Federico Ardila, born in Colombia and now a professor mathematics at San Francisco State University, is an expert in the field of combinatorics, the study of all the possible configurations of finite systems. This week, Ardila talks with host Steven Strogatz about the importance of feelings in mathematics, the music of collaboration, imagining higher dimensional spaces, and the art and science of exploring “the space of possibilities.” This episode was produced by Dana Bialek. Read more at Quantamagazine.org. Production and original music by Story Mechanics.
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Federico Ardila: It’s actually funny because then I had to go into the Portland underworld and start to say, “Hey, did anybody find these math notebooks? I’ll give you a lot of money for them, but please don’t take them.”
Steve Strogatz: Wait a second, that happened? You went to the Portland underworld?
Ardila: I just had to just go into the… Because this restaurant was next to one of the most … where people told me it was one of the more dangerous neighborhoods in Portland. So, I just went, and I learned that’s where you buy drugs and where you buy guns, and I was just trying to recover my math, but nobody gave it back.
Steve Strogatz [narration]: From Quanta Magazine, this is The Joy of x. I’m Steve Strogatz. In this episode, Federico Ardila.
Strogatz: Federico is a mathematician originally from Bogota, Colombia. In fact, that’s where he was at the time we recorded this, and you may hear some Bogota street sounds in the background.
I’ve actually never met Federico in person, but after talking to him, I felt like I knew him. There’s such a warmth and openness to him. And you know, mathematically, that was helpful because he introduced me to his subject. It’s a subject called combinatorics, which looks at all the different ways that something complicated can happen. It’s a really intricate and tricky and beautiful part of math, but honestly, it’s not something I knew much about before talking to him.
Strogatz: Let me ask you a little about the story of a paper that you wrote with Marcelo Aguiar. I was just looking at this paper. It was posted in this math pre-print archive in 2017, and one of the things that hit me about the paper is the acknowledgements. So, at the end of our papers, we often acknowledge the people who have helped us — and you did that — except you also did something I have never seen before, which is you made a point of not acknowledging someone, and so I’m gonna quote what you wrote. You have, “We do not thank the Portland, Oregon, thieves who set the project back in 2013.” And then you go on to say that you do thank them for not publishing your results as if they were something that the thieves had discovered. So, let’s first hear the story of the thieves and then more about the math that’s in this paper.
Ardila: Yeah, you know, it’s funny because this — so this story somehow made it to Wired magazine, and then Wired did this tweet about how this mathematician had 10 years of work stolen in Portland, Oregon. [Editor’s note: The story in question was a syndicated interview from Quanta.] And of course, all my fellow Colombians were like, “What? You got robbed in Oregon, of all places?” It’s like I lost my Colombian card, you know? Like, here we’re used to knowing how to take care of ourselves and not get robbed.
But yeah, so what happened is that I was helping my wife move a bunch of stuff from Vancouver, where her family home was, to California where we were. And so, we were just driving with a car full of stuff, and in Portland we stopped for a meal. And after like 12 hours of driving, I — she was driving at the moment — and I said, “Let me just jump out and put our name on the list of the restaurant and come back.” And so, I did that, and then she actually beat me to coming back, and I realized that my backpack was in the car. And when I went back to the car one minute later, the backpack was gone.
And you know, when your backpack gets stolen, usually people get worried about a computer or something. But in my case, what happened is that I had this folder of 10 years of work that I had done with Marcelo Aguiar on this project. And I’m kind of old-fashioned and maybe kind of artistic in that I just like to have my artist sketchpad by where I do all my math. And so, I just have these notebooks that I like to write down things in kind of old-school style, and so this stuff just is not on the computer anywhere. It was just gone, basically.
Strogatz: Oh boy, so this is 10 years of proofs, 10 years of examples, equations.
Ardila: Exactly, conjectures, counterexamples, pictures, all kinds of things. But you know, fortunately, at least in mathematics, everything takes place in our minds. At least it wasn’t like laboratory runs, laboratory experiments. With some time, we could recover, I think, pretty much everything. And it was very painful that — this is why this paper took so long to come out. It’s probably more like a book than, like, a paper.
Strogatz: One thing I liked about it, you know, being an outsider to your part of math, is that it felt very inviting. It seemed that you took great care to welcome people who were not exactly in your subcommunity.
Ardila: When younger students talk to me and ask me for advice on how to choose a research area or a research topic, I feel like my best trick has always been just to try to work with people that are not so close to me mathematically and find a way to talk to each other. And this is what happened with the paper with Marcelo: that he is much more of an algebraist, I’m much more of a combinatorist, and this paper’s basically about bridging together combinatorist geometry and algebra. And we found this nice object that lies right in the middle of these three things, something that I think neither me nor Marcelo would have discovered on our own. I think it took us talking to each other.
I feel like we have the sense that this point right in-between the geometry, the combinatorics and the algebra is valuable, and these ideas seem like they have a lot of applications. So, just to ensure that this paper would be read, we worked really hard to kind of bridge the gap between all these different fields and try to tell a story that, at least in the beginning, could be accessible. And then if people are interested, they can dig deeper.
