My Favorite Theorem

Evelyn Lamb: Hello and welcome to My Favorite Theorem, the math podcast with no quiz at the end. I like how we're always besmirching other math podcasts, which as far as I know, also don't have quizzes at the end. I am your host Evelyn lamb. I am a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: Hi, I'm Kevin Knudson, professor of mathematics at the University of Florida. Okay, full confession. I don't listen to other podcasts, so I don't know if they have quizzes at the end or not.

EL: Shame. They probably don't. I mean, how would you even administer that?

KK: That’s right. That's right. Yeah. Yeah. We don't need to find this out.

EL: Yeah. Well, we are, we should, we should just get right.

KK: Let’s do it. Yeah.

EL: We are very happy today to invite Daina Taimina onto the show. So Daina, please introduce yourself, tell us a little bit about yourself, and we'll get started.

Daina Taimina: Hello. Thank you for inviting me. And, actually, I didn't prepare much to tell about myself. Because usually, I tell my students, you know, just search for me on the internet. That knows more about me than myself.

EL: Maybe even some of it’s true.

DT: Maybe, yes. Sometimes it's true. Yeah. So well, okay. Well, I was teaching about 20 years, I was teaching mathematics at the University of Latvia. And at about the same time, I was teaching in Cornell, where I, now I have stopped teaching. And I officially count as a retired, so it means I have free time to do whatever I want.

KK: Nice.

DT: Yes. And then sometimes, well, it has been now for 25 years, I have crocheted hyperbolic planes. And I guess that's what people know most. Because sometimes I am introduced on people. “Oh, yeah.” And then this person who is introducing me says, “You know, she's the one with hyperbolic planes.” “Oh, yeah! Yes, yes, I know that!” Okay, I guess that's my other name.

EL: Yeah. And our in your multitalented our listeners can't see it, but when we were saying hi we saw that you have some of your very own paintings in the background on Zoom, where I'm seeing right now, and they look very lovely. So you do art in addition to crochet and math?

DT: Actually, that was before, and I and the reason why I was doing I was doing art, is I signed up for a watercolor lessons because I knew that I'm very bad at art. Because when I was in school, I was told I can do anything but that. And at that time in Cornell, I was teaching students who were very afraid of maths, and most most of them were actually architecture art students or music students, and I really wanted to experience how it is to take some subject where you are told, and you believe all your life, that you are bad and you can’t do it. So I did it. And so yes, that was interesting experience.

EL: Well, and it looks like with some practice, you gained skills. Amazing how that works.

KK: I did that to actually about a year ago. I cannot draw. I'm terrible. So maybe we have the same issues here.

DT: Yes, yes. Yes, exactly.

KK: And I took it I took like a just a two hour drawing workshop online where we draw birds, and I actually drew something that looked like a bird at the end. So you know, it can be done.

DT: Yes, because this is what you what you learn, is that — and then I was explaining to my students, too —I brought in one of my paintings, and I said, it's actually, what I realize is that it was things which I knew. I knew, well, perspective. I knew how to do composition from photography, you know, just like doing some photography. And all I needed was, you know, I did need some technical skills, and that is the same in math. You do need to learn some technical skills, and then you can then you can get on, so it's not that different.

EL: Yeah, that's a great, a great lesson to learn and to help share with your students like, “Hey, we're all learning various things. I have this background, you have this background, but we can all improve in various areas of our lives.”

KK: Yeah.

EL: Well, the name of this podcast is My Favorite Theorem. So, what is your favorite theorem?

DT: Well, as I told you, my favorite theorem is Desargues’ theorem. Yes, and then, well, actually it started with some more ancient theorem, which was Pappus’s theorem. And it was somewhere, I think I was in middle schoo, and I was reading something in a math history, and I read that there is this ancient theorem, where if you are having two lines, and then you choose three points on each of them and those lines are non-parallel. Well, if they parallel than they are, that’s a very simple case. But if you have like two lines at some kind of angle, and then you choose, and then you then you connect in pairs points from these lines, and you always have three points which are on the same line. And I know like I was just like, that's — okay, I'm so old that at that time, there was no Geometry Sketchpad or any of these programs, there was no computer. So I just kept drawing these lines and finding those points, then it was just amazing. And then of course, I was like, “Oh, what else is there?” And that's when I discovered this, this Desargues’ theorem, which said, okay, if two triangles are situated so that three lines joining their corresponding vertices all meet at a single point, then the points of intersection of the two triangles’ corresponding sides, if those intersection points exist, all lie on one line. I couldn't — I read it, and I couldn't believe that. So again, I took a pencil and took a straightedge and I started to draw, like, various ways, and it was really finding these points and having them and, and then later learning that the converse of this Desargues’ theorem is also true, and then that’s a converse, theorem, that's also a dual theorem. So it was just so fascinating. So that was something, like, different from the geometry, you know, like, exactly the geometry we were having going to school, and so that's kind of led to perspective. And yes, I was just like, really it was fascinating.

