What Does the Fourth Dimension Actually Look Like?

We know all about how things exist in the three dimensions of length, width, and height. Physicists often talk about time as a fourth dimension, but what if there were a fourth spatial dimension — another direction entirely? How on earth would we picture that? Mathematicians use topology to visualize abstract spaces in higher dimensions. Maggie Miller https://web.ma.utexas.edu/users/mhm799/ at the University of Texas at Austin explores what happens when knots encounter an extra dimension. Familiar tangled loops behave in unexpectedly complex and counterintuitive ways in 4D: Knots can always come undone, while ordinary surfaces like spheres can — surprisingly — become knotted in ways that can’t be undone. In this episode, Miller explains to co-host Janna Levin why 4D is the lowest dimension that mathematicians still don’t fully understand, how she and her collaborators resolved a question about knotted surfaces first posed by mathematician Charles Livingston in 1982, and how her interest in art has helped her develop the visual techniques she uses to picture 4D spaces.

• 0:00Speaker 00

  And we're off.

• 0:05Speaker 00

  I'm Janna Levin.

• 0:06Speaker 01

  And I'm Steve Strogatz.

• 0:08Speaker 00

  And this is The Joy of Why.

• 0:10Speaker 01

  A podcast from Quantum Magazine where we explore some of the biggest unanswered questions in math and science today.

• 0:18Speaker 00

  Steve, we have a really mathematical topic today. I'm glad to have a mathematician in cahoots.

• 0:23Speaker 01

  Okay, at your service.

• 0:25Speaker 00

  Excellent. We're speaking in this episode about a branch of mathematics known as topology. And that's not familiar to a lot of people. Do you want to help us a little bit just with the basics of topology? What makes topology different than regular geometry that we might have learned?

• 0:40Speaker 01

  Yeah, sure. Topology, very popular subject. You know, whereas geometry is ancient. Topology is only, depending where you date it, could be a few hundred years old, but it really took off in the 20th century. People may have run into it in childhood games. Like, did you ever take a piece of paper, like a strip, some kind of ribbon? And then you put a twist in it, half a twist, and then close it, put a piece of tape on it to make this shape called a Mobius strip or a Mobius band? Did you ever do it with your kids?

• 1:11Speaker 00

  No, I didn't. But I think about the Mobius band probably an unhealthy amount.

• 1:16Speaker 01

  Oh, really?

• 1:18Speaker 00

  Yeah. It comes up in my work. Yeah.

• 1:21Speaker 01

  Oh, okay. Well, there was a time when one of my kids was in elementary school and it was Bring Your Parent to School Day. So I had the kids make Mobius strips and then do this thing where you cut down the midline of the strip all the way around, the whole ring. And you think you're cutting it in half, but it doesn't fall apart. And so one kid started crying.

• 1:46Speaker 00

  Well, math is frustrating.

• 1:48Speaker 01

  This is the thing. It was an early introduction to what all mathematicians and math teachers are trying to do. Make people cry in math class.

• 1:55Speaker 00

  It's perfect, actually.

• 1:57Speaker 01

  Okay. But anyway, so that's our subject, topology. I mean, the Mobius strip, this property, that when you cut it down the middle, it doesn't fall into two pieces. That's a topological property about how things are connected. You could change a lot of geometrical things, and you wouldn't change this property. It's deeper than geometry.

• 2:15Speaker 00

  Right. Well, we usually think about geometry. We think about the Pythagorean theorem, which is precisely telling me how to measure distances. And topology is not about that. It's about this global connectedness. So it has this very abstract character. And our guest today is, in fact, Maggie Miller, an assistant professor at the University of Texas at Austin, who researches knots, surfaces, and four-dimensional spaces. So without further ado, let's hear what Maggie Miller has to say about topology.

• 2:48Speaker 00

  Welcome to The Joy of Why, Maggie. It's really a pleasure to have you on.

• 2:51Speaker 02

  Yeah, thanks for having me. I'm excited to be here.

• 2:53Speaker 00

  I'm really fascinated by your work. It intrigues me personally. There's a lot of questions I want to ask you, but let's just start with the Miriam Mirzakani New Frontiers Prize, which is within the Breakthrough Prize umbrella. You received that a few years ago. And you mentioned that early on in college, you were split between studying math and art, which I think is a really fascinating combination. What in particular led you to choose math in the end?

• 3:20Speaker 02

  As a kid, even before college, I was really invested in art. I took art classes for a long time. I mean, I know lots of people take art in school. It's a very common favorite subject. But I always participated in our state fair. It's just something I really like, that visual aspect of thinking and planning and the actual technical aspect of making something. When I went to college, actually, I chose to do math right away. And it was because math had always been my favorite subject as a student. I want to say in school, but I was actually homeschooled. So it's a little bit unclear what that means. But it actually felt similar to me, to art, in terms of it's a little puzzle. I guess it's more apparent with math.

• 4:03Speaker 02

  Maybe people can understand what I mean when I say solving a math problem is like a puzzle. But I feel like that about drawing something, too. I mean, anybody who's struggled with aspect ratios knows it's not always so easy to figure out where things are actually supposed to go.

• 4:16Speaker 00

  I agree. This is sort of a misunderstood aspect of art. People think art is structureless and without constraints. And that's really not the case, especially if the art is good.

• 4:28Speaker 02

  That's right.

• 4:29Speaker 00

  So I need to hear about the homeschooling.

• 4:32Speaker 02

  Yeah. My parents decided to homeschool me and my twin sister starting about halfway through third grade. It is very common in the state of Texas. I knew a lot of other homeschooled children growing up. I have two brothers who are quite a bit younger than me. And so they were both homeschooled all the way up until college, which, again, it's very common in Texas compared to some other states. And I think it's a little bit easier in terms of government restrictions. That produces its own challenges because it's not always so clear what you're supposed to be doing. Especially as a teenager, I knew I wanted to go to university. At the time, the University of Texas actually had a webpage that had a list of what they expected from homeschoolers in order to potentially be accepted.

