Episode 52 - Ben Orlin
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Kevin Knudson: Welcome to My Favorite Theorem, a math podcast. I'm Kevin Knudson, professor of mathematics at the University of Florida. And here is your other host.
Evelyn Lamb: Hi, I'm Evelyn Lamb. I'm a freelance math and science writer, usually based in Salt Lake City, but currently still in Providence. I'll be leaving from this semester at ICERM in about a week. So trying to eat the last oysters that remain in the state before I leave and then head back.
KK: Okay, so you actually like oysters.
EL: Oh, I love them. Yeah, they're fantastic.
KK: That is one of those, it’s a very binary food, right? You either love them—and I do not like them at all.
EL: Oh, I get that, I totally get it.
KK: Sure.
EL: They’re like, in some sense objectively gross, but I actually love them.
KK: Well, I'm glad you've gotten your fill in. Probably—I imagine they're a little more difficult to get in Salt Lake City.
EL: Yeah, you can but it’s not like you can get over here.
KK: Might be slightly iffy. You don't know how long they've been out of the water, right?
EL: Yeah. So there's one place that we eat oysters sometimes there, yeah, that's the only place.
KK: Yeah, right. Okay. Well, today we are pleased to welcome Ben Orlin. Ben, why don't you introduce yourself?
Ben Orlin: Yeah, well, thanks so much for having me, Kevin and Evelyn. Yes, I'm Ben Orlin. I’m a math teacher, and I write books about math. So my first book was called Math with Bad Drawings, and my second one is called Change Is the Only Constant.
EL: Yeah, and you have a great blog of the same name as your first book, Math with Bad Drawings.
BO: Yeah, thank you. And I think our blogs are, I think almost birthday, not exactly but we started them within months of each other, right? Roots of Unity and Math with Bad Drawings.
EL: Oh, yeah.
BO: Began in, like, spring of 2013 which was a fertile time for blogs to begin.
EL: Yeah. Well, in a few years ago, you had some poll of readers of like, what other things they read and, and stuff and my blog was like, considered the most similar to yours, by some metric.
BO: Yeah, I did a reader survey and asked people, right, what what other sources they read, and mostly I was looking for reading recommendations. So what else do they consider similar? Overwhelmingly it was XKCD. Not so much—just because XKCD, it’s like if you have a little light that you're holding, a little candle you're holding up, and you're like, what does this remind you of? And like a lot of people are going to say the sun because they look up, and that’s where they see visible light.
KK: Sure.
BO: But I think in terms of actually similar writing, I think Toots of Unity is not so different, I think.
EL: Yeah. So I thought that was interesting because I have very few drawings on on mine. Although the ones that I do personally create are definitely bad. So I guess there’s that similarity.
BO: That’s the key thing, committing to the low quality.
KK: Yeah, but that's just it. I would argue they're actually not bad. So if I tried to draw like you draw, it would be worse. So I guess my book should just be Math with Worse Drawings.
BO: Right.
KK: You actually get a lot of emotion out of your characters, even though they're they're simple stick figures, right? There’s some skill there.
BO: Yeah, yeah. So I tried. I tried to draw them with a very expressive faces. Yeah, they're definitely still bad drawings is my feeling. Sometimes people say like, “Oh, but they've gotten so much better since you started the blog,” which is true, but it's one of these things where they could they could get a lot better every five-year interval for the next 50 years and still, I think not look like professional drawings by the end of it.
EL: Right. You're not approaching Rembrandt or anything.
KK: All right, so we asked you on here, because you do have bad drawings, but you also have thoughts about mathematics and you communicate them very well through your drawings. So you must have a favorite theorem. What is it?
BO: Yeah. So this one is drawn from my second book, actually, the second book is about calculus. And I have to confess I already kind of strayed from the assignment because it's not so much a favorite theorem as a favorite construction.
KK: Oh, that’s cool.
EL: You know, we get rule breakers on here. So yeah, it happens.
BO: Yeah, I guess that's the nature of mathematicians, they like to bend the rules and imagine new premises. So pretending that this were titled My Favorite cCnstruction, I would pick Weierstrass’s function. So that you know, first introduced in 1872. And the idea is it's this function which is continuous everywhere and differentiable nowhere.
EL: Yeah. Do you want to describe maybe what this looks like for anyone who might not have seen it yet?
