The Joy of Why - Does Form Really Shape Function?

Episode Date: June 12, 2025

What links a Möbius strip, brain folds and termite mounds? The answer is Harvard University’s L. Mahadevan, whose career has been devoted to using mathematics and physics to explore the form and function of common phenomena.Mahadevan, or Maha to his friends and colleagues, has long been fascinated by questions one wouldn’t normally ask — from the equilibrium shape of inert objects like a Möbius strip, to the complex factors that drive biological systems like morphogenesis or social insect colonies.In this episode of The Joy of Why, Mahadevan tells co-host Steven Strogatz what inspires him to tackle these questions, and how gels, gypsum and LED lights can help uncover form and function in biological systems. He also offers some provocative thoughts about how noisy random processes might underlie our intuitions about geometry.

I'm Steve Strogatz. And I'm Jana Levin. And this is The Joy of Why, a podcast from Quantum Magazine exploring some of the biggest unanswered questions in math and science today. Hi, Jana. Hey, how you doing? Good. Great to see you.

Great to see you. Interested to hear what you have for me today? Good, I'm gonna start by talking about Mobius strips. Oh, you know, my favorite. They are? Oh, I have a whole thing about Mobius strips, yeah. What is that all about? Oh, you know, just connectedness of space time

and they're quite intriguing, but I don't wanna interrupt you. I wanna hear that. You're not. I was gonna ask you, maybe for anyone who needs reminding, what do you mean by a Möbius strip? Well, a Möbius strip to me is literally a strip, as though it was a piece of paper where you could glue two ends and make a cylinder.

But a Möbius strip, you twist it first and then glue it together. I would say you put a half twist. If I turn it 180 degrees and then glue it, then there's all kinds of funny implications. Right. That might not sound like a huge modification, but it's actually really peculiar. It means if you go around one time, you're now inside the strip. So it seems to suggest like there's no clear side of it. To me, the big moral is that notions like side and edge and surface are not as simple or intuitive

as we thought they were. All kinds of strange things can happen. You can take a left-handed glove on a trip and it comes back right-handed. It can really mess you up. Good if you've lost one of your gloves, I suppose. The reason I'm bringing up Mobius strip is because my guest today, his full name is Lakshmanarayanan Mahadevan. Professionally, he goes as El Mahadevan, but really his friends just know him as Maha. He's a professor at Harvard in physics, in applied math, and in organismal and evolutionary biology.

Yeah, that's surprising. But Maha, as a grad student, took a view of a Mobius band that was different from one I had ever seen anywhere. It's traditionally thought of as an object in topology. However, Maha is interested in questions that you wouldn't normally ask, like what's the equilibrium shape of a Mobius strip if it's made of an elastic material?

And you let it relax to its lowest energy. What would it look like? So in other words, what's its geometry? He's really interested in form, like that old architecture thing, form and function. How does form help constrain function? How does it enhance function? Does the function force certain forms to take place? So I want to introduce you to my friend, Maha.

Let's hear from him. Hi Maha, it's great to see you. Likewise. Unlike many of the guests on the show where I don't know them personally, we've known each other for, I don't know, something like 20 plus years, I'm guessing. I'm pretty sure that when I first met you,

you were telling me about supercoiling of DNA and your work with Art Winfrey, if I remember correctly. I think so. My memory of it is that you were still a graduate student or maybe just recently finishing PhD. I was very struck by work you did about the shape of a Mobius strip. That's further back than I remember, but yes. I mean, as someone who loves math, I've been thinking about Mobius strips since teenage years.

But you asked that question with your advisor, with Joe Keller, what's the shape of a Mobius strip made of a real material when it's at its lowest energy state or in equilibrium? Yes, I tried to ask that question, yeah. I feel like it's relevant for our discussion today because it seems like you've been interested in this interplay between shape and force, certainly from your early work. And it looks like you're still thinking about it these days.

Partly, yes. I would say a lot of what I'm trying to do nowadays is more biologically motivated and inspired, but flow, shape, increasingly now thinking about sentient systems, understanding how we learn, what we learn. I'm very much interested in embodiment as a way in which sentient organisms find their way and function in the world, and particularly with social insects, but a little bit with humans as well.