• 5:17Speaker 02

  And so I saw that when I was around 14. And I worked with my mom to make a list of, here's all the things that are sort of roughly equivalent. What counts as history?

• 5:26Speaker 00

  So Maggie, each time I ask you a question, you say something that I have to ask about? So first, it was the homeschooling. And then it was me and my twin sister. Twin sister and identical twin sister? And is she also a mathematician or an artist?

• 5:38Speaker 02

  No, we're fraternal. I think we have a lot in common. We're still sisters. She is not particularly interested in math or art in the traditional sense. She definitely has a lot of craft activities that she enjoys. My sister went to college and studied psychology. So I think she's a little bit more interested in maybe social science.

• 5:57Speaker 00

  Yeah. So I want to get into discussing your area of mathematics, which is one of my favorite subjects, topology. Can you explain what topology is?

• 6:07Speaker 02

  So topology is the study of abstract spaces, which could mean the space that we live in right now. We live in a space where we perceive three dimensions, X, Y, and Z. But it refers to a more general concept. There's what is a space is something that you start off defining in a course on topology. But for now, just imagine like a space you could live in. And we say that two objects or spaces are the same, equivalent, if you can continuously deform one into the other. So topology is the study of objects, like stretching and twisting, but you're never allowed to break things.

• 6:47Speaker 00

  So when you say X, Y, and Z, you mean the three dimensions we're used to occupying. But within topology, you also work on other dimensional spaces?

• 6:57Speaker 02

  That's right. It's very common to study spaces of any dimension or spaces that don't have a well-defined dimension because they can be really abstract. So for example, a very simple example of a space is something like the circle. So not the filled-in circle, but really just the edge of a disc, like if I took the edge of a plate or something. So that's an example of a one-dimensional object. If I'm standing on a circle, then there's one direction it can go, which is clockwise or counterclockwise, which is the same direction, but backwards. But then I could construct a really complicated space where what if I just took a million circles and glued them all together at one point? What if I took an infinite number of circles? And then we have to consider what kind of infinity? And I glued them all together at one point? And if I do something complicated like that,

• 7:42Speaker 02

  it's a lot harder to understand the object in whole.

• 7:46Speaker 00

  So if you glue them all together at one point, it's like a fan, almost.

• 7:50Speaker 02

  Yeah, it'd be a fan, yeah. Or sometimes people call it an earring.

• 7:54Speaker 00

  So some of these spaces are true to the word space. They represent where we can live or places that can be occupied. But some of them don't have to do with that physical dimensionality that we're used to.

• 8:07Speaker 02

  That's right. I think that the physical dimensionality motivates the name. And that's a common theme in topology and other areas of math. There's one easy example, which has an English name that makes sense. And then we analogize it and still use the same name, even when it feels like that's not what we mean in the English sense.

• 8:27Speaker 00

  So to gain some kind of perspective, let's start with the idea of a three-dimensional manifold. We live, presumably, in a three-dimensional manifold. Can you walk us through the sense in which we have three dimensions?

• 8:43Speaker 02

  Yeah. So when we talk about dimension, it means how many different perpendicular directions are there so that every direction is a combination of those? So we live in a three-dimensional space because there are three perpendicular directions, forward, backward, left, right, up, down. Those are all 90 degrees away from each other. Any direction that I would want to go is some combination of those three. So they take up all of the dimension as well. So that means it's exactly a three-dimensional space.

• 9:12Speaker 00

  So if people aren't used to think about 90 degrees and orthogonality, one way I like to think of it is I can move in one dimension, if it's orthogonal to the others, without moving in the other directions at all?

• 9:22Speaker 02

  That's right. I mean, if I wanted to tell you exactly where I was, it would be how many different pieces of information do I need to tell you? So I need to give you my coordinates in those three directions. The tricky thing is first, we have to all agree, what does it mean to be at point 000? So what is the, we call it the origin? And then from there, I can measure, well, how far in that direction am I from zero? And then how far forward, backward, and then how far up, down? Another tricky thing is that for topologists, we actually don't have a notion of distance. So I was being a little bit misleading when I said we could measure how far away I am from the origin.

• 10:02Speaker 02

  Because if I'm a topologist, then I actually don't know how far I am from any one point. I just know whether or not two points are the same. There's no notion of what it means to be farther away.

• 10:15Speaker 00

  What you care about is if things are smoothly deformable between one space and another space. So you don't care if it's made of rubber, and I literally stretch the distances or contract the distances? What you care about is if I walk far enough, will I come back to where I started? Some kind of global property. Would that be fair?

• 10:35Speaker 02

  That's exactly right. Yeah. So we talk about local structure versus global structure. For a topologist, everything is made of rubber. And as long as I can stretch two objects and have them come out looking the same, then they are the same. So a circle is an oval, but they're definitely not the same thing as a line.

• 10:54Speaker 00

  Right, because a line isn't connected in that particular way. So topology, you're saying, is that study of that large, that global structure. Now, you've, in particular, worked in four dimensions. Can you talk us through the fourth direction, using the time analogy? Because I think that helps people kind of imagine what we're thinking about when we're thinking about a dimension beyond the three spatial ones we're used to occupying.

• 11:23Speaker 02

  That's right. I mean, having a fourth dimension is a really abstract idea. People understand forward, backward, left, right, up, down. Those are three orthogonal perpendicular directions. So if we're thinking of those three directions as representing space, the usual three-dimensional space we live in, we have to think, what would it mean to move? But in a different direction, which means that my spatial coordinate does not change. And so we can think of that as time. So right now, I'm moving forward in time at a rate of, I guess I'm moving one second forward in time per second. But my position in three-dimensional space is not changing. So that means that I could think that, right now, I'm living in a four-dimensional space,

• 12:05Speaker 02

  where that fourth dimension is time, and it's perpendicular to the first three. It's a little bit more like a movie, where if I were watching a movie, then I could rewind or pause.