Strogatz: You’ve mentioned three words: algebra, geometry and combinatorics. People have all taken algebra and geometry in high school, so they might have an idea, although probably the algebra they took isn’t quite the algebra that you’re referring to. But combinatorics is not so familiar to many.
Ardila: I definitely agree that combinatorics is not familiar as a word, but I think it’s definitely familiar as a concept to people. I think everybody’s used to, for example — or not everybody, but a lot of people are used to shuffling cards. So, for example, if you have a deck of 52 cards, if any of the listeners are in their house and they reach to their deck of cards, they’re in some order. Who knows what that order is? And so, what combinatorics does is to study —. It’s the mathematics of studying the possibilities. What are the possibilities of ordering these cards? One thing that I find really shocking about this is that if one of the listeners really does have the deck of cards in their hand, I can almost guarantee to them that that order has never been seen before in the history of humanity. Nobody has ever seen the cards in the order that you have them in your hand right now.
Strogatz: That’s an amazing thought. Right, so assuming they’re not just coming right out of the pack where they’re in the standard order.
Ardila: Right. But if you look at how many different orderings there are, it’s larger than the number of atoms in the universe. And so even though it’s a very concrete question, it’s just incredibly huge, and you have to think harder about how you’re going to think about these possibilities. And so, a lot of what we do in combinatorics —
Strogatz: That’s an interesting point.
Ardila: And so that number’s just way too big for any person or any computer, and so that’s where we come in. So, these questions seem very innocent, but they’re just really huge, and so a lot of what we do is just think about the structure of possibilities. In the case of permutations, you have all these shufflings of the cards. And then you say there’s too many of them, but can I come up with some mental models for how this could be organized. So that I can think, for example, what’s the fastest way to take come random order and bring it back to alphabetical order? So maybe that’s — in a nutshell, what combinatorics is about is just the study of possibilities and how do you organize them, given that there’s too many of them to list them.
Strogatz: So, I love it. Combinatorics is not just the art of the possible, but the enumeration of the possible, the counting of the possible and the organizing of the possible.
Strogatz: It’s such a poetic image, actually: the space of possibilities. It’s actually really symbolic of Federico’s whole approach to math, how he opens up a creative space for his friends, his students, and his collaborators.
Ardila: We have these combinatorial problems that are very concrete, and usually you can play with them in your hands. But then when you’re going to organize them, there’s been two traditions in combinatorics that have been a little bit perpendicular to each other. One tradition that I’ve used a lot is to take these possibilities and somehow map them into some geometrical space. For example, as we were saying, if you have 52 cards, then there’s basically 52 positions that you need to fill, and so each one of those positions you can map into an axis.
Let’s say that I number the cards from 1 to 52, and then any permutation of them will just be some list of numbers between 1 and 52. And as you were saying earlier, when a mathematician sees a list of 52 numbers, we just have this useful mental picture that, oh, that’s just a point in 52-dimensional space. I have to admit, we cannot see that space. I can say this, but that does not mean that I can see in 52 dimensions. It’s just kind of a useful analogy.
Strogatz: That’s a good distinction to make because people often worry that we’re saying we’re actually picturing 52 dimensions. We’re not. It’s just we’re using the analogy with the one or two or three dimensions we can picture to say, well, it’s not that different in 52.
Ardila: Exactly. But then what ends up happening is that, even if I cannot see in 52 dimensions, I can see in three dimensions, and so I can say, okay, why don’t I try this with four cards? And then it turns out that if you do this exercise with four cards, you actually get a three-dimensional picture. It’s actually so beautiful. This picture, it turns out it’s the same as a molecule, zeolite. So, if you just Google “zeolite” and you look at what that molecule looks like, it’s exactly the shape made by the 24 permutations in three-dimensional space.
Strogatz: Well, this is great. I mean, because this was the one aspect of your paper that really caught my eye, having never done combinatorics at all. It’s funny how big mathematics is, that here I could be a professional, and you’re a professional, and I have never done anything in your part of math. So, I saw this thing that looked exactly like some kind of gem, but it had this beautiful — kind of reminds you of a jewel. It’s a gem, it’s a very pretty shape, this thing.
Strogatz: Zeolites are minerals that are filled with lots of little holes, little pores where water can get stuck, and so sometimes if you heat up a zeolite, steam will come out. That’s actually what the word means in Greek. It means a boiling stone. Mathematically what’s interesting is that they come up in connection with what Federico was talking about, that he was talking about the 24 permutations of a deck of four cards. Mathematically, he likes to think about that, by visualizing it, by putting one axis for each card. So now he’s got this wild four-dimensional space, and then each permutation is a dot in that space. And he kind of thinks of those dots almost like the corners of a crystal, like the sharp, pointy corners in a gem. And there’s a natural way mathematically to connect those permutations, to connect those dots in this four-dimensional space such that, when you do that, you end up making a shape that looks exactly like the crystal structure of a zeolite.
Now, okay, I realize that’s a little abstract, that’s hard to visualize. So, I asked Federico could he boil it down for us by … what about a deck of three cards? How would that work?