BO: Yeah, sure. So when you're picturing a graph, right, you're probably picturing—it varies. I teach secondary school. So students are usually picturing a fairly small set of possibilities, right? Like you're picturing a line, maybe you're thinking of a parabola, maybe something with a few more squiggles, maybe as many squiggles as a sine wave going up and down. But they all have a few things in common one is that almost anything that students are going to picture is continuous everywhere. So basically, it's made of one unbroken line. You can imagine drawing it with your pencil without picking the pencil up. And then the other feature that they have is that they—this one's a little subtler, but there will be almost no points that are jagged, or sort of crooked, or, you know, if I picture an absolute value graph, right, it sort of is a straight line going down to the origin from the left, and then there's a sharp corner at the origin, and then it rises away from that sharp corner. And so those kind of sharp corners, you may have one or two in a graph a student would draw, but that's sort of it. You know, like sharp corners are weird. You don't can't draw all sharp corners. It feels like between any two sharp corners on your graph, there's going to have to be some some kind of non-sharp stuff connecting it, some kind of smooth bits going between them.
KK: Right.
BO: And so what sort of wild about about Weierstrass’s function is that you look at it, and it just looks very jagged. It’s got a lot of sharp corners. And you start zooming in, and you see that even between the sharp corners, there are more sharp corners. And you keep zooming in and there's just sharp corners all the way down. It's what we today call it fractal. Although back then that word wasn't around. And it's just it's the entire thing. Every single point along this curve is in some sense, a sharp corner.
EL: Yeah, it kind of looks like an absolute value everywhere.
BO: Yeah, exactly. It has that cusp at every single point you could look at.
KK: Right? So very pathological in nature. And, you know, I'm sure I've seen the construction of this. Is it easy to say what the construction is? Or is this going to be too technical for an audio format?
BO: It’s actually not hard to construct. There are there whole families of functions that have the same property. But Weierstrass’s is pretty simple. He starts with basically just a cosine curve. So you sort of have cosine of πx. So picture, you know, a cosine wave that has a period of two. And then you do another one that has a much shorter period. So you can sort of pick different numbers. But let's say the next one that you add on has a period that's 21 times faster. So it's sort of going up and down much quicker. And it's shorter, though, we've shrunk the amplitude also. So it's only about a third, let's say, as tall. And so you add that onto your first function. So now we've got—we started with just a nice, gentle wave. And now we've got a wave that has lots of little waves kind of coming off of it. And then you keep repeating that process. So the next, the second one in the iteration has a period of 21 cycles for two units. The next one has 212 cycles. And it's 1/9 the height of the original.
KK: Okay.
BO: And then after that, you're going to do you know, 213 cycles in the same span, 214 cycles. And so it goes—I don't know if you can hear my daughter is crying in the background, because I think she she finds it sort of upsetting to imagine the function that's has this kind of weird property.
EL: Fair.
BO: Especially because it's such a simple construction. Right? It's just, like, little building blocks for her that we're putting together. And one of the things I like about the construction, is it at no step, do you have any non-differentiable points, actually. It's a wave with a little wave on top of it and lots of little waves on top of that, and then tons and tons of little waves on top of that, but these are all smooth, nice, curving waves. And then it's only in the limit, sort of at the at the end of that infinite bridge, that suddenly it goes from all these little waves to its differentiable nowhere.
KK: I mean, I could see why that would be true, right?
BO: Yeah, right. Right. It feels like it's getting worse. And you can do—Weierstrass’s function is really a whole family of functions. He came up with some conditions that you need, basically that’s the basic idea. You need to pick an odd number for the number of cycles and then a geometric series for for the amplitude.
KK: So what's so appealing about this to you? It's just you can't draw it well, like you have to draw it badly?
KK: Yeah, that's one thing, right. Exactly. I try to push people into my corner, force them to have to drop badly. I do like that this is something—right, graphs of functions are so concrete. And yet this one you really can't draw. I've got it in my book, I have a picture of the first few iterations. And already, you can't tell the difference between the third step and the fourth step. So I had to, I had to, you know, do a little box and an inset picture and say, actually, in this fourth step, what looks like one little wave is really made up of 21 smaller waves. So I do sort of like that, how quickly we get into something kind of unimaginable and strange. And also, you know, I'm not a historian of mathematics. And so I always wind up feeling like I'm peddling sort of fairy tales about about mathematical history more than the complicated truth that is history. But the role that this function played in going from a world where it felt like functions were kind of nice and were something we had a handle on, into opening up this world where, like, oh no, there are all these pathological things going on out there. And there are just these monsters that lurk in the world of possibility.
KK: Yeah.
EL: Right. And was this it—Do you know, was this maybe one of the first, or the first step towards realizing that in some measure sense, like, all functions are completely pathological? Do you know kind of where it fell there, or, like, what the purpose was of creating it in the first place?
