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

Episode Date: March 14, 2025

It’s March 14, or Pi Day, that day of the year where we celebrate the ratio that makes a circle a circle. The Greek letter that represents it is such a part of our culture that it merits our irrational attention.Joining Host Ira Flatow to help slice into our pi’s is Dr. Steven Strogatz, professor of math at Cornell University and co-host of Quanta Magazine’s podcast “The Joy Of Why.” They talk about how pi was “discovered,” the ways it’s figuring into recent science, and how AI is changing the field of mathematics.Transcripts for each segment will be available after the show airs on sciencefriday.com. Subscribe to this podcast. Plus, to stay updated on all things science, sign up for Science Friday's newsletters.

This is Science Friday. I'm Ira Plato. Today on the podcast, it's why we dedicate a whole day to celebrating pie. And I mean the math kind. You know, our world is filled with periodic, repetitive behavior. And as soon as you try to describe any of that with math, pie is going to pop up. Today is March 14th. Of course, that's 314 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 pies and why they matter so much is Dr. Stephen Stroggatz, Professor of Math at Cornell University and co-hosts of Quantum Magazine's podcast, The Joy of Why. Steve, welcome back to Science Friday. Always good to have you. Thanks so much, Ira. I'm always really happy to talk to you. Thrill to be here.

All right. So you like to talk about the joy. So why pie? Why does this number, this concept, have its own special day? Why is it so important? 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. You know, our world is filled with periodic, repetitive behavior. And as soon as you try to describe any of that with math, pie is going to pop up. All right. I'm glad you're brought up the idea of describing it with math. Because when you talk about pie, the 3.14, what's an easy way to visualize why it's 3.14?

Hmm. Well, I suppose one way, if you want to picture it, would be you could, you know, you could imagine measuring it. You could take any cylindrical object in your house, like a soda can or paint roller. 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, you know, 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, 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.

That's cool. I want to talk about the history of pie, because was pie invented? Was it discovered? Was there a big breakthrough someplace? Well, you do like your philosophy, don't you? I love the history of science and the philosophy, yes. 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 pie for a long time, you know, 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 pie. 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, like, 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 know, 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 pie was trapped between two fractions that he could calculate, which was three and 1070ths and three and 1070 firsts. Is that amazing? It is about, you know, just wondering what's going on in his head that he says, I'm going to go look into this. 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, you know, tighter and tighter around this mysterious number. You know, 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. It sort of

sounds like what he was doing with the circle with those, you know, putting up those little tangents to it, those little pieces. You're exactly right. I would say that this is maybe the first example of calculus that we know of. It's there, 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, pie, you know, kids love to recite all the digits of pie. Well, not all. Right. Because of course,

there's infinitely many. But they already get this feeling that there's something infinite and mysterious about pie, and that infinity can be traced back to what you just mentioned, that to approximate it, we kind of need to think about a polygon with. more and more aside. So infinity right there on our pie plate. Okay. So you brought up parents for parents listening, how would you suggest making pie interesting for their kids? Yeah. Well, that's a great question. There's all kinds of things you could do. There's some famous experiments that are like, you know, sort of tabletop ways of measuring pie.

And one of the strangest is you can drop a shape. like a needle or a little stick, anything that's straight. Toothpick, something like that? A toothpick. A toothpick. A toothpick would be great. Yeah. So if you drop toothpicks onto graph paper, there is a way of measuring pie 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-A-S. Bufant's needle is a kind of an experimental way of determining Pai using statistics. Yeah, I'd heard it once described a way to do is to drop toothpicks on a wooden floor where you have floorboards, you know, they go across. That's right.

Yeah. 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-exam. 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. I have to tell you, I saw that when I was 12 years old in the golden book of mathematics. Really? Those old golden books they used to have. So it's not my idea, but I've thought about it for years. I know the symbol of pie comes from,

of course, the letter in the Greek alphabet, but why is it that letter? And that's up there. like Ayota or something else. Yeah, it's a great thing to wonder about. You know, 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 around. You know, 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, right? So we speak of the perimeter of a polygon, and this is the limiting perimeter when you have an infinite polygon. 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? 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 pie, it's not to improve our engineering calculations. It's sort of the spirit of human adventure. You know, it's like Mount Everest, that you want to climb it because it's there. It's just geeking out. Yeah. We have to take a quick break, but don't go anywhere.

We've got lots more when we come back. I think the days when we will understand math may be numbered. Let's talk about math, because I know that's your favorite subject. You're right 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?

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 was 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 Langland's conjecture, the Langlands program. And what? What's kind of spooky about it is, you know, we tend to think of math as having, like, separate continents, like 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 Langland's 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, you know, cost for celebration. The champagne will be popping. Wow. Well, you know, I have to ask you because I so enjoy talking with you, Steve, about science and math. I thought, 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, you know, how it's affecting math. Is it changing mathematics? Is it affecting the future of mathematics at all? 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, you know, 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, you know, that won the Nobel Prize in Chemistry last year. And so now we know how to find the three-dimensional shapes of protein, 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... And there's no way to ask the AI to explain it?

That's the current bottleneck. They're much better at solving math problems than they are at explaining things to human beings. There is a real challenge here that is at the cutting edge of computer science, people call it explanatory AI or interpretable AI. And it's not, you know, like you may have noticed if you play around with chat GPT 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. Right. And that's because it's reassuring. If somebody's 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 like, especially in medicine, let's say, if we're going to have AI's 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, it's scary. Well, okay. So, but in math, I mean, we really are so proud of our ability to understand. You know, like 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 in 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 will just be sitting there with our mouth open. This is Science Friday from WNYC Studios. That is existentially scary. That's the question, is it or isn't it?

Yeah. You know, I mean, because it could be very valuable to have all those answers. They might give us wondrous things in engineering and medicine. And they're testable to be true. Yeah, that's the thing. There's other forms of evidence than human understanding. There's the real world. Right.

You know, 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, you know, so-called shock therapy. We don't really understand what that does to the brain,

not in detail, but we know that 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, then it worked. I mean, other people would say, there is a counter argument 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, you know, like a souped up version of your calculator. All right. One last question, because we're running out of time. And this, being March 14th, 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, right, to express his ideas mathematically. Is that, is that right? 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 name Marcel Grossman, 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 first wife was very smart at math and physics. And we think the Malava Merich also helped him. But no, he was darn good at math. You could be sure of that. Okay.

Steve, it's always a pleasure. Thank you, Irene. It's really fun to be able to chat with you. Happy Pye Day to you. Likewise. Dr. Stephen Strogatz, professor of math at Cornell University and co-hosts of Quantum Magazine's podcast, The Joy of Why. That's about all the time we have for now.

A lot of people help make this show happen. John Denkoski. Annie Niro. Jason Rosenberg. Rasha Reedy. I'm Ira Flato. Thanks for listening.