EL: Yeah, this is one that has come up a few times for me in things I've read, or people I've talked to in the past few years. But yeah, I loved geometry for a long time, and this is not a theorem that I was exposed to, in most of my geometry education.

KK: Yeah.

DT: And it's very interesting that you can you can prove this theorem using another ancient theorem, Menelaus’s theorem — okay, I'm not going to talk about that — but that sounds very algebraic, because that uses uses proportions, and it's totally in Euclidean geometry. But I like the way how Desargues himself saw, and he actually was thinking about it in three dimensions, and then it's simple. When you are cutting, like, a triangular pyramid with two planes, and then it's just totally obvious.

EL: Okay, I'll have to sit down and try to visualize that a little better.

KK: Right, isn't the simplest proof, don't you use three, you have to go into 3d and then it sort of, like you say, sort of becomes obvious?

DT: Exactly, yes, yes. Yes. That's one of those. Yeah, that's one of those cases, and that was so great! You know, you just jump out, and then it's obvious. And then it's also, the other thing is, if you are having — so you know, like you can imagine that you are having a book, or though now you're having a point, and then you are projecting a triangle, and then all you do is, you imagine that those lines, that it stretches, and then all you do is you open it up in one plane. And there's the theorem.

KK: There’s the proof. Yep, I wish our listeners could have just seen that.

DT: I don't know how to describe.

KK: So this actually came up for you in school in Latvia? Like, your instructor actually taught?

DT: No, I believe I was reading something from Martin Gardner, or something outside, but I did have a wonderful geometry teacher.

EL: So you were interested in math very early in your education?

DT: It was just one of my easiest subjects. I was interested more in literature and languages. That's what you are saying, you know, like an art. Math, it’s just something that comes by nothing. It’s just simple, just seeing things.

KK: I mean, well, so Evelyn, maybe you had this experience. I mean, I became a mathematician because I was always good at math, right? It was the thing that I could easily do. And so it's sort of interesting that it was sort of the easy thing you could do, but you liked something else more?

DT: Yes, that’s true. Yeah.

EL: Yeah. I I had sort of a similar experience to you Daina, I think, where I was, you know,“good at math” — good at arithmetic, basically — in elementary school. And I liked the proofs in geometry, although I didn't understand that those were “real math” also. I thought it was just a diversion.

KK: The two-column proofs?

EL: Yeah, I liked the logic part of it. You're working it through, but I thought arithmetic was real math. And so yeah, I wasn't as interested in that. I was more interested in — I really liked science, but I did a lot of music also and stuff. But eventually, it wasn't until college, that I really kind of fell in love with it, and decided to devote my life to it in at least some form.

KK: Including podcasts at this point. So yeah.

DT: Well, you've been successful.

KK: Sure. So, have you used this at all? I mean, Is this a theorem that you use in your own in your own mathematical work? Or is it just something that you just love?

DT: No, I this is something which I love. Yes. Because, ya know, it’s — well, using it in teaching, you know, and sometimes I have used it talking in schools, you know, you go meet students and show them something, like, here are some fun and some beautiful things. But no, also I was teaching history of mathematics for many years, too. And I loved that Desargues himself, he never published his theorem. It was it was published by his student, Abraham Bosse. And I think he mentions exactly that Desargues had this three-dimensional proof. But also there was an interesting thing: When Desargues — he belonged to Mersenne’s circle circle, and that is the same circle where there were also Descartes and Pascal. And there was all this mathematical writing and just that exchanging of ideas, so that's fun. I was trying to find out, I know there was a some, in discrete mathematics, there is this 10-line configuration, and then that can be used to solve some problem. And I remember there was some sports problem, but I didn't remember like, you know, just precisely what. It’s interesting.

And then the other thing which I like, about this, not only this perspective, but that you can exchange points with lines and lines with points.

EL: Yeah.