BO: Yeah, I think that's exactly right. I don't know the ins and outs of that story. I do know that, right, if you look in spaces of functions, that they sort of all have this property, right, among continuous functions, I think it's only a set of measure zero that doesn't have this property. So the sort of basic narrative as I understand it, leading from kind of the start of the 19th century to the end of the 19th century, is basically thinking that we can mostly assume things are good, to realizing that sometimes things are bad (like this function), culminating in the realization that actually basically everything is bad. And the good stuff is just these rare diamonds.
EL: Yeah, I guess maybe this slight, I don't know, silver lining, is that often we can approximate with good things instead. I don't know if that's like the next step on the evolution or something.
BO: Right. Yeah, I guess that's right. Certainly, that's a nice way to salvage some a silver lining, salvage a happy message. Because it's true, right? Even though, a simpler example, the rationals are only a set of measure zero and the reals, you know, they're everywhere, they're dense. So at least, you know, if you have some weird number, you can at least approximate it with a rational.
EL: Yeah, I was just thinking when you were saying this, how it has a really nice analogy to the rationals. And, and even algebraic numbers and stuff like, “Okay, start naming numbers,” you'll probably name whole numbers, which are, you know, this sparse set of measure zero. It’s like, o”h, be more creative,” like, “Okay, well, I'll name some fractions and some square roots and stuff.” But you're still just naming sets of measure zero, you’re never naming some weird transcendental function that I can't figure out a way to compute it.
BO: Yeah, it is funny, right? Because in some sense, right? We've imagined these things called numbers and these things called functions. And then you ask us to pick examples. And we pick the most unlikely, nicest hand-picked, cherry-picked examples. And so the actual stuff—we’ve imagined this category called functions, and most of what's in that category that we developed, we came up with that definition, most of what's in there is stuff that's much too weird for us to begin to picture.
EL: Yeah.
BO: Which says something about, I guess, our reach exceeding our grasp or something. I don't really know, but they are our definitions can really outrun our intuition.
EL: Yeah. So where did you first encounter this function?
BO: That’s a good question. I feel like probably as a kind of folklore bit in maybe 12th grade math. I feel like when I was probably first learning calculus, it was sort of whispered about. You know, my teacher sort of mentioned it offhand. And that was very enticing, and in some sense, that's actually where my whole second book comes from, is all these little bits of folklore, not exactly the thing you teach in class, but the little, I don't know, the thing that gets mentioned offhand. And you go “Wait, what, what was that?” “Oh, well, don't worry. You'll learn about that in your real analysis class in four years.” I don't want to learn about that in four years. Tell me about that now. I want to know about that weird function. And then I think the first proper reading I did was probably in a William Dunham’s book The Calculus Gallery, which is a nice book going through different bits of historical mathematics, beginning with the beginnings of calculus through through like the late 19th century. And he has the here's a nice discussion of the function and its construction.
KK: So when we were preparing for this, you also mentioned there are connections to Brownian motion here. Do you want to mention those for our audience?
BO: Yeah, I love that this turns out—so I have some quotes here from right when this function was sort of debuted, right when it was introduced to the world. You have Émile Picard, his line was, “If Newton and Leibniz had thought that continuous functions do not necessarily have a derivative, the differential calculus would never have been invented.” Which I like. If Newton and Leibniz knew what you were going to do to their legacy, they would never have done this! They would have rejected the whole premise. And then Charles Hermite? [Pronounced “her might, wonders if the pronunciation is correct]
KK: Hermite. [Pronounced “her meet”]
BO: That sounds better. Sounds good. Sure. Right. His line was, and I don't know what the context was, but, “I turn away with fright and horror from this lamentable evil of functions that do not have derivatives.” Which is really layering on I like the way people spoke in the 19th century. There was more, a lot more flavor to their their language.
EL: Yeah.
BO: And Poincaré also, he was saying 100 years ago prior to Weierstrass developing it, such a function would have been regarded as an outrage to common sense. Anyway, so I mention all those. You mentioned Brownian motion, right? The instinct when you see this function is that this is utterly pathological. This is math just completely losing touch with physical reality and giving us these weird intellectual puzzles and strange constructions that can't possibly mean anything to real human beings. And then it turns out that that's not true at all, that Brownian motion—so you look at pollen dancing around on the surface of some water, and it's jumping around in these really crazy aggressive ways. And it turns out our best models of that process, you know, of any kind of Brownian motions—you know, coal dust in the air or pollen on water—our best model to a pretty good approximation has the same property. The path is so jagged and surprising and full of jumps from moment to moment that it's nowhere differentiable, even though the particle obviously sort of has to be continuous. It can’t be discontinuous, I mean, it's jumping, like literally transporting from one place to another. So that's not really the right model. But it is non-differentiable everywhere, which means, weirdly, that it doesn't have a speed, right? Like, a derivative is a is a velocity.