Ardila: So, if you have three cards, and let’s just number them 1, 2, and 3, then we look at all the possible permutations, all the possible orders. And so, I’ll just say them slowly. So, there’s 1-2-3; 1-3-2; 2-1-3; 2-3-1; 3-1-2; and 3-2-1. So, these are the six possible orders, but I think we are used to if we have three numbers, we can think of it as a point in three-dimensional space. It’s the x-coordinate, the y-coordinate, and the z-coordinate. So, I just had six lists of three numbers, and so that gives me six points in three-dimensional space. If you draw them, you’ll find that they actually make this beautiful regular hexagon. Even though they’re in three-dimensional space, they make a hexagon.
Strogatz: You’re saying just literally graph them as the numbers? Like if I have, say, 1-2-3, I graph that as 1 on the x-axis, 2 on the y-axis and 3 on the z-axis, and then make a dot there?
Ardila: Exactly. That’s all I mean.
Strogatz: And then I just take these — that’s all you mean. And then you take those six points and connect them in the obvious way. What, connect them to the nearest neighbors, or what?
Ardila: So, what you have to imagine is kind of taking, like, a rubber band, and you just let it snap and see what shape it makes. We call that the convex hole, but it’s basically the shape enveloped by those points. And that’s what I mean that if you have four cards, then they actually make this beautiful zeolite. This is the picture that you saw in my paper that looks like a jewel.
Ardila: I think a little segue is that I love drawing nice pictures. Because people make connections, so I just drew this nice picture in my paper. And I posted the picture on Facebook, and a friend who’s a chemist told me, “Oh, that’s zeolite.” So that’s how I learned. These things, they feel like some kind of abstract thought experiment. Actually, they’re beautiful enough that nature wants to adopt those shapes.
Strogatz: Now that we have is this geometric insight, that there are shapes, jewellike shapes that are called polytopes, these shapes with flat faces in your world. So now that you have a geometry that corresponds to the combinatorial question of arranging things, what does that do for you? Why is it good to have this geometric picture?
Ardila: Because it turns out that people have thought about polytopes for many, many years. The people who pioneered this field are, at least for the kinds of things I do, are in combinatorial optimization. So, they’re people that are trying to optimize road systems or airport schedules or things like this, and they have found very useful techniques using polytopes and using linear algebra to basically say, “Okay, these problems that are very intricate, when you look at the possibilities, somehow the geometric shape itself is valuable,” because then you can use tools from linear algebra to do things.
For example, the question of, if you have your students’ exams, there’s 20 of them and they’re in an order that is not alphabetical, and you want to alphabetize them, what’s the fastest way to alphabetize them? It turns out that it’s useful to just imagine yourself walking around this high-dimensional shape.
So, the order of the exams is one point in 20 dimensions, and the alphabetical order is a different point, and these two points are part of this big crystal that happens to live in 20 dimensions, and we cannot visualize very well. But somehow, our intuition from three dimensions is good enough that we can imagine walking around this 20-dimensional shape and actually producing an algorithm to get from one order to alphabetical order. This is not just a beautiful analogy, but it turns out to be extremely powerful.
Strogatz: That’s fantastic. Yeah. It really sort of put the lie to this thing that many people might have gotten the impression from their school math, that math is a tower where one subject follows another. To us, math is this great jungle of interconnected ideas, and you can get from one place like geometry to another place like combinatorics, or to another place like optimization, you know, through all these different pathways in the jungle, and so you’re —. I see you as an explorer making new pathways into the jungle and finding new connections that help everybody.
Ardila: I’m really flattered. I liken it — I think it’s — I really like this analogy that you’re just walking around a forest, where maybe you know a little bit of it, but sometimes you just walk into this clearing that you know nothing about. I think this happens to almost anybody in mathematics, that sometimes you’re working in a field, and then you end up walking into a problem from a different field that you have to solve. I think at that point a lot of people get scared and stop. And I think the one trick that I have in my toolkit is to just try to keep pushing, and talk to people that don’t know about that field and try to make those connections, and try to build those bridges that maybe are not there yet.
Strogatz: One of the really unusual aspects of Federico, as both a teacher and a mathematician, is that he has a way of making people feel so comfortable emotionally in math, which is very important for creativity, for fostering a kind of safe space where people feel open to letting out their conjectures, their best intuition, their wild guesses.
And it goes beyond that. It’s not just safety. He brings out a feeling of celebration, like rejoicing in our own cleverness, our ingenuity. If we have ideas and they’re wrong, so what? They’re gonna lead us someplace cool.
Ardila: I think there’s something that I’m used to doing, which is start with combinatorics and go to polytopes, and Marcelo’s more used to starting with combinatorics and go to algebra, and then what we… We were having coffee one day, and we said, “You know, are we somehow able to put together these points of view and bring together the geometry and the algebra?” And so, the kind of big discovery of our paper is that you can take these polytopes that are geometric shapes and actually give them an algebraic structure. It’s actually not a terribly complicated idea or construction, but it turns out to be extremely powerful because of both the algebra and the geometry model, all these different kinds of combinatorics problems.
