In Donald E. Knuth’s 1974 novella Surreal Numbers, an Adam and Eve–ish couple come upon a tablet that reads, in Hebrew, “In the beginning, everything was void, and J. H. W. H. Conway began to create numbers.” The Conway in question is late mathematician John Conway (1937–2020)—whose discovery of surreal numbers in 1970 rocked the world of mathematics—and the godlike status conferred upon him is hardly hyperbolic. Over the course of his career, he did groundbreaking work in group theory, knot theory, coding theory, and the theory of finite groups. He was particularly revered among amateur mathematicians for his invention of the so-called Game of Life (which inspired the field of cellular automata), as well as the Doomsday Algorithm, a formula that calculates the day of the week for any date in history.
        When Esopus invited Conway to contribute something related to his working process, he came up with an extremely generous offer: An inveterate model-builder, he proposed creating a template for a three-dimensional geometric polyhedron that our readers could then assemble on their own, effectively allowing them direct insight into his working methods. In the Fall of 2006, we paid a visit to Conway at Princeton University’s math department, where we photographed him building a prototype of the model while he mused on aesthetics, symmetry, and the therapeutic benefits of drawing knots.

Tod Lippy: Is it common for mathematicians to build models?

John Conway: Actually, I’m not so sure it is, at least with professionals. But certainly it is when they’re young—I first started building them when I was in high school. I’ve just never outgrown the habit. You know, I don’t really like the process of doing it. I do like having the final things. It’s like when people say, “Do you enjoy traveling?” Well, who enjoys sitting on a plane for hours? But you like getting there. That’s the thing.

How many of these have you made?

Well, there might be a hundred or so in my office, and you know, double or treble it, maybe. These things, you see, are some of my friends. Like the guys on the wall, the famous mathematicians. Most of the models you see there were made between attacks of laziness.

Can you tell me a little bit about this particular polyhedron you’ve chosen for our readers to build?

This is the small stellated dodecahedron, which was named in the early 1600s by Johannes Kepler, the great astronomer. Kepler spent a good part of his life trying to understand the planets and how they are arranged. When he was trying to work out the ratios and distances between the planets, he thought they might have something to do with irregular polyhedra, and so he started studying these irregular solids—Platonic solids, they’re called.
   Along the way he discovered that if you prolong the edges of an ordinary dodecahedron, they meet again, and you get this thing with star-shaped faces. The faces of this particular one are called pentagrams. So he came up with this beautiful object, a star-shaped polyhedron. He also found another one, the great stellated polyhedron, and then a few hundred years went by. In the early 1800s, Louis Poinsot found the other two: the great icosahedron, which has an ordinary pentagon for its intersecting faces, and the great dodecahedron, which has triangles. These make up what are called the Kepler-Poinsot solids.

There are just four of them?

Yes. Augustin-Louis Cauchy, a really great mathematician of the 1800s, proved in the end that there are only four of these star polyhedra.

Were models of them made when they were discovered?

Actually, there were some models of polyhedra that were believed to have been made by Leonardo. You can see pictures of those in various places. The edges were made of wood, sort of hinged together in various ways.

How does model building relate to your theoretical work?

It’s sort of funny, really. As I said, this is a rather childish pursuit for a grown-up mathematician. I tell people that I make them for teaching purposes. That’s not really true. I make them because I think they are sort of inspiring. You know, the subject that I studied for a long time and made some progress in is called group theory, which is the study of symmetry. And the progress I made was all in high dimensions. I studied the symmetry of this wonderful object in 24-dimensional space, which was actually the first big discovery I made. Things like this stellated dodecahedron, they’re in three-dimensional space, because, unfortunately, we don’t live in 24-dimensional space. But, you know, they have the same sort of ring about them. So they’re a sort of substitute. Really not so interesting, but still quite beautiful.

So even if they only barely approximate these much more complicated objects, they can still prove useful?

Well, I should say that what I do is neither simple nor complicated; it’s sort of big—or maybe subtle is a better word. But yes, often they’re nice, easy examples. And people aren’t very familiar with these things nowadays. I really can’t rely anymore on a student knowing what I’m talking about when I mention a dodecahedron. So it’s nice to have a model around. But the theory of these three-dimensional things was really all finished off by the 19th century. So it’s old. But you know, what’s the difference if it’s old? It’s still beautiful. One could argue that the same thing is true with modern art. You sort of arty types frown upon the older, straight representational stuff. It’s no longer so much in vogue. So there’s a parallel in a way.

Well, that raises a question: Would you call these models abstract or representational?

No, these things are real things. You know, how can I say it? Some people like the human—or the earthly, let’s put it like that. And I must say I do, in a way. But ever since I was a teenager, I’ve been more interested in what I would call the outer problems. Things that concern our species, or our planet, I regard as parochial. Somebody might write a history of their own little town, but then who’s going to be interested in it? Well, I don’t know, but probably most of the people interested in it will be the town’s inhabitants. You know, Fifty Interesting Walks in New Jersey sells to people who live in New Jersey. Fine. I’m also a member of the human species, and live on earth, so I have a slight amount of interest in geography and the humanities. But I’m more interested in things that have a certain universality. I’m very intrigued by what would interest some creature I haven’t met yet. Like a Martian, or some other intelligent being from a long way away, who won’t be especially interested in human concerns. The fact that someone is human doesn’t especially turn me on.

