JAD ABUMRAD: Hey, I'm Jad Abumrad.

ROBERT KRULWICH: I'm Robert Krulwich.

JAD: This is Radiolab. Our topic today is ...

ROBERT: Mathematics, mathematics and mathematics.

JAD: [sighs] I suppose that is our topic, but actually we do have a gripping story for you coming up now from our producer Soren Wheeler. Hey, Soren.

SOREN WHEELER: Hey.

JAD: And this is about math, right?

SOREN: Uh, yeah. Well, math and friendship, really. And I heard it from Steve Strogatz. He's a mathematician at Cornell University, and he's been on the show once or twice.

JAD: Mm-hmm.

STEVE STROGATZ: Okay.

SOREN: And we sat down in the studio, and he told me about ...

SOREN: Why don't you back up and tell me a little bit about high school, and about ...

SOREN: ... his high school math teacher, Don Joffray.

STEVE STROGATZ: Well, there were several striking and peculiar things about him. I mean, probably the first thing is that he was physically incredibly impressive. When he would hold the chalk between his enormous fingers and write on the board, the chalk would pulverize with each stroke so that there would be this cloud of chalk dust all over him and his big sweater. Another thing that was very unusual about him, he'd be in the middle of a calculation, standing at the board, chalk dust all over him as usual. Then he would space out, and he'd get a look in his eye, a kind of far-away look, and then he'd say, "Oh, this reminds me," with a hushed tone. "This reminds me of the time Jamie Williams calculated the formula for the Nth term in the Fibonacci Sequence."

JAD: Who's Jamie Williams?

SOREN: Jamie Williams was a student.

STEVE STROGATZ: [laughs]

SOREN: He was just a couple of years ahead of Steve in Mr. Joffray's class.

STEVE STROGATZ: And that was part of the mystique, you know, that now he was graduated and it was as if the secret was lost to the ages. But the point was that he would talk about a student.

SOREN: With reverence.

STEVE STROGATZ: With reverence. What was very thrilling about that is that there was this kind of chain that we were now becoming part of. Yeah, so then I'm off to college, and it started very early. I started to write to him. It was like an annual tidbit. "Dear Mr. Joffray. Here's the gem that I learned this year in math."

SOREN: So Steve would write to him, Mr. Joffray would write back, add something, ask him a new question. And it went on like that for a while, with Steve kind of still being like a student and Mr. Joffray still like a teacher.

STEVE STROGATZ: There was one moment though where something new happened, where he wrote to me asking for help. He said a question came up in his class about an elliptical swimming pool. So, you know, picture a swimming pool. Often there's a little border on the edge of the swimming pool, like a piece of concrete that lines the pool. You stand on that part before jumping in. And so the question was: if you had an elliptical swimming pool with a one-foot border around it, is the outer edge of the border also an ellipse? Something about that really appealed to me. It was a very nice math problem. Probably there was a little bit of a show-off in me. Like, I thought if I could do this, he's gonna say something nice. [laughs]

SOREN: You'll become part of the pantheon?

STEVE STROGATZ: Yeah, maybe I'll enter the pantheon. They'll start talking about me like they used to talk about Jamie Williams. So I stopped whatever I was doing and I worked hard on that ellipse problem. And I figured out two or three different ways to—it turns out it's never an ellipse. It cannot be an ellipse.

SOREN: So Steve sat down and wrote back to Mr. Joffray about this puzzle, but ...

STEVE STROGATZ: I didn't just show him the answer. I wrote the answer in a very loving and gentle way that was meant to be empathetic. That is, I know where you're coming from, and I'm just gonna start from scratch to lead you from where you are to where you need to be to solve this problem.

SOREN: In other words, Steve acted like he was the teacher. And Mr. Joffray played along.

STEVE STROGATZ: This was such a generous thing in retrospect—the humility, the modesty, the kindness in playing the role of the student. It's like he knew that that's what I needed. And man, I loved it. I couldn't wait for the next question. [laughs]

SOREN: And as Steve went off to graduate school to become a math professor himself, he and Mr. Joffray kept writing to each other. In fact, they were writing to each other quite a lot.

STEVE STROGATZ: There was one sequence in March of 1989 where we wrote to each other almost every day. He sent me a puzzle, I worked on it. I showed him a really beautiful answer. He expressed a kind of ecstasy in seeing this answer.

SOREN: It was kind of a mathematician's dream correspondence of puzzles and equations. And Steve loved it, but every so often Mr. Joffray would break the routine.

STEVE STROGATZ: A little bit. He would say things about that he was doing some jazz piano gig. He would sometimes write about—he had three sons. He would talk about them a little bit. And, you know, I feel embarrassed, it feels mean, but I remember not liking those parts of the letters. And I didn't write about that. I mean, I would say maybe I was playing some tennis but I have lines in some of my letters that say after a few of those sentences, "Okay, enough stalling. Here's the math problem." [laughs] But then in later years he would almost pointedly ask me things. Like, there was a time when he said that, "Rumor has it that you're engaged. We wish you the best if this is true." [laughs] And guess what? In my letter back to him, I didn't say anything.

