Kevin Knudson: Welcome to My Favorite Theorem. I’m one of your hosts, Kevin Knudson, professor of mathematics at the University of Florida, and here is your jet-lagged other host.
Evelyn Lamb: Hi, I’m Evelyn Lamb, a freelance math and science writer in Salt Lake City. I’m doing pretty well right now, but in a few hours when it’s about 5 pm here, I think I will be suffering a bit. I just got back from Europe yesterday.
KK: I’m sure you will, but that’s the whole trick, right? Just keep staying up. In a couple of weeks, I’m off to that part of the world. I’ll be jet-lagged for one of the ones we have coming up right after that. That should be fun. I’ll feel your pain soon enough, I’m sure.
KK: So today we are pleased to welcome James Tanton. James, why don’t you introduce yourself and tell everyone about yourself?
James Tanton: Hello. First of all, thank you for having me. This is such a delight. So who am I? I’m James Tanton. I’m the mathematician-at-large for the Mathematical Association of America, which is a title I’m very proud of. It’s a title I never want to give up because who doesn’t want to be a mathematician-at-large, wreaking havoc wherever one steps? But my life is basically doing outreach in the world and promoting joyous thinking and doing of mathematics. I guess my background is somewhat strange. You can probably tell I have an accent. I grew up in Australia and came to the US 30 years ago for my Ph.D., which was grand, and I liked it so much here I am 30 years later. My career has been kind of strange. I was in the university world for close to 10 years, and then I decided I was really interested in the state of mathematics education at all levels, and I decided to become a high school teacher. So I did that for 10 years. Now my life is actually working with teachers and college professors all across the globe, usually talking about let’s make the mathematics our kids experience, whatever level they’re at, really, truly joyous and uplifting.
EL: Yeah, I’ve wondered what “mathematician-at-large” entails. I’ve seen that as your title. It sounds like a pretty fun gig.
JT: So I was the MAA mathematician-in-residence for a good long while. They were very kind to offer me that position. But then I’m married to a very famous geophysicist, and my life is really to follow her career. She was off to a position at ASU in Phoenix, and then off we moved to Phoenix four years ago. So I said to the folks at the MAA, “Well, thanks very much. I guess I’m not your mathematician-in-residence anymore,” and they said, “Why don’t you be our mathematician at large?” That’s how that title came up, and of course I so beautifully, graciously said yes because that’s spectacular.
KK: Yeah, that sounds like a Michael Pearson idea, that he would just go, “No, no, we really want to keep you.”
JT: It’s so flattering. I’m so honored. It’s great because, you know, actually, it’s the work I was going to do in any case. I feel compelled to bring joyous mathematics to the world.
KK: Right. Okay, so this podcast is about theorems. So why don’t you tell us what your favorite theorem is?
JT: Okay, well first of all, I don’t actually have a theorem, even though I think it should be elevated to the status of one. I want to talk about Sperner’s lemma. So a lemma means, like, an auxiliary result, a result people use to get to other big ideas, but you know what? I think it’s charming in and of itself. So Sperner’s lemma. This was invented slash discovered back in the 1920s by a German mathematician by the name of Emanuel Sperner, who was playing with some combinatorial thinking in Euclidean geometry and came up with this little result.
Let me describe to you in one way first, not really the way he did it, because then I can actually explain a proof as well of the result. Imagine you have a big rubber ball, just a nice clean rubber surface, and you’ve got a marker. I’m going to suggest you just put dots all over the surface of the rubber ball, lots of dots all over the place. Once you’ve done that to your satisfaction, start connecting pairs of dots with little line segments. They’ll be little circular arcs, and make triangles, so three dots together make a triangle. Do that all over the surface of the sphere. Grand. So now you’ve got a triangulated sphere, a surface of a sphere completely covered with triangles. Each triangle for sure has only three dots on it, one in each corner, so no dots in the middle of the edges, please. All right? That’s step one.
Step two, just for kicks, go around and just label some of those dots with the letter A, randomly, and do some other dots with the letter B, randomly, just some other dots with the letter C—why not?—until each dot has a label of some kind, A, B, or C. And then admire what you’ve done. I claim if you look at the various triangles you have, you have some labeled BBB, and some labeled BCA, and some labeled BBA, and whatever, but if you find one triangle that is fully labeled ABC, I bet if you kept looking, in fact I know you are guaranteed, to find another triangle that’s labeled ABC. Sperner’s lemma says on the surface of a sphere that if there’s one fully labeled triangle, there’s guaranteed to be at least another.