• 12:16Speaker 00

  How do we move to dimension four from dimension three?

• 12:21Speaker 02

  Well, one theme, I think, of four-dimensional topology is that you can't easily move from three to four-dimensional topology. So in dimension three, we have a lot of intuition for what should be true, because we live in a space that seems to be three-dimensional. That doesn't necessarily help with super abstract problems, but at least, simple behavior, you can probably guess what will happen just based on your lived experience. But dimension four is different and hard to conceptualize. I have a lot of papers where the format of the paper is, here's something that's true in dimension three, and here's an example of how it's not true in dimension four. And I think that's a fun thing to explore.

• 13:05Speaker 02

  There are many theorems or facts in topology that, for some reason, hold in other dimensions but not in dimension four, or perhaps we understand the answer in other dimensions but not in dimension four. The key reason tends to be, and I know this is like an answer that you hate to hear, but the key reason tends to be that 2 plus 2 equals 4, which is such an unhelpful thing to say. But the problem is that there are some theorems in topology where the proof goes through a step of, I have a curve or a loop inside of my abstract space, and I want to shrink it to a point. So I have like a really big circle, and I want to, over time, just have that radius of that circle go to zero

• 13:49Speaker 02

  without the curve breaking. And one issue is that, let's suppose that I could do that. One, one way that I would do that is I would find not just a circle, but a whole disk. So the whole, like, filled-in region. Because that would give me instructions for how to actually shrink the circle. I would just say, oh, I'll shrink that disk down to its center point, and that'll bring the circle down to the center point with it. So a disk is two-dimensional. And because 2 plus 2 is equal to 4, that tells me that it's possible that this disk actually intersects itself if we're in four-dimensional space. It would intersect itself in points. Because somehow, if I have four dimensions,

• 14:31Speaker 02

  so let's call them the x-direction, y-direction, z-direction, and now a w-direction, so I have four different perpendicular ways I can go. I have some point where it's in the disk, and the disk goes off in the x and y directions. But it's a self-intersection, and there's another piece of the disk, and it goes off in the z and w directions. And so there's no way I could push the disk off of itself and get rid of that intersection, because it's taking up the whole four dimensions. That stops me from necessarily being able to shrink curves, and it just shows up all the time in topology. It's like a key problem in the proof.

• 15:05Speaker 00

  So I've heard you say also that we shouldn't be so reliant on our intuition, because we can be incredibly surprised when we try to transfer our, really, common sense intuition from three dimensions to four dimensions. It just completely deceives us.

• 15:20Speaker 02

  Yeah, I like to tell students, there are three really important classes that students take, which are real analysis, algebraic structures, and topology. Real analysis is essentially a continuation of calculus. And most students who are taking that class probably have a lot of intuition from their experience in calculus, and probably their intuition is correct, and they're learning how to formalize it. And when you learn about algebraic structures, you probably don't have very much intuition at all, because it's very abstract. But students who take topology have intuition, because we're talking about objects that seem familiar. We can draw a lot of pictures. And the intuition is always wrong. It's very easy to get trapped.

• 15:59Speaker 00

  So what are the techniques that you use to not get trapped?

• 16:04Speaker 02

  Well, so if I am actually trying to understand objects through pictures, if I'm actually constructing an object visually, then I need to have a rigorous system in place for a picture to actually have meaning. It can't just be based on my intuition for how the object behaves. I need to formalize, if I draw a plane and a line intersecting it, what does that actually mean? And so there's some real analysis that goes into that. And so we have to all agree what these diagrams actually mean. And then we also have to develop, here's rules for what I can do to a picture of a space or a knot inside of a space, and have the picture, after I do something to it to change it,

• 16:46Speaker 02

  have that picture still describe the same space or object or knot. And so, once I have done the work, the analysis of understanding these diagrams, that gives me more flexibility to actually do constructions and know that I'm not accidentally losing information.

• 17:03Speaker 00

  So it's a combination of visual representation, but formalized mathematically?

• 17:09Speaker 02

  That's right. And not everything is done through visual representation. I like to work like that, because that's something that really draws me to mathematics. But there's lots of excellent topologists, even four-dimensional topologists, who really don't make use of visual representation in their work. It's not always important. But I do want people to understand, even when math has a lot of pictures in it, you'll hear this from students a lot. It's not proof by picture. It's not just a random picture that we make up. There's a lot of, that's right, formalization that goes into these arguments.

• 17:41Speaker 00

  With these four-dimensional spaces and your intuition faltering, what are the kinds of questions that you're asking? What are the hypotheses that you're really pursuing?

• 17:54Speaker 02

  So the thing that I study most commonly is the study of knotted surfaces. So in dimension three, the analogous object would be a knot. Knot, K-N-O-T. So a knot is a loop. If I took a piece of string, wait, I actually, I have a string. If I took a piece of string and I actually tangled it up, but then I glued the ends together to make one closed loop, then I would call that a knot.

• 18:20Speaker 00

  Right.

• 18:22Speaker 02

  Yeah, and so we, we study those objects up to continuous deformation. So I can stretch the knot around, twist it, but I can never pass it through itself. And this is an important object in three-dimensional topology. It's not immediately obvious why it's important. It takes some theory to understand why knots are important. But it turns out that every three-dimensional space can be understood in terms of knotted loops inside of just classical space. So in dimension four, I study knotted surfaces, which is everything one dimension up. Instead of a loop, it's something two-dimensional.