Do you think most mathematicians share that perspective?

No, not necessarily. And I don’t totally have it myself. For instance, I’m very interested in etymology, and particularly the etymology of the English language. That’s not a mathematical pursuit. But professionally, so to speak, the things that interest me are the ones that I can imagine interesting somebody from some other place.
   This model is an instance where rationality intersects with aesthetics. I mean, there is a beauty about this. It’s not everybody’s cup of tea, I’d imagine, but it’s undeniably aesthetic. I should say that mathematics is an aesthetic subject. A phrase you will constantly hear my colleagues applying to something is “It’s elegant.” That’s what we do. We study beautiful things. And there’s a subtlety to the beauty. And there are rules to the beauty, specifically. You can’t just sort of invent something and say, “I find this interesting.” Well, you can, but then you’re not doing mathematics. It’s a wonderful fact that there are only four of these Kepler-Poinsot solids. I can’t invent another star-regular polyhedron.

Can the process of building a model like this help people who know little about math to better understand it?

Well, you know, a former Princeton colleague, Bill Thurston, used to say that geometry is the user interface of mathematics. It’s “user-friendly.” The fact is that much of mathematics is really hard to learn or understand—and by the way, there’s no reason why everyone should do it. But people might like to know why we strange people are so interested in it. And the answer is, often, because of its beauty. Additionally, with geometry—unlike, say, algebra—the layperson can get some clue about it, because geometry is about shapes. You look at the shape and you don’t have to do any calculations to create the shape. You can just appreciate it. “Oh, that’s nice.” So you know, that gives you a little bit of a clue about mathematics without any pain.
   I think geometry is pain-free to its practitioners. Well, there may be some pain when you’re doing calculations. I did a little calculation when I was drawing this thing, to work out exactly how to draw the pentagon. So there was a little bit of pain there.

Do you find the assembly part to be therapeutic?

It can be. I remember when my first marriage was sort of breaking down, and it was a real mess, you know. At the time, I was planning on publishing a dictionary of knots, which would include about 1,000 different types, and every now and then I’d take out my drawing instruments, which included my special pen and india ink, and French curves to trace around with, and sort of draw knots for a half hour or hour at a time. There’s a certain sensuality to knots that polyhedra lack. This kind of geometry has straight lines. Knots have sinuous curves. So there was a certain pleasure and comfort in drawing those curves, arranging them.

When one finds templates for these various polyhedra in books or on the Internet, they tend to be one large form that you then fold together and glue down. Your assembly method is somewhat different—why?

You see, that’s what they call a “net” for polyhedra. It doesn’t actually save time. Well, it saves time if you don’t actually have to draw the thing in the first place. But the problem is, if you make a net, some of the edges are folded, and some of them have tabs and glue. So there are two different kinds of edges on your model. What that means is that it won’t be symmetrical. You will see round edges in certain places, and sharp edges in others. Basically, it doesn’t look nice.
   As far as I know, I’m the only person who’s ever assembled them in this interlocking way. It’s a bit tricky to do, but it has this great advantage in that the faces stay plane. Also, sticking the faces through each other—this intersecting jazz—is fun in a way because, at least to a certain perverse type of mind, it’s all about solving a puzzle. And it’s not at all obvious you can do it. There’s a sense of achievement in forcing the things through.

How much of your time is spent on process as opposed to, say, the completion of a theorem?

Well, the way I work is a bit peculiar, really. I sort of do the same thing over and over again. I’m not a systematic person; I don’t keep notes. And in fact, the only way I ever keep notes of anything is when I’ve published a paper. That’s my note. But you know, I’ll work out a piece of mathematics one year and then forget about it. And next year I sort of vaguely remember how I did it, but I don’t remember the answer, so I do it again. And then I forget about it again, and five years later I get back to it and do it again, and every now and then, I discover a sort of slight improvement. So by the 10th time I’ve done it, it’s actually getting pretty good. It’s just a perennial polishing. That’s not actually answering the question as you posed it, but that’s what I spend a fantastic amount of my time doing. Only every now and then do I actually sort of really discover anything that’s entirely new. Those are the great times. During those, I do not sleep. I’m awake for sixty hours in a row. When I discovered the surreal numbers, I was in a permanent daydream for six weeks, thinking how beautiful this thing was, and thinking, “How clever you are, John, to have discovered it.” I was lost in admiration of the discovery—the abstract things—and lost in admiration of my own accomplishment.
   I haven’t met anyone else who works quite in this way. It’s sort of like contemplating your navel, in a sense, except it’s not your navel, it’s whatever it is. I spend such an enormous amount of time just looking at this one thing from every conceivable point of view. I’ll count its vertices a thousand times. And you might go away and come back the next day and I’ll still be there, looking at it from a different angle.