Episode 88 - Tom Edgar

My Favorite Theorem

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