• 19:00Speaker 00

  Like a ribbon.

• 19:01Speaker 02

  There's more, well, less like a ribbon and more like a balloon. So I want to have an object that doesn't have any edges. But there's also more complicated surfaces. So a balloon is a sphere or a two-sphere. It's very common to say the dimension of the space. It's two-dimensional, because if I'm standing on a balloon, then I can go forward, backward, left, right on the balloon. But if I try to go up, I'll leave the balloon. Maybe instead of a balloon, I should say the surface of the Earth. Another example of a two-dimensional sphere. If I go up into space, then I leave the surface of the Earth. So it's only two-dimensional. But there are more spaces, like, for example, a torus, which is the two-dimensional space that I would get if an asteroid hit the Earth and drilled straight through. So there's a hole in it now, it's the surface of that.

• 19:46Speaker 02

  I think it'd be more normal to say it's the surface of a bagel or a donut. And then there's more spaces that I could get if I have more holes. So I could study surfaces embedded or knotted inside of four-dimensional space. So it's a little harder to imagine, but there's some version of tangling a two-sphere inside of four-dimensional space.

• 20:06Speaker 00

  I have a number of questions. One is, you said that you can understand all three-dimensional spaces in terms of these knots.

• 20:16Speaker 02

  That's right.

• 20:17Speaker 00

  That seems like a big leap. Can you help us make that connection?

• 20:21Speaker 02

  So this is a big theorem, I think, 1960s, due to Licorice and Wallace. Licorice-Wallace theorem. So to be more specific, I'm studying three-dimensional manifolds. And I want to say that these three manifolds don't have boundary, and they're compact, meaning I don't ever go out to infinity. So I can, maybe I'll say, finite volume. And I also want them to be orientable. So think about like a Mobius band as an example of something that's not orientable, because it has only one side, is what people often say. So I put some restrictions. I want to only study reasonable three-dimensional spaces. But once I do that, Licorice and Wallace proved that I could get to any three-dimensional

• 21:04Speaker 02

  space I want by starting out in the standard one, which is called the three-sphere. It's like R3, but if I try to go out to infinity, then eventually I come back to the point that I started, so I can't go out for forever. And all I have to do is take some knots, so knotted loops, inside of the three-sphere, and I'm going to do surgery on them. And this is confusing. It's a really important operation in topology. I think it's surprising the first time you hear it that it could do anything at all. But when I say surgery, I mean, I'm gonna delete those knots. So I'm deleting these knotted loops. I wanna think of them as being thick. It's like I'm cutting out a bunch of bagels from space.

• 21:45Speaker 02

  And then I'm going to just re-glue, like, or replace exactly the thing that I deleted. So I cut out a bunch of these, like, thickened circles, these solid bagels, solid tori, and just reattach them. But I'm going to reattach them in a different way that I found them, and I'll change the space because of that.

• 22:03Speaker 00

  I see. So if I'm understanding correctly, the knots themselves, you're saying, I can start with the simply connected finite space, which is the three-sphere. And I can construct any other topology in three dimensions by performing these kinds of surgeries on knots? Is that what you're saying?

• 22:23Speaker 02

  That's right. And so this is exactly right, and this is called Dain surgery.

• 22:25Speaker 00

  And so that way, you make more and more complicated spaces, far beyond the simplicity of this smooth, compact, connected, three-dimensional sphere. You make these very complicated spaces with handles and holes and kind of an origami. So now, take me up to four. Now we're trying to imagine the role of a knot in four dimensions, but the knots are no longer strings. They themselves are surfaces, complex surfaces. And you're trying to do the same thing. You're trying to start with the simplest space in four dimensions and find all the others on the basis of the knots.

• 23:02Speaker 02

  Yeah, and so it's almost true now that I can start out with the simplest space, which is called the four-sphere, which is harder to imagine because it's four-dimensional. But it's some standard four-dimensional space that we all agree to start with. And the surfaces that I'm going to take are tori, which are these surfaces of bagels or donuts, sitting inside of four-dimensional space. And I'm doing surgery, which means that I'm deleting, one at a time, I'm gonna delete a torus, re-glue it back in, but I'm allowed to, like, re-glue it in a funny way, and I'll change the four-dimensional space. And so I said it's almost true that I can get to anything.

• 23:41Speaker 02

  There are two really basic numerical invariants that I have to start out with the right values. But so at least there is a very simple family of four-dimensional spaces that I start out with. Depending on what space I wanna get, I know which one to start with, and all I have to do is do surgery on tori, and I can get to whatever space I wanted.

• 24:02Speaker 00

  I see. And has that been a successful program?

• 24:04Speaker 02

  Well, so that's a fact about dimension four. This is a cool theorem, I think it's of Iwase several years ago. It's not exactly a program. It's more motivation for why it's worthwhile to study surfaces. Any question about four-dimensional spaces, in principle, I could rephrase as a question about surfaces. And that's just one example. There's other aspects of four-dimensional spaces that we tend to study using surfaces. So, for example, let's move back down to a lower-dimensional space. Let's actually look at surfaces again. I gave two examples of surfaces before, which were the two-sphere and the torus.