And so, what we found is that the combinatoric — the geometric model and the algebraic model were actually very close related to each other. And so, then we’ve got to do, like, 120 pages’ worth of consequences of this. But I think somehow the key insight is that one, that again, just kind of bringing together two different lines of thought that maybe seem disparate, and just try to learn how to talk to each other and see if there’s something — there’s a bridge in-between. I think it’s — yeah, I think it’s been a super exciting process, and — but again, just the beauty of doing it with somebody else is just so cool when you see that there’s something that…
It’s like being in a band where there’s some things that you just cannot do on your own, but if you bring this to… I think of it… I play a lot of kind of Afro-Latin percussion where you have these —. Every drummer or every musician is doing one rhythm, and maybe that one rhythm is not that complex. But when you put all these sheets of rhythm together, it makes these beautiful polyrhythms that are so much richer than what one person could do.
And I think this project was one of those things where I feel like Marcelo’s voice was very different from mine, and somehow when you put them together, it was just really beautiful and really exciting, and certainly something that I couldn’t have done on my own.
Strogatz: Perfect. I love that, the polyrhythm of mathematical collaboration.
Strogatz: I saw a video of Federico playing percussion, more specifically marimba de chonta. It’s music from the Afro-Colombian Pacific Coast, and it’s just such rich, beautiful, complex music, and Federico’s been learning it from the masters. What you’re hearing now is a recording from a tiny show that Federico did in the Bay Area with his band, Neblinas del Pacifico, and a couple of special guests.
Strogatz: There’s something so inviting about Federico’s music, and watching him on the marimbas is just so joyful. If you see him, he’s smiling ear-to-ear, he’s moving, he’s grooving. Man, it’s infectious. It’s honestly — it’s got me jiggling in my chair. I feel like getting out of my seat and dancing.
I wasn’t surprised to learn that Federico is also part of a DJ collective, La Pelanga. The collective was born in Colombia in 2008 and is now based in Oakland, California. Federico told me how they build a really positive and welcoming dance floor, essentially celebrating music’s constant migration between Africa, the Caribbean, the Americas, crossing every border and, in effect, disrespecting every border. What you’re hearing now is a clip from a La Pelanga mix tape that Federico sent me.
Strogatz: After the break, more of Federico’s unique rhythm. And as for those positive and welcoming vibes he creates on the dance floor, well, you know what? He also creates them in the classroom. That’s ahead.
[MUSIC PLAYING FOR BREAK]
Strogatz: I was curious about how Federico got into this particular part of math, so I asked him to tell me a little bit about his childhood.
Ardila: I first got really interested in math through this math program. I remember being eight or nine years old, and these exams arrived in my school, and they were just these really cool, interesting questions that I did not really understand were math, and I really liked them. There’s actually a question that I really liked that maybe I can tell you, because it’s one of these memorable questions that I encountered 35 years ago, and I still remember.
You have this snail, and the snail is trying to climb up a 10-meter pole, but the snail is very slow, and it can only go up 3 meters in a day. But then it gets tired and it falls asleep, and then it slides down 1 meter overnight.
And then the question is how many days does it take for the snail to go up the pole? Maybe for the listeners I won’t say the answer. Maybe you can think about it, but maybe I will say that the answer is not five, which is what you might first thing that it is. And I just remember discovering that the answer was not five, which is what I — it first seems like it is. I just thought this is so cool and just so funny, and I didn’t realize that math could be creative and even funny.
I’m the kid of an engineer who really loved thinking about human relations and spent most of his time thinking about that. And my mom was a sociologist and community organizer who also kind of understood that, sometimes to convince people, you have to be quantitatively proficient. So, I think she also understood the power of science. So, I grew up very much kind of appreciating the sciences, but also understanding that we have a social responsibility to try to leave the world a little bit better than when we got to it.
Strogatz: Yeah. That’s already very interesting that both parents had these both sides of their brain and soul operating.
Ardila: It’s really fascinating. Two people that you love, and you sometimes wonder how they ended up together, but they really complemented each other in really cool ways. And I think they learned a lot from each other, starting in different kinds of careers and learning from each other. This is something that I think I learned early on, that the more that you talk to people in fields very different from yours, you’re gonna get a lotta lessons that are really original and different from what you are gonna get in your field.
Strogatz: I’m especially struck by you mentioning your dad, the engineer with the sensitivity to human relationships, because that goes against the stereotype for engineers. Why do you say that about him?
Ardila: He started out as a very technical engineer and doing kind of hydroelectric engineering and things like this. But then as time went on, he started getting more interested — he joined companies and ended up being more interested in working and just how people work together. How do you create — a working space where people feel comfortable, and where they give the best of themselves, and they’re really committed to being a part of it, that kind of thing? This was something that always interested him, and he ended up doing what actually — I think it was very pedagogical interventions that he would do.
Strogatz: So, you feel that there’s a lot of teaching in the family too.
Ardila: Yeah, yeah, definitely.
Strogatz: A spirit of a teacher.
Strogatz: Did you think of being a teacher yourself from an early age?
Ardila: You know, it’s interesting because I think in Colombia, like in many other parts of the world, teaching is very stigmatized as kind of a negative thing to do. It’s seen as kind of a lower-level kind of work. I tend to think that this is one of the biggest problems in our society, that we don’t value our educators. Growing up, you were told that if you were gonna be a mathematician, then you were gonna be a teacher, and somehow, we all knew that we didn’t want to be teachers, that this was not interesting work, that it was kind of at the lowest level of work.
