My Favorite Theorem

This transcript is provided as a courtesy and may contain errors.

Evelyn Lamb: Welcome to My Favorite Theorem. I’m your host Evelyn Lamb. I am a freelance math writer usually based in Salt Lake City but currently based in Paris. And this is your other host.

KK: I’m Kevin Knudson, professor of mathematics at the University of Florida.

EL: Every episode we invite a mathematician on to tell us about their favorite theorem. This week our guest is Dave Richeson. Can you tell us a little about yourself, Dave?

Dave Richeson: Sure. I’m a professor of mathematics at Dickinson College, which is in Carlisle, Pennsylvania. I’m also currently the editor of Math Horizons, which is the undergraduate magazine of the Mathematical Association of America.

EL: Great. And so how did you get from wherever you started to Carlisle, Pennsylvania?

DR: The way things usually work in academia. I applied to a bunch of schools. Actually, seriously, my wife knew someone in Carlisle, Pennsylvania. My girlfriend at the time, wife now, and she saw the list of schools that I was applying to and said, “You should get a job at Dickinson because I know someone there.” And I did.

KK: That never happens!

EL: Wow.

DR: That never happens.

KK: That never happens. Dave and I actually go back a long way. He was finishing his Ph.D. at Northwestern when I was a postdoc there.

DR: That’s right.

KK: That’s how old-timey we are. Hey, Dave, why don’t you plug your excellent book.

DR: A few years ago I wrote a book called Euler’s Gem: The Polyhedron Formula and the Birth of Topology. It’s at Princeton University Press. I could have chosen Euler’s Formula as my favorite theorem, but I decided to choose something different instead.

KK: That’s very cool. I really recommend Dave’s book. It’s great. I have it on my shelf. It’s a good read.

DR: Thank you.

EL: Yeah. So you’ve told us what your favorite theorem isn’t. So what is your favorite theorem?

DR: We have a family joke. My kids are always saying, “What’s your favorite ice cream? What’s your favorite color?” And I don’t really rank things that way. This was a really challenging assignment to come up with a theorem. I have recently been interested in π and Greek mathematics, so currently I’m fascinated by this theorem of Archimedes, so that is what I’m giving you as my favorite theorem. Favorite theorem of the moment.

The theorem says that if you take a circle, the area of that circle is the same as the area of a right triangle that has one leg equal to the radius and one leg equal to the circumference of the circle. Area equals 1/2 c x r, and hopefully we can spend the rest of the podcast talking about why I think this is such a fascinating theorem.

KK: I really like this theorem because I think in grade school you memorize this formula, that area is π r2, and if you translate what you said into modern terminology, or notation, that is what it would say. It’s always been a mystery, right? It just gets presented to you in grade school. Hey, this is the formula of a circle. Just take it.

DR: Really, we have these two circle formulas, right? The area equals π r2, and the circumference is 2πr, or the way it’s often presented is that π is the circumference divided by the diameter. As you said, you could convince yourself that Archimedes theorem is true by using those formulas. Really it’s sort of the reverse. We have those formulas because of what Archimedes did. Pi has a long and fascinating history. It was discovered and rediscovered in many, many cultures: the Babylonians, the Egyptians, Chinese, Indians, and so forth. But no one, until the Greeks, really looked at it in a rigorous way and started proving theorems about π and relationships between the circumference, the diameter, and the area of the circle.

EL: Right, and something you had said in one of your emails to us was about how it’s not even, if you ask a mathematician who proved that π was a constant, that’s a hard question.

DR: Yes, exactly. I mean, in a way, it seems easy. Pi is usually defined as the circumference divided by the diameter for any circle. And in a way, it seems kind of obvious. If you take a circle and you blow it up or shrink it down by some factor of k, let’s say, then the circumference is going to increase by a factor of k, the diameter is going to increase by a factor of k. When you do that division you would get the same number. That seems sort of obvious, and in a way it kind of is. What’s really tricky about this is that you have to have a way of talking about the length of the circumference. That is a curve, and it’s not obvious how to talk about lengths of curves. In fact, if you ask a mathematician who proved that the circumference over the diameter was the same value of π, most mathematicians don’t know the answer to that. I’d put money on it that most people would think it was in Euclid’s Elements, which is sort of the Bible of geometry. But it isn’t. There’s nothing about the circumference divided by the diameter, or anything equivalent to it, in Euclid’s Elements.

Just to put things in context here, a quick primer on Greek mathematics. Euclid wrote Elements sometime around 300 BCE. Pythagoras was before that, maybe 150 years before that. Archimedes was probably born after Euclid’s Elements was written. This is relatively late in this Greek period of mathematics.

KK: Getting back to that question of proportionality, the idea that all circles are similar and that’s why everybody thinks π is a constant, why is that obvious, though? I mean, I agree that all circles are similar. But this idea that if you scale a circle by a factor of k, its length scales by k, I agree if you take a polygon, that it’s clear, but why does that work for curves? That’s the crux of the matter in some sense, right?