• 24:46Speaker 02

  It's actually pretty hard, the first time you learn about these objects, to really verbalize what is the difference between them. When I give a talk that's a little bit more general audience, I'll say something like, "Oh, well, the torus has a hole in it, and the sphere doesn't have a hole." But it's pretty hard, on average, if I asked a student, "Okay, now tell me what a hole is." Like, what does it mean that the torus has a hole? That's a tricky question. And so the way that I like to think about it is that, well, if I draw a loop on a two-sphere, so think like the equator of the Earth, it will always divide the two-sphere into two pieces. The equator divides the Earth into the northern hemisphere and the southern hemisphere, and

• 25:30Speaker 02

  you can't go between them without crossing the equator. The torus doesn't have that property, because I could take a loop on the torus that goes around the hole. And if I cut, it actually doesn't cut the torus into two pieces. But I can only fit one loop like that. If I try to cut along two loops, then I'm gonna end up cutting the torus into two pieces. So that tells me that the torus has one hole, and I can formalize it in terms of these subspaces, these circles that are smaller in dimension that sit inside of the surface. So I can do that in higher dimensions, too. If I'm trying to distinguish two four-manifolds from each other, sometimes I can rephrase

• 26:12Speaker 02

  that as a question about, well, what kind of surfaces sit inside of these four-dimensional spaces?

• 26:18Speaker 01

  So when I listen to that with my topologist hat on, I'm thinking that there's a gambit that Maggie is using here that we're trained to use in topology all the time, to use, as she puts it, these smaller-dimensional objects to probe higher-dimensional objects. And since it's not something we all do every day, I thought, rather than talking about the four-dimensional usage, what if we just talked about something like the kitchen table? You know, so like, if I put a piece of string on the table and just make it into the shape of a circle, there's a very deep topological thing that has happened as soon as I do that, which is that the loop of string separates the table into two parts, one of which we

• 27:02Speaker 01

  call the inside, and one of which we call the outside. And that's kind of interesting. I mean, you might say, "No, it's not. That's so totally obvious." But this is the kind of thing a mathematician ponders. Like, okay, let me be more abstract. Rather than a tabletop, how about the infinite plane? You know, the x-y plane, going out to infinity in both directions? If you put a circle in the x-y plane, it separates the plane into an inside part that's bounded, and an outside part that includes infinity. And that's true, even if I deform the loop, there's always one inside and one outside.

• 27:38Speaker 00

  Yeah. Well, look, we have skin, two-dimensional surface, and that separates, in three dimensions, our inside from our outside. But it wouldn't if we lived in 4D. Our skin would not be sufficient to completely enclose our internal organs.

• 27:54Speaker 01

  Very convenient, then, that we don't live in 4D, because we'd be spilling.

• 27:58Speaker 00

  Right. We'd just, you'd be able to just, like, go in and pluck out, you know, without surgery.

• 28:03Speaker 01

  Surgery would be a lot easier.

• 28:05Speaker 00

  Yeah. Well, she also talks about something that's similar to your classroom experience, which is, in some sense, the connectedness of spaces can be categorized by thinking about how you can divide things, how many cuts you have to make, essentially, to really cut the thing in half.

• 28:20Speaker 01

  Right. The other thing that hit me in her discussion was the difficulty of saying what you mean by a hole. So, you know, where this really became a live discussion on the internet was, if I look at a pair of pants, so just ordinary pants, how many holes are in there?

• 28:35Speaker 00

  Okay, so there's one, two.

• 28:39Speaker 01

  But some people wanna say three, because, you know, like, there's a loop at the bottom of your leg, and then on your other leg, and then there's your belt. It seems like there's three.

• 28:50Speaker 00

  Well, okay, wait. So in contractable loops, I have one, two, I guess, three.

• 28:59Speaker 01

  I'm gonna let the audience ponder this one. Where are you taking this next?

• 29:04Speaker 00

  Well, there is something, Steve, that I wanted your help with. There's a topic that comes up after the break, and that is ciphered surfaces. I think you've thought about this.

• 29:13Speaker 01

  I have. They've actually come up in some of my applied math work. And probably an easy way to picture them is to imagine a game you might have played as a kid, where you take a loop of wire to make bubbles outside, and then if you dunk that in the soapy water, you get a kind of soap film that spans that loop. Well, that disc is an example of a ciphered surface, because it's a surface whose boundary is a given curve, in this case, a loop. But you could also have ciphered surfaces for knotted curves, where the boundary of that surface is the knot. It's a really useful construction in knot theory.

• 29:49Speaker 00

  That's actually a helpful visualization, and we're gonna get into that and more just after the break.

• 30:07Speaker 00

  Welcome back to the Joy of Why. We're joined today by Maggie Miller, a topologist at the University of Texas at Austin. Let's talk about this work you did to answer a longstanding question that was posed by Charles Livingston in 1982 with you and your collaborators that received a lot of attention. And that had to do with the Seifert surfaces. What's special about those surfaces?

• 30:31Speaker 02

  Yeah, that's right. And I'll just say that was with Kyle Hayden, Sungwon Kim, Jong-han Park, and Isaac Sundberg. A Seifert surface is a surface where it's not just an abstract surface. It's a surface that actually lives inside of three-dimensional space. So I really want to make this distinction. Even though I know donut lives inside of our three-dimensional space, I'm treating it as its own abstract object, like outside of space. It's in some weird place. But a Seifert surface is really here. It's important that they live inside of three-dimensional space because when I study the surface, I'm allowed to continuously deform it, but it has to stay inside of three-dimensional space.

• 31:14Speaker 02

  So I could have two different tori that are both tori, but I can't actually turn one into the other inside of three-dimensional space. A simpler thing is just even knots, knotted loops. If I take a loop that's really a circle, just a nice round circle, we call that the unknot. If I take a loop and actually tie it into some complicated thing, it won't be the unknot. And they're still both circles, but they're not the same knot. I can't actually turn one into the other without passing it through itself. Seifert surfaces are the same thing. But for Seifert surfaces, I do want them to have boundary. They should have an edge, and the edge will be a knot, a knotted loop.

• 31:52Speaker 00

  So the idea is you're imagining, at least with some of the knots, that if you increase the number of dimensions in which they live, that you might be able to unknot them, one into the other, without cutting, or breaking, or gluing?

