind of sitting in the background saying, like, some interval along this has to — or sorry, some point in this interval, has to do this precise thing.

KA: Exactly. And you can prove that via, I think it's a result of the intermediate value theorem, because your function is continuous.

KK: Brouwer fixed point theorem, essentially.

KA: Right, exactly. You're essentially applying that and then you're just walking along this graph, and all you need to do is make sure that you start and end at the same point, in order to get that periodic-ness. And so then the punch line is that once you have this graph, now I can create, you know, a path, a closed path, of any length that I want, just from the existence of that period 3 orbit.

KK: Yeah. Okay.

KA: And so I really love that,— you know, I'm not a graph theorist. For a long time, I was calling closed paths loops, and my graph theorist friends got really mad at me, told me I had to stop doing that. So now I'm like, okay, closed paths. But I love that it builds — you look at the structure of the graph, and you kind of just read it off from there. And so the proof of the entire Sharkovskii’s theorem basically constructs these graphs with these intervals, and looks at which intervals map to each other, and then you draw the edges as they are needed. And then you, again, you just read off, like, what length closed paths can you get from this. And it’s so cool.

EL: Yeah, there's something so appealing about this, because, you know, you see this, and you're like, oh, man, I'm going to have to find a fixed point. Okay. Is it going to be, like, 3/4 of b plus? Like five? But no, you just have to lose all of this.

KA: You don't need to find it. You just know it's there yet?

EL: You can almost forget the whole dynamical system!

KA: Exactly.

EL: The whole actually nitty gritty details of what's happening with this dynamical system and just say, like, oh, look, I made my little lollipop.

KA: You just take all that information, and you put it in the graph, and then you can kind of almost forget about the dynamical system and just look at the graph.

KK: Well, until you want to actually find this the orbit, right? I mean…

KA: Yeah, well, sure. And that is a much more challenging question, right? Because these orbits are often what we call unstable, which means that other orbits don't converge to them. And so in order to find them, you kind of have to start, like, exactly on top of them, right? And that's very hard to do.

KK: Right, because because of the chaos business?

KA: Exactly, exactly. There’s zero room for error, literally.

EL: And so was this a love at first sight kind of theorem for you?

KA: Oh, absolutely. Once I learned it in grad school, I was just really taken away with — I started by being really captivated by this ordering on the naturals, because I never thought to, like, rearrange the natural numbers — you know, the usual ordering seemed good enough to me. But like, oh, we can rearrange in this ordering, like, maybe is — sorry, my dog is groaning behind me. I hope that that's coming through. We can take this different ordering, and it still like, makes sense. I can still compare any two natural numbers. But it just completely like violates my intuition. And then I saw the proof and I was like, oh, that's just beautiful. Taking all of this information and reducing it down to this directed graph and then just reading off the graph is just so cool to me.

EL: Yeah, so I've kind of maybe opening Pandora's box here, but, so we just did the period 3 implies the all other periods kind of thing. So I — so is the proof, if you wanted to show that, like, period 7 implies period 9 or something, do you do the same kind of thing?

KA: It’s the same idea. Yeah, you draw again, you, you think about a period seven orbit, you label the intervals in a very specific way, and you have to be a bit careful about the way that you label the intervals to make sure which intervals are going to cover which ones. And then again, you draw the graph, and then you can just read things off the graph.

EL: Okay.

KA: So that's exactly how it works. And so there is kind of a generalized graph that has structure that's a little bit too difficult for me to explain right now. But, you know, we'll say, like, take an arbitrary natural number k, we can generally describe the structure of what that graph is going to look like, and then we can describe what the lengths of the closed paths are going to be.

KK: Right, right.

EL: Yeah. Okay. So cool. So, like I said earlier, well, before we started recording, I've heard this “period three applies chaos,” it’s such a great tagline and stuff. And I always nod, like, yeah, yeah, it totally implies chaos. You know, I have looked at this a little bit, but I'm really happy to get to know a little more about what this theorem of actually means so I don't have to just pretend I actually understand what’s going on anymore.

KA: Yeah, it is a really cool, like you said tagline to this theorem. It's very punchy, and succinct.

EL: Yes.

KK: Yeah. Very cool. So okay, now, I’ve got to know. So the other part of this podcast is, we ask our guests to pair their theorem with something. So what what pairs well with, with Sharkovskii’s theorem.

KA: Okay, so I gave this a lot of thought. And I think I came up with a really great pairing. I don't know if you have all ever watched taffy pulling videos?

KK: Absolutely.

EL: Yeah, I actually wrote an article.

KK: Right. I remember this.

EL: Yeah. Dynamical systems in taffy pulling at one point.

KA: Yeah, so I will say I don't personally particularly enjoy taffy. I think it's sweet and sticky and makes my teeth feel kind of weird. But I could watch videos of taffy being pulled for hours. So the idea, and I you know, I don't make candy. So I don't necessarily know what I'm talking about. But my understanding is, they have this big mass of sugar that they've boiled. And they need to basically aerate it, they need to introduce air into it somehow. And so you can do this either on a machine or by hand. But basically, the taffy has to be stretched and then folded back on itself and stretched and folded back. And they just do this over and over and over again. And I think about like, well, that's exactly what we're talking about doing with the unit interval when we talk about a continuous function on the unit interval. We're kind of deforming the unit interval somehow and then putting it back on top of itself. And then just doing that over and over and over and over again.

KK: Right? Yeah, so.

KA: Go ahead, Kevin.

KK: No, I was just going to say, this reminds me, so Evelyn, I think had this diagram of the taffy pulling machine in your article, right? It is really fascinating. That's how these sort of it's 3 things. Yeah. 3 shows up everywhere, right?

KA: Yeah, and now again, I don't know anything about this, but I wonder if that's intentional. And what's also really cool is sometimes they'll put, if they’re going to dye the Taffy a certain color, they put a little bit of coloring just somewhere on the taffy, and then as the taffy gets pulled, that color works its way through the entire thing. And I like to think of that as an analogue to the dense orbit: you start in a very concentrated little area, but slowly this dye works its way throughout the entire the entire mass of candy.

KK: No, I think that's actually an excellent analogy. That's exactly what's happening. And so I wonder if — the inventor didn't know this theorem, of course. But somehow taffy pullers intuitively knew that 3 would do the trick.

KA: Three is the one that would work.

KK: Two’s not enough to just going to do what we want, but just sort of flip it over itself, but three will really you know, yeah, braid it at least.

EL: Yeah, it's cool. It was so long ago at this point that I wrote that that I can't remember the punchline of my article. I might need to go find find my article and read it again. Be like, oh, that person really wrote so well! But yeah, that's fun, and actually as a kid I loved taffy. But as an adult with, you know, various tooth and jaw problems, it's not the very friendliest candy.

KA: No, it's not. But at least you can watch videos of taffy being pulled without having to actually eat it.

EL: That really is the fun part.

KA: And actually, there’s one more thing that I should tell you about Sharkovskii, that is very cool about it. There's this thing that's called the — maybe this is not quite correct what they call it, but they call it the converse Sharkovskii — which says that, so we have this ordering on the natural numbers. There is, for every tail of that ordering, there is going to be a continuous function that has exactly periods of those exact — periodic orbits of those exact periods.

EL: And, like, none before it.

KA: And none before it, every single one. So it's kind of like this sharp, you know, it goes both ways.

KK: So for every natural number, there is a dynamical system that has that many of that order, but then everything less than that in the Sharkovskii ordering, but nothing before.

KA: Exactly, but nothing that came before. So it goes both ways, in a way, right.

KK: So there's something with an orbit of order 57, but not 55.

KA: Exactly. Exactly.

KK: All right. Interesting.

KA: So I think that's also a very cool result.

KK: Dynamics is hard!

KA: Yeah, it is. And I feel like I'm constantly having to, like, I feel like my spatial sense is never very good. But maybe it's gotten better over the course of me studying dynamics more. I’m, like, constantly having to turn things around in my brain and, like, fold things over. And yeah, I love it a lot.

KK: So we also like to give our guests a chance to plug anything. Where can we find you on the worldwide intertubes?

KA: So I am on Twitter. My handle is @kimdayers. Some other cool projects going on is I've gotten I've been doing a fair amount with LGBTQ people in STEM. And so I did an interview over the summer talking to this organization called LGBT Tech, and you can find that on YouTube, just talking about my experience as a mathematician and my identity as a queer woman. And so that's something that I'm very passionate about. And you know, if anybody ever wants to talk more about that, they're welcome to reach out to me on Twitter.

EL: Nice.

KK: Excellent.

EL: Yeah, we'll include a link to that in the show notes. Check those out. Yeah. Thanks for joining us.

KK: Yeah, this has been great. Really great.

KA: Thank you so much. This was a lot of fun.

KK: Good. Take care.

[outro]

On this episode, we were happy to have Kimberly Ayers of California State University San Marcos on the podcast to talk about Sharkovskii's theorem. Here are some links you might enjoy perusing after you listen to the episode.
Ayers' website and Twitter account
Her interview with LGBT Tech
Tien-Yien Li and James A. Yorke's article Period Three Implies Chaos
Our "flash favorite theorem" episode, where Michelle Manes also professed her love of Sharkovskii's theorem
Evelyn's Smithsonian article about the mathematics of taffy pullers

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

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I like how we're always besmirching other math podcasts, which as far as I know, also don't have quizzes at the end. I am your host Evelyn lamb. I am a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. Okay, full confession. I don't listen to other podcasts, so I don't know if they have quizzes at the end or not.

EL: Shame. They probably don't. I mean, how would you even administer that?

KK: That’s right. That's right. Yeah. Yeah. We don't need to find this out.

EL: Yeah. Well, we are, we should, we should just get right.

KK: Let’s do it. Yeah.

EL: We are very happy today to invite Daina Taimina onto the show. So Daina, please introduce yourself, tell us a little bit about yourself, and we'll get started.

Daina Taimina: Hello. Thank you for inviting me. And, actually, I didn't prepare much to tell about myself. Because usually, I tell my students, you know, just search for me on the internet. That knows more about me than myself.

EL: Maybe even some of it’s true.

DT: Maybe, yes. Sometimes it's true. Yeah. So well, okay. Well, I was teaching about 20 years, I was teaching mathematics at the University of Latvia. And at about the same time, I was teaching in Cornell, where I, now I have stopped teaching. And I officially count as a retired, so it means I have free time to do whatever I want.

KK: Nice.

DT: Yes. And then sometimes, well, it has been now for 25 years, I have crocheted hyperbolic planes. And I guess that's what people know most. Because sometimes I am introduced on people. “Oh, yeah.” And then this person who is introducing me says, “You know, she's the one with hyperbolic planes.” “Oh, yeah! Yes, yes, I know that!” Okay, I guess that's my other name.

EL: Yeah. And our in your multitalented our listeners can't see it, but when we were saying hi we saw that you have some of your very own paintings in the background on Zoom, where I'm seeing right now, and they look very lovely. So you do art in addition to crochet and math?

DT: Actually, that was before, and I and the reason why I was doing I was doing art, is I signed up for a watercolor lessons because I knew that I'm very bad at art. Because when I was in school, I was told I can do anything but that. And at that time in Cornell, I was teaching students who were very afraid of maths, and most most of them were actually architecture art students or music students, and I really wanted to experience how it is to take some subject where you are told, and you believe all your life, that you are bad and you can’t do it. So I did it. And so yes, that was interesting experience.

EL: Well, and it looks like with some practice, you gained skills. Amazing how that works.

KK: I did that to actually about a year ago. I cannot draw. I'm terrible. So maybe we have the same issues here.

DT: Yes, yes. Yes, exactly.

KK: And I took it I took like a just a two hour drawing workshop online where we draw birds, and I actually drew something that looked like a bird at the end. So you know, it can be done.

DT: Yes, because this is what you what you learn, is that — and then I was explaining to my students, too —I brought in one of my paintings, and I said, it's actually, what I realize is that it was things which I knew. I knew, well, perspective. I knew how to do composition from photography, you know, just like doing some photography. And all I needed was, you know, I did need some technical skills, and that is the same in math. You do need to learn some technical skills, and then you can then you can get on, so it's not that different.

EL: Yeah, that's a great, a great lesson to learn and to help share with your students like, “Hey, we're all learning various things. I have this background, you have this background, but we can all improve in various areas of our lives.”

KK: Yeah.

EL: Well, the name of this podcast is My Favorite Theorem. So, what is your favorite theorem?

DT: Well, as I told you, my favorite theorem is Desargues’ theorem. Yes, and then, well, actually it started with some more ancient theorem, which was Pappus’s theorem. And it was somewhere, I think I was in middle schoo, and I was reading something in a math history, and I read that there is this ancient theorem, where if you are having two lines, and then you choose three points on each of them and those lines are non-parallel. Well, if they parallel than they are, that’s a very simple case. But if you have like two lines at some kind of angle, and then you choose, and then you then you connect in pairs points from these lines, and you always have three points which are on the same line. And I know like I was just like, that's — okay, I'm so old that at that time, there was no Geometry Sketchpad or any of these programs, there was no computer. So I just kept drawing these lines and finding those points, then it was just amazing. And then of course, I was like, “Oh, what else is there?” And that's when I discovered this, this Desargues’ theorem, which said, okay, if two triangles are situated so that three lines joining their corresponding vertices all meet at a single point, then the points of intersection of the two triangles’ corresponding sides, if those intersection points exist, all lie on one line. I couldn't — I read it, and I couldn't believe that. So again, I took a pencil and took a straightedge and I started to draw, like, various ways, and it was really finding these points and having them and, and then later learning that the converse of this Desargues’ theorem is also true, and then that’s a converse, theorem, that's also a dual theorem. So it was just so fascinating. So that was something, like, different from the geometry, you know, like, exactly the geometry we were having going to school, and so that's kind of led to perspective. And yes, I was just like, really it was fascinating.