DT: Yeah, so in some ways, maybe this is why I was getting more interested in mathematics, like what's beyond what we learned in school. This was this was like, I guess a very first example that I found that is something more about math, you know, more fun than you'll learn in school — which I wouldn't say that I was bored, but still, that there was something more.

EL: Yeah, well, and you mentioned exchanging points for lines, which, I always feel like I've sort of pulled one over on someone if you could do that kind of duality thing. And you know, move intersections of lines to a different line, and then the points intersect at where the lines were, something like that, and so yeah, that's always very satisfying, I think. So part of this podcast is that we ask our guests to pair their theorem with something the way you might pair food and wine or food and your favorite jazz CD or something like that. I’m not appealing to the youths if I say CDs. I don't know if Gen Z knows what that is.

DT: Yeah, it's like floppy disks.

KK: You have to say vinyl now.

EL: Or Spotify. But anyway, what have you chosen to pair with this theorem?

DT: With travel.

KK: Okay, yeah.

DT: So for me like it is exactly because you can change these points and lines. And then if we go back to that, what I was talking about ,projecting the triangle through that one point. And that's what I was imagining if I was that point, and then I'm looking on someplace on this first triangle. but when I travel, it really expands what I'm seeing when I get to that second triangle and see it in reality, and then I can go back. And that would be like, exchanging, so that would be this duality. From that place, now I can see myself differently.

EL: Oh, I love that.

KK: This is a very thoughtful pairing.

EL: Yeah. And where are some of your favorite places that you've traveled?

DT: Well, there are places like, I always like to travel to Sicily. So this is a very, very significant place for me, because that's where I met my husband, and then we had returned back there, and it's never too much to go back there. So I liked that, well, we managed to travel, and I liked my travel to South Africa. And actually, what I like in my travels is to meet people. And I think that's what those those big triangles when you project is, you meet the people and you talk with people, and I don't travel like without some purpose. And mostly it can be like really meeting people and doing some workshop or talk, and then of course learning about the place where I am.

EL: Yeah. And I always love how you learn both that the way you're used to doing things isn't the only way to do things, and that there are similarities across cultures, that we're all kind of the same in some way. So they're kind of those two contrasting ideas, but maybe they're dualistic in some way as well.

DT: Yeah, it’s like one of the things, since I was in Europe, and I have been knitting a lot. And I come here, and I see that there is another way of knitting, and I'm just staring and it’s just so totally different. But at the end, well, we get we get to same thing.

EL: Yeah, well, and since you bring that up, I did want to mention, as you said, you might be best known in the math community for your hyperbolic crocheting. And I know that Desargues’s theorem is from Euclidean geometry, not hyperbolic. But I would love to talk a little bit about kind of where hyperbolic crochet came from and how you got the idea to do that. Everyone loves it.

DT: No, I'm glad. It's just because I had to teach hyperbolic geometry, and, well, it happened. It's like this. So it’s summer, so in a month, it's 25 years since I crocheted my very first plane. And I'm really surprised people are still interested in that. And then it's kind of now a usual thing. It’s also like with Desargues’s theorem, you go into 3-d and then the hyperbolic plane, you can see only in 3d, because if you if you project then those are maps, like on in 2-d, but in 3-d and then particularly, you can just touch it. And you can do a tactile exploration. My main purpose was exactly for my students, so that they could touch and explore it. Because I saw a paper model, which was done, but one thing is to glue a paper model, which I did, but then once you fold it, for the next class, you have to do it again. And it was like, okay, no, I need something more durable. Yeah, so it was really for teaching. And then it's interesting that this hyperbolic plane started to teach me. When I got suddenly, unexpectedly for me, invited for the first art show, you know, now I had to learn, okay, what does that mean? So finding colors and ways to express, so and it has been going on and on.

KK: So, my wife actually crocheted me one once. So she, she can crochet really well, and it is remarkable to hold that thing in your hand. You can sort of begin to really understand how distances work in this weird, floppy, hyperbolic plane. It's really beautiful.

DT: Yeah, because it's another way to get our knowledge, because we need to feel it. It's not like when you are reading — well, okay, if you are an experienced cook, you can read a cookbook and taste a recipe, you know, like field tasting. But actually the best is that you try to cook it, and then you taste and then, you know, like, that's when you really learned about it.