• 32:09Speaker 02

  That's right. I mean, that happens very often. So in the case of knots, like a knotted loop, it's only knotted because we're in three-dimensional space. If I had a fourth dimension of freedom, if I moved into a four-dimensional space, then my knot would actually become trivial. It's so small in dimension compared to the ambient space that it can't be complicated anymore.

• 32:32Speaker 00

  So it would untie effectively? It would unknot effectively?

• 32:35Speaker 02

  That's right. It would just untie. So in four dimensions, in the standard four-dimensional space, there is exactly one knotted loop. They're all the same.

• 32:44Speaker 00

  Ah, I see. But that's not the case if they're not... Wait, you're saying if there are knots in three dimensions and you up the dimension to four, then they're all the same?

• 32:55Speaker 02

  Then they're all the same.

• 32:56Speaker 00

  But if you have a knot that lives in four dimensions, so it's a surface knot, and you go up a dimension, are they also then all the same?

• 33:07Speaker 02

  That's not so clear. But if I go up two dimensions to dimension six, then they're all the same. If I increase the dimension enough, eventually they'll all be the same.

• 33:17Speaker 00

  So when I learned a little bit about this, you can take kind of a two-dimensional example, which might be easier for people to visualize. If you imagine one small rubber band inside a bigger rubber band, just on a table. You can't get them past each other, right? But if you have three dimensions, it's real easy. You lift one up, you stretch it out, and then you can lay it outside the other. So this idea of the number of dimensions allowing you to unknot things or to separate things without cutting and pasting is kind of intuitive. We use it all the time.

• 33:54Speaker 02

  That's right. That's exactly the right idea. That's the same reason as why knots in dimension three become trivial or all the same in dimension four. Like somehow I just need that one extra dimension to pull the rubber band.

• 34:06Speaker 00

  So let's go back to the Seifert surfaces. What was the question that you addressed about their behavior in four dimensions?

• 34:15Speaker 02

  Yeah, so we were really interested in this paper by Charles Livingston, Chuck Livingston, where he was studying Seifert surfaces for the unknot and the unlink, where the unlink is a bunch of unknots that are all unlinked from each other. So just a bunch of round circles that have nothing to do with each other. Yeah. So even though the unlink is somehow not very interesting, I could still find complicated Seifert surfaces for the unlink that are different from each other. So the surface boundary, trivial, not interesting, but the surface itself is complicated. And Livingston proved that if I have two, let's say, connected Seifert surfaces for

• 34:56Speaker 02

  the same unlink, and I know that those two surfaces have the same genus, which is our measure of how many holes the surface has, even though those surfaces are probably not the same inside of three-dimensional space, if I add a fourth dimension of freedom, then they become the same. I have this extra direction that I can pull them in. In the paper, he was very specific about the boundary being the unlink. And he pointed out that it's surely not true if the boundary is not the unlink, but he didn't have a proof of that, primarily because it's really hard to come up with weird examples of higher-dimensional objects. You have to come up with some crazy pair of surfaces with the same boundary in the

• 35:40Speaker 02

  first place before you can even then try to prove that they aren't the same as each other. That's always the problem in topology.

• 35:46Speaker 00

  Yeah. I think we could do a little more justice to the idea of these are the same. That is really one of the core principles of topology, is what do you mean by the same? Can you clarify that?

• 35:59Speaker 02

  Yeah, it's a core principle. What does it mean for two objects to be the same? It's also a core principle that there are lots of different notions of the sameness. So this is always, whenever you define an object, you have to say, what is it going to mean for two objects to be the same? So far, we've talked about manifolds, which included surfaces, and knots. So for manifolds, we typically use the definition homeomorphic or homomorphism. So we say that two manifolds are homeomorphic if there is a homomorphism from one to the other. And what that means, if I have my two manifolds, call them X and Y, they're homeomorphic if I can find a function F that goes from X to Y, and the function needs to be continuous,

• 36:42Speaker 02

  and it also needs to be a bijection, which means that it's one-to-one and on-two. So any two points in X have to go to two different points in Y. They can't go to the same point. And I also have to hit every single point in Y. Something from X has to map to it. And then the last thing is that I want F to have a continuous inverse as well. So it's just a set of continuous instructions to go from X to Y, but these three bullet points make it a rigorous definition. So in the context of knots, I can use the same definition, actually. I could say that two knots are equivalent if there's a homomorphism, so this kind of restricted function, from three space to itself.

• 37:26Speaker 02

  But when I plug in the first knot, it outputs the second knot.

• 37:30Speaker 00

  If I were to give this formalized mathematical description to a child with some Play-Doh, I would say, the game is, can you make this shape into that shape without breaking it, without puncturing, or re-gluing? As long as you're just smoothly mushing dough, then topologically, we're going to consider those homeomorphic.

• 37:56Speaker 02

  Right, yeah. So it's okay to stretch things. It's okay to twist things. If I have two knots, if I have one knot and I'm trying to turn it into another knot, I can make the string longer, and I can add twists, but I can never pass the string through itself, which would break it.

• 38:10Speaker 00

  No, I think the question of games is not totally irrelevant because for people outside of mathematics, when they're listening to the problems that mathematicians are addressing, there's an attempt to assess, is this just a game that's being played? And I think the sort of philosophical way to phrase the question is, do you consider mathematics to be something that is discovered? That it exists in some sense out there? Or is this, or is mathematics sort of a series of different games that are invented by human beings?

• 38:44Speaker 02

  Oh, that's interesting. I think, hmm, I think the fundamental, or the most important results are things that are discovered, but the way that we get to them is often by winning a game. I think that's exactly right. If I were trying to solve a problem, let's say I have a conjecture that I think is true, and I need to prove that it's true for all knots, then I might phrase it as a game. Well, if I give you a picture of a knot, and I tell you, okay, you can start off with this picture, but you're allowed to change the picture in the following ways, and I tell you exactly what's allowed. Is it possible to do only those things and eventually get to whatever it is that I've conjectured is true?