EL: Interesting! I don’t think I knew that, or at least I don’t know that formulation of Sperner’s lemma.
JT: And the reason I said it that way, I can now actually describe to you why that is true because doing it on the surface of a sphere is a bit easier than doing it on a plane. Would you like to hear my little proof?
KK: Let’s hear it.
JT: Of course the answer to that has to be yes, I know. So imagine that these are really chambers. Each triangle is a room in a floor design on the surface of a sphere. So you’re in a room, an ABC room around you. You’ve got three walls around you: an AB wall, a BC wall, and an AC wall. Great. I’m going to imagine that some of these walls are actually doors. I’m going to say that any wall that has an AB label on it is actually a door you can walk through. So you’re in an ABC room, and you currently have one door you can walk through. So walk through it! That will take you to another triangle room. This triangle room has at least one AB edge on it, because you just walked through it, and that third vertex will have to be an A, B, or C. If it’s a C, you’re kind of stuck because there are no other AB doors to walk through, in which case you just found another ABC room. Woo-hoo, done!
JT: If it’s either A or B, then it gives you a second AB door to walk through, so walk through it. In fact, just keep walking through every AB door you come to. Either you’ll get stuck, and in fact the only place you can possibly get stuck is if there’s exactly one AB door, in which case it was an ABC triangle, and you found an ABC triangle. Or it has another door to walk through, and you keep going. Since there’s only a finite number of triangles, you can’t keep going on indefinitely. You must eventually get stuck. You must get stuck in an ABC room. So if you start in one ABC room, you’ll be sure to be led to another.
EL: Oh, okay, and you can’t go back into the room you started in.
KK: That was my question, yeah.
JT: Could you possibly return to a room you’ve previously visited? Yes, there’s a subtlety there. Let’s argue our way through that. So think about the first room that you could possibly—if you do revisit a room, think of the first room you re-enter. That means you must have gone through an AB door to get in. In fact, if you’ve gone through that room before, you must have already previously used that AB door to go into and out of it. That is, you’ve used that AB door twice already. That is, the room you just came from was a previously revisited room. You argue, oh, if I think this is the first room I’ve visited twice, then the room you just came from, you’re wrong. It was actually that room that you first visited twice. Oh, no, actually it was the one before that that you first visited twice. There can be no first room that you first visited twice. And the only way out of that paradox is there can be no room that you visit twice.
JT: That’s the mind-bendy part right there.
EL: I feel like I need a balloon right now and a bunch of markers.
JT: You know, it’s actually fun to do it, it really is. But balloons are awkward. In fact, the usual way that Sperner’s lemma is presented, I’ll even not do it in the usual way. Sperner did it on a triangle. I’ll do it on any polygon. This time, this we can actually do with markers, and it’s really fun to actually do it. So draw a great big polygon on a page and then triangulate it. Fill its interior with dots and then fill in edges so you’ve got all these triangles filling up the polygon. And then randomly label the dots A, B, or C in a random, haphazard way. Make sure that you have an odd number of AB doors on the outside edge of that polygon. If you do that, no matter what you do, you cannot escape creating somewhere on the interior a fully labeled ABC triangle. The reason is, you just do this thing. Walk from the outside of the polygon through an AB door, an outside AB door, go along on a journey. If you get stuck, bingo! You’re on an ABC triangle. Or you might be led out another AB door back to the big space again. But if you have an odd number of AB doors on the outside, you’re guaranteed to have at least one of those doors not leading outside, meaning you’ve been stuck on the inside. It’s guaranteed to lead to an ABC triangle in the middle of the polygon.
EL: Okay, and this does require that you use all three—is it a requirement that you use all three letters, or does the odd number of things…I don’t know if my question makes sense yet.
JT: There’s no rules on what you except on the outside, please give me an odd number of AB doors.
JT: And there’s nothing special about the letters A and B. You could do an odd number of BC doors or an odd number of AC doors.
JT: What you do on the interior is up to you. Label them all A, I dare you, and you’ll still find an ABC triangle.