EL: Yeah, this is one that has come up a few times for me in things I've read, or people I've talked to in the past few years. But yeah, I loved geometry for a long time, and this is not a theorem that I was exposed to, in most of my geometry education.

KK: Yeah.

DT: And it's very interesting that you can you can prove this theorem using another ancient theorem, Menelaus’s theorem — okay, I'm not going to talk about that — but that sounds very algebraic, because that uses uses proportions, and it's totally in Euclidean geometry. But I like the way how Desargues himself saw, and he actually was thinking about it in three dimensions, and then it's simple. When you are cutting, like, a triangular pyramid with two planes, and then it's just totally obvious.

EL: Okay, I'll have to sit down and try to visualize that a little better.

KK: Right, isn't the simplest proof, don't you use three, you have to go into 3d and then it sort of, like you say, sort of becomes obvious?

DT: Exactly, yes, yes. Yes. That's one of those. Yeah, that's one of those cases, and that was so great! You know, you just jump out, and then it's obvious. And then it's also, the other thing is, if you are having — so you know, like you can imagine that you are having a book, or though now you're having a point, and then you are projecting a triangle, and then all you do is, you imagine that those lines, that it stretches, and then all you do is you open it up in one plane. And there's the theorem.

KK: There’s the proof. Yep, I wish our listeners could have just seen that.

DT: I don't know how to describe.

KK: So this actually came up for you in school in Latvia? Like, your instructor actually taught?

DT: No, I believe I was reading something from Martin Gardner, or something outside, but I did have a wonderful geometry teacher.

EL: So you were interested in math very early in your education?

DT: It was just one of my easiest subjects. I was interested more in literature and languages. That's what you are saying, you know, like an art. Math, it’s just something that comes by nothing. It’s just simple, just seeing things.

KK: I mean, well, so Evelyn, maybe you had this experience. I mean, I became a mathematician because I was always good at math, right? It was the thing that I could easily do. And so it's sort of interesting that it was sort of the easy thing you could do, but you liked something else more?

DT: Yes, that’s true. Yeah.

EL: Yeah. I I had sort of a similar experience to you Daina, I think, where I was, you know,“good at math” — good at arithmetic, basically — in elementary school. And I liked the proofs in geometry, although I didn't understand that those were “real math” also. I thought it was just a diversion.

KK: The two-column proofs?

EL: Yeah, I liked the logic part of it. You're working it through, but I thought arithmetic was real math. And so yeah, I wasn't as interested in that. I was more interested in — I really liked science, but I did a lot of music also and stuff. But eventually, it wasn't until college, that I really kind of fell in love with it, and decided to devote my life to it in at least some form.

KK: Including podcasts at this point. So yeah.

DT: Well, you've been successful.

KK: Sure. So, have you used this at all? I mean, Is this a theorem that you use in your own in your own mathematical work? Or is it just something that you just love?

DT: No, I this is something which I love. Yes. Because, ya know, it’s — well, using it in teaching, you know, and sometimes I have used it talking in schools, you know, you go meet students and show them something, like, here are some fun and some beautiful things. But no, also I was teaching history of mathematics for many years, too. And I loved that Desargues himself, he never published his theorem. It was it was published by his student, Abraham Bosse. And I think he mentions exactly that Desargues had this three-dimensional proof. But also there was an interesting thing: When Desargues — he belonged to Mersenne’s circle circle, and that is the same circle where there were also Descartes and Pascal. And there was all this mathematical writing and just that exchanging of ideas, so that's fun. I was trying to find out, I know there was a some, in discrete mathematics, there is this 10-line configuration, and then that can be used to solve some problem. And I remember there was some sports problem, but I didn't remember like, you know, just precisely what. It’s interesting.

And then the other thing which I like, about this, not only this perspective, but that you can exchange points with lines and lines with points.

EL: Yeah.

DT: Yeah, so in some ways, maybe this is why I was getting more interested in mathematics, like what's beyond what we learned in school. This was this was like, I guess a very first example that I found that is something more about math, you know, more fun than you'll learn in school — which I wouldn't say that I was bored, but still, that there was something more.

EL: Yeah, well, and you mentioned exchanging points for lines, which, I always feel like I've sort of pulled one over on someone if you could do that kind of duality thing. And you know, move intersections of lines to a different line, and then the points intersect at where the lines were, something like that, and so yeah, that's always very satisfying, I think. So part of this podcast is that we ask our guests to pair their theorem with something the way you might pair food and wine or food and your favorite jazz CD or something like that. I’m not appealing to the youths if I say CDs. I don't know if Gen Z knows what that is.

DT: Yeah, it's like floppy disks.

KK: You have to say vinyl now.

EL: Or Spotify. But anyway, what have you chosen to pair with this theorem?

DT: With travel.

KK: Okay, yeah.

DT: So for me like it is exactly because you can change these points and lines. And then if we go back to that, what I was talking about ,projecting the triangle through that one point. And that's what I was imagining if I was that point, and then I'm looking on someplace on this first triangle. but when I travel, it really expands what I'm seeing when I get to that second triangle and see it in reality, and then I can go back. And that would be like, exchanging, so that would be this duality. From that place, now I can see myself differently.

EL: Oh, I love that.

KK: This is a very thoughtful pairing.

EL: Yeah. And where are some of your favorite places that you've traveled?

DT: Well, there are places like, I always like to travel to Sicily. So this is a very, very significant place for me, because that's where I met my husband, and then we had returned back there, and it's never too much to go back there. So I liked that, well, we managed to travel, and I liked my travel to South Africa. And actually, what I like in my travels is to meet people. And I think that's what those those big triangles when you project is, you meet the people and you talk with people, and I don't travel like without some purpose. And mostly it can be like really meeting people and doing some workshop or talk, and then of course learning about the place where I am.

EL: Yeah. And I always love how you learn both that the way you're used to doing things isn't the only way to do things, and that there are similarities across cultures, that we're all kind of the same in some way. So they're kind of those two contrasting ideas, but maybe they're dualistic in some way as well.

DT: Yeah, it’s like one of the things, since I was in Europe, and I have been knitting a lot. And I come here, and I see that there is another way of knitting, and I'm just staring and it’s just so totally different. But at the end, well, we get we get to same thing.

EL: Yeah, well, and since you bring that up, I did want to mention, as you said, you might be best known in the math community for your hyperbolic crocheting. And I know that Desargues’s theorem is from Euclidean geometry, not hyperbolic. But I would love to talk a little bit about kind of where hyperbolic crochet came from and how you got the idea to do that. Everyone loves it.

DT: No, I'm glad. It's just because I had to teach hyperbolic geometry, and, well, it happened. It's like this. So it’s summer, so in a month, it's 25 years since I crocheted my very first plane. And I'm really surprised people are still interested in that. And then it's kind of now a usual thing. It’s also like with Desargues’s theorem, you go into 3-d and then the hyperbolic plane, you can see only in 3d, because if you if you project then those are maps, like on in 2-d, but in 3-d and then particularly, you can just touch it. And you can do a tactile exploration. My main purpose was exactly for my students, so that they could touch and explore it. Because I saw a paper model, which was done, but one thing is to glue a paper model, which I did, but then once you fold it, for the next class, you have to do it again. And it was like, okay, no, I need something more durable. Yeah, so it was really for teaching. And then it's interesting that this hyperbolic plane started to teach me. When I got suddenly, unexpectedly for me, invited for the first art show, you know, now I had to learn, okay, what does that mean? So finding colors and ways to express, so and it has been going on and on.

KK: So, my wife actually crocheted me one once. So she, she can crochet really well, and it is remarkable to hold that thing in your hand. You can sort of begin to really understand how distances work in this weird, floppy, hyperbolic plane. It's really beautiful.

DT: Yeah, because it's another way to get our knowledge, because we need to feel it. It's not like when you are reading — well, okay, if you are an experienced cook, you can read a cookbook and taste a recipe, you know, like field tasting. But actually the best is that you try to cook it, and then you taste and then, you know, like, that's when you really learned about it.

EL: Yeah, and I think in my experience making hyperbolic crocheted planes, one of the the lessons, the math lessons I learned the most from it is just exponential growth is even more than you think. Because I think the first one I made, I started with something like 10 stitches, and I did it, maybe it was a five to four increase ratio. And, you know, five or six rows down, I was like, am I ever going to finish this row? Even as, you know, five to four to me, it doesn't seem like a big number. Like it's just barely above one. But if you, you know, multiply it by itself a few times, it takes a while.

DT: My hope is when the pandemic started, and then people were told that this virus is spreading exponentially, I hope that at least those who had crocheted hyperbolic plane instantly knew what it is what it means.

EL: Yeah.

KK: And then, of course, this also spawned this whole, like hyperbolic crochet reefs that they would do. So our librarians here actually organized an exhibition, they put it in the this display window in the front of the library. They got people to crochet coral, but it's still the same basic thing that you you came up with.

DT: well, it just it came like that, it all came up. So yes, it's actually that came up when — that was from my very first lecture to the general audience. And I was just thinking, Okay, well, one thing is to show the mathematical object, but then I need something, you know, in real life, too. And I remember that before that, first off, that's when I really was finding, like, lettuce leaves. And it was finding like, some curly things, and that gave an idea to see, you know, like, okay, here we go. It’s in a nature and then that just span us by not just the spin off. Idea. Yeah.

EL: Yeah. And I have to put in an advertisement for your book. Is it Crocheting Adventures in the hyperbolic plane? Is that the title?

DT: With.

EL: With hyperbolic planes.

DT: Crocheting Adventures with Hyperbolic Planes. Yes. And then there is another one, Experiencing Geometry, 4thedition, is open source. And there are a lot of hyperbolic planes in that one, too. It's on Project Euclid. And yeah, that’s open source. That was my husband's wish, that it would be open source. So that's — when I finished this fourth edition, that’s it.

EL: Yeah, and I just can't recommend — I haven't looked, I think I've looked at Experiencing Geometry, but not spent as much time with it as the crocheting one because around the time I, you know, a friend gave me a little crochet kit (not because of math at all, just because like, Oh, you like crafty things, you might like this), I happened to see your book. So it’s, like, this confluence, and your book has more than just making a plane. There are a lot of other interesting math ideas that you put in there. So I just can't can't recommend it enough. And it won an award for weirdest title, didn't it?

DT: Yeah, yes. Well, of course, that that's when — it was the strangest book title, the Diagram Prize — that went around the world. Of course, two years later, when I got an Euler Prize, which is much more serious, from the Mathematical Association of America, that press wasn't interested at all.

EL: Yeah, you know, the strangest title, I guess is a little little more headline-grabbing or something like that. But yeah, yeah. Anyway, I know we sort of went on a diversion from triangles, but I'm glad we got to talk about the hyperbolic crochet a bit because really, I think, so many people have had positive experiences with math and with experiencing geometry in a different way than just looking at a flat on a piece of paper thanks to you.

DT: Thank you.

EL: So, we've plugged your book, but we do like to give our guests a chance to share anything else that they'd like to share, other resources or websites or anything like that for people to, to look at?

DT: Well, when the book came out, I started to write a blog which is called hyperbolic-crochet.blogspot.com. But I'm not very good at keeping it up, so I’m not sure how many people are reading.

KK: Is anybody?

DT: But there is still lots of material and trying to answer a set of questions. And then of course, as I said, on the Project Euclid forum, it's on Project Euclid, look for Experiencing Geometry. So that is my newest book, which I’ve finished. And yeah, so as I said, a lot of interesting geometric things, too. If you are interested in geometry, that would be a good thing to look up, particularly for teachers, because I added one of the appendices, suggestions from various geometry projects, which you can do in class. Because those were questions people were sending me, like, okay, so what can we do in class and what can be some fun things we can do? So yes, I was trying to help. I hope it's helpful.

EL: Yeah. Wonderful.

KK: All right.

EL: Well, thank you so much for joining us. I was really glad to get to talk with you.

DT: Okay, thank you.

[outro]

On this episode of My Favorite Theorem, we were pleased to talk with Daina Taimina, recently retired from Cornell University, about Desargues's theorem. Here are some links you might find interesting after you listen.

Her website, blog, and Twitter account
Desargues's theorem on Wikipedia
Our episode with Annalisa Crannell, who also loves Desargues's theorem
Taimina's book Crocheting Adventures with Hyperbolic Planes, which won the Diagram Prize for oddest book title and the Euler Prize from the Mathematical Association of America
Experiencing Geometry by Taimina and David Henderson on Project Euclid

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Evelyn lamb, one of your hosts. I'm a freelance math and science writer in Salt Lake City, Utah, currently enjoying very beautiful spring mountains, which my guest and my cohost can see behind me in my Zoom background. And this is my co host.