And so, I always kind of thought that I was never gonna do this. But then when I — I don’t know, when I think of stories of my childhood, I think I was a teacher from very early on. When I was a kid, I was really, really into football, like soccer, like most kids in Colombia. And I think when I was about eight years old, I gathered all the kids in the neighborhood, and I started a football camp, and I would just line them up and make them do, like, improve everybody’s technique. It makes me think that I was kinda meant to be a teacher.
Strogatz: Yeah, the kids want to just get out there and play, run around, kick it at each other, and you’re saying, “Let’s do dribbling exercises and stuff, shooting.”
Ardila: Absolutely. I used to say, we can have fun, but we also have to build our foundations, and so it was very structured. We would start with drills. Every day, we would do a different drill, and then go and incorporate it and actually play for the rest of the afternoon.
Strogatz: Well, but wait a second. It sounds like you must have had some leadership skills in addition to teaching skills that you could convince everyone to do this.
Ardila: Yeah. You know, it’s interesting because as a kid, I was an incredibly shy kid. And so, I always kind of wonder how this happened because clearly, I have some kind of leadership with the younger kids. And I do know that I always gravitated toward people younger than me, and somehow this was a kind of a social attraction that I was very comfortable with. But otherwise, I was just a very quiet kid that just kind of kept to himself. I was really seriously into football, I was really seriously into music, art, math, and I think that I actually just kind of liked listening more than talking. And so, I would just — I think I was just a pleasant kid smiling in the corner and not saying very much. It’s interesting because, you know, you take that personality type and you put us in front of a classroom, and I think that transforms us. And I’ve actually become a much more talkative person, and I’m used to talking to hundreds of people at a time, and I’m not fazed by that anymore.
Strogatz: People who teach math often think of it as a one-way flow of information from the teacher to the student, and so that style of teaching focuses on really clear handwriting, nice board work, excellent explanations. Sure, that’s all great, but something I’ve found is that that doesn’t always translate into good comprehension on the part of the student, or even good engagement. So, I got inspired, actually, by hearing how Federico describes how his teaching style has evolved into something where the information is flowing in both directions, and that means a lot less talking by the teacher and a lot more listening.
Ardila: What I try to shift toward is to work with my students to kind of collectively understand that the more that we actually think in front of each other, and communicate in front of each other and learn to communicate our partial understandings, then we understand more what’s happening. And I just think it makes for a much better educational experience. It’s a much harder one for me, because it’s much easier to prepare a beautiful lecture and give it as planned. I’ve actually shifted a lot to this kind of metaphor of classical music versus — I play some more kind of improvised forms of music where you’re supposed to just kind of set some initial conditions and then you just start improvising, like jazz or a lot of kind of music from the African diaspora. And this is actually what I think of my teaching now more as just I give you some initial conditions and then you jam together. It’s more of like an improvisation where you actually get to understand each other better and express better, and also I think it makes for a little bit more human experience for everybody.
I’m a professor at San Francisco State. You know, San Francisco State is, I think, one of the most diverse campuses in the nation. Our math department is one of the most diverse departments in the nation, I think, and we have people from very different walks of life, very different backgrounds. And so, I just find people work very differently from each other, think very differently from each other. And so usually what I’ll do is, that maybe I’ll set up a scenario and then they’ll get into small groups and just, you know, play with the material before we try to state any theorem or anything. Just try to kind of play with the scenario and try to understand what is this about, what are the questions that we would ask about this? So, then they’re kind of jamming with this material a little bit.
Strogatz: Federico is truly a man of fusion. If I had to sum him up in one word, that would be it. He is all about collaboration and harmony. We’ve already heard him do this with his music, bringing together sounds from all over the world, and now we’re hearing about the fusion in his teaching. I wanted to know a little bit more about his students at San Francisco State and where they come from.
Ardila: More than half of the students on my campus are first-generation college students, so they’re the first ones in their family to go to college. More than half of them, English is not the language spoken at home, which is basically the demographics of San Francisco. So, we’re kind of the city university, and it reflects that. Also, ethnically, there’s no majority group, actually, so I think our campus is about 30% white, 30% Latino, 25% Asian, 10% black, something like this.
Also, gender-wise, there’s — you’ll see the whole gender spectrum, and there’s definitely not a very… Our math department is not male-dominated the way that most departments are. In terms of age, I’ve had students who are in their 80s, which is so lovely, right? They retired and they… I had one student who was taking math as his fourth major after retiring. He had already graduated from philosophy, from English, from — I forget what the third one was, and then he was studying math.
Strogatz: That’s such a beautiful model for everybody. This is about the pleasure of learning and the privilege of learning.
Ardila: Exactly. Or for example, we have a lot of students that — they graduated from high school and had to go work, had to earn some money for the family. And maybe after five, ten years, they realize, “You know what, I think it’s time to go back to school.”
Strogatz: What about math phobia?
Ardila: I think if we’re honest, even we as professional mathematicians suffer from math phobia sometimes, no? Sometimes you’re working on a problem that is so hard that it paralyzes you a little bit, and it’s scary to go back to it. I feel like it’s more like we’ve gotten used to dealing with that. And I think the question is more — okay, there are always moments where we’re scared of the problem that we’re thinking about, so the question is, at that point, what do you do?