EL: Yeah, and I think in my experience making hyperbolic crocheted planes, one of the the lessons, the math lessons I learned the most from it is just exponential growth is even more than you think. Because I think the first one I made, I started with something like 10 stitches, and I did it, maybe it was a five to four increase ratio. And, you know, five or six rows down, I was like, am I ever going to finish this row? Even as, you know, five to four to me, it doesn't seem like a big number. Like it's just barely above one. But if you, you know, multiply it by itself a few times, it takes a while.

DT: My hope is when the pandemic started, and then people were told that this virus is spreading exponentially, I hope that at least those who had crocheted hyperbolic plane instantly knew what it is what it means.

EL: Yeah.

KK: And then, of course, this also spawned this whole, like hyperbolic crochet reefs that they would do. So our librarians here actually organized an exhibition, they put it in the this display window in the front of the library. They got people to crochet coral, but it's still the same basic thing that you you came up with.

DT: well, it just it came like that, it all came up. So yes, it's actually that came up when — that was from my very first lecture to the general audience. And I was just thinking, Okay, well, one thing is to show the mathematical object, but then I need something, you know, in real life, too. And I remember that before that, first off, that's when I really was finding, like, lettuce leaves. And it was finding like, some curly things, and that gave an idea to see, you know, like, okay, here we go. It’s in a nature and then that just span us by not just the spin off. Idea. Yeah.

EL: Yeah. And I have to put in an advertisement for your book. Is it Crocheting Adventures in the hyperbolic plane? Is that the title?

DT: With.

EL: With hyperbolic planes.

DT: Crocheting Adventures with Hyperbolic Planes. Yes. And then there is another one, Experiencing Geometry, 4thedition, is open source. And there are a lot of hyperbolic planes in that one, too. It's on Project Euclid. And yeah, that’s open source. That was my husband's wish, that it would be open source. So that's — when I finished this fourth edition, that’s it.

EL: Yeah, and I just can't recommend — I haven't looked, I think I've looked at Experiencing Geometry, but not spent as much time with it as the crocheting one because around the time I, you know, a friend gave me a little crochet kit (not because of math at all, just because like, Oh, you like crafty things, you might like this), I happened to see your book. So it’s, like, this confluence, and your book has more than just making a plane. There are a lot of other interesting math ideas that you put in there. So I just can't can't recommend it enough. And it won an award for weirdest title, didn't it?

DT: Yeah, yes. Well, of course, that that's when — it was the strangest book title, the Diagram Prize — that went around the world. Of course, two years later, when I got an Euler Prize, which is much more serious, from the Mathematical Association of America, that press wasn't interested at all.

EL: Yeah, you know, the strangest title, I guess is a little little more headline-grabbing or something like that. But yeah, yeah. Anyway, I know we sort of went on a diversion from triangles, but I'm glad we got to talk about the hyperbolic crochet a bit because really, I think, so many people have had positive experiences with math and with experiencing geometry in a different way than just looking at a flat on a piece of paper thanks to you.

DT: Thank you.

EL: So, we've plugged your book, but we do like to give our guests a chance to share anything else that they'd like to share, other resources or websites or anything like that for people to, to look at?

DT: Well, when the book came out, I started to write a blog which is called hyperbolic-crochet.blogspot.com. But I'm not very good at keeping it up, so I’m not sure how many people are reading.

KK: Is anybody?

DT: But there is still lots of material and trying to answer a set of questions. And then of course, as I said, on the Project Euclid forum, it's on Project Euclid, look for Experiencing Geometry. So that is my newest book, which I’ve finished. And yeah, so as I said, a lot of interesting geometric things, too. If you are interested in geometry, that would be a good thing to look up, particularly for teachers, because I added one of the appendices, suggestions from various geometry projects, which you can do in class. Because those were questions people were sending me, like, okay, so what can we do in class and what can be some fun things we can do? So yes, I was trying to help. I hope it's helpful.

EL: Yeah. Wonderful.

KK: All right.

EL: Well, thank you so much for joining us. I was really glad to get to talk with you.

DT: Okay, thank you.

[outro]

On this episode of My Favorite Theorem, we were pleased to talk with Daina Taimina, recently retired from Cornell University, about Desargues's theorem. Here are some links you might find interesting after you listen.

Her website, blog, and Twitter account
Desargues's theorem on Wikipedia
Our episode with Annalisa Crannell, who also loves Desargues's theorem
Taimina's book Crocheting Adventures with Hyperbolic Planes, which won the Diagram Prize for oddest book title and the Euler Prize from the Mathematical Association of America
Experiencing Geometry by Taimina and David Henderson on Project Euclid