Really? Okay. When we spoke at that Miller meeting at Berkeley, you were already pretty deeply immersed in thinking about biology, it seemed, at that point. Yes. It's been a very slow process. But macroscopic biology particularly fascinates me, trying to understand how from the constituents you can get complex behaviors, shapes, flows, sentience, cognition, function in particular. Are there ways to explain or steer or quantify complexity, which has a wonderful cognate complicate is to fold together. Let's talk about folds. Maha 1.0 that I knew a long time ago was thinking about things like folding of paper or crumpling. But Maha 2.0 is thinking about folding of brains, brains of creatures like ferrets as a guide to thinking about other organisms. Can we start by talking about convolutions and folds and cortices?

Yeah. So it started out by actually just staring at the palm of one's hand. People try to predict the future by looking at the creases in the palm of one's hands, but don't succeed. And so the question was, how do singular structures form when you have very soft systems? You see these in the context of folds that you mentioned, the inner part of the brain, the outer part of the brain where the cortex essentially is folded together. And then you have the gyri, which are the bumps between the folds,

and it seems like they're interesting singular structures. And so about 15 years ago, we started trying to unpack what happens with a single one of them, literally inspired by the creases in the palm. We did experiments with little pieces of gel and you fold them. From there, it was not a very long walk, so to speak, to try and use similar ideas to explain salsification or gyrification in the brain. You know, you can construct a physical model and do an experiment with a piece of gel where you just soak it in a solvent and as the gel starts to absorb the solvent, the outer layers start to swell, not dissimilar to the way the gray matter is

expanding relative to the white matter in your brain or a ferret brain. That combination of growth with constraints leads to patterns. And it's a very simple idea, but iterations and variations of that seem to be able to explain a substantial range qualitatively and even quantitatively of how the brain folds. Interesting. So I should know how to think about a gel, but I'm not so sure. Of course, I know what Jell-O is. Is Jell-O an example of a gel? A Jell-O is a very good example of a gel. So a long chain molecule, a polymer, which may be cross-linked or it may be so long that if you essentially try to deform it quickly,

it doesn't have time to flow and so consequence of it is it behaves elastically. That's the simplest sort of molecular picture of a gel. It may have solvent within it, it doesn't have to. It does not have to be ordered. Our own cortices, our grey matter is ordered, so the cells there tend to be roughly columnar and we have multiple layers. And how do they form? Well, you have neurons which are migrating from the interior towards what eventually will form the grey matter. Then you have cells which divide, change their shape or size, all of which effectively in the end lead to the surface expanding. And when the surface expands and the bulk is not able to accommodate that, then the

surface is put into compression and you start getting these very deep folds. And that's what you said earlier, that we should think of the gray matter as roughly the surface or the outer part. The gray matter is expanding due to, as you say, the cells dividing, they may be increasing in size. And then as a result of them spreading or growing compared to the bulk underneath, it leads to the formation of these grooves or what you're calling folds or salsai.

I don't mean the sort of gentle undulations that you would see on a beach. I mean the very deep furrows. And what sets up the initial conditions is the initial shape of the brain when in a fetal state it's not spherical, it's a little more ellipsoidal. And that seems to set up the first few folds and then the next few folds follow. And, you know, we literally created a small physical model using some gel, cross-linked polymer, and then we just dipped it in a solvent and the solvent started to get absorbed by the surfaces of the gel and it mimics the surface growth of the brain. And from there you can see things which are quite similar to what happens in the fetal brain and calculate them.

And then like you said, you can do this for a human, you can do this for a ferret, you can do this for a monkey, and you can start this for a human, you can do this for a ferret, you can do this for a monkey, and you can start comparing the shapes. You can just use simple geometrical ideas to understand what the statistics are, what the patterns are relative to each other, what the orientations are. Now, maybe you should help us understand, for people who haven't thought about folded brains before, why is that of interest? Are those folds associated with something we care about? Ah, very good.

So to understand morphogenesis, of which brain folding is one aspect, you have to sort of unpack at what level these patterns, which seem to both enable, but also constrain physiology, arise. Your lungs have a lot of surface, so your gut has a lot of surface, it's all folded up. Your brain also has a lot of surface, your gut has a lot of surface, it's all folded up. Your brain also has a lot of surface and it's folded up. Each one

of them is doing something different. In one case, maximizing the area for exchange of gases. In another case, maximizing the area for nutrient uptake. In another case, minimizing the wiring length for maximizing information processing. But in all of them there is in one form or the other a kind of packing problem that seems to be solved for a functional reason. So one can ask how do these things come about? Now you can essentially say well it comes about because of molecular processes, there are gene regulatory networks which say which cells should divide, where they should move, what fate should they take. You can also think about at a macroscopic level and say well these are consequences of simple things to state, but not so simple to sort

