A Number Theorist Who Connects Math to Other Creative Pursuits

“There are many different pathways into mathematics,” said Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison. “There is the stereotype that interest in math displays itself early. That is definitely not true in general. It’s not the universal story — but it is my story.”

That account was backed up by a biostatistician at the University of Pennsylvania — his mother, Susan Ellenberg. “Jordan recognized numbers before he could walk,” she said. “We’d be going someplace with him, and he’d start to call out numbers, and his father and I would have to figure out where he was seeing them. Each night, he’d ask me to teach him something new about math.” When he was in second grade, a local teacher began taking him through the high school math curriculum. Ever since, he’s been preoccupied with mathematics — though not exclusively so.

After graduating from Harvard University in 1993, Ellenberg completed a one-year master’s program in fiction writing at Johns Hopkins University, where he wrote a novel that was published a decade later, titled The Grasshopper King. But he always felt that he would eventually return to mathematics, and in 1994 he entered a doctoral program back at Harvard, pursuing research under the supervision of Barry Mazur, a number theorist.

“Barry was a great adviser and a very learned guy,” Ellenberg said. “One of the things he showed me is that it’s OK to be interested in things other than math. Through him I saw that being in a university isn’t just about being in the math department, but rather being part of a whole world of scholarship.”

Ellenberg has taken that view to heart, finding mathematics to explore in everything from internet fads to voting rights. He has interacted and even collaborated with colleagues from many different fields and departments, while keeping up his writing — academic papers for math journals, and popular articles for newspapers and magazines. In 2001, he started writing a column for Slate called “Do the Math.” Many entries are not typical mathematician fare, such as “Algebra for Adulterers,” “Cooking the Books on Virginity,” and “What Broadway Musicals Tell Us About Creativity.”

His latest book, Shape, is all about geometry — though, as you might expect, it departs significantly from the traditional geometry of your high school days. Proving the congruence of triangles and the like, he said, bears little resemblance to the work of modern geometry. In the book’s introduction, Ellenberg confesses that it was a curious subject for him to have taken up: “Reader, let me be straight with you about geometry: at first I didn’t care for it.”

Quanta spoke with Ellenberg earlier this month about geometry, electoral math and creativity. The interview has been condensed and edited for clarity.

When did you first realize there was something special about math?

When I was 6 years old, I was in the living room, gazing at the rectangular pattern of holes on a speaker where the sound comes through. I noticed there were 6 rows of 8 holes and, equivalently, 8 columns of 6 holes. I knew that 6 × 8 equals 8 × 6. But at that moment, I grasped that this was a fact about the world, not just a fact from the multiplication tables. Mathematical knowledge, I realized, was something that existed on its own — something you could directly apprehend — and not just something you were taught.

That, for me, offered an early glimmer of the power of mathematical thinking — and the emotional force that comes with it. As teachers, we aspire for every kid to have that kind of experience of mathematical knowledge.

Mathematics is a diverse field. How did you decide to focus on number theory?

I went to graduate school not really knowing what I would work on. It was just after Andrew Wiles proved Fermat’s Last Theorem. There was so much energy and enthusiasm about number theory at that time. It seemed to be the most exciting thing going on.

Students often ask me: “How do I figure out what area of math is right for me?” I tell them that it’s all interesting. Every field of research has deep wonderful ideas. You just have to see what you fall into. And wherever you fall, there is excitement to be found.

Of all the possible subjects in math, why did you write a book on geometry, especially when you admit to having a mental block when it comes to visualizing things?

It’s true, I didn’t really take to high school geometry. There was a certain style — the Euclidean “theorem, statement, proof” approach — that did not vibe with me. That approach is certainly a part of geometry, but it happens to be a tiny part.

It’s also true that I have difficulty with some geometric things. For example, when you have to put a credit card into a machine, I can’t follow the diagram and instead end up trying all four possibilities. If I’m on the first floor of my house and am asked about the layout of things in the room above me, I can’t really picture that. But it turns out that those skills aren’t so important when it comes to doing geometry.

Even though I steered clear of geometry when I was young, I later learned that you can’t maintain a dislike for any part of mathematics because all of its branches touch each other.

You also like to find mathematical connections even among ideas that don’t seem too mathematical, like pondering how many holes a straw has. Why bother answering that?