DR: Yeah, that’s it. I think one mathematician I read called this “inherited knowledge.” This is something that was known for a long time, and it was rediscovered in many places. I think “obvious” is sort of, as we all know from doing math, obvious is a tricky word in math. It’s obvious meaning lots of people have thought of it, but if you actually have to make it rigorous and give a proof of this fact, it’s tricky. And so it is obvious in a sense that it seems pretty clear, but if you actually have to connect the dots, it’s tricky. In fact, Euclid could not have proved it in his Elements. He begins the Elements with his famous five postulates that sort of set the stage, and from those he proves everything in the book. And it turns out that those five postulates aren’t enough to prove this theorem. So one of Archimedes’ contributions was to recognize that we needed more than just Euclid’s postulates, and so he added two new postulates to those. From that, he was able to give a satisfactory proof that area=1/2 circumference times radius.

KK: So what were the new postulates? DR: One of them was essentially that if you have two points, then the shortest distance between them is a straight line, which again seems sort of obvious, and actually Euclid did prove that for polygonal lines, but Archimedes is including curves as well. And the other one is that if you have, it would be easier to draw a picture. If you had two points and you connected them by a straight line and then connected them by two curves that he calls “concave in the same direction,” then the one that’s in between the straight line and the other curve is shorter than the second curve. The way he uses both of those theorems is to say that if you take a circle and inscribe a polygon, like a regular polygon, and you circumscribe a regular polygon, then the inscribed polygon has the shortest perimeter, then the circle, then the circumscribed polygon. That’s the key fact that he needs, and he uses those two axioms to justify that.

EL: OK. And so this sounds like it’s also very related to his some more famous work on actually bounding the value of π.

DR: Yeah, exactly. We have some writings of his that goes by the name “Measurement of a Circle.” Unfortunately it’s incomplete, and it’s clearly not come down to us very well through history. The two main results in that are the theorem I just talked about and his famous bounds on π, that π is between 223/71 and 22/7. 22/7 is a very famous approximation of π. Yes, so these are all tied together, and they’re in the same treatise that he wrote. In both cases, he uses this idea of approximating a circle by inscribed and circumscribed polygons, which turned out to be extremely fruitful. Really for 2,000 years, people were trying to get better and better approximations, and really until calculus they basically used Archimedes’ techniques and just used polygons with more and more and more and more sides to try to get better approximations of π.

KK: Yeah, it takes a lot too, right? Weren’t his bounds something like a 96-gon?

DR: Yeah, that’s right. Exactly.

KK: I once wrote a Geogebra applet thing to run to the calculations like that. It takes it a while for it to even get to 3.14. It’s a pretty slow convergence.

DR: I should also plug another mathematician from the Greek era who is not that well known, and that is Eudoxus. He did work before Euclid, and big chunks of Euclid’s Elements are based on the work of Eudoxus. He was the one who really set this in motion. It’s become known as the method of exhaustion, but really it’s the ideas of calculus and limiting in disguise. This idea of proving these theorems about shapes with curved boundaries using polygons, better and better approximations of polygons. So Eudoxus is one of my favorite mathematicians that most people don’t really know about.

KK: That’s exactly it, right? They almost had calculus.

DR: Right.

KK: Almost. It’s really pretty amazing.

DR: Yes, exactly. The Greeks were pretty afraid of infinity.

KK: I’m sort of surprised that they let the method of exhaustion go, that they were OK with it. It is sort of getting at a limiting process, and as you say, they don’t like infinity.

DR: Yeah.

KK: You’d think they might not have accepted it as a proof technique.

DR: Really, and maybe this is talking too much for the mathematicians in the audience, but really the way they present this is a proof by contradiction. They show that it can’t be done, and then they get these polygons that are close enough that it can be done, and that gives them a contradiction. The final style of the proof would, I think, be comfortable to them. They don’t really take a limit, they don’t pass to infinity, anything like that.

EL: So something we like to do on this podcast is ask our guest to pair their theorem with something. Great things in life are often better paired: wine and cheese, beer and pizza, so what’s best with your theorem?

DR: I have to go with the obvious: pie, maybe pizza.

KK: Just pizza? OK?

EL: What flavor? What toppings?

KK: What goes on it?

DR: That’s a good question. I’m a fan of black olives on my pizza.

KK: OK. Just black olives?

DR: Maybe some pepperoni too.

KK: There you go.

EL: Deep dish? Thin crust? We want specifics.

DR: I’d say thin crust pizza, pepperoni and black olives. That sounds great.

EL: You’d say this is the best way to properly appreciate this theorem of Archimedes, is over a slice of pizza.

DR: I think I would enjoy going to a good pizza joint and talking to some mathematicians and telling them about who first proved that circumference over diameter is π, that it was Archimedes.