of unpack. You're not adding material at the same rate everywhere. You have flowing material, but you're not flowing everywhere at exactly the same rate. And the consequence of both these is that you get these patterns. So now in the context of the brain, we do know that different parts of the brain are functionalized differently. You have a visual cortex, you have an auditory cortex, speech areas and so on. So where are these functionalizations set up? Are they set up before the folding patterns? Are they set up after the folding patterns or simultaneously with the folding patterns? We don't know. I suspect that it's associated with a combination of both these. And what seems to be happening, at least in the systems I'm familiar with, is that both folding and functionalization are happening at similar timescales. There's

always this question in many of these biological morphogenetic processes. Is there self-organization? Is this pre-patterning? And pure pre-patterning will give you very robust patterns, but it will not allow for any mistakes. Pure self-organization will give you enormously robust patterns, but very hard to guide. Could it be a combination of some pre-patterning, give you enormously robust patterns but very hard to guide. Could it be a combination of some pre-patterning, some genetic signature, some molecular signals

and processes associated with self-organization? I like to think of the genes as providing the logic and the rest is calculus. Well, I'm very happy with that description. Now, are there developmental disorders that could happen in the baby if these folds are not occurring in the way that they're expected? Yeah, so we do know there are a whole class of these morphological pathologies, so lisencephaly, which is smooth brain, pachygyria, which is thick skin like an elephant, like a pachyderm, and so you have few folds, or polymicrogyria, many folds in some locations. Each one of these, depending on where it happens and when it happens, seems to be correlated strongly

with certain kinds of disorders. They have been best studied in the ferret, but there are homologues associated with what seems to also happen in humans. And so we've been talking to some of our colleagues in the medical school, trying to understand how much of this is genetic and how much of this is macroscopic and how much of this is a combination of both. It's unusual to hear about ferrets as model animals. We're used to hearing about laboratory mice or rats.

Mice don't have brains which fold. Amazing. So mice can't work for this. Mice can't work. They're cerebellum. So the part of the brain associated with controlling movements is folded, but the cerebrum, the part that we typically think of as an icon of biology, is not folded for the mouse or for rats and the ferret it is.

Interesting because those are clever creatures. It's not like they're not smart. Oh, yeah, sure. So you don't need a super folded cortex to still be smart. It depends on what kind of smartness you're thinking about. I'll give you an example. Which part of the brain do you think is extraordinarily folded for an elephant? I'll take the bait.

I mean, elephants are known for memory, but still, I don't know, is there a part of the brain associated with like something that controls their trunk? Yeah. Something in their cerebellum? Massive cerebellum. After we finish, just take a look and Google the elephant brain, you will see the cerebellum is hypertrophied. And it's almost certainly associated with this incredible task, which is to control its trunk, which can pick up a peanut, but can also uproot a tree.

Unbelievable. Which sort of leads to this interesting question. I don't like to think about the brain on its own. You've got to think about what the animal is trying to do. It's the brain, the body, and the environment. Like Richard Lewontin, he has this perspective of the triple helix. The triple helix, okay. Well, let's shift gears a little bit because you're so incredibly broad in your interest.

We only talk about the brain and folds. I think that'll be misleading. So let's talk about social insects a little bit. Yeah. It might seem initially disconnected with things like morphogenesis, but at least in my mind, I think there are remarkable similarities, except a large number of insects are coming together to create a society. Instead of a society of cells, it's a society of organisms. For the social insects, they would like to solve problems much larger than themselves, problems associated with preventing predation,

trying to create environments which are not hostile, even if the environment is changing. How do they do that? They create architectures, they build mounds or nests. So what sets the size, what sets the shape, what's the function, and how is this essentially built? Is there a plan? Is there a planner? Is there a design? As far as we can say, the answer to all the latter questions is there isn't. So now to come back to this analogy, in us, our soma are different from our germline, right? The cells which are responsible for transmission are fundamentally different from the ones that make up our body. That's true actually in social insights.

