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 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. Listen on Apple Podcasts, Spotify, TuneIn or your favorite podcasting app, or you can stream it from Quanta. JANNA LEVIN: And we’re off.

STROGATZ: A podcast from Quanta Magazine where we explore some of the biggest unanswered questions in math and science today. LEVIN: Steve, we have a really mathematical topic today. I’m glad to have a mathematician in cahoots.

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 wanna help us a little bit just with the basics of topology? What makes topology different than regular geometry that we might have learned?

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 Möbius strip or a Möbius band?