Pi, Anyone? A Celebration Of Math And What’s New

IRA FLATOW: This is Science Friday. I’m Ira Flatow. Or should I say Science Pi Day. Because today is March 14. Of course, that’s 3, 1, 4, Pi Day, the day of the year where we celebrate the ratio that makes a circle a circle.

Now, if you think about it, that Greek letter is so part of our culture that it merits our irrational attention. And here to help slice into our pis and why they matter so much is Dr. Steven Strogatz, professor of math at Cornell University and co-host of Quantum Magazine’s podcast, The Joy of Why. Steve, welcome back to Science Friday. Always good to have you.

STEVEN STROGATZ: Thanks so much, Ira. I’m always really happy to talk to you. Thrilled to be here.

IRA FLATOW: All right, so you like to talk about the joy of why. So why pi? Why does this number, this concept have its own special day? Why is it so important?

STEVEN STROGATZ: It’s important in geometry, as we all know from what you mentioned about circles. But I like to think that the real importance has to do with what circles represent, which is they represent anything that goes around and around and repeats itself.

So think of your heartbeat. Think of the cycles in the seasons of the year, the orbits of the planets. Our world is filled with periodic repetitive behavior. And as soon as you try to describe any of that with math, pi is going to pop up.

IRA FLATOW: All right, I’m glad you brought up the idea of describing it with math. Because when you talk about pi, the 3.14, what’s an easy way to visualize why it’s 3.14?

STEVEN STROGATZ: Mm. Well, I suppose one way if you want to picture it would be you could imagine measuring it. You could take any cylindrical object in your house, like a soda can or–

IRA FLATOW: Paint roller.

STEVEN STROGATZ: Yeah, you could do a paint roller. Sure. If you wanted to paint your wall and you pulled out your paint roller, get it all full of paint– and then imagine rolling it on the wall one complete revolution. So like you could make a little tick mark on your roller and then start rolling.

And the next time that tick mark is pointing straight up, you know you’ve done one exact revolution. And at that point, you will have rolled out a distance of paint on the wall that is a little more than three times the width of the roller. That’s what it means, circumference to diameter, 3.14 approximately. So that’s what the paint will show you.

IRA FLATOW: That’s cool. All right, I want to talk about the history of pi because was pi invented? Was it discovered? Was there a big breakthrough someplace?

STEVEN STROGATZ: Haha. Well, you do like your philosophy, don’t you?

IRA FLATOW: I love the history of science and the philosophy.

STEVEN STROGATZ: No, that’s a question that people are still wrestling with. Is it invented? Is it discovered? Certainly, human beings didn’t know the value of pi for a long time, for a millennia. And we often give credit to the Greek mathematician Archimedes around 250 BC for giving the first really principled way of estimating pi. He figured out a way to measure it as accurately as you would want.

And his trick was to think of a circle as a limiting shape where if you imagine putting a square inside of a circle so that its four corners touch the circle and then– well, that’s not a very good approximation to a circle. But you could make something that looks like a stop sign. You could do a hexagon or an octagon. And as you add more sides to the polygon, it starts to look rounder and rounder.

And using that kind of thinking, both putting polygons inside the circle and putting it outside, Archimedes was able to prove that this number pi was trapped between two fractions that he could calculate, which was 3 and 10/70 and 3 and 10/71. Isn’t that amazing?

IRA FLATOW: Yeah, it is. And I’m just wondering what’s going on in his head that he says, I’m going to go look into this?

STEVEN STROGATZ: Well, I wish I knew. He knows that he can’t get it exactly. But he found this kind of numerical vice. Like he could tighten the screws and get tighter and tighter around this mysterious number.

IRA FLATOW: It seems to me like when you learn calculus, first, you talk about making tiny little rectangles under an area. And then you add more and add more until you get an infinite amount. That sort of sounds like what he was doing with the circle, with those, putting those little tangents to it, those little pieces.

STEVEN STROGATZ: You’re exactly right. I would say that this is maybe the first example of calculus that we know of. You could argue, if you’re really a historian of math, there’s an even earlier ancient Greek named Democritus, who we often give credit for the atomic theory to Democritus.

But he had the idea of slicing up shapes into smaller and smaller pieces to approximate a curved shape. But it’s really the virtuosity of Archimedes that translates that ancient idea of Democritus into a workable calculation.

And so, yeah, I think– to my mind, pi– kids love to recite all the digits of pi. Well, not all because, of course, there’s infinitely many. But they already get this feeling that there’s something infinite and mysterious about pi. And that infinity can be traced back to what you just mentioned. To approximate it, we kind of need to think about a polygon with more and more sides. So infinity, right there on our pi plate.

IRA FLATOW: Huh. Hey, so you brought up parents. For parents listening, how would you suggest making pi interesting for their kids?