You know, the queen is the only part of the colony that can reproduce and the rest of the colony does not. There is a similar separation. But now if you want to sort of protect yourself against all these different environmental variations, temperature, humidity variations, oxygen variations, or mechanical loading, how do you solve these problems on scales much larger than the individual? So a termite the size of a fingernail, three or four millimeters can create a mound

which is six, seven meters in length. So if you were able to do the same thing, then we would be creating things which are twice the size of the Burj Khalifa, the tallest building without a designer or a designer. How do they do it? And so we started trying to first probe what their function is and then working from there, trying to understand what the rules are. Would you like to wager? Well, by analogy with a house, I would tend to think of it as a place of safety from the

torments of the environment. Yeah, exactly. So you want to build a barrier, which is neither perfectly insulating nor perfectly open, because perfect insulation, you can't exchange anything. And if it's perfectly open, you've not created a barrier. So there seems to be a Goldilocks zone of how to create barriers. I see. So this is going to be some kind of Frank Lloyd Wright indoor-outdoor house. But it precedes him by a few hundred million years. That's okay. Frank wouldn't mind. And so we went to Namibia, we went to India, and we measured how airflow arise inside the mound.

We measure carbon dioxide concentrations, humidity concentrations, and we found that they change as a function of dienyl variations in temperature across climes, across species, and across continents. What was happening was that during the day, the outside of the mound starts to heat up, therefore the air inside starts to heat up. We know through some wonderful experiments that you can pour gypsum or calcium and magnesium sulfate and let it solidify and then you wash out all the

clay and you get the skeleton of the holes. And when you do that, you find that there is a porous structure and the holes in the mound are small near the boundary and as you go deeper and deeper they become larger and they also start orienting mostly vertical up and down. And so we knew that and we wanted to know how they work and to cut a long story short the mound works like a lung. So during the day, the air inside the walls heats up and therefore it starts to rise. There isn't a chimney, the mound is closed. And so if it rises up near the boundary, it has to go down through because it cannot go anywhere else. And so during the day, you have air flows hugging the walls on the inside moving upwards and

in the interior moving downwards. But then at night, the outside cools fastest, the air sinks near the walls, the whole thing reverses. So it breathes twice a day. Oh, very poetic. And so in our bodies, our lungs are powered partly by a diaphragm and we breathe roughly once a second or once in two seconds. It's essentially serving as a ventilation system, but unlike in physical

engines, the sources are changing. During the day, the outside is warm. At night, the outside is cold. And so this is a very interesting thermodynamics question, if you will, associated with harnessing energy from a periodically varying environment. And you're converting the temporal variations into a spatial. Oh, that's a pretty way of saying it. The temporal oscillation from the day and the night, or the hot and the cold, becomes this spatial variation in the mound itself. Then, if you have a small mound, it should be relatively spherical, but large mounds

cannot remain spherical because gases go up and down in response to buoyancy and buoyancy breaks the symmetry. If you look for small mounds in some parts of West Africa, you will see the mounds are roughly spherical. But if you look at the mounds in India, the mounds are more conical. And if you look at the mounds in Namibia, they're tall. And the reason they're even taller is because the temperature variations between day and night in the savannah is much more than the temperature variations close to the tropics. You cannot separate the environment from the organisms. The organisms change their behavior in response to the environmental changes in this case associated with temperature variations. When they build,

they change the air flows. But when they change the air flows, the pheromones, the small molecules which change the behavior of things like insects, change the way they move. And so the behavior changes the environment, which then changes the behavior, which then changes the environment. So you cannot separate the living from the non-living anymore. Well, there's a number of things that are really intriguing. I feel like we used to think of organisms and animals as these isolated entities, and we're backtracking on all of that. Now we have a gut microbiome that's so important and contributes to us as an organism, and then you see these colonies where their physical environment is breathing with them.

It's so fascinating, and they're in this feedback loop with the system. So I think this kind of idea to separate the individual is fading from fashion. It does feel like that, doesn't it? And his metaphor of the mound as a lung is so striking. It really is a poetic thing. I mean, this mound, if you haven't ever seen, and I have to admit, I have not seen a termite

mound, it have not been to Namibia or India, these are gigantic structures compared to the size of the termites. The termites are just a millimeter or two, they're like your tiniest fingernail. These mounds could be a few meters high, so we're talking about something relative to the organism that's taller than the tallest building on Earth compared to a person. But here's what's so crazy, it's mostly empty. They're not living in there. So biologists understood that it must have something to do with ventilation, that the organisms are making carbon dioxide that will build up and they want to be breathing oxygen.

It's not going to be good to have too much carbon dioxide. So how are you going to have a ventilation system? That's what this enormous mound is for. That's insane. Okay, but these tiny creatures don't know anything. This is a case of swarm intelligence. No individual termite knows what it's trying to do. Right.