Well, it’s kind of an internet craze . It goes viral all the time, and you may wonder why people are so captivated by such a weird question. I’d say it’s actually a deep mathematical question, not a triviality nor a cut-and-dried matter. You could say one hole or two holes — or zero holes if you think about taking a rectangular piece of paper (with no holes in it) and rolling it up. It’s a way of getting people to understand topology and homology groups, which involves classifying objects based on the number of holes they have.

It turns out there is a mathematical answer to this: Topologists would say the straw has just one hole. But the point is not just to give people an answer, but rather to show them why it’s an interesting problem. Although this question is settled, many of the things that are now settled in mathematics may not have been settled 100 or so years ago. People have fought hard over almost every single conceptual advance.

In 2019, you and 10 other mathematicians signed a brief about gerrymandering that was submitted to the Supreme Court. What does math have to do with that?

The gist of our brief came down to this: How do you make maps fair? It would be useful to compare the real-life map to one drawn at random. Unfortunately, the number of possible random maps is insanely huge — so huge you simply can’t list them all. But the miracle of the Markov chain is that it allows you to sample pretty well from a set of things that is way too long to exhaustively list.

For example, if you wanted to put the numbers 1 to 100 in random order, there are 100! possibilities . You can’t write them all down and randomly pick one. Instead, you can write down the numbers 1 to 100 and pick two numbers and switch them. If you do that 1,000 times, you end up with a pretty random ordering. That’s an example of a Markov chain and a random walk. I wrote my senior thesis in college about that. It’s something I’ve been thinking about for almost my entire mathematics life.

At the center of our Supreme Court brief was this kind of analysis of the random walk. What’s sad is that the court did not end up invoking this argument at all. The justices instead took up other issues of fairness, which was kind of a bummer.

Do you ever worry that you might hurt your standing among colleagues by dwelling on topics that may seem frivolous?

I worried about that in the beginning — especially when I started blogging in 2007. I called the blog Quomodocumque and didn’t put my name up front, though I didn’t keep it secret either. What I have found is that, rather than anyone being upset that I spend time not proving theorems and writing papers, people in the community seemed to value what I was doing in pretty much the same way that teaching is valued. My colleagues seemed to be glad that I was reaching out to people in this way, perhaps because they wouldn’t have to.

What drives you to explore the mathematical underpinnings of straw holes and elections in the first place?

I always wanted to be a writer when I was young, and I came to see that I could do math as a profession and write as a hobby. I was happy to find out later that I could combine these two interests. Putting English sentences together into appealing sequences was always something I’ve wanted to do, and I was lucky to find a way to do that within my existing work.

But as for what really drives me — the motivation to see everyday life through a mathematical lens — well, I think every mathematician does that. I talk with my colleagues on Facebook every day, and there is a unique cultural way in which we tend to look at things. I won’t say it’s the right way to look at things, but it can be useful.

You’ve written that “some substantial proportion of people think they hate” geometry. Why do you think that is?

I’d say, based on many years of teaching and writing about math, that people don’t really hate mathematics or geometry, even if they have had some unpleasant experiences in the classroom. In fact, people use geometry all the time without realizing it. If you look at a map and try to figure out if two things are distant from each other or close to each other, you’re doing something fundamentally geometric right there. Geometry is the way we measure the world. That’s what the word means.

Do you think that better approaches to math education could change people’s opinions of the subject?

When it comes to improving math education, there is no single answer. Humanity is a great big variegated tapestry. People are very different in what they respond to. You could, for instance, introduce mathematical concepts through games like checkers or Nim. I think that would be great, but it wouldn’t work for everybody.

As teachers, we have to be open-minded and try lots of different things. That’s easy to say from my perspective , because I’m not a fifth grade teacher who may not have the option of trying 15 different things. Yet it’s definitely true that the spirit of play is present in real mathematical thinking, and I think that should be present in the classroom too.

What gets every mathematician’s goat is the notion that there’s a tension between the mathematical world and the creative world. That bothers us because mathematics is part of the creative world. We create things all the time. In fact, it’s hard for me to imagine being a mathematician and not believing that the spirit of creativity and play and rigor can all be present at the same time. How the hell could I be a mathematician if that were not true?