Kevin Knudson: Hi, I’m Kevin Knudson, professor of mathematics at the University of Florida. My wife and I are off to California this weekend. So, you know, she's she's a book artist, and there's a there's a big biannual, every two years is biannual, right? Or is that semiannual?

EL: Maybe?

KK: Who knows?

EL: Biennial is also a word. Maybe they’re the same word? [Editor’s note: dictionary.com says biannual can mean the same thing as biennial (every two years) or semiannual (twice a year). The Oxford English Dictionary, on the other hand, says biannual means twice a year and biennial means every two years.]

KK: Every two years. Except that this two years is this two years is three years, because two years ago was, well, anyway.

EL: Right.

KK: So yeah, so it's called CODEX, and she is exhibiting there, and I am tagging along because I like Berkeley.

EL: Fun!

KK: And we're going to do stupid things like spend too much for a meal at Chez Panisse.

EL: That sounds great.

KK: Yeah, we're really looking forward to it. So anyway, but let's talk math.

EL: Yes. And we are excited today to be talking with Tien Chih. Tien, would you like to introduce yourself?

Tien Chih: Hi, my name is Tien Chih, and I'm currently an assistant professor of mathematics at Montana State University Billings, which is a comprehensive teaching school in the middle of Montana. But later on this fall, I will be joining the faculty of Oxford College at Emory University. Oxford is college separate from the main campus of Emory that's a small liberal arts spin off, so students can do a small liberal arts experience with the first two years of their undergraduate degree at Emory and then go to the main campus to finish, so I'm really excited for that.

KK: Yeah. And that's that that's a little bit outside of town, right? So Emory itself is in Decatur, correct? And the Oxford College is where? I know it's not right there.

TC: Oxford, Georgia, about a half hour or 45 minutes or so east, I think.

KK: Okay. Nice.

EL: Yeah. Well, I am excited that we got you while you're still in Montana, because I just love having guests who are also in the Mountain Time Zone because that they don't think I'm, you know, a layabout because I never want to do anything before 11am.

KK: This is a common problem we have.

EL: Yeah, I am excited, I'll be going to Glacier, or I'm working on planning a trip to Glacier National Park for this summer, which I know isn't actually that close to Billings, because Montana is enormous. But I think it will be very beautiful. Some of the pictures that you post sometimes are really beautiful with the scenery up there in Montana. I'm sure, maybe you're a little behind us on the spring timeline, but similar mountain beauty.

TC: Yeah, I went to graduate school, actually at the University of Montana in Missoula, which is much closer to Glacier. So while I was a grad student I managed to go up there a couple of times, and you're in for a really good time this summer.

EL: Yeah, I'm excited.

KK: Excellent.

EL: Yeah, well, let's dive into the math. What math would you like to talk about today?

TC: Okay, so my favorite theorem is not exactly a theorem, or is a theorem, depending on your point of view. But my favorite math concept that I'm going to talk about today is mathematical induction. There are a couple of reasons why I chose this. One is that I am a combinatorialist/graph theorist. In our field, we don't have a lot of big foundational theorems or theories. In our line of research, we tend not to build skyscrapers, we tend to sprawl. And so because of that there aren't like these big foundation things, so it's hard to point at, like, one theorem and say this is a key theorem in our discipline, but the idea of induction is always present in our work, and especially in my work. Another reason I like induction is because I do a lot of math outreach kind of things. I'm involved in the Math Circle community quite heavily. I run a student circle here at MSU Billings. And mathematical induction is one of those things that almost all students, even children, intuitively understand. The actual mechanisms and formal logic, of course, is not something most people are familiar with. But this idea of if you have something that's true, and you know, you can do this thing, and it's still true, that this means that it just keeps being true, that’s something that students inherently just grasp right away, especially if you compare it with visuals. So I think there's a lot of concepts in math that are like this, that are technically kind of difficult results to articulate, but intuitively, everyone already understands them to some extent.

KK: Right.

EL: Yeah. And you mentioned visuals. So what are some of the visuals you might use to describe induction?

TC: So one of the classic induction proofs that you would give as an example or an exercise in an intro to proofs class is the proof that the sum of the first n odds is n squared. And very often with as presented, that's done with the usual algebra, and the flip, and then the 2k+1 and all of that. But you can easily show that if you take a square, and you take a two by two square, you have to draw a one by one by one extra L around the original square to get the two by two square. And then you get the three by three square by drawing a two by two by one L around the two by two square. And then you just say, Okay, you keep doing that, right. And most students or people without formal mathematical training will recognize, oh yeah, that just keeps happening. But the “keeps happening” is induction, right? That's what we don't say out loud, but that's inherently in this reasoning.

KK: That’s a good visual, because as you say, the algebra — I mean, so when we teach our sort of intro to proofs course, this is where students really get their first taste of induction. And I think it's kind of cold, right? So you say, okay, yeah, so it works for the base case: check. You know, so 1=1. All right, we understand that. And then you say, assume it's true for k and then show it's true for k+1. And I think students learn this mechanically, but I'm never sure that they really grasp what's going on.

EL: Yeah, well, and you if you've learned it by manipulating, you know, like k+1, and, you know, multiply that out to square it or something, that sometimes does remove you from actually thinking about the concepts. I mean, it's an important way to be able to work, but if you have pennies that you're arranging on a desk, or cards, or something in a square, that can be a lot more like, yeah, look what's happening. Here's the next odd number, here we go. Yeah. So just maybe to back up for a moment. Maybe we should actually, state, you know, what induction is? Because it is I mean, it as you said, it's an idea that a lot of people will intuitively feel is correct, but might not have have actually seen as you know, this like packaged, you know, with a little definition bow on the top kind of thing.

TC: Right. So the idea of induction is you need two prerequisites. One is a statement, at least one typically integer value for which that statement is true. So you have a statement, let's say capital P, at an integer k, and we verify that P(k) is true. And then we couple that with an argument that shows that anytime something is true for, say, an n, it must also be true for n+1. So P(n) implies P(n+1), then starting at k, the statement is true for all n greater than or equal to k, so for k and then k+1, and then the implication gives you k+2, and so on. And then you keep going.

KK: And so the visual that I sometimes use with students is you know, it's imagine you have an infinite line of dominoes. If you knock down the first one, they're all going to fall down, right? Which isn't exactly correct, but it's a reasonable visual.

EL: There’s this TV show — not to derail totally, but I'm about to — I think, I don't remember if it's called Domino Wars, or Domino Masters, or something like that it [Editor’s note: It is called Domino Masters, and mathematician Danica McKellar is one of the hosts]. We were in a hotel room flipping through channels, and we saw this and they make these domino things, and in fact, sometimes not all the dominoes fall down when you push the first domino because there's some sort of problem in the the domino line of implications there. But in mathematics, the dominoes are all set up at you know, just the right distance or angle that they do fall down.

KK: Ideal dominoes, right?

EL: Yes.

TC: Although sometimes the dominoes, you can find arguments that that end up skipping a step in some of the dominoes. And so I don't know if you've ever seen these induction, like non-proofs. But I give this one as a challenge to my discrete math students: All cows have the same color.

KK: Yes.

TC: Yes.

EL: I don't think I — so this sounds familiar, but I don't remember what it is.

KK: Well, I’ll let Tien tell us, but I have a story about this. So when I was an undergrad at Virginia Tech and MAA members almost universally heard of Bud Brown, who was very well-known among the MAA community and really entertaining and won the Polya award several times for the things he wrote. And this was his standard joke about induction. He used horses instead of cows. But yeah, let us let us hear the theorem that all cows are the same color.

TC: So let's prove via induction that all cows are the same color. So we start with a base case, n=1, and we say, all right, if we have one cow it’s the same color as itself, ergo for a size of set n=1, all cows are the same color. And then, all right, we assume that it’s true for a set of size k and say, all right, so let's take a set of k+1 cows. Well, by induction, we know the first k cows are the same color. And by induction, we also know the last k cows have the same color. And so ergo the k−1 cows in the overlap, they're also the same color. And more importantly, the first cow has to be the same color as the k−1. The last cow must also be the same color as the k−1. So all k+1 cows are the same color, right?

EL: Yeah. Convinced me.

KK: Sure.

EL: We only see one color of cow.

KK: Right? But you know, for Bud’s joke, it was somehow he made this work where it's like, well, that's a horse of a different color. And now the trick, of course, is to get your students to understand why that proof doesn't work. Because it seems convincing. Right?

TC: Yes.

KK: It feels pretty good. So why don't you explain to our readers why it doesn't work.

TC: So it is a true statement at a certain point that this first cow — in a sense, a true statement that the first call has to be the same color as the middle k−1, as does the last cow. But of course, if k is 1 as in the base case, then that’s zero cows, so it's vacuously true. So the first and the second cow are not not the same color as the middle k−1. But that’s not useful information. So this is what what pops into my — visually, how I interpret this, is as that missing domino, right? We actually took away the second domino, right, the n=2 case, and so because that domino is missing, that induction obviously does not carry through and we have more than one color of cow on this planet.

KK: Yeah. And what's fun about induction is you never stop using it. So you know, you're a combinatorialist, you're clearly going to use it all the time. I'm teaching graduate algebraic topology this term and just maybe two weeks ago, I used induction. I was computing the cohomology of the Eilenberg-MacLane spaces of type K(Z,n). And it's a double induction, because you know case 1 and so you pull yourself from something odd to something even. And then it's a different argument for something even to something odd. So, it just never stops.

EL: So I guess I'm going to betray my naivety about like what combinatorialists do, but as a combinatorialist are you kind of mostly using like the regular induction, not transfinite or you know, some spicier flavor of induction. I don't know, do you have to induction — I’m even forgetting, there are a few different ones that you know go for, like different types of sets. So rather than the integers, you can do induction on real numbers, right, if you set things up right. But are you basically just doing it on the integers mostly? That's the only place I would want to do it.

TC: Yeah. Don't feel bad if you don't know what combinatorialists are doing. Most of us don't know what each other are doing. Right? Again, the sprawl that we talked about. Yes. So in my case, I try not to even deal with infinite things that aren't countable. I have to for some of my constructions, but I would prefer not to have to do anything uncountable. And so yeah, I'm only doing, me in particular, I'm only doing induction on integers. I'm certain if you define graphs on uncountable sets, or this sort of thing, then it would make sense possibly to invoke Zorn's lemma or those other kinds of induction-ish things, but that's not what I do.

EL: So well, the base version is powerful enough, I mean, I don't want to besmirch the fine name of induction. So you know, of course, this is important in your work. My big experience, really learning about induction in my first heavy-duty proofs class in college was a transformative moment, for me, in really being able to work with axioms and stuff. So it's also really important to me, maybe from a pedagogical — I mean, I was the one being pedagog’d at — but, you know, from a pedagogical point of view, I think it is also really important. Do you find that too? Is this a really important moment for your students?

TC: I think so. I think — again, like I said, the idea of induction is something that’s inherent in us, I think, but it's like this idea that you're able to keep doing something. And so when I cover induction, I kind of also use it as maybe one of the first examples where you try to you try to formalize what your intuition told you to do anyway. And there is often a tendency I notice when teaching entrance to proofs type classes that because the idea of writing proofs is such a new thing, that somehow, it's totally like — when they write these proofs, they want to use a lot of jargon, and symbols and theorems maybe that they learn. And it sometimes can get away from their actual understanding of what it is they're trying to prove. So when we start with induction, it’s like, look, you tell me why this is true without trying to prove it, just tell me why it's true. And our goal is to turn that into math, and then also clean up any loose bits or cases or whatever, along the way. But that's really what we're trying to do. So if we're trying to improve that, you know, again, the sum of the first n odds is n squared. It's like, all right, why is that? Tell me why I keep adding these rows, and it still keeps being a square. Okay, now what we need to do is just make that concrete. And that's such an important skill throughout, right, because you can't reasonably prove anything unless you have some idea of why this thing might be true. And then all we're trying to do with the formal writing is just say that. Say exactly what you're thinking, but very, very, very, very carefully.

KK: Yeah. So another thing we like to do on this podcast is ask our guests to pair their theorem with something. So what pairs well with induction?

TC: Let’s see if there's a couple of different pairings I could go with.

KK: You’re allowed more than one.

TC: Okay, sure.

EL: And if you have one, and then you have n of them, you're allowed to have n+1 of them.

TC: I don't think I want to go through the gamut of naturals. So when I first learned induction, I was an undergraduate, and I think I was drinking a lot of whiskey at the time.

KK: That sounds illegal, but we’ll allow it.

TC: And so that's one possible pairing. In the math outreach stuff that I do, back when we used to have it in person, I would often buy little granola bars and little single-serving bags of chips for the students to snack on while we're working on this stuff. And so that's a pairing I would give for working with induction with middle school students, which is the target audience for our circle here. And then, in my current work now as a grownup mathematician, I am making a lot of homemade noodles recently. And I’ve been really enjoying them. And so maybe that's what I would pair for for the induction that I'm doing right now in my own stuff.