But somehow collectively, they built this Burj Khalifa scale thing out of dirt and dung, their own excretions out of dung. Their own dung. And saliva. Put spit and you know what, something that rhymes with spit and dirt. And they make this skyscraper out of it. It makes me wonder, well many things. First of all, how did they strike on this through the process of Darwinian natural selection?

It just seems quite amazing that through a series of environmental pressures, they figured out what they're doing, but how that led to this successful structure, and also, I'm really fascinated by biomimicry. Do we have something to learn from them on how to regulate carbon dioxide in our atmosphere? It's a wonderful question, actually,

because the biomimicry is something that Mahai and his colleagues have suggested, not so much in connection with climate change, but just with smart buildings. Because see, this is a passive structure. The termites aren't building any engines. They don't have any air conditioners. They're just letting naturally occurring processes

in the environment. That sets up these convection currents that circulate the air in their house. And it doesn't cost them anything. processes in the environment, that sets up these convection currents that circulate the air in their house. And it doesn't cost them anything. I mean, it's not only carbon neutral. They're not using any energy at all. This is energy for free. Right. Talk about zero carbon footprint. Well, we're going to hear more from Maha about the wonders of biology and form and function after the break.

Welcome back to the Joy of Why. We're here with Professor L. Mahadevan from Harvard discussing form and function, especially in biological systems. You focus here on behavior and environment. You haven't mentioned genes, particularly in this part of the discussion. Anyone who's taken molecular biology, you think that's the story in biology. Do you want to bring them back in? So, without saying that the genes are unimportant, I want to say that they're only part of the story.

Genes set up, if you want, the pre-patterning associated with how the structures are built, the anatomy is built, and how the function arises. If you open that hood, then you will find there are a set of circuits associated with how the olfactory signals are first processed and even under that, how the circuits themselves are put together. But that has to be now selected for through interactions of the organism with the environment. Sure. And so this long arc or even a loop is associated with starting with the gene moving towards the circuits, moving towards the organism, understanding how the organism interacts with the environment, and then seeing all the different possibilities that might arise initially through neutral mutations

and once in a while through mutations which give rise to an advantage or a disadvantage. But that's one way. We could have gone that way. We didn't. Instead, we said, let's think about robotics and then ask, oh, can I essentially now build a system of robotic ants or termites? There are four large classes of social insects, ants, termites, wasps and bees. We started with termites and termites are very hard to maintain in a lab and also not very popular for obvious reasons,

but ants we can. We started working with ants and we saw similar kinds of building and breaking behavior. And then we said, can we recreate that with these robots which we can build, which can essentially sense each other just by proximity. And instead of pheromones, we call the analog with light, because it's easy, photormones. So we put robots on a TV screen, and then wherever the robots are, they secrete equivalent of a pheromone, but it becomes a photormone, so there's a spreading aura of light. So do they have little LEDs or something on their bellies? What are they doing to spread

the light? We have one camera which is detecting each robot. When it detects the robot at that location, it starts spreading. And now they don't interact with each other except through the photomode, through the environment. Through the light that is driven by this external light source, but that that light source is queued to their locations. Exactly. So again, it's the environment which is malleable. The environment serves as a substrate on which you can write information, in this case, chemical concentration. That chemical concentration is changing as a function of space and time.

That concentration changing causes other ants to respond. They change that environmental signal, and so you don't ever interact directly. This is called stigmurgy in entomology and again don't think of just the organism, don't think of just the environment but think of them together and things which you might not have imagined or which you thought were very complex turn out not to be so complicated once you're able to unpack them. I like it and I'm just having this vision with the pedestrians on the Millennium Bridge in London

on its opening day when it started to wobble. The pedestrians pay attention to other pedestrians directly, how close they're standing to them. But they also pay attention indirectly as the pedestrians jiggle the bridge. I don't necessarily feel you directly, I feel your effect on the environment, in this case, the wobbling bridge. So that is a theme that seems to be around a lot these days. I'm wondering a little bit how people like us trained in math and physics, we like to make problems as simple as possible because the math can be difficult. But in biology, they're very interested in interplay among different levels of organization, like from genes up to populations. And it

seems like you're absorbing that lesson in the way that you're describing the interplay in the triple helix that you mentioned. It's interesting you say that I think it's very close to things that I have played with in both mathematics and physics, which is systems with multiple scales and how you handle that without completely ignoring any of the scales, but recognizing that dynamics on some scales is fast and dynamics on other scales is slow. Give you an example. So the environment changes in these cases relatively slowly, while the organismal behavior