JT: Isn’t that crazy?
KK: Okay, so why did Sperner care?
JT: Why did Sperner care? Well he was just playing around with this geometry, but then people realized, as one of your previous guests mentioned, the fabulous Francis Su, that this leads to some topological results, for example, the Brouwer fixed-point theorem, which people care about, and you should listen to his podcast because he explains the Brouwer fixed-point theorem beautifully.
EL: Yes, and he did actually mention to us in emails and stuff that he is actually quite fond of Sperner’s lemma also, so I’m sure he’ll be happy to listen to this episode.
JT: In some cases, Sperner’s lemma is kind of special because people knew Brouwer’s fixed-point theorem before Sperner, but they had very abstract nonconstructive proofs of the theorem. Fixed points, when you crumple pieces of paper and throw them on top of themselves, fixed points exist, but you can know that and not know to find them. Sperner’s lemma, if you think about it, is giving you a way to possibly find those ABC triangles. Just start on the outside and follow paths in. So it gives you a kind of hope of finding where those fixed points might actually lie, so it’s a very sort of constructive type of thinking on this topological result that is proved abstractly.
One thing that Francis Su did not mention is the hairy ball theorem, which I think is a lovely little application of Sperner’s lemma, which goes back to the spheres. Spheres—in my mind, this is how I was first thinking about Sperner’s lemma. So I don’t know if you know the hairy ball theorem. If you take a tennis ball with the little fur, the little hairs at, ideally, every point of the sphere, but that’s not really possible. But we can imagine in our mind’s eye a hairy ball. If you try to comb those hairs flat, tangent to the surface all the way around—well, maybe there would be a little angle, something like that. But as long as you don’t do anything crazy, you know, it’s a nice, smooth, continuous vector field on the surface of the sphere, just these hairs, close hairs go towards the same direction, very smoothly, nothing abrupt going on, then you are forced to have a cowlick, that is, one hair that sticks straight up. That is, you are forced to have a tangent vector that is actually the zero vector. You can actually prove that with Sperner’s lemma.
JT: Yeah, and the way you do that is: choose one point, like the North Pole, and imagine a little magnet there, and you can imagine all the magnetic fields make these two circular, the magnetic field of a dipole, sorry, I have to think back to my physics days. So you’ve got these natural lines associated with that magnet all over the sphere, so I suggest just triangulate the sphere. Just draw lots of little triangles all over it. And then at each vertex of the triangle, you’ve got this vector field, and you’ve got these hairs all over the vector field. At any point on the triangle, look at the direction the hair is pointing compared to the direction of the magnetic field. And you can label that either A, B, or C by doing the following. Basically you’ve got 360 degrees of possible differences of directions between those things. So if it’s in one of the first 0-120 degrees of counterclockwise motion, label it A. If it’s between the 120 and 240 mark, label it B. If it’s between the 240 to 360 mark, label it C. There is a way to label that triangulation based on the direction of the hairs on the surface of the sphere. Bingo! So we’ve just now proved that in any triangulation, you can argue that you arrange things at the pole as an ABC triangle, there’s this little thing you can arrange, then there has to be some other ABC triangle somewhere on the sphere. That is, there’s a little small region where you’ve got three hairs trying to point in three different directions. And do finer and finer triangulations. You actually argue the only way out of that predicament is there’s got to be one hair that’s pointing three directions at the same time, that is, the zero vector.
KK: That’s very cool.
JT: I just love these. These things feel so tangible. I just want to play with them with my hands and make it happen. And you can to some degree. Try to comb a fuzzy ball. You have a hard time. Or look at a guinea pig. They’re basically fuzzy balls, and they always have a cowlick. Always.
KK: Are there higher-dimensional generalizations of this? This feels very much two-dimensional, but I feel there’s an Euler characteristic lurking there somewhere.