EL: I like that. I mean, I would like to try your homemade noodles also. That’s always a lot of work, I think, to to make those at home. But yeah, I like the potato chips idea a lot too. Because, of course, there's the Lay's slogan. I don't know if it's still in their branding, “You can't eat just one.”

KK: Yep.

EL: Which works really well. Plus, for me personally, every potato chip tastes like one more potato chip.

KK: That’s right. Yeah, it truly is. You have wide and then you have to have another one. Forever.

EL: Yeah, but noodles. Noodles, I think are good too. Because, like, it’s hard to count the number of noodles there are. Plus delicious. Is there a noodle dish that isn’t delicious?

KK: What do you do with your noodles? How do you prepare them?

TC: So far I've stir-fried them and I've also made a beef noodle soup with them. Which are both excellent. And they're actually much easier to make than I would have thought.

KK: Do you have one of those pasta maker things where you crank them out? Or?

TC: No, I just roll them into a rectangular-ish sheet. And then I just dust it and fold it up and then

KK: You just cut it?

TC: Cut it with a knife.

KK: Okay, excellent.

TC: Yeah. So they come out a little thick, but I do prefer it that way anyway, so that works for me.

KK: All right. So I think Evelyn I already looking up flights to Billings.

EL: I’ll just drive. Yeah, I could be there—

KK: You’ll “just” drive? I mean, look.

EL: I'll be there in eight hours. I don't know how long it is from where I am.

KK: It’s got to be more than that.

TC: I’m closer to one of you now, but in a few months, I will be closer to the other one.

KK: That’s right. It’s like five hour to Atlanta.

EL: Yeah, well, this has been great. I really glad that we've we got to talk about induction which really has deserved an episode of My Favorite Theorem for quite a while. Would you like to let people know how to find you online or any anything you're involved in that you'd like to plug to make sure people know about it, anything like that?

TC: You can find me on Twitter at @TienChihMath.

EL: We’ll include a link to that.

TC: As you both know, I am a co-organizer for the online seminar Talk Math With Your Friends, which has collaborated with this podcast previously in the past for a live taping. And I don't have too much going on right now. Again, I'm in that very odd phase where I'm in between things, but I am very involved, like I said, in math outreach, in teaching, undergraduate research, that sort of thing, a very teaching-focused kind of career. And so I'm certain that I will have things along those lines happening soon in the future, once I get settled at my new position, so I'm really looking forward to all that.

KK: Alright, sounds great. Well, Tien, thanks for joining us.

TC: Thank you for having me. It was a blast.

[outro]

In this episode, we talked with Tien Chih, who will soon be starting a position at Emory University's Oxford College, about mathematical induction. Here are some links you might enjoy with the episode.
Chih's website and Twitter profile
Talk Math With Your Friends, the online math colloquium series he co-organizes (and with which My Favorite Theorem has collaborated!)
A wikipedia page dedicated to the proof by induction of the statement that all horses are the same color
Domino Masters, a TV show about dominoes

Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida, and I am joined by your other co-host person.

Evelyn Lamb: Hi, I am Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. And I was actually thinking we should have a quiz at the end of this one.

KK: We really should.

EL: It’s just so jam-packed. There's gonna be so many different things floating around. So, like, be prepared…actually don't because we haven't prepared a quiz for you, so we don’t want you to be disappointed.

KK: I’ll start writing the quiz now. Yeah, today we have an interesting new experiment that we're going to try. So Mike Krebs from Cal State University in Los Angeles reached out to us with an idea. Mike, why don't you just introduce yourself and explain?

Mike Krebs: Hi, my name is Mike Krebs. I'm a professor of mathematics at California State University Los Angeles. Graduation is tomorrow, and I think our students have had enough of quizzes, so thank you for passing on the quiz. Yeah, I listen to a lot of podcasts, and my origin story of finding your podcast is sometimes to find a new one, I will go to Wikipedia and click the “random article” button, and then whatever comes up, search to find a podcast on that.

KK: Okay.

MK: I found various things that way like the story of Sylvia Weiner, an octogenarian marathon runner, and so on and so forth. And then one time, I clicked “random article,” and up came a page on differential geometry of surfaces.

KK: Okay.

MK: And one Google Search later, I started screaming at my laptop, “There’s a podcast called My Favorite Theorem!” So, yeah, I discovered that at the time I was teaching, this past semester, a capstone course for our math majors, in which students select a topic and then have to write about it and present about it. And I said, “Oh, I wonder if the good folks at my favorite theorem would be interested in doing something like that with students.” So I recruited some students from that class, as well as a bunch of other students from our university. And here we are now.

KK: All right.

EL: That’s amazing. And so you're mostly graduating seniors about to graduate and you're spending the morning before your graduation with us? I feel so honored.

KK: I really do. This is something else. Yeah.

EL: Well, let's get to it.

KK: We have nine students. And so as Mike pointed out, there are nine factorial or 362,880 possibilities here. And we have chosen one of those orders.

EL: Yes. You know, if you so choose, you can always divide this into tracks and listen to them in every possible order and then get back to us and tell us what the optimal order would have been. But for now, it's the order in which they appeared on my Zoom screen. So our first guest today is Pablo Martinez Gutierrez. Great to have you. Would you like to say a little bit about yourself and let us know your favorite theorem?

Pablo Martinez Gutierrez: Hi, thank you for having me on the show. Yes, I'm Pablo. I'm currently a math undergraduate at Cal State LA, hoping to complete my Bachelor's, not this semester, but hopefully next fall next semester. And my favorite theorem that I'm covering for today is Euler’s formula and Euler’s identity. It's something that I got exposed to back in Professor Krebs’s class when I took his class for differential equations. He was teaching us about second order linear homogeneous differential equations. And in one class session, he introduced the topic of Euler’s formula and identity as a side gem. And I was like, “Oh my goodness, this thing is so incredibly beautiful.” The way that I learned it in his class was he introduced the mathematical expression ex as a Taylor series, and he expanded it out as a series. And then when plugging in eix, then you spat out that series and because of the i, something interesting happens where it starts to be, you could split it up into two individual, or two smaller series, so to speak, of cosine, and i sine x. So you would have the expression eix equal to cosine x plus i sine x. And that to me just seemed that for me, it was like I was gobsmacked. It was just baffling. It was incredible.

EL: Yeah, everything just falls out after that, right?

PMG: Yeah, you're seeing all these terms that come from math, you have e, that comes from compounded interest back when you're learning about it in algebra, you have sine and cosine, that are coming in from the unit circle and trig. And then you have i from complex numbers. So all those just coming in together is is like mind-boggling, right? And then if that wasn't amazing in and of itself, something interesting and amazing, even more amazing, happens when you plug in π for x, right? So you have eiπ is equal to cosine π plus i sine π. And so the cosine π just becomes negative one. And the i sine π becomes zero, which just goes away. So then you have eiπ equal to negative one. And then if you add one to both sides, you get eiπ+1 is equal to zero. And that's just — when I saw it, I was in awe. And I was just like, how do these things align and assemble so beautifully and neatly and concisely? It doesn't seem like, it seems crazy that it would happen that way.

EL: I have an unpopular, or possibly controversial opinion about this, which is, it's cooler to leave it with the minus one on the other side, instead of doing the plus one equals zero. Don't cancel me for my controversial Euler formula takes, but I’ve just got to put that out there.

KK: My favorite part about complex exponentials like this is that you can forget all of those sum formulas, right? Like, if you want to know the cosine of three theta, you just use the complex exponential. It makes your life so much simpler. So that's my fun thing. Okay, this is a really beautiful fact. So what have you chosen to pair with this fact?

PMG: So my pairing for this formula and identity is this. I don't know if anyone's seen the Stephen Hawking movie Theory of Everything. The ending scene of that movie has this musical score that I like to listen to, that evokes a similar feeling of elegance and beauty, and awe about the universe, which is the same feeling I get from this identity and formula. It's called the Arrival of the Birds by the Cinematic Orchestra and the London Metropolitan Orchestra. You can give it a listen on YouTube. And any you as you listen to it, it elicits that feeling of awe.

EL: Yeah, listen to it while you do some complex integrals, maybe. I like it. Yeah. Thank you.

KK: All right.

PMG: Thank you.

EL: Yes, well, the bar has been set high. But yeah, we will see — no, I won’t pit anyone against each other. Our next guest is Holly Kim. So yes, Holly, if you'd like to tell us about yourself, and tell us your favorite theorem.

Holly Kim: Hi. So my name is Holly. And I'm currently a grad student at Cal State Los Angeles. And I'm not graduating this semester, so I still have about, like a year or year and a half before I graduate. But I'm happy to be here. So thank you for having me on the show as well.

KK: Absolutely.

HK: My favorite theorem is currently Ore’s theorem from graph theory, which states that for a given graph that’s simple and finite, and for two vertices that are distinct and non-adjacent, if the sum of the degrees of those two vertices is greater than or equal to the total number of vertices of your graph, then the graph is Hamiltonian, meaning you can find a Hamiltonian cycle, meaning you can find a spanning cycle that reaches every vertex once it's in the graph. So that one is my favorite. And it's interesting because I was not a math person when I got my bachelor's. So when I took my first proof-based course, the professor quickly mentioned Hamiltonian graphs. And I had not seen graph theory, I think in that form, at least, ever. So it was really interesting at that time. And he had made a joke about like, “It's not the Hamilton that they made the musical about.” And around that time, I thought that was so funny because I was also listening to Hamilton the musical, or had started listening to it, even though it had been out for a while by the time I'd taken that course. But it just sort of stuck with me, and I thought Hamiltonian graphs, and Hamilton the musical, they’re just sort of like, every time I thought about it, I thought, oh, how fun and how interesting and how funny Hamiltonian graphs are. And then what makes them even more interesting is that unlike Eulerian graphs, where you can tell a graph is Eulerian quickly by looking at the — you know, it's if and only if every vertex has an even degree. So then you know that graph is definitely learning. The Hamiltonian graphs don't have sort of a defining characteristic, like Eulerian graphs. So Hamiltonian graphs are sort of elusive, like there are some theorems that will work for certain families, or types of graphs, but nothing that quite, I think, captures, yes, for sure every graph — or this graph is Hamiltonian if and only if these conditions are satisfied. So it's not been discovered or found out yet. So that's my current favorite theorem.

KK: I don't know this theorem. So this is sort of interesting, right? So it basically says that if you have two vertices in the graph that have enough edges out of them, basically, you're guaranteed a Hamiltonian cycle. That's just, that's pretty remarkable, actually.

HK: Yeah, and the proof has like a funny — it's like a proof by contradiction, but there are certain edges, like you cannot have, as you construct this proof, otherwise, you will end up having a Hamiltonian cycle. So it's like, you’ve got to have just enough, but not too many, or where you actually end up with another, like a Hamiltonian cycle kind of embedded in, in your graph. And so it's very fun. The converse is of course not true. You can see, the other direction would not.

EL: I'm sitting here trying to doodle myself a graph and see, but I think I think I need to do it a little bigger graph, because there weren't enough vertices in this one. And I can't doodle and talk at the same time.

KK: I can’t either. What’s that about?

EL: So yeah, no multitasking for me.

KK: So what pairs well, with this theorem?

HK: Well, it might be on the nose, but I'm going to do it anyway. I paired it with Hamilton the musical. And I've mentioned it before, but beyond just, it being a good soundtrack to listen to with just about everything, I thought, well, certainly, there must be a deeper connection I can draw between the musical and Hamiltonian graphs. And if you listen to the musical, a motif of it is that oh, Alexander Hamilton is just like never satisfied in terms of his goals and ambitions always wants to do more. And he's never at a point where he's like, Oh, I'm, I'm good. And I don't need to keep going. Just based on the musical. And I kind of thought that Hamiltonian graphs, they aren’t personifications of Alexander Hamilton, but that you know, there is nothing that quite satisfies them at this time, or at least as a whole, like Hamiltonian graphs as a whole. So, there is nothing that that would satisfy like, oh yeah, for sure, I am a Hamiltonian graph if these conditions are met. And so that was my connection. And if you want to go further, that symbol, like the iconic logo of Hamilton, or the Hamiltonian graph of Hamilton the musical, there is a star, which is isomorphic to a C5, a five-cycle, so my pairing was Hamilton the musical.

EL: I like that. Well, just in case you have not seen it, there is an excellent parody of it. (singing) William Rowan Hamilton (end singing) of the the Alexander Hamilton song, done by, I'm forgetting the name of the YouTube channel [Editor’s note: it’s A Cappella Science]. I think it's it's run by a guy called Tim Blais, B-L-A-I-S. So, yeah, check that out. I was actually, I was trying to write a parody, I just would always get in my head that (singing) William Rowan Hamilton (end singing).

HK: That’s so funny.

EL: Then I discovered some other person did it already. But they've got a bunch of people coming in. They've got, like, someone playing Emmy Noether, and a few other math contemporaries. So yes, definitely check that out. We'll include a link to that in the show notes. Yes, excellent. We're doing great here.

KK: Yeah. Who’s next?

EL: So thank you, Holly. Bryce Van Ross, welcome to the podcast!

BVR: Hi, thanks for having me. Excited to be here. I guess I'll introduce myself.