SOREN: Do you remember, like, thinking not to respond? Or just ...

STEVE STROGATZ: Well, I can tell you what was going on, which is that I was already in couple's therapy with my fiance. You know, like, in that time, the letters were a kind of refuge from all that. That is, we could go into this pristine world of math where things are simple and logical and well-ordered. There may have been part of me that felt like, "Oh, come on. This is the one place where it's all perfect."

SOREN: But over the years, that perfect world got a little less perfect.

STEVE STROGATZ: Because his oldest son died. Marshall died. Marshall died when he was only 27, and I didn't ask about it. Can you believe this? I feel so sick about this when I think about it now.

SOREN: So you would just write back," Oh, I've got another puzzle."

STEVE STROGATZ: "Got another math problem for you. Look at this." Um, yeah.

SOREN: And then more than 20 years into this relationship of letter writing, Mr. Joffray retired, and now that he couldn't teach anymore ...

STEVE STROGATZ: He'd write to me. He'd show me these beautiful math problems that he would make up for himself, usually about hawks flying over the Earth and, you know, how much spherical area can the hawk see if it's at such and such altitude. And what is happening at this time is that now I have just gotten married and we've started having kids. And I'm not answering his letters anymore. They're sitting in their envelopes stacking up. He's writing them faster than I can answer them—a lot faster. And then at one point, I got one more letter from Mr. Joffray, except as soon as I looked at the envelope, I could see that something was really very wrong. His handwriting didn't look normal.

SOREN: Huh.

STEVE STROGATZ: My address, my name was written in a craggy ...

SOREN: Like shaky.

STEVE STROGATZ: Shaky. And I knew what that looked like because my dad wrote like that when he had Parkinson's. So I thought, "What's this?" And I opened the letter, and the first sentence is, "Eek! I just had a mild stroke." I didn't write back to him right away. I didn't call him.

SOREN: And then just a couple months later ...

STEVE STROGATZ: My brother died very suddenly. And he heard about it from someone else, and immediately wrote to me how—you know, that he and his wife had heard and they were very sorry to hear that my brother had died. That to me was—you know, I still had never said I'm sorry about Marshall all those years ago. And it kept nagging at me. "Why won't you talk to him?" Beckoning. I obviously care about him. It's sort of like in math there's this concept of bifurcation, which really means a fork in the road, a splitting. When the forces on a system get too large, there can be a moment when the dynamics of that system change abruptly and qualitatively.

STEVE STROGATZ: This was a moment of bifurcation. I should have just said how sorry I was to hear about Marshall. So I thought, "I gotta go talk to him," and I asked him, "Can I come to your house?" You know, he seemed a little reluctant about it, but said, you know, "Okay, fine." So I bought a little pocket tape recorder, just a cheap thing. [laughs] Drove up Route 95 to his house in Connecticut on the shore. I knock on the door, I hear the piano that was playing inside stop. He comes and rushes to see me. And we give each other hugs, take out a big plate of cold cuts and say, "Let's sit out on the porch."

[ARCHIVE CLIP, Steve Strogatz: Does that work? Hello?]

STEVE STROGATZ: And so we're eating, and ...

[ARCHIVE CLIP, Steve Strogatz: It seems to be recording.]

STEVE STROGATZ: And then he takes out his journal ...

[ARCHIVE CLIP, Don Joffray: I decided that I would keep a journal since I was retired.]

STEVE STROGATZ: ... where he's drawn pictures of all kinds of birds.

[ARCHIVE CLIP, Don Joffray: Here's a picture of me doing an eagle watch out in the Connecticut River.]

STEVE STROGATZ: And there's a lot of stuff about ...

[ARCHIVE CLIP, Don Joffray: Hank and his typical way of discovering things.]

STEVE STROGATZ: ... people I don't know.

[ARCHIVE CLIP, Don Joffray: This is a bird that's moved up from the south, too. You never saw these—well, what some people call buzzards.]

[ARCHIVE CLIP, Steve Strogatz: Yeah.]

[ARCHIVE CLIP, Don Joffray: They've moved up here. This is one of my favorite birds. It's a marsh hawk, and it flies low over the meadows. More about that. Hank says, "I'm gonna take you over to see a rough-legged hawk." Now he didn't say, "We're going to see if we can see a rough-legged hawk ..."]

[ARCHIVE CLIP, Steve Strogatz: Yeah.]

[ARCHIVE CLIP, Don Joffray: He produced.]

[ARCHIVE CLIP, Steve Strogatz: Uh-huh.]