Strogatz: That’s what people say, right? That courage is not about not being afraid; it’s about what you do when you’re afraid.
Ardila: Exactly. Exactly. This is something that I try, to be very vulnerable in this way in front of my students, and just be honest about what it’s like to do this kind of work. And I think what they go through — all of us go through to some extent, but it’s definitely true that, for example, another part of student population that I teach is, that I teach a lot of future schoolteachers. There’s a statistic that we never know if it’s correct or not, but the San Francisco Unified School District claims that 70% of the public schoolteachers in the Bay Area are graduates of our university.
You know, when I realized that, I was thought, “Wow, this is a really serious responsibility, that we have to train the teachers of most of the Bay Area.” And so, I started asking to teach those classes and, you know, I think that’s a class where you sometimes see people who are going to be math teachers, and they’re scared of math. And so again, I think it’s really important to expose that and to kind of say it out loud. And I think it can be a valuable tool, actually, when they go teach and they can be honest with their students also, who are also scared of math. But I think a lot of what I try to do is try to offer opportunities.
In the kinds of activities I organize, I always try to make sure the activities are — have a very accessible entry point, so that everybody can engage. Even if you’re scared, you can at least do some examples. To give you an example, one of the classes that I teach for this population of students is a geometry class. One activity that I like to do with them is just, we all go outside, and we look at the tallest tree in the main quad of our building.
And the question for the day is, just how tall is that tree? That’s actually a much harder question than a lot of the questions we put on exams. It requires creativity and it requires having some ideas, and that’s an activity that’s… Pretty quickly, every student has some idea. They do it in groups, and it’s also fascinating how each group has a different approach to it. And I think after we do the activity, they also recognize, “Wow, that was actually harder than proving the congruence of two triangles by side-angle-side” or something like this. It’s a richer activity.
Strogatz: Can you remember any attempts that the groups made at addressing this question about the tall — the height of the tree?
Ardila: Yeah, one thing they did is to just take a photo of one of the students in front of the tree, just side-by-side with the tree. The student knew how tall he was, and then they could just kind of transpose —. Actually, they put this on Photoshop, and they put, like, a stack of copies of the student on top of each other and they realize, “Oh, you know, the tree is 4.6 times as high — as tall as Carlos,” and then they multiply it.
Strogatz: That’s beautiful.
Ardila: I guess if you’ve done a lot of problems like this, then you think of just… Like, you know, have somebody lie down on the ground and then have Carlos stand in front of the tree, so that the top of Carlos’ head lines up with the top of the tree, and then use some similar triangles. Some students did that also.
Strogatz: Oh, I see. Let me see if I get the picture. The person lying on the ground is looking up at Carlos’ head until it’s just lined up with the top of the tree from that angle of viewing from the ground. Is that what you’re saying?
Ardila: Yeah, that’s right. And then they can do some proportions so they can say it’s the height of Carlos divided by the height of the tree is the same as the distance from the point of view like my point of view to Carlos to my point of view to the tree, and that they can actually measure because it’s on the ground. So, that was another solution that I thought was really cool.
Strogatz: I mean, it’s an interesting thing because I’m sure there are some colleagues who will think, well, you know, you’re taking a lot of time to teach proportions, and you’re not covering that much, blah, blah, blah, right.
Ardila: I think a student that comes up with this solution, they’re never gonna forget similar triangles, and they’re actually going to internalize this in a way that they wouldn’t if they’re just kind of memorizing “angle-angle-angle” criteria or things like this. And I think it’s also kind of a psychological thing, especially for people who are not going to necessarily be professional research mathematicians, to really see that math is a tool to understand your own world, and to ask your own questions about it. I think once you do something like that, it also changes… This is one thing that I see in student evaluations is that a lotta students just say that their relationship to mathematics changed and that they see it more as a tool that’s useful to them instead of something that is forced upon them.
Strogatz: It’s amazing to me that Federico keeps his teaching so accessible when the math he does seems so abstract.
Strogatz: But does that come naturally to you? Because I think of your own mathematical work as not particularly about the real world.
Ardila: It’s interesting, because that’s what I thought initially, but I’ve had three projects that started very, very abstract. I think they were my most abstract research projects. They actually became quite applied. It’s kind of a luxury. I think a lot of pure mathematicians understand that maybe one day our stuff could be applicable to something, but I’ve actually had that happen within my lifetime, which is really exciting and really shocking too.
Strogatz: Can you tell us about any of those?
Ardila: Yeah, I think so. Maybe I’ll give you one example. I started studying these geometric spaces. So, these spaces are made out of cubes, except that for mathematicians we have a cube in every dimension, right? So, we have a three-dimensional cube, which is what we usually call a cube. But for us, a square is a two-dimensional cube, a line segment is a one-dimensional cube, and then we also have a four-dimensional and five-dimensional cube, and so on. And the kinds of spaces that I was studying are spaces that you obtain by just gluing cubes next to each other. Maybe an example of this is, like, I don’t know, the map of Manhattan, for example. It’s just a lot of squares — a lot of little squares that are right next to each other, and they form some larger geometric shape, but the building blocks are squares.