JT: Absolutely you can do this in higher dimensions. This works in any dimension. For example, to make this three-dimensional, stack all these tetrahedra together. Take a polyhedron, triangulate it. If there’s an odd number of ABC faces on the outside, then there’s guaranteed to be some ABCD tetrahedron in the middle. And higher dimensions. And people of course play with all sorts of variations. For example, I’ll go back to two dimensions for a moment, back to triangles. If three different people create their own labeling scheme, so you’ve got lots of ABC triangles around the place, then there’s guaranteed to be one triangle in the middle, so if you chose one person’s label for the vertex, the second person’s for the second vertex, the third person’s label for the third vertex, according to their labels, which are all different, that’s an ABC triangle in this sort of mixed labeling scheme. So they call these permutation results of Sperner’s lemma and so forth. Just mind-bendy, and in higher dimensions.
EL: So was this a love at first sight kind of theorem for you? What were your early experiences with it?
JT: So when did I first encounter it? I guess when I studied the Brouwer fixed-point theorem, and when I saw this lemma in and of itself—and I saw it in the light of proving Brouwer’s fixed-point theorem—it just appealed to me. It felt hands-on, which I kind of love. It felt immediately accessible. I could do it and experience it and play with it. And it seemed quirky. I liked the quirky. For some reason it just appealed to me, so yes, it appealed to all my sensibilities. And I also have this thing I’ve discovered about me and my life, which is that I like this notion that I’m nothing in the universe, that the universe has these dictates. For example, if there’s one ABC triangle, there’s got to be another one. I mean, that’s a fact. It’s a universal fact that despite my humanness I can do nothing about it. ABC triangles just exist. And things like the “rope around the earth” puzzle: if you take a rope and wrap it around the Equator, add 10 feet around the rope and re-wrap, you’ve got 19 inches of space. What I love about that puzzle, if you do it on Mars 10 feet from its Equator, it’s 19 inches of space. Do it for Jupiter: it’s 19 inches of space. Do it for a planet the size of a pea: it’s 19 inches of space. You cannot escape 19 inches. That sort of thing appeals to me. What can I say?
KK: So you are a physicist?
JT: Don’t tell anyone. My first degree was actually in theoretical physics.
KK: So the other fun thing we do on this podcast is we ask our guest to pair their theorem, or lemma in this case, with something. So what have you chosen to pair Sperner’s lemma with?
JT: You know, I’m going with a good old Aussie pavlova.
JT: And I’ve probably offended all the people from New Zealand because they claim it’s their pavlova. But Australians say it’s theirs, and I’ll go with that since I’m an Aussie. And why that, you might ask?
EL: Well first can we say what a pavlova is in case, so I only learned what this was a couple years ago, so I’m just assuming—I was one of the lucky people who learned about this in making one, which was delicious, so yeah.
JT: First of all, it’s the most delectable dessert devised my mankind, or invented, or discovered. I’m not sure if desserts are invented or discovered. That’s a good question there. So it’s a great big mount of meringue, you just build this huge blob of meringue, and you bake it for two hours and let it sit in the oven overnight so it becomes this hard, hard outer shell with a soft, gooey meringue center, and you just slather it with whipped cream and your favorite fresh fruits. And my favorite fruits for a pavlova are actually mango and blueberries together. That’s my dessert. But you know, well maybe to think of it, there are actually two reasons. I happen to know it was invented in the 1920s, the same time Sperner came up with this lemma, which is kind of nice. But any time I bake one—I bake these things, I really enjoy baking desserts—it kind of reminds me of a triangulated sphere because you’ve got this mound of meringue, and you bring it out of the oven, and it’s got this crust that’s all cracked up, and it kind of looks like a triangulation of a polyhedron of some kind. So it has that parallel I really like. So pavlovas bring as much joy to my life as these quirky Sperner’s lemma type results, so that’s my pairing.
EL: So they’re not, so I went to this Australia-themed potluck party a couple years ago, and I decided to bring this because I was looking for Australian foods, so I got this. I was pretty intimidated when I saw the pictures, but it’s actually, at least I found a recipe and it looked good, and it worked the first time, more or less. I think you can handle it.
JT: It is a showstopper, but it’s so easy to make. Don’t tell anyone, it’s ridiculously easy, and it looks spectacular.
KK: Yeah, meringues look like something, but really, you just have to be patient to whip the whites into something, and then that’s it. It works.
JT: Then you’re done. It kind of works. You can’t overcook it. You can undercook it, but then it’s just a goopy delicious mess.
KK: Right. So we also like to give our guests a chance to plug various things. I’m sure you’re excited to talk about the Global Math Project.