EL: Yes, please do.

BVR: My name is Bryce. Come tomorrow, I'll be a master's graduate in math from Cal State LA. So I'm pretty excited about that.

KK: Congratulations!

BVR: Thank you. Yeah, so the theorem today, I like learning a lot and learning new things. So in the process of choosing a theorem was like, find something new. I found the Hales–Jewett theorem, I believe that's how it's pronounced. The general idea is it's a combinatorial theorem, usually applied to game theory, and you start off with two positive integers, N and C. N would correspond to, for example, like a grid, right, like how many columns or how many rows you have in a grid, and C would correspond to the amount of players alternating in some game. So, the Hales–Jewett Theorem states that for any two positive integers N and C, there will exist some H-dimensional cube corresponding to that N and C. So by cube I mean, like an N by N by N by N H-many times, grid. Now, more than that, that theorem says that corresponding to that cube, there has to exist some, at least one, row, column, or diagonal that is all of the same color, which is pretty impressive. Yeah. And a great way to conceptualize that is, for example, like generalizing the notion of Tic-Tac-Toe, where you have a bunch of like Tic-Tac-Toe, grids as faces of some very big cube. This theorem tells us that, at some point, you're going to find a very long line that's either a row, a column or a diagonal where someone wins, guaranteeing that also someone loses. So that's the theorem I picked.

KK: Wow, okay.

EL: Yeah, so what what kind of — so you gave the example of Tic-Tac-Toe. I must admit I'm fairly ignorant in game theory. It's like, I sort of get, I like the part where you're like, Oh, you've got an N-dimensional, or H-dimensional cube. It’s the part where it's actually the games, that’s why I don't know that much about game theory. Because like, I'm a little bored by my that kind of thing. So what other kinds of games can show up in this?

BVR: Yeah, I'm also unfamiliar with game theory, never even like read a book on game theory or taking a course, but from my understanding, you could extend it towards notions are familiar games like Connect 4, any turn-based game such that it requires somebody satisfying a line of something of the same color.

EL: I know Game Theorists always have these super tricksy ways of like, oh, yeah, a chess game you could just make into this or, you know, Nim you can represent by this or, you know, any game you want, whether it's like a real game, like chess, or a math game like Nim. Yeah. Sorry, hashtag not all math games. But yeah, that's interesting. So you just kind of stumbled on this theorem in a some sort of curiosity rabbit hole?

BVR: Lots of googling at 2am. And Wiki-ing at 2 am.

KK: That's what happens. This sort of reminds me of Sperner’s lemma, right, where you try to, if you're coloring the vertices, you start coloring vertices, then you have to get a triangle with all the same color vertices. I wonder if it's sort of related to that in some way. Or maybe — I'm a topologist, so I’m always trying to think of ways to just turn this into a topology question. It feels like it should be one. But anyway, yeah, all right, so this is a good theorem. What pairs with it well?

BVR: Yeah, so I was trying to think of something where audiences can very much relate to. I'm a big fan of games and media. And I think several people in the world last year, watched the show Squid Game. And season two is upcoming. So Squid Game that's my pairing. The reason why it's my pairing is not just because game is in the title there, but also because for those familiar having seen season one, at certain points in the game, I think it was with marbles or something, they were like, oh, make your own game using these marbles. And I was like, if I were to like change up and impose my own ideas of season two, what if a player gets to choose a game, any game and make it and compete against any number of players. I think that the Hales–Jewett theorem is ideal because everyone would be intimidated by a giant cube of Tic-Tac-Toe. But if you know the theorem, then technically you have an advantage because you know, hey, someone's got to win and ideally, it's you.

KK: Okay.

EL: I must admit, I am too much of a weenie to watch Squid Game.

KK: Yeah, I never watched it either.

EL: From just — from even the SNL sketch that was based on Squid Game, I was like, Nope, not gonna watch that one. It looks too scary for me.

KK: Yeah, my wife really doesn't like anything violent. So we're watching a couple of things right now that there is some violence and she has to you look away. It's just really pretty rough. But I know it’s very popular.

EL: Our very brave listeners, I think, will enjoy that pairing.

KK: Okay, excellent.

EL: Well thank you. Oh, and Bryce, I know that you wanted to share an exhibit that you've got up right now.

KK: That’s right.

BVR: Yeah. Yeah. So I really value, for example, like the themes of this podcast, to give an opportunity for people of any background to get a different perspective on math. And I like that for a variety of topics within STEM. So I work as a library archivist at Cal State LA, and I'm developing a STEM exhibit for all faculty, students, etc, to just visit and change their minds. So there are going to be interesting math related artifacts, as well as just unconventional things. You're going to see, like, dinosaur bones and whatnot. And it will be debuting in the fall. So just wanted to give a hype for anyone interested, they could reach out to Cal State LA Special Collections and Archives to find out more, and it'll be open for everyone.

EL: Awesome. Yeah. I hope our LA-based listeners will check that out.

KK: Sounds very cool.

EL: Okay, thank you, Alvin. Alvin Lew? Would you like to join us? Sorry, I don't want to introduce people differently. Yes, please introduce yourself and let us know about your favorite theorem.

Alvin Lew: Hello, everyone. I'm very glad to be here. I'm Alvin I'm currently a third-year computer science major and math minor. Not directly just in math, but I've been doing machine learning research and kind of being at the intersection of two different subjects there. And so today, I'd like to share not so much a theorem, but I guess the actual theorem is that the cardinality of the real numbers is greater than the cardinality of natural numbers. And specifically, I really, really enjoyed the proof of Cantor's diagonalization.

KK: Class.

AL: And so I don't know, has anyone played the game in elementary school where each person is trying to come up with a bigger number?

KK: Sure.

EL: Yeah.

AL: And then someone would play the trump card of infinity, you know.

EL: And then infinity plus one.

AL: But I think what people don't realize is that there are actually different levels of infinity. And I remember watching a TED Talk video back in middle school, and my mind was so blown by the fact that there could be an infinity greater than a different one. And so having taken a more formal class last semester, talking about Cantor's diagonalization, I think it's such a beautiful proof, because it's so easy for even non-math majors to understand. And it just goes something like: suppose you can list everything out so that you can match each natural number to a real number. And what you'll find is that the real number you can write out as an infinite decimal, so a number with infinite decimal places. And so you just write all of them out, you pair each of them up. And what you ended up doing is you take one of the digits from every single real number, and then you create a new number based on that by changing up all the digits, and you'll find a contradiction that you actually didn't list out that number. And so just that very simple idea there leads to actually quite a few paradoxes. I think there's like a Hilbert hotel paradox if anyone's heard of that, and some other ones, but something so simple like that, just one idea, and it opens the kind of, you know, mathematics world where there's different levels of infinity. And for a normal person who's not too involved in math, I think it's actually very interesting that it's accessible. It's a simple idea that anyone can go ahead and check out.

EL: Yeah, yeah, that such a great one. And I remember that is the one when I was in undergrad, when I saw that and finally understood it, it was it was one of those like, mind blown kind of moments for me, a very special place in my heart.

KK: Yeah, yeah. And then you want all these other weird things like, you know, like the cantor set is also uncountable. It's kind of the same argument. Yeah, it definitely blows blows everyone's mind the first time. And but the second time, sometimes too.

EL: I’ll just think about it every once in a while be like, is that really true? Do I just need to add that number to the list? And then I'll fix everything.

KK: Well, yeah, it doesn't. You can you can prove that too. But you know, I saw some Twitter thread the other day, that was always trying to argue that there's some models out there where the or the reals are countable, you know, like, you can change your rules and get different answers. But I don't like that. I like my real numbers to be what I think they are. Right? Although, who knows what they are really. Anyway.

EL: So what what is your pairing for?

AL: So I was really actually thinking quite hard about how to match some concept of infinity with real life. And so I think some people might have heard the analogy of, like, you put a box in a box in a box or whatever. And so to me, that's actually very hard to actually even imagine, right? Because in real life, you could never actually make an infinitely small box. It's not practical. Theoretically, it's possible. So what I was thinking is I pulled the old mathematician trick and pair it with alphabets. Because mathematicians, every time we struggle to match numbers to something, we just slap a variable. We take the Greek alphabet, we take everything. And in terms of uncountable and countable infinities, we actually take from the Hebrew alphabet, right, the Aleph and the Bet characters to represent the two different sets. And so I decided, you know, by pairing it with different languages, I figure, if we ever do come up with infinitely, or more more discoveries like that, we can just slap another letter on it and hope it pairs up. And we'll have a new method of explaining something without without the numbers with the more general letter from from a random alphabet.

EL: I like that. Yeah, I've seen, you know, the Latin alphabet, the Greek, Hebrew, and Cyrillic. But yeah, we need to just the next time when one of you comes up with with some new concept that just needs you know, Greek letters are not sufficient for the amazement of this, we need to start using other alphabet.

KK: Oh, yeah. How about like those Southeast Asian ones? Like the like the Thai alphabet? That’s really beautiful and completely different. Right?

EL: Yeah, we’ve got a lot of options.

KK: We do.

EL: New goal. I like it. Thank you, Alvin. Very good. Okay, next up, we have Judith Landau. Judith, please tell us about yourself and share your favorite theorem.

Judith Landau: Well, thank you for having me. My name is Judith Landau, thank you for introducing me. I'm also graduating as an undergrad in math tomorrow.

KK: Congratulations!

JL: Thank you. And I'll be moving on to an interdisciplinary biology program, as a Ph. D. program. And I've been doing some research modeling biological systems. So my favorite theorem is actually the fundamental theorem of Markov chains, which is from the field of probability theory and statistics. And I'm very interested in Markov chains, because they are a way of modeling systems based off of probability instead of deterministically. So instead of saying we know what's going to happen next, we're going to say it's based off of probability. Markov chains can be represented by a directed graph. So a set of vertices with edges that are directed pointing to a specific vertex in one direction or the other. The outgoing edges of each vertex are actually, the that's the probability — sorry, the random variable associated with that vertex. So each vertex has its own random variable, its own probability distribution that tells you how likely you are to go to any other vertex, and so in a very simple weather model for down here in Southern California where we could only might only consider sunny, rainy and cloudy days, no snowy days, those could be our three vertices or our three states. And we could talk about the probability of moving between them from day to day. And so the fundamental theorem of Markov chains can be stated in many different ways. There are long ways and short ways to state theorem. But the simple way to state it is that an aperiodic irreducible Markov chain has a stationary distribution. So an aperiodic Markov chain for our sunny rainy cloudy model would basically mean that the chain is set up so that there is no regular period for any of those states. So there is no regular period between the sunny days, cloudy days, or rainy days. And the irreducibility just means that there is some path, directed path, obviously, between every vertex in that directed graph for this Markov chain, so you're able to reach every vertex from any other in some number of steps. And the stationary distribution: basically, for the Markov chain, if I were to give you a longer version of the theorem, the longer version of a theorem says that the stationary distribution for the Markov chain is actually the long-term probabilities of ending up on any given state. So in my example, sunny, rainy and cloudy. My pairing, if I can move on to that, is actually in biology. It's kind of a pun, because my pairing is proteins. Proteins are actually chains themselves, they’re chains of amino acids. And so since there's this dependency going on in Markov chains, where the next day's weather is dependent on today's weather, and that's a really characteristic idea in Markov chains is that there's this dependency from one state to the next, proteins, the order of those amino acids, is dependent on a gene, because genes are what code for proteins.

KK: Okay.

EL: I love that pairing. That's very nice. It it kind of makes me wonder, like if — I do not know anything about biology, the last biology class I took was in, probably you weren’t born yet.

KK: Ninth grade.

EL: But it makes me wonder like, are there things where like, it's a little more likely that you'll have a tryptophan after a lysine, than something else?

JL: Yes, we actually study that in bioinformatics heavily, what amino acids are more likely to be next to each other and things?

EL: Oh, cool! I’d never thought about that. But it's kind of like, of course there's going to be some sort of tendency, because it would be extremely improbable that it would all be exactly equal all the time.

KK: Right.

JL: And according to our theories, in biology, it's all related to the structure of that protein and how that structure will affect the function of the protein. And that's why proteins evolve as they do.

EL: That is so cool. Well, thank you for sharing that theorem and that biology connection, and I'm so glad that you are going on to study this more.

KK: Yeah, that's very cool. Yeah. Good luck.

JL: You too.

EL: All right. Next up, we've got Kevin Alfaro. So yes, welcome. Please let us know about yourself. And I see that you've got the Golden Gate Bridge behind you in your Zoom background. Are you from the Bay Area?

Kevin Alfaro: No, it’s just my favorite bridge.

EL: It’s a good bridge.

KK: I mean, I've stood in that spot and taken that picture. I think I think a lot of us have, right?

KA: Yeah. Thank you for having me. It's a pleasure to be here. I'm Kevin and I'm majoring in math at Cal State Los Angeles. And for my theorem, it's actually Archimedes’ theorem on his approximation of pi. So I'm going to be taking us back a couple thousand years.

EL: Yeah, I love it. I love the variety.