STEVE STROGATZ: And I'm thinking to myself, "I'm not really interested in this. I want to talk about him, about all these things that we never talked about, that are emotional hard things like what happened? How did your son die?"

[ARCHIVE CLIP, Don Joffray: ... a lot of work. They're just trying to make guys put in extra hours to pay guys extra hours.]

STEVE STROGATZ: There was a fidgeting feeling inside me.

[ARCHIVE CLIP, Don Joffray: ... pay them benefits and all of the other stuff. I don't know.]

STEVE STROGATZ: And there was a pause. My heart was beating fast. Then I thought, "I'm gonna ask him now."

[ARCHIVE CLIP, Steve Strogatz: So I don't think we ever talked about Marshall, but I wanted ...]

STEVE STROGATZ: And I did. I asked what ...

[ARCHIVE CLIP, Steve Strogatz: I didn't really know him either, but I know that he died very young. What happened?]

STEVE STROGATZ: You know, what happened to Marshall?

[ARCHIVE CLIP, Don Joffray: Well, you know, that's unreal, really.]

[ARCHIVE CLIP, Steve Strogatz: You don't want to talk about that? Okay.]

STEVE STROGATZ: And I think he was gonna say, "That's something we don't talk about."

[ARCHIVE CLIP, Don Joffray: Well, it was—he had ...]

[ARCHIVE CLIP, Steve Strogatz: I remember him as a star.]

SOREN: And did he?

STEVE STROGATZ: He did.

[ARCHIVE CLIP, Don Joffray: He had a wonderful 27 years. Music was gonna be his thing.]

STEVE STROGATZ: It was so beautiful and so uplifting and sweet.

[ARCHIVE CLIP, Don Joffray: He'd be at home, and we'd sit around the piano. And I'll get out the Cole Porter songbook, and just turn to a page—something that he'd never seen. He could sight read it, play it and sing it all at one time with us. And I thought, "God, this guy has got a multichannel mind that I wish I had."]

STEVE STROGATZ: You know, he talked about what a great life he had in his 27 years.

[ARCHIVE CLIP, Don Joffray: Even in his waning moments, he'd stay up all night long playing the piano. Just the house would just fill with beautiful music. And he had made plans to get a job at the New England Conservatory and things like that, but the fates were wrong for him. Oh yeah, we miss him.]

SOREN: Was that—I mean, in that moment, did it change the way you see him?

STEVE STROGATZ: Well, I have to tell you how that day ended. So we talked more, and I asked him at one point, "Do you think Marshall had a religious feeling?" And he said ...

[ARCHIVE CLIP, Don Joffray: Yeah. I think he felt close to having to come to terms with somebody out there. That was a good thing, that I think he went peacefully.]

STEVE STROGATZ: Then actually, the conversation drifted to easier things like calculus problems. And we talked some more about math, and then he said, "How about a swim?" Or "Let's go to the beach."

[ARCHIVE CLIP, Don Joffray: How would you like to go out to the beach and relax?]

[ARCHIVE CLIP, Steve Strogatz: Yeah, I'd like to do something where I get outdoors a little bit.]

STEVE STROGATZ: So we did go to the beach. And it was a beautiful evening, and there were waves coming in from Long Island Sound. And in fact, we were talking about a math problem about waves, about Fourier analysis.

SOREN: Which is really about—well, infinity, and the fact that if you take an infinite number of simple waves, you can create any shape of wave you want.

STEVE STROGATZ: As long as it's a wave that repeats.

SOREN: But then Mr. Joffray asks, "How do you create waves that don't repeat, waves that change?"

STEVE STROGATZ: Sometimes waves don't exactly repeat. They can grow or die out.

SOREN: And Steve told him that the deal with those kinds of waves, you need a different kind of infinity. Not the kind where you just keep adding and adding and adding numbers, but the kind that sits in the space between two numbers.

STEVE STROGATZ: This higher kind of infinity than Don had thought about before.

ROBERT: Thanks to Soren Wheeler, our producer who interviewed Steve and produced that story.

JAD: Thanks to Steve Strogatz, who has a book out now which tells this very story, called The Calculus of Friendship. I'm Jad Abumrad.

ROBERT: Three seconds to go.

JAD: You are?

ROBERT: Robert Krulwich.

JAD: Bye!

[PAUL HOFFMAN: Okay, I'm gonna leave some credits for you. This is Paul Hoffman. Radiolab is produced by Soren Wheeler and Jad Abumrad. Our staff includes Michael Raphael, Ellen Horne, Jonathan Mitchell and Lulu Miller, with help from Adi Narayan and Tim Howard. Special thanks to Ben Calhoun, Steven Vitello, Ron Graham, Ike Sriskandarajah, Ted Gibson, Amanda Aronczyk, Rob Replay, David Krulwich, Ken Baum and Eric Cherries. Thanks.]

[ANSWERING MACHINE: End of message.]

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