EL: So that means maybe an average speed but not a speed at any time.
BO: Yeah, well, actually, even—I think it depends how you measure. I’d have to looked back at this, because what it means sort of between any two moments according to the model, between any two points in time, is traversing an infinite distance. So I guess it could have an average velocity, but the average speed I think winds up being infinite rates. Over a given time interval, you can just take how far it travels that time interval and divide by time, but I think the speed, if you take the absolute value of the magnitude? I think you sort of wind up with infinite speed, maybe? But really, it's just that you can’t—speed is no longer a meaningful notion. It's moving in such an erratic way. that n you can't even talk about speed.
KK: Well, because that tends to imply a direction. I mean, you know, it’s really velocity. That always struck me as that's the real problem, is that you can't figure out what direction it's going, because it's effectively moving randomly, right?
BO: Yeah, I think that's fair. Yeah. The only way I can build any intuition about it is to picture a single—imagine a baseball having a single non-differentiable moment. So like, you toss it up in the air. And usually what would happen is that it goes up in the air, it kind of slows down and slows down and slows down. There's that one moment when it's kind of not moving at all. And then it begins to fall. And so the non-differentiable version would be, like, you throw it up in the air, it's traveling up at 10 meters per second, and then a trillionth of a second later, it's traveling down at 10 meters per second. And what's happening at that moment? Well, it's just unimaginable. And now for Brownian motion, you've got to picture that that moment is every moment.
KK: Right. Yeah. Weird, weird world.
BO: Yeah.
KK: So another thing we like to do on this podcast is ask our guests to pair their, well in your case construction, with something. What does the Weierstrass function pair with?
BO: Yeah. So I think, I have two things in mind, both of them constructions of new things that kind of opened up new new possibilities that people could not have imagined before. So the first one, maybe I should have picked a specific dish, but I'm picturing basically just molecular gastronomy, this movement in in cooking where you take—one example I just saw recently in a book was, I think it was WD-50, a sort of famous molecular gastronomy restaurant in New York, where they had taken, the comes to you and it looks like a small, poppyseed bagel with lox. And then as it gets closer, you realize it's not a poppyseed bagel with lox, it's ice cream that looks almost identical to a poppyseed bagel with lox. So that's sort of weird enough already. And then you take a taste and you realize that actually, it tastes exactly like a poppyseed bagel with lox, because they've somehow worked in all the flavors into the ice cream.
KK: Hmm.
BO: Anyway, so molecular gastronomy basically is about imagining very, very weird possibilities of food that are outside our usual traditions, much in the way that Weierstrass’s function kind of steps outside the traditional structures of math.
EL: Yeah, I like this a lot. It's a good one. Partly because I'm a little bit of a foodie. And like, when I lived in Chicago, we went to this restaurant that had this amazing, like, molecular gastronomy thing. I’m trying to remember one of the things we had was this frozen sphere of blue cheese. And it was so weird and good. Yeah, you’d get you get like puffs of air that are something, and there’s, like, a ham sandwich, but it was like the bread was only the crust somehow there's like nothing inside. Yeah, it was all these weird things. Liquefied olive that was like in inside some little gelatin thing, and so it was just like concentrated olive taste that bursts in your mouth. So good.
BO: That sounds awesome to me the the molecular gastronomy food. I have very little experience of it firsthand.
KK: So you mentioned a second possible pairing. What would that be?
BO: Yeah, so the other one I had in mind is music. It's a Beatles album, Revolver.
KK: Great album.
BO: One of my favorite albums, and much like molecular gastronomy shows that the foods that we're eating are actually just a tiny subset of the possible foods that are out there, similarly what revolver did for for pop music and in ’65 whenever it came out.
KK: ’66.
BO: Okay. 66 Alright, thank you for that.
EL: I am not well-versed in albums of The Beatles. You know, I am familiar with the music of the Beatles, don’t worry. But I don't know what's on what album. So what is this album?
BO: So Kevin and I can probably go to track by track for you.
KK: I’d have to think about it, but it's got Norwegian Wood on it, for example.
BO: Oh, that's rubber sole, actually.
KK: Oh, that’s Rubber Soul. You're right. Yeah, I lost my Beatles cred. That's right. My bad. I mean, some would argue that—so Revolver was, some people argue, was the first album. Before that, albums had just been collections of singles, even in the case of the Beatles, but Revolver holds together as a piece.