KA: Yeah, I'm a big daydreamer. So this is all something I could daydream about all day. And so what he was trying to do, he was trying to approximate pi. And what he did was since it’s included using a circle, he placed the hexagon inside the circle. And this hexagon had already included the radius of the circle as one of its sides. And this is because the hexagon is made of six equilateral triangles. So each side ends up being the same, and each side also corresponds to the radius. So he uses this to calculate the circumference of the hexagon. And since it’s inside the circle, he knows it has to be smaller than the circumference of the circle. So he, he knows that the circumference of the hexagon is 6R since it's six radiuses each, each part of the triangles. So already has a bound on that. He knows that it has to be greater than 6R. So now what he does is a series of calculations to try to get a bound. He tries to squeeze pi in two hexagons, a hexagon outside of the circle and a hexagon inside of the circle. So he knows that that circle has to be between these two hexagons. But the way he does it is so so laborious. Yeah, so for the for each of these. So he starts with the first hexagon, right? And it's a six-piece hexagon. So he has to build that up each time, he has to create a longer hexagon that corresponds to the actual circle. So he has to cut up each arc in half. And he does that. So he cuts every one in half, and he creates a midpoint. So he has to do the Pythagorean theorem for each one. And yeah, so he's just doing it like every day. So if you guys have like, a couple of days of free time, you guys could do this yourself.

KK: And you didn't have algebra either, right? I mean, he didn't actually have algebra. So he was — and these numbers didn't even exist to him like square root of two or whatever.

EL: Yeah, it’s amazing. I’m trying to remember, is the last level, was it, like, a 96-gon, or a polygon with 96 sides or 100-something sides?

KK: I think that's how far he got. Yeah, yeah.

EL: And it's just like, I immediately give up on hearing that. And I've got a little calculator in my phone that I carry in my pocket all the time. And like, I've just like, I've noped right out of doing that. But yeah, it's amazing. Yeah. So is that called the method of exhaustion, which I think is perfectly named?

KA: Yeah.

EL: Yeah, it’s so amazing to think about 1000s of years ago, the people like like Archimedes, and other people like having, having the wherewithal, the persistence, to go through and be like, Well, I don't have a calculator, and I don't know what pi is yet, so I guess I'm just going to, you know, figure out the length of this 96-sided polygon. I don't have anything to do for the next week. So did you first encounter this theorem? Or this idea?

KA: I actually got this from a really great book. It's called Infinite Powers by Steven Strogatz. So I recommend reading that book to anyone listening. It is good. Yeah.

EL: And so yes, what is your pairing with Archimedes exhaustive proof of — or exhaustive approximation of pi.

KA: My pairing is actually more of an abstract pairing. And it's more of an idea, because I think around the context of math around this time, too, and the context is always changing. And around this time, it was, it was more like a spiritual level of math, like people were more into it. And it was more spiritual. So then if we connect it to now, and how advanced math is, and we know what Archimedes was doing, since he could never find out the numerical value of pi, since it's an irrational number. And he couldn't do that back then. But he still knew that it was between two fractions, between 3+10/71, and 3+10/72, I believe. And that's all he needed to know. So he knew that it was between two numbers, but he can never really find out what exactly this number was. And I think that really speaks a lot about what math is, or what people do, in terms of knowing what we don't know. So in this time, he knew that he didn't know, which is a lot what, what's happening on today. And I think that's just a cycle of what we do for maths or for anything, really. We try to get to the point where we know that we don't know, and then we're done.

EL: That can be such a difficult part of basically anything in life. This whole pandemic thing we've been living through, like knowing what we do and don't know about what's happening with that has been such a challenge. It's like, oh, that really affects my life in a way that maybe knowing the exact value of pi doesn’t.

KK: Right.

EL: But yeah, it's so hard. And sometimes you think you know what, you know, what you don't know, and you don't realize what you don't know. So yeah, I like that kind of, from the very concrete polygons to this sort of abstract. Thank you, Kevin.

KA: Thank you.

KK: That’s a good one.

EL: All right. Next up, welcome to the show, Francisco Leon. Would you like to tell us a little bit about yourself and share your favorite theorem? Yeah,

FL: Thank you for the welcome. Tomorrow, along with Bryce, I'm graduating from Cal State LA through the master's program, so I'm really excited about that.

KK: Congratulations.

FL: Yeah, thank you. Yeah. Today, I want to mention, when I was asked, what's my favorite theorem, at the time, I really had one on mine from my topology class. I’ll state the theorem now. It says a topological space is regular if and only if for every element in the set and open set containing the element, that there exists another open set that also contains elements whose closure is in the first mentioned set. So maybe to label some of this, to get a better visualization, so we have some element, say P. And then say, some open set that contains it, U. Now in between P and U, this alternate characterization of regularity says there's going to have to exist another open set V, that also contains P inside of U whose closure is also inside of U. So it goes, maybe nested: P, V-closure, V, and then U.

KK: Yup.

FL: And that's equivalent for regularity. And you know, at first, this was a homework assignment from the class. And one of the key tools to do this problem is to understand that if you illustrate an element in open set, so you draw it ,right, you kind of draw a little point P, and a dashed circle around an open set, that's going to be equivalent to having P outside that complement. So you can draw P outside of a box and a box, just kind of know that, since of the complement of open sets are closed, that's why it's represented as a box, to be able to comfortably go between those two perspectives of the same situation, I found difficult to get at the time. And so, you know, I fostered an appreciation for this theorem because it made me resolve that difficulty. So it led me to some trains of thoughts that were really interesting for me at the time. And it's been a while since a theorem held my attention for so long. So just to share some lines of thoughts I was having when working through this exercise: okay, I was like, how do I go from being an element in this open set to being outside its complement? I started imagining myself inside of a room, just as the element would be inside the open set. And I'm like, how do I push the walls of this room so that suddenly the outside is contained inside these walls, right? Because you want to go from being inside the open set to being outside the complement. So how can I move this boundary, so that the outside is contained? I felt like I couldn't do it. I look outside, and I know the universe is huge. How can I push the walls of this room to contain everything else outside and suddenly be the one outside? So I had great difficulty. And then as I would think about this problem and look outside, I would see the window. And then I realized that the illustration of the element inside the open set, you know, when you have that point P and those dashed lines around it, those dashed lines aren't necessarily representative of a border, it's really representative of a window, because what you do at the illustration, is you look at what's within those dashed lines to see the elements. You are really looking into the set to see what's inside of it. So what I really needed was a window in this room for me to be able to understand the outside. So now if I reimagine the situation, I'm in a room, which is representative of the open set, and there's some element in here. What I need is a window so that I can see what's on the outside. And then if I change my perspective, if I pass through the window and then turn back around, then I see through the window, the element inside the room. And now I have the outside perspective that I want it. So now I see the element inside the room. So now I'm seeing the complement of the open set. And that really helped me change that perspective from being inside the open set to being outside the complement, which is a closed set. And then I was able to do that exercise pretty straightforwardly, but I was really struggling before that. And I never really think about theorems as visually as I just described, so that's why I really appreciate this theorem and that's why it's one of my current favorites.

KK: You know, when I was an undergrad, I was always going to be a math major. That was always what I was going to do. But that class, point-set topology, is when I really fell in love. I mean, I was already planning to go to graduate school, but my professor for that, Peter Fletcher, who passed away a couple of years ago, really put me on this path to being a topologist. I didn't know what I was going to do until I actually hit that. And of course, you know, that flavor of topology is pretty well settled and has been for quite a while. But it's still really beautiful to think about these basic properties of topological spaces. And I love your visual description of this sort of thing, like regular spaces. You know, I haven't thought about them in a long time, but I think you really nailed it right there. So nice. Yeah.

EL: And I like, I like seeing that little glimpse into how you were thinking this through, even if that isn't what you wrote down in your proof in the end. But it seeing — so often, when we read math, we don't, we don't see those little glimpses of how the person actually thought about it. We see the cleaned up version that, you know, is presentable for professional company or something, but I really liked seeing that. So what what is your pairing for this theorem?

FL: Oh, yeah, so I'm thinking maybe a hot apple pie, just because we typically do place it on a windowsill to cool down and I just figured that this theorem is cooler with a window.

KK: That’s good.

EL: And yeah, who can argue with a nice warm slice of apple? It's delicious. Yeah. Excellent. Well, thank you so much.

FL: Thank you guys.

EL: All right. Next up, we have Marlene Enriquez. Marlene, would you like to introduce yourself and tell us about your favorite theorem?

Marlene Enriquez: Yes. So hi, my name is Marlene, and my favorite theorem has to be the ham sandwich theorem.

KK: Yes!

EL: You've got company there with Kevin.

KK: So in episode zero, Evelyn and I had our favorite theorems. That was mine. Here we go. All right.

EL: So please tell us what you like about it.

ME: What I like about it is just, it's the first time that I encountered a theorem that really didn't have an exact answer to how to solve it. So the idea of basically having a line cut something and split it into equal volumes, per se. So when I first read it, I was like, oh, that sounds cool. When I read it some more. I was like, oh, no, this is really interesting. And it's actually the the theorem that I used to write the my research paper for my senior seminar class with Dr. Krebs. So the theorem, when I read it, I read it like this is like, you have a sandwich made out of bread, ham, and another piece of bread and like, they don't have to be exactly aligned. Or even touching each other. But there exists, a line, or a cut, a straight cut that will split the bread, ham, and the other piece of bread in equal volumes. And then when I saw that, like, wait, you don't really have to be next to each other, or like on top of each other, or even in the same room as each other? And you can still be cut and split into equal volumes? I was like, Oh, wow. I can read more about this, it’s pretty remarkable.

EL: Yeah, it can be the sloppiest made ham sandwich ever, and it’ll still work.

KK: Like you can have one slice of bread in Los Angeles. I have one here in Gainesville and a piece of ham could be in Salt Lake with Evelyn, and there is — it will take a big knife — there is a big knife that can cut them in half. That's right.

EL: Where did you first see this theorem?

ME: I was actually searching for a topic for my paper and I was online Google searching, every time at 2am, it always happens. And I kind of stumbled upon it. And I think I saw a YouTube video and I was like, Oh, this is interesting. Then I clicked to another one. I'm like, okay, this is very interesting. And then I just started searching and searching and reading and yeah

EL: Excellent. Yeah, I guess this podcast brought to you by the time 2am: the perfect time to be just finding weird math stuff to learn about.

KK: So what's what pairs with a ham sandwich there?

ME: Well, it's not on the nose, not a ham sandwich. But actually, when I was typing my paper, I was in my living room, and I have siblings. And they were fighting over a chocolate bar. They were saying, oh, like, let's split it equally. And they're like, No, you got the bigger piece or you got the bigger piece. And just the idea of sharing. Even when you go out with friends, you split the bill and stuff like that. That kind of just kind of brought it to, like, that's kind of like this ham sandwich. But like, in like, money, or a chocolate bar, who got the biggest piece, right?

KK: That’s a hard problem, actually, the whole equal division problem is extremely difficult. As I'm sure you found out with your siblings already over a chocolate bar.

EL: The answer is probably just giving up all earthly desires.

KK: Yeah. Right. The Buddha, the Buddhists understand.

EL: Well, thank you so much Marlene. And our final guest is Daniel Argueta. Thank you for joining us. And please tell us a little bit about yourself and your favorite theorem.

Daniel Argueta: Well, thank you, Kevin, and Evelyn, for having me and Evelyn, don't worry, you said my last name correctly. So I'm also graduating tomorrow. I'm getting my bachelor's in math.

KK: Congratulations.

DA: It’s a big thing. I’m first gen. So when Dr. Krebs approached me about this, I was like, What am I going to talk about? What's my favorite theorem? And when I was doing research for our capstone class, I actually stumbled across you guys’ podcast. And the one, I think it was episode 17 with Dr. Naomi Joshi or something like that.

EL: Nalini.

KK: Nalini Joshi.

DA: And it was the Mittag Leffler theorem. And that theorem was just way too complicated for me. Dr. Krebs helped me bring it down a little bit. And I stumbled across the infinite product convergence theorem. And I was like, okay, maybe that’s what I’m going to talk about. I wrote a whole paper on that. And then I thought about it and I was like, you know what, that's not my favorite theorem. So I started doing some googling, and I came across the headline “The theorem shook math to its core.” So maybe that might give you guys some ideas as to where I'm going to. And I'm actually doing I'm going to talk about Kurt Gödel’s, incompleteness theorems.

KK: Oh, yeah.

EL: Okay, yes, is a great way. I’d just like to say, of all the however many thousands of possibilities, I like that, at least that this one was at the end, to sort of be like, okay, we talked about all this math that we can do. Now let's finish it off with: Oh, yeah, we can't do math. That's probably not the best summary of the Incompleteness Theorems. So why don't you tell us about the incompleteness theorem?

