SCOTT SIMON, host:

This is WEEKEND EDITION, from NPR News. I'm Scott Simon.

Most mathematicians prove many theorems in their lives, but the process by which their name actually gets branded onto one of them is haphazard. For instance, there is Oler(ph) Gauss and Ferma(ph), they all have theorems. But each proved hundreds of theorems, aside from their important ones. Their names got attached to just a few.

From Stanford now to join us to talk about All Things Theorem is our math guy Keith Devlin.

Professor KEITH DEVLIN (Stanford University): Hi, Scott. Good to be here again.

SIMON: And he has a guest in the studio, which surprises us, because I think a lot of people are reluctant to go into the study with you, Keith. This is Danica McKellar. Now I think it's safe to say that many people know her as Winnie from The Wonder Years as well as guest spots on The West Wing. So thanks very much for being with us, Ms. McKellar.

Ms. DANICA MCKELLAR (Actress and Mathematician): My pleasure. Thank you for having me.

SIMON: And you're with us because you have a theorem.

Ms. MCKELLAR: Yes, I do. I took a break from acting for four years to get a degree in mathematics at UCLA, and during that time I had the rare opportunity to actually do research as an undergraduate. And myself and two other people co-authored a new theorem. And the name of the theorem is Percolation and Gibbs States Multiplicity for Ferromagnetic Ashkin-Teller Models on Two Dimensions, or Z2. That's the name of the paper. But nobody calls it that because it's hard to say. And when these journalists in People magazine and Star magazine and whatever the magazine is that decided to do an article about me and this math research, they just would rather call it the Chayes-McKellar-Winn Theorem. And that's why it's sort of become known as that.

SIMON: Could the two of you perhaps explain this theorem to us in a way that a substantial portion of even our audience would understand?

Professor DEVLIN: I'm not going to make a fool of myself in a different way on the air this time. I'm going to let the author of this paper talk about it. So I'm just going to hand it over to Danica.

Ms. MCKELLAR: We took a two-dimensional mathematical model of magnetic material and proved a theorem about it. It's called the Ashkin-Teller Model. And it's a pretend two-dimensional lattice grid representation of this material, and it has properties based on temperature. One of them is called percolation, and another one is called whether it has multiple Gibbs States or not. And each of these two properties, percolation and Gibbs States multiplicity, they each have a crucial temperate above which they do not happen and below which they do happen. And we proved that for this model, for the Ashkin-Teller Model, those two temperatures are the same.

SIMON: Now, let me follow up a little bit. Why percolation as opposed to drip or French press?

Ms. MCKELLAR: That's a really good question. I've got to take that up with someone.

Professor DEVLIN: I'll tell you, Scott, if I said that Danica has just said, you would have cut me off halfway through. I know you would have done it.

SIMON: We're polite to guests.

Ms. MCKELLAR: I could go on, but you probably don't really want me to.

SIMON: Keith, what has this theorem opened the door to?

Professor DEVLIN: It's part of the whole business of taking mathematics and trying to apply it to the real world by constructing models of the real world, and then proving things about those models. And then occasionally you work with these models, and sometimes you find that there were insights that you get about the real world, and you can take the results and feed them into science and engineering and [unintelligible] so forth.

Ms. MCKELLAR: One of the most amazing things about mathematics is the people who do math aren't usually interested in application, because mathematics itself is truly a beautiful art form. It's structures and patterns, and that's what we love, and that's what we get off on. And so we write papers because we discovered a new pattern of sorts. Then it's up to somebody else to figure out if it has a use or not. And maybe it will be like a hundred years from now, and who cares. But that's really what it's all about for us.

SIMON: Ms. McKellar, may I ask, is it a real chore for you to do like a Burger King commercial or something? I mean, just in terms of your interest level.

Ms. MCKELLAR: I love acting. Acting is a true love of mine, acting and math. Although they are both creative, they use very different sides of your brain. And I love both. Acting is my first love, and that's my main career, it really is. After I did the paper, I really did go back to acting. So it's not hard to go back to it. I absolutely love it. And it has its own dignity and intelligence and everything else.

However, I recognize that I have a unique position to be a role model to young girls because I am doing something that they consider glamorous, which is acting, and yet I took a time to really get my education and study mathematics, and I think math is the cat's meow. And I share that I do. I have my website, Danicamckellar.com, where I've been answering math questions for years and years now. I mean, that makes me so happy, I just don't even know what to do with myself. It makes me happier than when I get an acting job.

SIMON: You're in acting for the long haul. Math, however, is a passion for you?

Ms. MCKELLAR: Exactly. And I believe that, you know, even though there is that chance that if I stayed in mathematics that I would have proven some incredibly groundbreaking theorem, I'd rather use my powers in math to be an example that makes people question their stereotypes about what a mathematician looks like.

SIMON: A mathematician, for example, can be a droll and distinguished looking man from Yorkshire, right, in theory?

Ms. MCKELLAR: Absolutely.

Professor DEVLIN: There's only one of those, Scott.

Ms. MCKELLAR: Absolutely.

SIMON: They did break the mold.

Professor DEVLIN: I'll tell you, Scott, this is, as a professor of mathematics, this is a substantial piece of mathematics. And any undergraduates who gets their name on a paper like this deserves, justifiably, to be very proud of what they have done. This is real mathematics.

SIMON: Our math guy, Keith Devlin, who's the executive director at the Center for the Study of Language and Information at Stanford University. His most recent book is the Math of Instinct: Why You're a Mathematical Genius. I don't think that applies to all of us, along with lobsters, cats and birds. And also, Danica McKellar, the actor from the West Wing and who's also author of Percolation and Gibbs State Multiplicity for Ferromagnetic Ashkin-Teller Models on Z2.

Ms. MCKELLAR: Very good. Very good.

SIMON: Thank you very, well, I'm very familiar with it. I read it all the time. Vulgarly known as the Chayes-McKellar-Winn Theorem. COST: $00.00

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