• 39:28Speaker 02

  Like, the part that it's the discovery, I think, is coming up with the game in the first place. It's really hard to make a conjecture because we don't always have good intuition. And then there's lots of papers that are just, well, here's a big problem that we're all interested in, and my theorem is that I can't actually answer it, but I can prove that if you can always win this game, then the theorem is true.

• 39:52Speaker 00

  For people who are outside of mathematics, who are trying to understand what's at stake...

• 39:59Speaker 00

  Why are these different games important and compelling? And I know that's an existential question for mathematics.

• 40:06Speaker 02

  The games that we were mentioning before are always going into, well, if I could win this game, then I proved this conjecture. So at the heart of it, it's really why is pure mathematics important? The one issue is that it's not always clear to anybody working in the field why it's important for the real world. People like to joke that there's like a 50-year time gap between pure math being invented or discovered, or whichever verb you want to use, versus actually being applied in real life. So for example, in topology, there's a relatively recent field called topological data analysis. You can tell from the name immediately is much more applicable to real life.

• 40:51Speaker 02

  It's the idea of how can we understand very large data sets efficiently, or understand some patterns in them using topology? So maybe I have some data set which is way too massive for me to do a search through, but I can understand global patterns. Like, it's meaningful that somehow this data set I can approximate is this space. And the fact that it's this space tells me something about the data. But the actual topology, the pure math part that goes into topological data analysis, is, at this point, relatively old in terms of just literally the pure math component of it. Maybe we're using things like classification of surfaces, or some classical invariance in topology.

• 41:35Speaker 02

  And so it's still somehow catching up. It's important that we have those things, but the stuff that we're doing in topology this week isn't being implemented in applied math yet.

• 41:45Speaker 00

  Yeah, and I understand that there's a sense in which it's an unfair question. It's more sort of giving mathematicians an opportunity to address it, because all the time physicists face similar questions. What is the significance of your work to my daily life? And the real answer is, that's not what we care about. And that's okay.

• 42:05Speaker 02

  Yeah, the real answer is, that's not the goal. In fact, I think most topologists, if you ask, they'd probably try to link it back to physics, and then hope you don't ask why the physics part matters.

• 42:15Speaker 00

  Right. Well, you know what? Let's play that game, because that's interesting and fun. There is a really valuable sense in which topology has huge application to theoretical physics. There's lots of complex ways in which topology applies, but there's one that's kind of intuitive, which is that the universe is probably a three-dimensional manifold. It might be a four-dimensional manifold if we include time in the real sense, right? Not just another spatial direction, but time in which measures are different on that manifold for time than they are for space. Do you think about those implications for physics?

• 42:54Speaker 02

  Well, so actually, what's interesting is that I think most topologists would go the opposite direction and think, well, I never think about what physicists need, sorry, but I'm very interested in what physicists bring to topology. So when I was in grad school, a professor I was taking a course with made this joke, which I still think about all the time, he was totally right. Whenever topologists want a new way of distinguishing two spaces from each other, what they do is they walk across the hall to the physics department, which is surely in the same building, and ask them, oh, can you please give me a new partial differential equation that tells me something about spaces? And then you bring it back, and you totally forget that it had anything to do with physics, and that's just your new way.

• 43:37Speaker 02

  Yeah, so we like to say, oh, well, physicists told me that there's this partial differential equation that describes heat flow, and I can count how many stable solutions there are, or whatever. And I'm just going to remember that number from now on. I'm not going to think about heat flow, but it's different for these two spaces, so they must be different.

• 43:55Speaker 00

  I love that. I love that it goes in both directions. I mean, of course, physics also has to grapple with the question of why three dimensions? And so a lot of physicists are interested in higher-dimensional spaces, because we think the universe may well be higher-dimensional. And so when we're trying to understand what are the possible universes, what are the geometries of the possible universe, we have to look to the mathematicians and say, well, if we're in four dimensions, these are unique universes in some sense, and these, although they might seem different, are actually the same. And this is, of course, very relevant for what we're trying to do.

• 44:34Speaker 02

  Yeah, that totally makes sense. And of course, there's lots of mathematicians who study higher-dimensional spaces. A lot of people would say, oh, I'm a three-manifold topologist, I'm a four-manifold topologist. It's not so common to have a specific higher number. Like, oh, nobody would say they study eight-dimensional topology, I think.

• 44:51Speaker 00

  No. Well, come, because it's too generalizable after eight? It's just N?

• 44:55Speaker 02

  They tend to be techniques that are special to three, special to four, and then higher. But there are lots of people who work in higher dimension, especially algebraic topology tends to be higher arbitrary dimension.

• 45:07Speaker 00

  Just out of curiosity, is there something special about three dimensions and four dimensions?

• 45:13Speaker 02

  Yeah, there is something special, which I think in dimension three, in some ways it's similar to dimension two. In dimension two surfaces, I can enumerate. I can totally list out all of the different two-dimensional objects. And there's really not very many pieces of data that you would have to give me to totally determine which two-dimensional manifold I have. In dimension three, it's a little bit more complicated to state, but there is a complete classification of three-dimensional objects. And this is much more modern. This is going from maybe 70s to 2000s work of Thurston and Perelman. We totally, well, maybe I want to be careful with the word totally, but I would say that we totally understand three manifolds.

• 45:58Speaker 02

  At least, we have a complete characterization. There's still a lot of interesting questions, but it's generally not, are these two spaces different from each other? In dimension four, that's a harder question, but there is a lot of interest, not necessarily in a classification, that's not possible, but it's special because of the failure of theorems from higher or lower dimension in dimension four. So it's a little bit different in motivation. But in higher dimension, we don't have that anymore, and we also don't have the classification. And then also, there's this kind of psychological thing of, it's hard enough to imagine a four-dimensional space, but at least with the time analogy, you can kind of get there.