is happening on much faster timecales, unless you come close to a bifurcation. I would say the most important thing for me was not being afraid of the messiness that biology is and just embracing it. Don't necessarily break the problem down completely, but understand the organism in context. Small does not mean negligible. This is a famous aphorism. But it is true, actually. I did misspeak there, and I'm glad to be corrected, that certain cultures of math and physics are concerned with multiscale phenomena,

famously fluid dynamics. Let me shift gears a little bit to talk about your process as a person. Is there a method that you use for problem selection based on certain criteria or is it just pure curiosity, what hits you at the moment? How do you know what to work on? I'm asked this question. Tell us the true answer.

Yeah, I have some answers. I don't know whether I have one. I tend to be very curious about everything, which is a bad habit, but now that I have a job, it's okay. And so then the question becomes what to work on and if to work on something. When I was finishing graduate school and the internet was just becoming capable of allowing you to essentially explore, that for me eye-opening. I didn't have to know the right kinds of people or to be part of the right groups and so you could just study anything. And so that allowed my curiosity to go wild.

Most of the time I fail, but science is a very forgiving human endeavor. The integral counts much more than the derivative, thankfully. Meaning there are errors which eventually correct themselves. People make mistakes and over time we think that we have a slightly better understanding or at least a sharper way of asking questions, which for me is perhaps the most important thing. It's easy to just look around you and construct questions. Nowadays, we do our own experiments when we can, but early on it was not necessary. You just walk around and you just see things and you can ask questions.

And I still find that a source of much pleasure. Most of the time I can't answer them. So it's just sort of a general wandering and wandering. I've been very lucky to be in institutions which more or less have been accepting of that. I had a previous guest mention this quote of Francis Crick that haunts me. He said approximately, it's just as easy to work on an important problem as a trivial problem. Does that ever factor into your thinking? How would you react to that comment?

I do have a reaction. I think it was Landau who said something to this effect, that one must have a ridiculous sense of immodesty to decide to only work on what the most important problems are, that a scientist should not allow vanity to be part of what they decide to work on. And I think what he was trying to say is that it's not clear when you look at a problem, whether it is or it is not important. Landau himself worked on things which determine

how we understand things like phase transitions, but he also tried to look at problems associated with how liquid climbs up a slide when you dip it into liquid. And it's hard for me to imagine that one didn't somehow inform the other. In other words, it's very hard to unpack, except post facto, what is important and what isn't.

My inspiration has been, I think, that kind of an approach. You mentioned my advisor, Joe Keller. I don't think he would have disagreed. He was curious about things and who knows what is and what is not important. I consider it a privilege to be a wanderer. You pick up a little stone and you find a little worm under it. You pick up another stone and you find something else worth studying, that's enough. You know how in art they sometimes talk about the concept of negative space, that if I were drawing you, I could also draw the space around you. Is there a negative space concept for problem selection?

Is there something that would rule a problem out for you? Oh, ignorance. Your own ignorance? Yes. You mean if I know nothing about the area I won't work on that? Because you wouldn't know where to start. That's touching on another aspect of this I wanted to bring up with you. Since we're quoting all these luminaries, is it Peter Medawar who describes science as the art of the soluble?

Yeah. So is that part of your selection criteria that you want something that as you look at it, it's not only fascinating, but it seems like it might be solvable by you. I would replace importance by interest. So something which is not intractable, but something which is not trivial. Now, these notions are subjective because if I don't know something, it might look intractable and then when you know about it, you feel like, why did I waste so much time on it? Well, I'll give you an example of something that we recently studied,

which is a problem in high school physics. How does a ball roll down an inclined plane? What happens if the ball is not perfectly spherical, or what happens if the cylinder is not perfectly spherical? So now you have a little bit of dynamics, but a little bit of stochasticity because of the noise, but this noise or the shape is irregular and that irregular shape is not changing, it's just fixed.

