What Does the Fourth Dimension Actually Look Like?

Steve Strogatz (0:06)

And I'm Steve Strogatz.

Janna Levin (0:08)

And this is The Joy of Why.

Steve Strogatz (0:10)

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

Janna Levin (0:18)

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

Steve Strogatz (0:23)

Okay. At your service. Excellent.

Janna Levin (0:25)

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?

Steve Strogatz (0:40)

Yeah, sure. Topology, very popular subject. You know, whereas geometry is ancient, topology is only, depending where you dated, 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?

Janna Levin (1:11)

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

Steve Strogatz (1:17)

Really? Yeah.

Janna Levin (1:19)

It comes up in my work. Yeah.

Steve Strogatz (1:21)

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.

Janna Levin (1:46)

Well, math is frustrating.

Steve Strogatz (1:48)

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.

Janna Levin (1:55)

It's perfect actually.

Steve Strogatz (1:57)

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.

Janna Levin (2:15)

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.

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

Maggie Miller (2:51)

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

Janna Levin (2:53)

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 Mirzakhani 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?

Maggie Miller (3:20)

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 don't 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. 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 struggled with aspect ratios knows, it's not always so easy to figure out where things are actually supposed to go.