• 46:40Speaker 02

  But it's a little bit hard to conceptualize, you know, as a 14-dimensional space. I don't think there's that same level of motivation.

• 46:47Speaker 00

  We all know that it's impossible, even for the most accomplished mathematician, to literally visualize higher-dimensional spaces. Is there a sense in which you feel you have a kind of a mental theater?

• 47:00Speaker 02

  Yeah, I think in dimension four, I really, literally imagine and will draw what I described before as a movie. I really like to just draw a sequence of three-dimensional spaces. And so, okay, here's three-dimensional space at time zero, and a little bit to the right of it, I'll draw another three-dimensional space, be like, this is time one. And a little bit to the left, I'll draw the same three-dimensional space, be like, this is time negative one. And in some sense, that's a literal picture of a four-dimensional space. There's some interpolation of, it's not really time straight from zero, straight to one. There has to be like, there's one half in the middle. You can't draw all of them. You can, in principle, do that in higher dimensions, too.

• 47:42Speaker 02

  I have a paper, or maybe a few papers, where I draw five-dimensional spaces as, well, I draw four dimensions, but drawing a row of three-dimensional pictures. So if I stack rows and have a grid, now it's five-dimensional.

• 47:54Speaker 00

  You could go up one more.

• 47:56Speaker 02

  I have this like, sort of secret goal of, I would really like to draw like a six or maybe seven-dimensional picture, where that one of the dimensions is you turn the page.

• 48:06Speaker 00

  Right. Exactly. Or you have to build it in a room.

• 48:11Speaker 02

  Oh, that'd be interesting, too.

• 48:13Speaker 00

  I wanted to ask you a question that we like to ask here at The Joy of Why, which is, what brings you the most joy in your work?

• 48:21Speaker 02

  I mean, I think that I have probably the same answer that a lot of mathematicians, where we tend to work on problems that are really hard and abstract, and maybe you think about something for several years before you understand the answer. Of course, we're working on more than one thing at a time. I know very few people who are stuck on something for several years, and just that's the only thing they're doing. But every once in a while, you actually solve a problem. It feels really good. I mean, I think everybody can relate to that feeling of you're working on a hard puzzle of some kind, even like a crossword, and just this aha moment where you finally get an answer. It's like a big relief.

• 49:04Speaker 00

  Yeah, I think it's interesting to be in a subject where you're always working at the very edge of your own abilities and maybe what's even possible in the field. And it can be very frustrating.

• 49:17Speaker 02

  Yeah, it's so frustrating. All the time, I'm just like, why can't I have a hint button, you know?

• 49:22Speaker 00

  Right. Well, Maggie, thank you so much. It's such an incredible field, and I love this connection with the visual. Thank you so much for taking time with us.

• 49:30Speaker 02

  Yeah, thank you for having me. It was interesting to talk about math.

• 49:35Speaker 01

  It's very charming to listen to the two of you talking about what is really a data visualization problem. And I'm reminded of the great book by Edward Tufti, Visual Display of Quantitative Information.

• 49:48Speaker 00

  Gripping title.

• 49:49Speaker 01

  Well, okay. Do you know the book?

• 49:51Speaker 00

  I don't.

• 49:52Speaker 01

  Oh, come on. You're teasing me. Okay, fine. The title may not be the selling point for the book. But he has one diagram of Napoleon's march into Russia. So you see the thickness of a line that represent how many soldiers Napoleon has at a given time. And as this line starts moving across the page, this black band starts getting thinner and thinner as Napoleon's army is getting depleted. Meanwhile, there's other data showing the temperature, how cold it was, and what the date was when they attacked this or that place. So you're actually getting not only geographical 2D information, you're getting the size of the army, that's a third dimension. You're getting a fourth dimension for the date and so on.

• 50:36Speaker 01

  So it's really clever that people have been thinking about how to visualize multiple dimensions, even on a two dimensional picture, for a long time.

• 50:44Speaker 00

  Yeah, visualizations in mathematics is fascinating. I personally really appreciate visualizations. And yet I understand it's a limitation. But what she described is actually a really fun way to imagine a higher dimensional space is to unpack it in some sense. Just draw it here, then draw it there. You know, this idea of a sphere passing through, let's say, flatland, a world in which all creatures only live in two dimensions. And what do you see? You can understand the three dimensional sphere as a series of concentric circles that widen as the sphere enters and then shrink as the sphere leaves. And it really helps you say, well, I could probably do that with a four sphere passing through three dimensions. And indeed, you can.

• 51:29Speaker 00

  It really sort of helps connect with the physicality of these spaces.

• 51:34Speaker 01

  Wow.

• 51:34Speaker 00

  Well, we have to unfortunately leave this abstract realm behind. Back to reality. But Steve, thank you so much for being a partner in crime.

• 51:43Speaker 01

  Very good. I will see you next time.

• 51:47Speaker 01

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• 52:04Speaker 00

  The Joy of Why is a podcast from Quantum Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation 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 Falls, Jade Abdul Malik, Genevieve Sponsler, and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzalez. Edwin Ochoa is our project manager. From Quantum Magazine, Simon France and Samir Patel provided editorial guidance

• 52:46Speaker 00

  with support from Samuel Velasco, Kit Sudal, Simone Barr, and Michael Conyongolo. Samir Patel is Quantum's editor in chief. The episode art is by Chanel Nibblink, and our logo is by Jackie King and Christina Armitage. Special thanks to Garth Avery at the Cornell Broadcast Studio. I'm your host, Janna Levin. If you have any questions or comments, please email us at [email protected]. Thanks for listening.