So, I was studying these things called cubical complexes, and I was studying the ones that are negatively curved. So, this is something that, when we talk about curvature, we think, okay, if we look at a floor, a floor is flat. If we look at a sphere, like the surface of the earth, we think of that as being positively curved. And if we think of a saddle, like a horse-riding saddle, we think of that as a negatively curved space.
The objects that I was studying were the kinds of spaces that you can make using little cubes that happen to have negative curvature. One way that I like to think about it is if you draw triangles. So, if you take the globe, for example, and you draw the triangle from, I don’t know, I’m in Bogota. Are you in Ithaca right now?
Strogatz: I’m in Ithaca, New York. Yeah, Ithaca.
Ardila: Okay, and then let’s go to San Francisco. So those are three points, Bogota, San Francisco, and Ithaca. And if you take the globe, and if you just draw the shortest path between Bogota and San Francisco, between San Francisco and Ithaca, between Ithaca and Bogota, if you’ve never done this, you’re gonna be surprised that this triangle looks really fat. And if you add up the angles, it’s more than 180 degrees, which is what we’re used to triangles being. Whereas if you try to do this on the surface on a Pringle, take three points on a Pringle and draw the — but yeah, so basically you can tell from the angles of a triangle. If the triangle is too fat, if the angles add up to more than 180, then that’s positive curvature. And if the angles add up to less than 180 degrees, then it’s negatively curved.
But the reason that I got to this thing is that I was actually doing something in geometric group theory, which — I don’t think necessarily there’s a lot that we need to say about that except that it’s some kind of abstract field of mathematics that I was investigating. And somehow from here, I would like to study in these spaces, and I thought, okay, if I want to study these groups, then I’m gonna have to learn some of this geometry. And so, this is what I mean when I say that this was one of the most abstract projects that I ever started with.
But then what I learned with some time is that, actually, there’s a field of robotics that uses these kinds of spaces. If you have some kind of robot that you’re trying to get to perform some task, you know, like get it to move from Point A to Point B, and if you want to just figure out what’s the fastest way to do that, we realize with some of my students that these very abstract spaces that I was studying are very useful to study this question.
So, if you take a robot and just imagine what are all the possible positions of the orbit, like what’s the world of possibilities, what’s the world of all possible positions of the robot, it turns out that this is one of these spaces. The world of possibilities of the robot is one of these spaces made of cubes with negative curvature. And then these ideas that I had for a very different reason ended up being useful to actually move a robot from Point A to Point B, which was really fascinating to me.
Strogatz: Okay, in a nutshell, what Federico is trying to do is figure out how to move a robot’s arm from one configuration to another in the most efficient way. What’s the best way to reorganize it? Now, he has a way of translating that problem into geometry, where he thinks of a configuration of a robot arm as a kind of a point in a space, and then that’s true for both the starting and the ending point. So, you would think the most efficient way to get from Point A to Point B is just connect ’em with a straight line. That’s how you should move the robot’s arm through all those intermediate configurations.
But there’s the rub. It may not be possible. The straight line may not exist in that space of possibilities ’cause it’s a weird space. It’s not what we’re used to thinking of. It’s not like a tabletop. It’s more like the surface of a saddle. That’s his discovery, that the space of these robot arm configurations has to be thought of as a negatively curved surface.
Strogatz: I’m picturing a robot driving around in an Amazon warehouse for the book — that’s a big issue nowadays, right, in manufacturing. They have all these robots driving around on the floor, grabbing stuff, and then dropping it in some bin somewhere, but there are lots of things in the way. You can’t just drive on a straight line to where you need to go because there are people there and there are stacks of all kinds of other things in the way, and possibly other robots to watch out for too, but let’s ignore that maybe for now.
So that could be one space, the allowed places that this robot can drive on the factory floor, or the warehouse floor. But a different possibility is a robot with an arm that’s an articulated arm, like with a wrist and an elbow and — so which kind — that has its own limited set of possibilities for where the hand or the fingertip of the robot can be.
Ardila: Exactly. Exactly. So actually, I mean, this —
Strogatz: Do you mean both or what?
Ardila: To some extent I mean both, but in the first model, you’re paying attention to where the robot lives and looking at what are the constraints that the space makes on your robot. But intrinsically, you can also ask, okay, you have a robot that has, let’s say, two arms and two legs, and it has some limited range of motion, and then you can ask what are all the possibilities. To make it more concrete, let’s say that you have the right arm, and the right arm can move either up or down, and then you can just imagine kind of a slider where if you slide up or down, then the arm of the robot moves up and down.
Now let’s do that with the other arm of the robot, so you have a different slider that controls the left arm of the robot and now because you have two different sliders, you can imagine that actually you could make a square out of those two. Does that make sense?
Strogatz: Oh, I see what you mean. So, there’s a square sort of like one axis tells me where the slider is for the left arm. That’s like sort of a horizontal axis in a high school x-y graph. And the other slider is moving on the vertical y-axis. That tells me where the other arm is.