So what kinds of trajectories will it have? And you find interesting connections to stopping and starting, periodic orbits of different periodic solutions and so on. Why did we study this? Because I was curious. Did I expect all the answers? Some, but not all. Did we learn something? I learned something. A couple of students who worked with me did. I think it's okay. You know, you mentioned art, and I want to come back. Do you always question what art is for? Or what is important

in music? I don't know that I would use a word like important, but I do have a sense of taste. For instance, just to give an example, if someone is singing a song that's already a fantastic song in terms of lyrics, melody, everything, and they over-embellish it to show off their verbal pyrotechnics,

that really bugs me. I think that person may be a great singer technically, but they have such bad taste, I don't respect them as a musician. Would you say that of a scientist or a mathematician? I would, actually. There are people who are technique-driven, who have wonderful technique, and they don't really care what they work on as long as they can show off their technique. Yeah, I feel similarly. So, I think it's not importance or interest. Interest is potentially a matter of taste.

What drives me towards problems is some measure of I would learn something. It's a very personal decision. And if it happens to interest a few other people, that's wonderful. And if it doesn't, it's okay, because I've still learned about some tiny part of the world. I feel very privileged to have a chance to participate in that. As we come in for a landing here, are there some budding interests that you're willing to share with us, things that you're currently curious about?

So yes, one area is how we learn geometry and how do we learn probability, two areas of mathematics which I love very much. Notions of probability, some of them are very intuitive, but as you I'm sure know, many of them are extremely non-intuitive. If those are hard, and they're already problems where you have a finite number of outcomes, so the alphabet is finite.

Now you can ask what happens when the alphabet starts to be continuous, so to speak, it's not an alphabet anymore. I'll give you an example, which is Buffon's needle. There's a version of that, a slightly different whole class of questions associated with what's called Bertrand's paradox. In a nutshell, the notion of randomness is not well defined unless there is a process associated with it or mathematically what is the measure associated with that random variable. In the context of Buffon's needle, for example, we're asking a question which is geometric, asking about what the probability is that a needle crosses a set

of one-dimensional tilings. But we're asking it from a probabilistic point of view. So we're asking a geometrical question from a probabilistic point of view. And I love this because geometry to me is about relationships. probabilities associated with asking what would happen if I had multiple instances, if you will. Now, in this case, they're both colliding with each other. We did an experiment in that spirit, which is, do you know Euclid or do you have to be taught Euclid? Oh, boy, Immanuel Kant would like this question. Euclid's Axiom 32 is that the sum of the angles,

a triangle, add up to two right angles. Do you know this? Do you have to be taught this? So we did an experiment, post-doc working with me, Yuval Hart, Elizabeth Spelke, who was a wonderful colleague of mine in psychology, a student of hers, at that time Molly Dillon. So here's the experiment.

I show you two of the vertices of a triangle and ask you where is the third vertex? And I also ask you what is the angle at the third vertex, which is really asking about do you have a sense of what a triangle is and what the relationships are between the angles? And you wouldn't be surprised if I tell you that people don't perfectly pick the third vertex or the angle. So we find that it's noisy and then we try

to ask how can you unpack that. The answer turns out to be that they have a small propensity to have the point below where it normally is. So how do you solve this problem? Well, you have to solve this problem by recognizing that you're an embodied agent, that you have eyes, that you're trying to essentially track what might seem like an initial direction, but then you kind of lose your way, then you go back and do it again and again, and eventually you come to say, I'm going to make that decision. So there is clearly a probabilistic process of iteration, but there is also a geometric aspect. So how do you make sense of that? So to get perfectly the vertex, you need two pieces of information. One is the initial orientation, and another one

is that when you move along the path, you move along a straight line. So I have to make sure that the line that I'm moving along is not got curvature, but my natural tendency is to wander away because there's only one way to draw a straight line. There's an infinite number of ways to not draw a straight line. That's for sure. Yeah. Okay. So now that I recognize, there is an interesting parameter, which is the ratio of how much you weight the straightness and how much you weight the initial orientation. That means there is a hidden scale, which means that you're going to make more or less

errors depending on the size of the triangle if you think of the penalty to move along a straight line is to minimize the curvature mm-hmm and Then a penalty to move along the initial orientation is something else the penalties that are the ratio of these two must have a scale Well, they must by dimensional ground since the curvature is one over a distance and an orientation is a dimensionless quantity. So there is a length scale involved here. That length scale, initially, I thought, wow, this could be very cool because that means our intuition of Euclidean geometry is probably not Euclidean at all because again, there's only one way to have Euclidean geometries. There are an infinite number of non-Euclidean geometries. Our view of geometry, effectively what I'm saying is, it's statistical. And what we teach is deterministic.