DA: Well, what Gödel pretty much set out to prove, everybody at the time was trying to prove that math is perfect. And they were trying to say, hey, math is perfect. They were trying to find like this theory of everything, and Gödel’s all like, hmm, I don't know about that. And so pretty much what his proof did, and not to get super technical or anything, was he showed that in any formal system, there are certain truth statements about the system that can not be proven by said system. So a way to think of this is our classic logical problems with the knights and the knaves. In this case, you only need the knight, where you have a knight in shining armor who can only tell you the truth. So so you ask yourself, what is a sentence that this knight cannot say if he can only tell the truth? And one sentence is: I cannot say that sentence. You know, because if he can say the sentence, then it's true. But if he can't say the sentence, then you know, it's not gonna work. So I just really like this theorem, because to me, you know, we usually tend to think of math as perfect. It's, a lot of people call it the universal language, because math across everywhere is perfect. But to me, it's super interesting that the fact that we don't know everything about it, and we will never know everything about it. So that's why I really like this theorem. And it actually brings me to my pairing, which is — I know usually you guys do food on the on the podcast, but in this case, my pairing is our Final Frontier, which is space. I think I think it super coincides with it. Because, you know, we're going to always keep discovering more and more about space as our technology advances. But at the same time, our universe is ever-expanding. So will we ever completely know everything about space? There's so much to learn and discover, and the same can be said for math. And that's my favorite theorem.

EL: Excellent. I love that. Yeah, it's almost like okay, yeah, math is like, perfect, and you can do everything. Just don't look too hard. This incompleteness theorem stuff is just saying, oh, yeah, you think you could do calculus? Just don't look too hard.

KK: Right, right. Yeah. Well that’s everybody.

EL: This has been so fun. I’m so optimistic about the future of math right now. And biology. And whatever else you all do.

KK: And machine learning, and whatever it is yes.

EL: Yeah. Good luck to all of you in your next steps. It’s so appropriate that we got to do this right before many of you are graduating and, you know, taking those next steps. So thank you so much for sharing part of your, Monday morning with us.

KK: And thanks to your professor, Mike, for reaching out to us and making this happen.

MK: Well thank you for hosting everybody. This is great.

EL: Yes. I don't know if you all want to unmute it and do a goodbye. I don't know. How do you end a podcast with 12 people? (I just did multiplication there.)

KK: Is there a CSULA kind of cheer? Like I'm at the University of Florida so it'd be like a Go Gators kind of thing. They’re looking at me like, no, we don’t do that.

EL: Math majors might not be the best choice for for you know, knowing all the sports things, not to invoke any stereotypes.

KK: I know every word of the Virginia Tech fight song.

EL: Well, it may not be universal.

KK: Maybe not. All right. Well, you’re all unmuted. Maybe say goodbye.

(General chaos of 12 unmuted people on Zoom.)

[outro]

In this Very Special Episode of My Favorite Theorem, we were excited to welcome nine students from California State University Los Angeles, along with their professor Mike Krebs. Each student shared their favorite theorem and a pairing with us. Below are some links to more information about their favorites.

Pablo Martinez Gutierrez talked about Euler's formula and identity, which connect trigonometric functions with the complex exponential.

Holly Kim told us about Ore's theorem about Hamiltonians in graph theory. Check out A Capella Science's Hamilton parody video!
Bryce Van Ross shared the Hales–Jewett theorem from game theory.

Alvin Lew chose Cantor's diagonalization proof of the uncountability of the reals, which was also a favorite of our previous guest Adriana Salerno.

Judith Landau shared the fundamental theorem of Markov chains, which relates to her work in bioinformatics.
Kevin Alfaro talked about Archimedes' approximation of pi.

Francisco Leon chose a theorem from point-set topology about regular spaces.
Marlene Enriquez chose Kevin's favorite, the ham sandwich theorem.
Daniel Argueta finished off the episode with Gödel's incompleteness theorems.

Kevin Knudson: Welcome to my favorite theorem, a math podcast with no quiz at the end. I'm Kevin Knudson, a professor of mathematics at the University of Florida. And I'm joined by my other host.

Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah, where we just got a lovely few inches of snow last night. So I've developed a theory that podcasts cause snow here. Although it could be the other way. Maybe snow causes podcasts.

KK: Maybe.

EL: It’s hard to tell.

KK: I don't know. It's 85 degrees for today. Sorry.

EL: Yeah. I meant to say don't tell me what the weather is in Florida.

KK: It’s very nice.

EL: It’s too painful.

KK: It’s very nice. Yes, speaking of painful, we were having our pre-banter about I had a little hand surgery yesterday, and I have this ridiculous wrap on my right hand, and it's making me kind of useless today. So I have to do anything left handed, and to control everything on the computer left handed. But you know, it'll resolve my issue on my finger. That'll be good. So anyway, today, we are pleased to welcome Dave Kung. Dave, why don't you introduce yourself?

Dave Kung: Hi, there. I'm Dave Kung. I'm a mathematician by training. I spent 21 years at St. Mary's College of Maryland, and I've recently moved on from there, and I work at the Dana Center, the Charles A. Dana Center, at the University of Texas at Austin. I work with Uri Treisman down there on math ed policy.

KK: It’s very cool. So you got a really serious taco upgrade.

DK: Definitely. I'm still living in Maryland, but I get to visit to Texas every few months.

KK: Are you gonna relocate there? Or are you just gonna stay in Maryland?

DK: I’ll be here for now. Yeah.

KK: I mean, it seems like the sort of work that you could do remotely, it’s policy work mostly, right?

DK: A lot of policy work. There's going to be a fair amount of travel once that's more of a thing, but not all of it will be to Texas. Some of it will actually be in DC, in which case, I'm pretty close.

EL: Yeah, you’re right there.

KK: Yeah. Yeah. Well, they do really great work at the Dana center. I've been involved a little bit with the math pathways business, and it is really vital stuff. And of course, Uri is, like, well, he's the pied piper or something else. I don't know. But when you hear him talk about it, he's an evangelist, you really you can't help but like him.

DK: Yeah. And making sure that we know that students have the right math at the right time with the right supports, we're far from that goal right now, but we can get closer.

EL: Yeah, very important work for all mathematicians to care about.

KK: Yeah, yeah. Okay, but I think we're going to talk a little bit higher-level than math pathways today. So we asked you on to have a favorite theorem. What is it?

DK: My favorite theorem is the Banach-Tarski theorem, which is usually labeled the Banach-Tarski paradox.

KK: Yes. Yeah. So what is let's hear it. Well, let's let our listeners know.

DK: So the Banach-Tarski paradox says the following thing: that you can take a ball, think of a sort of a solid ball, and you can split it up into a finite number of pieces — we’ll come back to that word pieces in a bit — but you can split it up into a finite number of pieces, and then just move those pieces and end up with two balls the same size and the same shape as the original. It is incredibly paradoxical. And I remember hearing this theorem a long time ago, and it just sort of blew my mind.

EL: Yeah, it was one — I think I was an undergraduate, I don't think I'd even taken, like, a real analysis class. But I heard about this and read this book, there's this book about it that I think it's called The Pea and the Sun. Because another statement of — I mean, once you can make two of the same size things out of one, you can make kind of anything out of anything.

DK: Just repeat that process. That's the other statement, you take a pea and you do it enough times. And if you do it, well, you can reassemble them to form a sun. Absurd.

EL: Yeah, I just, I mean, reading it, there was a lot that went over my head at the time, because of what my mathematical background was, but at first, I was like, Okay, this means that math is irrevocably broken. And then after actually reading it, it’s like, okay, it doesn't mean math is broken. So maybe, maybe you should talk a little more about why it doesn't mean math is broken. If you have that perspective. Maybe you do think math is broken.

DK: It certainly feels like. I mean, I think my first reaction was: Cool. Let's do that with gold, right? We'll just take a small piece of gold and split it up and keep doing that. I think there's a lot in this theorem, right? And so I mean, it's one thing to understand it, sort of at a deep sense why it's not absurd. And I think it helps me to think about just sort of, you know, once you know a little bit about infinity and the fact that there are different sizes of infinity. And once you know that somehow the even numbers, the even integers, have the same size as all the integers, which is already sort of weird, this feels a little bit like that. I mean, you could sort of somehow take the odd integers and the even integers, and each one of those is the same size as the full integers. But some of the integers themselves, it's weird to be able to split the integers up into two things, which are the same size as itself.

EL: Right.

DK: And so fundamentally, this is about infinity. And the reason this is a little bit more than that, well, first of all, obviously, the ball here is not countable, right? We're not dealing with a countable number of points. This is uncountable. So now we're talking about the continuum in terms of the cardinality of the points. But I think then the surprising thing is that it works out geometrically. So it's not just about cardinality, but you can do this geometrically. And so you can actually define these sets. And, you know, the word pieces, when we say you can split it up into a number of finite number of pieces. I think the record is somewhere under 10 pieces.

EL: It might be just like five or something. It’s been a while.

DK: But the word pieces is doing a whole lot of work in that statement.

EL: Yes.

DK: And these pieces are not something you could ever do with like a knife and fork or, you know, even define easily. It requires the axiom of choice to define these pieces.

KK: Uh-oh!

DK: So it's really high-level mathematics, to understand how to do these pieces, but the fact that you could do these pieces, and then geometrically it works to just reassemble them, to just rotate and translate these pieces, and get back two balls the size of the origina, that’s just astonishing.

KK: Yeah, that's why I've always had a problem with this. I mean, I, I can read it and understand it and go, yes, you can follow every logical step. But you're right, it doesn't work visually, if you think about it. So can you describe these pieces at all?

DK: They are screwed up. So the analogy I like to make is, you know, if you're working on the interval from zero to one, you know, so first of all, the Banach Tarski paradox does not work in in one dimension. But in terms of these pieces, if you're thinking about the interval from zero to one, you could think of the rational numbers as a piece of that, right? And so it's a piece in the sense that it's part of the whole, it's not a piece in the sense that you could cut it out with a butter knife, or you could model this with a stick of butter or something like that, right? You have to hit 1/2, 1/3, 2/3, you have to hit, you know, 97/101 in there, right? All of those are rational numbers, but you can certainly think of all of the rational points in the interval from zero to one as a single piece. You can talk about them, you can define them, and then you can talk about moving them. And so you can think about it that way, right? So you have all those rational numbers, and you can think of that as one piece. And it certainly is a lot more complicated than that. You know, when you think about Banach, Tarski, one of the things I love to do is with students is to go back and think about what it would mean for a set to be non-measurable. So we can measure things like intervals, but it doesn't take too much mathematics to sort of dive into the fact that there have to be sets — if you have a sense of measure, which works on intervals and things like that, and you want it to have other properties, like when you take two disjoint sets, the measure of the two disjoint sets together should be the sum of the two measures, like really basic properties, it doesn't take too much to be able to prove that there are sets that are non measurable. And once you can prove that, like Oh, then then the world gets really screwed up. Because in the world, in our everyday living, everything seems measurable, in some sense. Like even if you have some screwed up sculpture, you could measure its volume. You dunk it in water and see how the water level rises, right? I mean, it is, it has a measure. And the idea that there is no way to measure something is just incredibly counterintuitive. But once you get that, then it's it's it's a little bit more of a leap, but to understand at a fundamental level that you can define pieces that are so screwed up that you could just rearrange them and get two copies of the original, it’s fantastic.

EL: It’s interesting that you, when you first introduced this, you said it's something about infinity and I remember what — so now I'm thinking I might have taken a real analysis class before I had read this book, because I remember when I read the book, thinking, Oh, this is telling me something about non-measurability and, like, really giving me a concept of what non-measurability does to things. And that's that's how I viewed it. It's a statement about how important measurability is.

DK: And just to be really clear, if these pieces were measurable, right, then we would just have some volume or something like that. And there's no way to double the volume. Right? So you can't you just can't do that. So clearly these pieces, at least some of them have to be non measurable.

KK: Right? So much for your alchemy idea, right?

DK: No more doing this with gold. Okay.

EL: It won’t work on atoms.

DK: There’s this fundamental idea that's permeates mathematics that we can continue to divide things, right? You can get things as small as you want. You see this — this is basically what calculus is all based on, infinitesimally small things. And it's just a reminder that the real world does not work like that. You take a gold atoms, you could keep splitting something up. But eventually, you’ve got one gold atom in each piece and you can't go any smaller than that.

KK: Well, you could, but then

DK: It wouldn't be gold anymore.

KK: Right. All right. So you sort of hinted at this, but why do you love this theorem so much?

DK: I love that it makes you question so many things. I mean, I love paradoxes in general, right? paradoxes are these moments when there's, there's so much cognitive conflict going on and cognitive dissonance, that it forces you, in order to resolve that cognitive dissonance, you have really have to question some other fundamental aspect of the world. So it's something that you were thinking before, is not true, or this paradox is like totally crap, right? So something like that. But in this case, the theorem is true, right? The Banach-Tarski paradox is true. And so it forces you to just go back and question some fundamental ideas about the world. And I love that there are statements like that, that can force you to go back and question so many things. I think we as humans need to do better at this. There are so many things that we just accept, as, as we take for granted, right, and we take them for granted as if they are true with a capital T. And in this case, it's about, like, you can measure all things. But of course, in our in our everyday lives, it's things about the world, it's things about people, it's things about politics, it's things about, you know, topical issues. And we grow up, and it's so normal that we think these things are just part of the world with a capital T on truth. And I love those moments that force us to go back and question those those fundamental “truths,” which all of a sudden turn out to be assumptions, some of which may not be right, or some of which we might want to reassess.

KK: So maybe you're arguing that we should study more mathematics to make the world better.

DK: We certainly should do that, Kevin.