Ardila: Exactly. But then you can imagine, okay, there’s also the leg, and so you need a third axis, and so this is why all of a sudden you have a three-dimensional cube made by the three different sliders for the right arm, left arm and right leg.
Strogatz: It’s great. Just to interrupt for a second, because earlier I was thinking, some people listening to this would wonder why did you say also a four-dimensional cube or a five-dimensional cube? But it’s now becoming clear because this robot, its configuration of where its arms and legs and ankles and fingers are could be represented by all these different numbers, which could make a very high-dimensional cube or set of cubes or space in general.
Ardila: Exactly. If your robot has 10 joints, then you need 10 dimensions to describe its position, and so this is… I really like this because it makes it clear that high dimensions are not some kind of crazy abstraction that mathematicians came up with. They’re really about modeling our world. If you’re trying to model a system that can move in 10 different ways, you actually need to go to 10 dimensions, and so it’s worthwhile to.
Strogatz: There is a conceptual leap there that we’re used to making that I think a lot of people don’t make, which is if you said to people, “I need 10 numbers,” they would say yeah, fine. But if you say, “I need 10 dimensions,” rather than 10 numbers, it’s one point in a 10-dimensional space. And that’s elegant to say one point instead of 10 numbers.
Ardila: Exactly. And this is exactly the kind of exercise that I was alluding to earlier. I’ve done this activity with 12-year-old kids and, you know, once they start playing with the robot, they start seeing the need for the fourth dimension.
Again, I said conceptually that is not immediate, but if you have a reason to make it, if you’re just drawing these pictures, you realize, “I drew this thing. What is this thing called?” And it turns out they drew a four-dimensional cube. So, it’s kind of a nice gateway to some deep mathematics. This is exactly the kind of gateway exercise that I really like.
Strogatz: Should I take that literally? Have they actually drawn a picture that is like a projection of a four-dimensional cube?
Ardila: Yeah, yeah, absolutely. I mean, I’ve done this three weeks ago. I did a workshop for something called the Escuela Robotica del Choco, and this is a robotics school for middle schoolers in Chaco. And you might not know this, but if there’s any Colombians listening to this, they will know that Chaco is in the country’s imaginary, it’s seen as the place that has the lowest indices of education and has the highest indices of poverty. It’s an area of Colombia that has just been ignored by the government. It’s a historically black area, and for that reason, just because of the history of it, the government has never paid much attention to it. So, I think within Colombia’s imaginary, this would be the last place where students would do something like this.
But of course, we know that that talent’s everywhere, and mathematical potential is everywhere, and so I asked to go and do these workshops with these 12-year-olds about these high-dimensional spaces, and they got it. I mean, I didn’t have to explain it. It became natural for them.
Strogatz: This is fantastic. What a story.
Ardila: I used to teach at MIT, so I know what it’s like to teach MIT students, but I have to be honest with you that I learn much more from the students in Chaco than from my MIT students because they just think so much differently from how I do, and they just have much… Also, I think that they’re less indoctrinated into our ways, and so that can make for a really fresh, original perspectives. It’s interesting because every time I tell people that I went there, they tell me, “Oh, it’s great that you taught them stuff,” but I actually feel like I learn much more from them.
And there’s one thing that I found really beautiful, which is that we finished this robotics workshop and then we got in a circle, and everybody started sharing what was their experience. And then this 14-year-old student just stands up and asks me and the other teachers, “Prof, I really enjoyed this workshop. I learned a lot from you. I hope you learned from us also, and what I wanted to ask you is, how did you feel? How did you feel teaching us?” And I just thought that was such a beautiful question because we never talk about feelings in mathematics, yet it’s such an emotional pursuit, actually.
Strogatz: It is an emotional pursuit. And Federico is such a wonderful example of that in everything he does. What it’s like to bring feelings into math. It’s inspiring. I mean, I think of that paper that he did with his collaborator, Marcelo Aguiar, laboring over that thing for 10 years, struggling, but along the way sharing joy with each other and the pleasure of figuring things out. It wasn’t just a mathematical journey for them. It was a journey of shared feelings.
Ardila: I think that it’s such a — you know, when you do these research projects that are 10 years old, it’s such an emotional rollercoaster, and sometimes you just wanna throw everything in the trash can, especially if your backpack gets stolen with all your notes. But also, sometimes it’s so exhilarating, where you’ve been pushing on something for several months and you finally get somewhere.
Strogatz: Next time on The Joy of x, computer scientist Rediet Abebe and I discuss the intersection of the personal, the political, the mathematical, and a few other things she cares about.
Rediet Abebe: I was like I don’t trust these Americans. I don’t know what kind of coffee they have, so I’m, like, bringing my own. It’s funny because at the border, they were like, “Why do you have so much coffee?”
Strogatz: The Joy of x is a podcast project of Quanta Magazine. We’re produced by Story Mechanics. Our producers are Dana Bialek and Camille Peterson. Our music is composed by Yuri Weber and Charles Michelet. Ellen Horne is our executive producer. From Quanta, our editorial advisors are Thomas Lin and John Rennie. Our sound engineers are Charles Michelet and at the Cornell University Broadcast Studio, Glen Palmer and Bertrand Odom-Reed, who I like to call Bert.
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