JT: Of course I’m going to talk about the Global Math Project. Oh my goodness. You know, when I mention I’m kind of a man on a mission to bring joyous, uplifting mathematics to the world, I’m kind of trying to live up to those words, which is kind of scary. But let me just say something marvelous, really marvelous and humbling, happened last October. We brought a particular piece of mathematics to the world, a team of seven of us, the Global Math Project team, not knowing what was going to happen. It was all volunteer, grassroots, next to no funding, we’re terrible at raising funding, it turns out. But it really was believing that teachers, given the opportunity to have a real joyous, genuine, human conversation about mathematics with their students, that’s actually classroom-relevant mathematics. Classroom mathematics is a portal to the same mystery, delight, intrigue, and wonder, they will. Teachers are our best advocates across the globe for espousing beautiful, joyous, uplifting mathematics. So we presented a piece of mathematics called Exploding Dots, and we invited teachers all around the globe to do that, to have just some experience on this topic with their students during Global Math Week last October, and they did. We had teachers from 170 different countries and territories, all of their own accord, reach out to about 1.77 million students just in this one week. Phenomenal. And this is school-relevant mathematics. So we’re doing it again! Why not?
EL: Oh, great!
JT: So this year, 10/10, we chose that date because it’s a universal date. No matter how you read it, it’s the tenth of October. We’re going to go up to 10 million students with the same story of Exploding Dots. So I invite you, please look up Global Math Project, go Google Exploding Dots. See what we’re bringing to the world. And on its own accord, in the last number of months, we’ve now reached 4.6 million students across the globe, so 10 million students sounds outlandish, but you know what? We might actually do this. And it’s just letting the mathematics, the true, joyous mathematics, simply shine for itself, just get out of its way. And you know what? It happens. Math can speak for itself. Welcome to Global Math Project.
EL: Yeah. We’ll include that in the show notes, for sure.
KK: In fact, this is June that we’re taping this, recording this. Taping? I’m dating myself. We’re recording this in June. So just this weekend Jim Propp had a very nice essay on his Mathematical Enchantments blog about this, about Exploding Dots. I’d seen some things about it, so I knew a little about it. It’s really very lovely, and as you say relevant.
JT: I’m glad you mentioned Jim Propp. I was about to give a shout out to him as well because he wrote a beautiful piece, and it’s this Mathematical Enchantments blog piece for June 2018. Worth having a look at. Absolutely. What I love about this, it really shows, I mean, Exploding Dots is the story of place value, as simple as that. But it really connects to how you write numbers, what you’re experiencing in the early grades. It explains, if you think of it in one particular way, all the grade school algorithms one learns, goes through all of high school polynomial algebra, which is just a repeat of grade five, but no one tends to tell people that. Why stop at finite things? Go to infinite things, go to infinite series and so forth, and start getting quirky. Not just playing with 10-1 machines with base 10 and 2-1 machines with base 2, start playing with 3-2 machines and discover base 1 1/2, start playing with 2-negative 1 machines and discover base -2, and you get to unsolved research questions. So here’s this one simple little story, just playing with dots in boxes, literally—like me playing with dots on a sphere; I seem to be obsessed with dots in my life—takes you on a journey from K through 5 through 8 through 12 to 16, and on, all in one astounding fell swoop. This is mathematics for you. Think deeply about elementary ideas, and well, it’s a portal to a universe of wonder.
KK: That’s why we’re all here, right?
JT: Indeed. So let’s help the world see that together, from the young’uns all the way up.
KK: All right. Well, this has been great fun. I knew Sperner’s lemma, sort of in the abstract, but I never really thought of it too closely. So I’m glad that I can now prove it, so thank you for that.
EL: Yeah, I’m going to sit down and make sure you’re not pulling my leg about that. I think the odd number of AB’s is key here.
JT: Absolutely. The odd number of AB outside edges is key.
JT: Because you could walk through the door and out a door, so pairs of them could cancel each other out. So play with them.
EL: I bet if I start drawing, I’ve been restraining myself from going over here to the side.
JT: Well you know, sketches work so well on a podcast.
KK: That’s right.
JT: Absolutely, play. That’s all mathematics should be, an invitation to play. Go for it.
EL: Yeah, thanks a lot for being here.
JT: My pleasure. Thank you so much.