No, this is a radical idea to think that there's anything statistical about geometry. We never hear that idea. Yeah. So it turns out, not surprisingly, Newton already was thinking about statistical geometry of which Buffon's needle is the simplest problem. So I am starting to think about this way of how do we learn geometry by doing psychology experiments. You could, of course, go the other way. You could start with the circuits and the brain, but I prefer to essentially switch it around and say, no, I want to work with the environment and the organism and then ask how does the organism learn how

to navigate. So this is a very important task, right? Navigation is something that all organisms have had to learn. Another class of problems is how do we learn physics? For example, I put a child on a swing, the child will wiggle its body and eventually learn to swing even if nobody gives instructions. So how do organisms and humans learn about the world? Well, there are so many more things I would love to discuss with you, but maybe we better

wait till the next time we have a chance to talk. As we wrap up here, Mahaj, there is one question that I'm dying to ask you. Can you put your finger on what about being a mathematician and scientist brings you joy? I think asking questions, I'm not sure that it's always possible to answer these questions, but it's almost always possible to sharpen the question a little bit. And I think that is something that I find exhilarating because that means that there is another layer of the onion that one can peel without becoming too teary-eyed. Sounds like you are always going to be peeling that onion. I hope not crying too much. No, or making anyone else around you cry too much.

So, Maha, it's really been fun. Thank you so much for joining us today. Thank you so much, Steve. Well, this idea of geometry being statistical is either crazy or brilliant. You know, I wonder if it's just leaning too much into how human beings do things. I mean, specifically the argument was if you're not taught it, what happens? And the idea that, of course, there's going to be a spread of approaches to the solution of something is just not at all surprising, that that's going to be statistical in something that's

not taught, right? But then I wonder if our sense of, oh, no, we have flat Euclidean geometry and then we have these kind of geometry on curved surfaces, that is also human in a way, because we're separating things into parameters and constraints that we can understand one at a time. And that kind of organizational impulse

is kind of also distinctively human. I'm just not really sure what to make of it. It's really a psychology result. I don't think the claim is that geometry in some God-given sense is inherently statistical. It's that human perception of geometry, independent of whether people have taken formal classes in geometry or not, you can do studies with people in Amazon rainforests who have never had a geometry course and ask them the same question.

These studies have been done. Remember how it goes. He says, I'm going to give you the two base angles of, it turns out, an isosceles triangle. So you have a little picture with these two V shapes at the base. And then there's like an unfinished triangle at the top. And you're supposed to point where is the third corner? Where's the top vertex? And you're also supposed to say, what's the angle at the vertex. They did it with little triangles, they did it with massive triangles. And the finding was, whether the people were educated or not about triangles, they always tended to place the third vertex

a little bit lower. I mean, think about the task, right? You should just extrapolate those straight lines. What's the problem? Why don't you just draw the lines and see where they meet? Well, people can't do that in their head very well, it turns out. People kind of inch forward dynamically. They simulate the lines in their head. That's what he's talking about statistical. He imagines a random walk where you're taking little line segments from each vertex until you hit.

Right. And that's the statistical process that they show in their work leads to the observations that they see in real data on real people performing this task. It makes us think that our human perception of geometry is not as innate as we thought. It's a more dynamic simulation thing that we're doing in our head. Hmm. Yeah, I'm not sure I can see the conclusion so cleanly. It seems to me there's lots of possible areas in which something could participate in the results, besides just our raw perception of geometry.

I think an interesting aspect of this is that it's always in the same direction, the error that's made. Yeah. That by and large, that's made in a certain direction. That's a little surprising. I would have imagined it more sort of statistically random around the right answer.

It is. The bias is always towards the base. They always put the corner closer to the base than where it really is. And the mean size of that bias turns out to be proportional to how big the triangle was. The bigger the triangle, the bigger the error. How does it fit into Maha's larger work? I think he says at one point that he's interested in embodiment. That we're not just brains in a vat thinking. We're brains inside a body that gets to live in the world and think about the world by moving around in it, by tasting it, by sampling it, by seeing it. And so how does that affect our geometry, our architecture?

Just one more way in which we're inferior to AI. Hey, they don't even have bodies. Yeah, exactly. All right, Jana, this was really fun. Look forward to catching up with you on the next one. So interesting. Till next time. If you're enjoying The Joy of Why and you're not already subscribed, hit the subscribe or follow button where you're listening.

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have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quantum Magazine. The joy of why is produced by PRX Productions. The production team is Caitlin Falds, Livia Brock, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzalez.

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