Actually, I was saying to Kevin before we started recording that I actually have a funny story about this, that I started investigating this. I wanted to know who first proved that circumference over diameter is a constant. I did some looking and did some asking around and couldn’t really get a satisfactory answer. I sheepishly at a conference went up to a pretty well-known math historian, and said, “I have this question about π I’m embarrassed to ask.” And he said, “Who first proved that circumference over diameter is a constant?” I said, “Yes!” He’s like, “I don’t know. I’d guess Archimedes, but I really don’t know.” And that’s when I realized it was an interesting question and something to look at a little more deeply.

EL: That’s a good life lesson, too. Don’t be afraid to ask that question that you are a little afraid to ask.

KK: And also that most answers to ancient Greek mathematics involve Archimedes.

DR: Yeah. Actually through this whole investigation, I’ve gained an unbelievable appreciation of Archimedes. I think Euclid and Pythagoras probably have more name recognition, but the more I read about Archimedes and things that he’s done, the more I realize that he is one of the great, top 5 mathematicians.

KK: All right, so that’s it. What’s the top 5?

DR: Gosh. Let’s see here.

KK: Unordered. DR: I already have Archimedes. Euler, Newton, Gauss, and who would number 5 be?

KK: Somebody modern, come on.

DR: How about Poincaré, that’s not exactly modern, but more modern than the rest. While we’re talking about Archimedes, I also want to make a plug. There’s all this talk about tau vs. pi. I don’t really want to weigh in on that one, but I do think we should call π Archimedes’ number. We talk about π is the circumference constant, π is the area constant. Archimedes was involved with both of those. People may not know he was also involved in attaching π to the volume of the sphere and π to the surface area of the sphere. Here I’m being a little historically inaccurate. Pi as a number didn’t exist for a long time after that. But basically recognizing that all four of these things that we now recognize as π, the circumference of a circle, the area of a circle, the volume of a sphere, and the surface area of a sphere. In fact, he famously asked that this be represented on his tombstone when he died. He had this lovely way to put all four of these together, and he said that if you take a sphere and then you enclose it in a cylinder, so that’s a cylinder that’s touching the sphere on the sides, think of a can of soda or something that’s touching on the top as well, that the volume of the cylinder to the sphere is in the ratio 3:2, and the surface area of the cylinder to the sphere is also the ratio 3:2. If you work out the math, all four of these versions of π appear in the calculation. We do have some evidence that this was actually carried out. Years later, the Roman Cicero found Archimedes’ tomb, and it was covered in brambles and so forth, and he talks about seeing the sphere and the cylinder on Archimedes’ tombstone, which is kind of cool.

EL: Oh wow.

DR: Yeah, he wrote about it.

KK: Of course, how Archimedes died is another good story. It’s really too bad.

DR: Yeah, I was just reading about that this week. The Roman siege of Syracuse, and Archimedes, in addition to being a great mathematician and physicist, was a great engineer, and he built all these war devices to help keep the Romans at bay, and he ended up being killed by a Roman soldier. The story goes that he was doing math at the time, and the Roman general was apparently upset that they killed Archimedes. But that was his end.

KK: Then on Mythbusters, they actually tried the deal with the mirrors to see if they could get a sail to catch on fire.

DR: I did see that! Some of these stories have more evidence than others. Apparently the story of using the burning mirrors to catch ships on fire, that appeared much, much later, so the historical connection to Archimedes is pretty flimsy. As you said, it was debunked by Mythbusters on TV, or they weren’t able to match Archimedes, I should say.

KK: Well few of us can, right?

DR: Right. The other thing that is historically interesting about this is that one of the most famous problems in the history of math is the problem of squaring the circle. This is a famous Greek problem which says that if you have a circle and only a compass and straightedge, can you construct a square that has the same area as the circle? This was a challenging and difficult problem. Reading Archimedes’ writings, it’s pretty clear that he was working on this pretty hard. That’s part of the context, I think, of this work he did on π, was trying to tackle the problem of squaring the circle. It turns out that this was impossible, it is impossible to square the circle, but that wasn’t discovered until 1882. At the time it was still an interesting open problem, and Archimedes made various contributions that were related to this famous problem.

EL: Yeah.

KK: Very cool.

DR: I can go on and on. So today, that is my favorite theorem.

KK: We could have you on again, and it might be different?

DR: Sure. I’d love to.

KK: Well, thanks, Dave, we certainly appreciate you being here.

DR: I should say if people would like to read about this, I did write an article, “Circular Reasoning: Who first proved that c/d is a constant?” Some of the things I talked about are in that article. Mathematicians can find it in the College Math Journal, and it just recently was included in Princeton University Press’s book The Best Writing on Mathematics, 2016 edition. You can find that wherever, your local bookstore.

EL: And where else can our loyal listeners find you online, Dave?

DR: I spend a lot of time on Twitter. I’m @divbyzero. I blog occasionally at divisbyzero.com.

EL: OK.

DR: That’s where I’d recommend finding me.

KK: Cool.

EL: All right. Well, thanks for being here.

DR: Thank you for asking me. It was a pleasure talking to you.

Show notes (click here)