STEVEN STROGATZ: Mm. Yeah. Well, that’s a great question. There’s all kinds of things you could do. There’s some famous experiments that are sort of tabletop ways of measuring pi. And one of the strangest is you can drop a shape like a needle or a little stick, anything that’s straight–

IRA FLATOW: Toothpick, something like that.

STEVEN STROGATZ: A toothpick. Toothpick would be great. Yeah. So if you drop toothpicks onto graph paper, there is a way of measuring pi from the data that you collect having to do with how many lines the toothpick intersects. It’s not so easy to say off the top of my head. But if people wanted to look it up, the web search you would do– it’s called Buffon’s needle, B-U-F-F-O-N apostrophe S. Buffon’s needle is an experimental way of determining pi using statistics.

IRA FLATOW: Yeah, I heard it once described– a way to do it is to drop toothpicks on a wooden floor where you have floorboards that–

STEVEN STROGATZ: Nice.

IRA FLATOW: –go across.

STEVEN STROGATZ: That’s right. That’s right. If the slats in the floorboard were exactly the length of the toothpick, then you can imagine if you happen to drop the toothpick so that it went lengthwise, it might just fit within a slat and not cross any edges of the slat, whereas if it goes crosswise, it might cross one line. Or you make it so that it can’t cross two. It’s not a long enough toothpick.

IRA FLATOW: I have to tell you, I saw that when I was 12 years old in the Golden Book of Mathematics.

STEVEN STROGATZ: Really?

IRA FLATOW: Those old Golden Books they used to have.

[LAUGHTER]

So it’s not my idea. But I’ve thought about it for years. I know this symbol of pi comes from, of course, the letter in the Greek alphabet. But why is it that letter and not something like iota or something else?

STEVEN STROGATZ: Yeah. It’s great thing to wonder about. It’s tempting to think it has to do with circles. So why not use a round letter like omicron? But in math, we already have something else that represents a round– we use the symbol for zero. So you don’t want anything that looks too much like zero or that’ll be confusing.

So I think why we use pi is it’s supposed to make you think of the word perimeter, the distance around the shape. So we speak of the perimeter of a polygon. And this is the limiting perimeter when you have an infinite polygon.

IRA FLATOW: All right, now we know that it goes on forever. But in mathematics and science or even in engineering when you’re using it to talk about building bridges, how far out do you really go? How far do you need to go in that?

STEVEN STROGATZ: That is a curious point because, yes, you’re right. There’s infinitely many digits. They don’t repeat. They don’t show any pattern. And yet in practice, in engineering or any part of physics or other parts of science, we never really need more than something like 10 digits after the decimal point.

So to me, when we set our supercomputers to calculating trillions of digits of pi, it’s not to improve our engineering calculations. It’s the spirit of human adventure. It’s like Mount Everest that you want to climb it because it’s there.

IRA FLATOW: It’s just geeking out–

STEVEN STROGATZ: Yeah.

IRA FLATOW: –on– Yeah. Let’s talk about math because I know that’s your favorite subject. You write about mathematics all the time. And I know that in physics, there’s been these efforts over the last decades, century to try to unify all the branches of physics. Is there something like that going on in mathematics, and has it been successful?

STEVEN STROGATZ: That is one of the most exciting developments in modern math. So you’re right, the physicists love to unify. They had electricity unified with magnetism in Maxwell’s era, Einstein with space and time, energy and mass. So that’s been a great trend in physics. And in math, we’re doing something similar. People call it the Langlands program or the Langlands conjecture, which hard to describe precisely.

But roughly, it connects the world of numbers, questions about prime numbers and all the subtleties of whole numbers, with what seems like a totally different universe in math, the world of waves, like the world of sound waves that we use in audio engineering, or waves on the ocean, or waves for light propagating through space. So the math of waves and the math of numbers are being connected in really profound ways through this Langlands conjecture, the Langlands program.

And what’s kind of spooky about it is– we tend to think of math as having separate continents. Here’s the world of geometry here. And here’s the continent of algebra there. But we know that there are land bridges that connect those continents. And they may be hard to find. But they’re very valuable in math when we do find them because we can shuttle ideas from one part of math to another and sometimes solve a problem that was intractable on one side using techniques from the other side.

So this Langlands program is still not complete. But there’s all kinds of tantalizing clues that there is a kind of grand unified theory of math that will gradually be getting fleshed out. And it’ll be cause for celebration. The champagne will be popping.

IRA FLATOW: Wow, wow. I have to ask you because I so enjoy talking with you, Steve, about science and math– while I have you here, I like to run all these ideas by you. And the idea I want to ask about is artificial intelligence. Because AI is affecting so many different fields of science. And I want to know how it’s affecting math. Is it changing mathematics? Is it affecting the future of mathematics at all?

STEVEN STROGATZ: That’s the biggest thing going on right now. And I’m excited to talk to you about it. You will find mixed reactions in our community about it. Some people see what’s happening in AI as an existential threat to math, not just to us as a civilization. I mean, there are people worrying about the singularity terminator scenarios.