KK: To build a better citizenry, right?

DK: Certainly if we all understood mathematics better, we would have been better off during this pandemic. That's certainly true.

EL: I have a question. So as you mentioned, this is called a paradox often. Do you think it is a paradox?

DK: Yeah, that's a great question. You know, paradox has a couple of different meanings. One is sort of this deep philosophical meaning, like, is it really something which is somehow both true and false simultaneously? And, and this is not in that sense, a paradox. It is a paradox in a sort of weaker sort of more everyday sense of that word where it really throws us into into some cognitive dissonance that forces us to question other things. We can't hold both true that like everything can be measurable and things have volume and you can take a ball, split it into six pieces, rearrange them and get two balls. Those are fundamentally in conflict, and one of them has to go.

KK: It does rely on choice, though, right? So there's something to argue about there. You know, there are those people who deny the axiom of choice.

DK: They're few and far between in the math community, but they're out there. Yeah, they're not measure zero, so to speak. So yeah, so it does, it does use the axiom of choice. So this idea that you can, you know, you can make infinitely many choices. And you can see in the proof where you have to do that. So you end up with uncountably many sets and you choose one point from each one of those uncountably many sets, and that's part of the way you get one of the sets that creates the the Banach-Tarski paradox.

KK: Yeah.

EL: I guess I'm a little surprised that the Banach-Tarski paradox hasn't made more people reject the axiom of choice, to say like, Okay, well, clearly, you can't do this. So, therefore, what is this relying on? Well, it's relying on the axiom of choice.

KK: Well, there's a lot of things about like that, like the Tychonoff product theorem. So that's equivalent to the axiom of choice. So do you want your product of compact spaces to be compact or not, you know, I mean, I don't know.

DK: Yes, I think Evelyn, you're hitting upon this fundamental thing: at a very deep level, math gets weird. Yeah. Right. And you can have things. I mean, you see that in, you know, in Gödel’s work, you see like, well, is that true? Well, you know, it can either be true or false. What's your pleasure today? Right? To the axioms or we can reject it. And, and fundamentally, it kind of doesn't matter. But you know, go ahead like, is there an infinity between the integers and the real line? Like, oh, you know, take your Take your pick?

EL: Yeah, what seems more more useful to you right now? Or what sounds like more fun to play with?

KK: Sure.

DK: And in some sense, that that is kind of the beauty of math, because so much of what mathematicians do is based on wherever you want to start. It’s theoretical, like, oh, well, if we start here, this is where we get. If we start at a different place, we get this other thing. And so you can hold those both in your head, despite the fact that maybe you can't have the hold them simultaneously. Like, either the axiom of choice, you take it to be true or you take it to be false, but you can't take it to really be both. Then things break down. But you can you can do a lot of mathematics either way.

KK: Yeah. So another thing we like to do on this podcast is invite our guests to pair their theorem with something. So I'm curious, what have you chosen to pair with this paradox?

DK: So there's this piece that I played, I think I first played this when I was in high school. I'm a violinist and Evelyn and I share this as string players, but it's called the Enigma Variations. It's by Edward Elgar. And it's a really interesting piece with sort of a fascinating story behind it. And the story tells, or explains the name, enigma. So the idea is that Edward Elgar was sitting there, and he played this little melody, and he was sitting at the keyboard — this is in, like, 1898, late 19th century. And he started riffing on this on this melody, right? And he's like, oh, okay, like, we could play it this way, or this way. And then he started putting names on it, like, Oh, my friend would play it this way. Or, you know, my wife, she would play it in this way. And then he sort of just kept taking that further and further, and he ended up writing a bunch of different movements based on how different people in his life, he imagined would play this particular melody. And one of the it's sort of ingenious if you want something to go viral, I guess. But he never, he never lays out what the what the melody was. And so that's the enigma part of it. Somehow out there somewhere is this melody, and all you're hearing is different people's take on it. He does tell you who the people are, and so some of these movements have initials telling you who some of them are. Some are explicit with names. But all you have is that, right? And it's beautiful music and when you listen to it, some of the some of the movements are really fast loud and really exciting. And others are just incredibly slow and languishing. And it's hard to imagine that all of that could sort of in some sense be based on a single melody, and somehow it is. That's what I'd like to pair it with, the Enigma Variations.

KK: Do you have a favorite so that we can insert a clip?

DK: Um, let's see. I do. So probably the most famous of these is called Nimrod. It's an incredibly slow movement. I've played it a bunch of times. It's the sort of thing that, you know, if you want to get people to cry in a movie you could play a little bit of Nimrod. I don't think I've ever successfully been on stage playing this without crying. It is that emotional come and it's also, for me it's more emotional if you play it slower, so it's incredibly evocative. [Clip of the Nimrod variation]

KK: Do you still play?

DK: I do. I do. Well, in theory. We’re in this pandemic where I haven’t been able to play much. But yeah, I play in a local community orchestra and play with my daughter and yeah, I've had some had some fun over during the pandemic, by myself playing, picking up some old pieces and playing them.

KK: Very cool.

EL: Yeah, I was thinking probably everyone has heard Nimrod without realizing it, because it's in the background. If it's not Adagio, for strings, it's Nimrod in the background of that, you know, swelling, emotional scene, your farewell or someone’s dying or whatever. You know, it’s there.

DK: And when you hear it, you can hear little hints of Pomp and Circumstance, which we hear all the time during graduations. And so you can see like, yeah, those are, those are similar sort of in structure, in the melodies and the harmonies and how they fit together,

KK: Right, plus it’s public domain by this point, so you can just throw it into movies pretty easily. Right? That's true, too.

EL: Yeah, I guess you're the practical one here.

DK: Yeah, that's important when you're doing things like podcasts. Yeah, yeah.

KK: Yeah. All right. So um, that's a really good pairing. I like that a lot. So we also like to give our guests a chance to promote anything they want to promote. So So we've talked about the Dana Center a little bit. Anything you want to pitch? Where can we find you online?

DK: You can find me at the at the date of the Charles Dana center, it's UT Dana Center. dot.i.com. I think maybe .edu I'm not sure actually sure. [Editor’s note: it’s neither! It’s actually https://www.utdanacenter.org/], but it’s a pretty easy web search to find us. You know, we're trying to make sure that mathematicians, that everybody has an opportunity to see themselves as a mathematician, and that everybody has access to the right math for them at the right time with the right supports. And so much of the math community, so much of our curriculum is grounded in things that are really old. T algebra, geometry, algebra II sequence was decided in 1892 by a group of 10 white men in the northeast who decided that that would be the right thing. And this focus on calculus comes out of the Cold War, it comes out of the need to produce a small number of engineers who are going to work pencil and paper, and then with massive computers that they could program, to win the Cold War, to beat the Soviets into space and to do all of that. That's no longer the world we live in. And we need to expand what we think of as mathematics. And it's not that calculus isn't important, it still is, but maybe not sending everybody in that direction would be better. And so some students would be much better off if they took statistics. Other students, Quantitative Literacy would be great. We're swimming in so much data, we don't know what to do with and the careers out there that deal with data are taking off like you wouldn't believe. What are the tools we need to give students so that they can deal with that?

KK: Yeah.

DK: All right. So all of these things are just updates that we need to do for their math curriculum. And and updating a system that's so complicated, with so many moving parts is difficult. And so that's part of what we work on at the Dana Center. And if anybody's out there who wants to help work on those things and at the same time, make sure all of these systems are equitable, because we know they haven't been in the past, and make sure all students have access to high-quality math instruction independent of your zip code, or who your parents are your economic circumstances. Those are the sorts of things that we work on at the Dana Center.

KK: Yeah, that's really important work. I mean, it's funny to hear you talk about this, because I've been saying the same thing. You know, the math degree that we still give out more or less is what I got 30 years ago, which isn't different than what they were handing out 30 years before that. I mean, things have changed. And we're still sort of stuck that way. It's unfortunate. And they're still doing it on the high schools. You know, my son when he was going through high school, they just marched him in lockstep through algebra, geometry. And he's a musician. I mean, it's great, you know, it's good for him, but I don’t know.

DK: And even some of these topics that are still important, like, I think algebraic thinking is still important. I think a lot of people do algebraic thinking out there all the time. It's just, it's not pencil and paper. It's not symbol manipulation. The most popular place that people use it, their algebraic thinking is when they're working with spreadsheets, right? Writing formulas in spreadsheets is incredibly algebraic. You have these placeholders that are really just variables, things change, some things stay the same. It's highly algebraic. And we could be teaching algebra using that so that everybody, every student coming out of algebra would have a basis for understanding how algebra is going to be used. You know, it's something like 98% of employers want their employees to be able to use spreadsheets. Well, there's a perfect example. We could we could tweak the curriculum to take something that right now is seen as kind of useless and make it very useful and in fact vital. And it's still teaching essentially the same core ideas, but it's really approaching them from a very different perspective.

KK: Yeah, yeah. All right.

DK: And as long as you give me the space Kevin, you know talking about mathematics and music, so I I have to take a little space. I did this set of 12 lectures for the Great Courses on math and music. And it's a sort of a tour through the listening of music. So like when when we record something, and that music eventually gets into your ear, like every step on that process involves mathematics. And over the course of 12 lectures, we talk about rhythm, we talk about harmonics, we talk about tuning, we talk about lots of, we even talk about the digital side of it. So there's a ton of mathematics that goes into CDs, there's a lot of mathematics that goes into compression that we're using now, so that we can, you know, actually hear each other from this far away. And all of that, there's just lots of mathematics embedded in that. Yeah, so it's really fun, it was just a real honor to be able to do that series. So you can find that out there in the Great Courses.

KK: Yeah. Do you talk about why you can't have perfect tuning on a piano or anything like that?

DK: Yeah, in fact, I got to demonstrate tuning. So they brought in a baby grand for me to do this on. And so we tuned an octave and pulled it out of tune a little bit and you could hear the beats. Again, like these things that I saw in high school and in a trig class, but I had no idea they were applicable, right? So I mean, you can actually hear — there are trig identities that tell you, if you play two notes that have really similar frequencies, that's kind of equivalent to something that goes a wah-wah-wah, it has beats in it. And you can actually do that. And so I played that and tuned it, so that was perfect. And then I took a fifth and we did the mathematics to figure out if you want everything to be sort of in tune, how out of tune do your fifths need to be right, which is, you know, fascinating thing. And I know, Evelyn, you've written about this, what was the column called, like, the saddest thing you know about the natural numbers?

EL: Yeah. Something like that.

DK: Like three is not a power of two or something like that.

EL: It is so sad.

DK: It is sad. And, you know, there are mathematical facts at the heart of that, and so much that we hear in music. And I get notes every week from somebody who has had these nagging questions in the back of their head about how music and math are related. And they watch these 12 lectures, and they're just so thrilled to sort of unpack some of that.

EL: Yeah, it's neat. And it's also it's not just, like, the sound waves and the math, which is definitely part of it. But there's also this extra perception in our ears and our brains that is involved and has some math, but that has some, basically, I don't know, wizardry that our brains do to be like, Oh, when I hear this kind of thing, I often hear this kind of thing with it, so it probably came together, I just missed something there.

DK: There’s a there's a whole section on auditory illusions, which are really fascinating, ways in which our brain can trick itself or, or like, because it's been useful in the past, like, our brain does certain things. And so you can intentionally use that to have these auditory illusions, which are really just fascinating. And my favorite fact about this is that is that essentially, your ear is doing a Fourier transform. When you are listening, your ear is doing this Fourier transform. And you know, just sort of physically doing it and breaking down the sound into its constituent frequencies. And that is just a phenomenally cool idea.

KK: That our brains are that sophisticated.

DK: That our brains somehow involved evolved to do something that we didn't describe mathematically until the 19th century, right? Like, oh, now we know what our brains did just sort of by, you know, random chance and little tweaks to random changes in a DNA sequence. And somehow we got to like, oh, oh, yeah. And that's now called Fourier transform.

EL: Yeah. All right. It's a lot of fun. I feel like we could almost do a whole nother podcast episode about one of these facts.

KK: Probably.

DK: That would be so much fun.

KK: Yeah. Okay. Yeah. Well, maybe we’ll have you back for part two sometime.

DK: That'd be great. Yeah.

KK: Well, this has been great fun, Dave. I'm surprised Banach-Tarski made it this long without being somebody's favorite. This is pretty good.

EL: Yeah. Yeah.

KK: Because we've had we've had repeats before, but but this one, I'm surprised. So thanks for joining us. And yeah.

DK: Thank you guys so much. I love the work that you do, and I really appreciate it.

On this episode of My Favorite Theorem, we welcomed Dave Kung from the Dana Center at the University of Texas at Austin to talk about the Banach-Tarski paradox/theorem. Here are some links you might enjoy:
Kung's website and Twitter account
The Dana Center website
Leonard Wapner's book The Pea and the Sun about the Banach-Tarski paradox
A shorter article by Max Levy explaining the theorem
A primer on the axiom of choice from the Stanford Encyclopedia of Philosophy

The Tychonoff product theorem
Kung's course How Music and Mathematics Relate from the Great Courses

Evelyn's article The Saddest Thing I Know about the Integers, mourning the fact that no power of 3 is also a power of 2