BO: Yeah, that’s one thing. Which again, there's probably some an analogy to Weierstrass’s function there. Also, it begins with this kind of weird countdown where, I don’t remember if it's John or George, but they’re saying 1234 in the intro into Taxman.
KK: Yeah. Into Taxman, which is probably, it's not my favorite Beatles song, but it's certainly among the top four. Right.
BO: Yeah. So that one, already right there it’s a pop song about taxes, which is already, so lyrically, we're exploring different parts of the possibility space than musicians were before. Track two is Eleanor Rigby, which is, the only instrumentation is strings. Which again is something that you didn't really hear in pop. You know, Yesterday had brought in some strings, that was sort of innovative. Other bands have done similar things but, but the idea of a song that’s all strings, and then I’m Only Sleeping as the third track, which has this backwards guitar. They recorded the guitar and just played it backwards. And then Yellow Submarine, which is, like, this weird Raffi song that somehow snuck onto a Beatles album. Yeah, and then For No One has this beautiful French horn solo. Yes, every track is drawn from sort of a distant corner of this space of possible popular music, these kind of corners that had not been explored previously. Anyway, so my recommendation is, is think about the Weierstrass function while eating, you know, a giant sphere of blue cheese and listening to Taxman.
EL: Great. Yeah. I strongly urge all of our listeners to go do that right now.
BO: Yeah, if anyone does it, it'll probably be the first time that that set of activities has been done in conjunction.
EL: Yeah. But hopefully not the last.
BO: Hopefully not the last. That's right. Yeah. And most experiences are like that, in fact.
KK: So we also like to let our guests plug things. You clearly have things to plug.
BO: I do. Yeah. I'm a peddler of wares. Yes, so the prominent thing is my blog is Math with Bad Drawings, and you're welcome to come read that. I try to post funny, silly things there. And then my two books are Math with Bad Drawings, which kind of explores how math pops up in lots of different walks of life, like, you know, in thinking about lottery tickets or thinking about the Death Star is another chapter, and then Change Is the Only Constant is my second book, and it's all about calculus, and it’s sort of calculus through stories. Yeah, that one just came out earlier this year, and I'm quite proud of that one. So you should check it out.
KK: Yeah, so I own both of them. I've only read Math with Bad Drawings. I've been too busy so far to get to Change Is the Only Constant.
EL: And there were there been a slew of good pop—or I assume good because I haven't read most of them yet—pop math books that have come out recently, so yeah I feel like my stack is growing. It’s a fall of calculus or something.
BO: It’s been a banner year. And exactly, calculus has been really at the forefront. Steve Strogatz’s Infinite Powers was a New York Times bestseller, and then David Bressoud [Calculus Reordered] and others who I'm blanking on right now have had one. There was another graphic, like, cartoon calculus that came out earlier this year. So yeah, apparently calculus is kind of having a moment.
EL: Well, and I just saw one about curves.
KK: Curves for the Mathematically Curious. It's sitting on my desk. Many of these books that you've mentioned are sitting on my desk.
EL: So yeah, great year for reading about calculus, but I think Ben would prefer that you start that reading with Change Is the Only Constant.
BO: It's very frothy, it's very quick and light-hearted and should be—you can use it as your appetizer to get into the the, the cheesier balls of the later books.
KK: But it's highly non-trivial. I mean, you talk about really interesting stuff in these books. It's not some frothy thing. I mean it's lighthearted, but it's not simple.
BO: I appreciate that. Yeah, the early draft of the book I was doing pretty much a pretty faithful march through the AP Calculus curriculum. And then that draft wasn't really working. And I realized that part of what I wasn't doing that should be doing was since I'm not teaching, you know, you had to execute calculus maneuvers. I'm not teaching how to take derivatives. I can talk about anything as long as I can explain the ideas. So we've got Weierstrass’s function in there. And there's a little bit even on Lebesgue integration, and other sort of, some stuff on differential equations crops up. So since I'm not actually teaching a calculus course and I don't need to give tests on it, I just got to tell stories.
EL: Well, yeah, I hope people will check that out. And thanks for joining us today.
BO: Yeah, thanks so much for having me.
KK: Yeah. Thanks, Ben.
[outro]
Our guest on this episode, Ben Orlin, is a high school math teacher best-known for his blog and popular math books. He told us about Weierstrass’s construction of a function that is continuous everywhere but differentiable nowhere. Here is a short collection of links that might be interesting.
Ben’s Blog, Math with Bad Drawings
Math with Bad Drawings, the book