But even leaving aside that, the question is, what is it we’re trying to do when we do math? Are we looking for answers? Are we looking for proofs? Do we want to solve problems? Those are all very commendable things. But what if, for instance, the AIs can solve problems for us, but they can’t explain what they did?

So take the recent work in protein folding that won the Nobel Prize in chemistry last year. And so now we know how to find the three-dimensional shapes of proteins, super important for medicine and biology, using AI.

But the AI is not yet smart enough to explain to us what principles it discovered that enabled it to solve that problem. It doesn’t work like that. It doesn’t think in terms of principles. It thinks in terms of pattern recognition. So we’re in this weird situation where we now understand much better how to fold proteins, how it works in nature, but we don’t understand why. And in math–

IRA FLATOW: There’s no way to ask the AI to explain it.

STEVEN STROGATZ: That’s the current bottleneck. They’re much better at solving math problems than they are at explaining things to human beings.

IRA FLATOW: [LAUGHS] Oh, I see.

STEVEN STROGATZ: There is a real challenge here that is at the cutting edge of computer science. People call it explanatory AI or interpretable AI. You may have noticed, if you play around with ChatGPT right now or the other competing large language models, they’re all now trying to give you chains of reasoning so you can see what they’re thinking.

IRA FLATOW: Right.

STEVEN STROGATZ: And that’s because it’s reassuring. If somebody is going to give you an answer, you’d like to know, yeah, but why? And then if the thing can tell you why, you might trust it more. And so especially in medicine, let’s say, if we’re going to have AIs making claims about what the right diagnosis is, why they think that, or what the right treatment should be, you’d like to understand the reasoning. Otherwise–

IRA FLATOW: Absolutely.

STEVEN STROGATZ: –it’s scary. Well, OK, so but in math, I mean we really are so proud of our ability to understand. To us, the aesthetic pinnacle of our subject is when we have a proof that goes from axioms all the way to the theorem. We know the reasoning at every step in between.

And so the idea of doing math without understanding, just answers without insight, that’s what I mean by the existential threat. I really feel like– personally, I’m this camp. I think the days when we will understand math may be numbered, that it will not be far in the future when computers are producing really impressive math that we will not understand. And it will be correct. But it’ll be like they’re oracles just telling us the truth and we can’t understand it. We’ll just be sitting there with our mouth open.

IRA FLATOW: This is Science Friday from WNYC Studios. That is existentially scary. I mean–

STEVEN STROGATZ: That’s the question. Is it or isn’t it? I mean, because it could be very valuable to have all those answers. They might give us wondrous things in engineering and medicine.

IRA FLATOW: And they’re testable to be true.

STEVEN STROGATZ: Yeah, that’s the thing. There’s other forms of evidence than human understanding. There’s the real world. Like does it work? And if it keeps working, maybe you’ll come to trust them that way.

And it’s not like it’s unheard of. Think about people with intractable depression. And the doctors will give them electroconvulsive therapy, so-called shock therapy. We don’t really understand what that does to the brain, not in detail, but we know that is sometimes a technique of last resort. And it’s very helpful for some patients.

So doctors have kind of gotten used to the idea of doing certain things that they don’t understand because they just need to do it. And it worked. I mean, other people would say– there is a counterargument to all this, that there’s still a special place for human creativity and imagination. Don’t get carried away. These are just machines.

They’re doing pattern recognition, but they’re not thinking. They’re not smart. So there are plenty of my friends who know a lot more about AI than I do, who say, don’t get all worked up. This is just like a souped up version of your calculator.

IRA FLATOW: All right, one last question because we’re running out of time. And this being March 14, it’s also Albert Einstein’s birthday. And it’s been said about Dr. E that mathematics was not his strong suit. And he sought out the help of others for the equations he needed to express his ideas mathematically. Is that right?

STEVEN STROGATZ: It’s a complicated thing. He was certainly really good at math. No doubt. It’d be an exaggeration to say he wasn’t great. But what he wasn’t great at was being a diligent student. He sometimes would skip class. And so there were times when his more studious classmates, like a friend of his named Marcel Grossmann, clued him in on some things that he missed by sleeping late or doing whatever he was doing.

So yeah, Einstein did have help. Also, his wife was– his first wife was very smart at math and physics. And we think that Mileva Maric also helped him. But no, he was darn good at math. You could be sure of that.

IRA FLATOW: OK, Steve, it’s always a pleasure.

STEVEN STROGATZ: Thank you, Ira. It’s really fun to be able to chat with you.

IRA FLATOW: Happy Pi Day to you.

STEVEN STROGATZ: Likewise.

IRA FLATOW: Dr. Steven Strogatz, professor of math at Cornell University and co-host of Quantum Magazine’s podcast The Joy of Why.

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