## Tatiana Shubin

Q: To me, mathematics and SF have one essential thing in common: they both grow from the big “What if?” question. What's your take on this? A: One thing we do in mathematics is to investigate the consequences of constraints or assumptions. You might, for instance, add a new axiom of set theory and then see if any nice theorems come out of this. Or you might make a definition, such as “an Archimedean solid has regular polygons for its faces (not necessarily all the same) and has the same arrangement of polygons meeting at each vertex,” and then carry out a search, partly empirical and partly theoretical, to characterize the objects satisfying your definition. Science-fiction can be carried out in this vein. Thus I might ask what would happen if people had “femtotechnolgy” wands that would turn dirt or air into whatever kinds of objects they wanted. Or what would happen if people could make hundreds of copies of themselves. Or what it would be like if we had a mountain as tall as all the transfinite ordinals. Science fiction can be thought of as a laboratory for carrying out thought experiments. The bare idea of a femtotechnolgy wand doesn’t tell you much. You need to do some work to investigate the consequences. In effect, you have to carry out a simulation of a society with your additional assumption. This is in some ways similar to what we do in mathematics. Note that just thinking about a question often isn’t enough. You need to write something down. The paper does part of the work, that is, the act of writing elicits further ideas and fills in details, regardless of whether you’re writing literature or math. Something I learned from mathematics was to never turn back from an idea just because it seems too counterintuitive. Logic can take you to some very strange places. All this said, I need to point out that science-fiction is also quite different from mathematics. SF is a form of literature, after all, and literature involves creating realistic human characters and using words to capture one’s sensations and emotions. Personal human experience isn’t something that mathematics directly deals with. Q: In your short story “A New Golden Age” you speak of mathematics being translated into music in order to make its beauty apparent to non-mathematicians. Isn't this defeating the purpose of mathematics? Could it be that the beauty of math appeals to special “taste buds”, to a special sense that needs and deserves special cultivation? A: Most people do indeed have mathematical taste buds, if only in an untrained form. But of course they’ll run screaming from the room if you show them an equation. So how do you get them to appreciate math? If you look at “A New Golden Age” again, you’ll see that my idea was not at all to turn mathematics into music. My idea was to tape or simulate the brain activity of some mathematicians and project these thought patterns into people’s brains so that they would feel what it’s like to do math. The punch-line of my story is that, just as people tend not to like the most intellectual music, they might tend not to like the most elegant math. The public at large could prefer a somewhat shallow and selfimportant work to a profound and modest one. They might like, say, G. Spencer Brown’s Laws of Form better than Paul J. Cohen’s Set Theory and the Continuum Hypothesis. Q: Speaking of the beauty of math, I distinctly remember the very first moment when it struck me with an intensity that was almost painful. It happened in an undergraduate linear algebra lecture on the Cayley-Hamilton theorem, which asserts that a linear operator satisfies its own characteristic polynomial. What are your favorite examples of beautiful theorems? A: There are different kinds of mathematical beauty. The result you mention is maybe a kind of “fixed point” situation where you find the answer inside the question. A higher-order language wraps around to the standard level of discourse. Gödel’s wonderful incompleteness theorem is like this in that it’s based on a sentence G which means “G is not provable.” It’s also nice when mathematics establishes objective truths about external reality, such as Plato’s proof that there are only five regular solids. In this context, I also think of the Frenet formulas using curvature and torsion to express the derivatives of the moving trihedron of a space curve. Still another form of mathematical beauty involves discovering that two seemingly quite distinct concepts turn out to be the same. A classic example is the proof using Taylor series that e to the i pi plus one is zero. It’s also beautiful to discover interesting features in previously unheard-of territory. Here I think of Cantor’s proof of the uncountability of the continuum and Mandelbrot’s work on the gorgeously gnarly Mandelbrot set. Q: Your mathematical training was as a set theorist. Do you have a favorite set theory SF story? A: My novel White Light is my favorite tale about set theory. It’s subtitle is in fact taken from the title of a paper by Kurt Gödel: “What is Cantor’s Continuum Problem?” In my novel, a disgruntled math professor with a bad job at a state college in upstate New York leaves his body and visits an afterworld where all of the infinities of set theory are real. As chance would have it, I wrote this novel after losing my job at SUCAS Geneseo. But Mother Mathematics provided for me, I obtained a Humboldt fellowship to visit the University of Heidelberg. (I might mention as an aside that I didn’t manage to get tenure until I was fifty, so take heart, all you unemployed young mathematicians.) In Heidelberg I’d hoped to make some formal, mathematical progress on the continuum problem, but instead I wrote a novel about an unsuccessful math professor who meets Cantor and discovers that continuous objects in our physical world have aleph-two “aether” particles each. In other words, my novel became a thought experiment demonstrating that the continuum hypothesis is false! For me, writing science fiction is a lot easier than proving results in set theory. Q: To quote from a blurb on the back of Spaceland, “Rucker gives us a tour of higher mathematics”. Could you elaborate on this statement: what particular sort of mathematics, and how much of it? A: Spaceland is primarily about four-dimensional space, and it’s an exaggeration to say it’s a tour of higher mathematics. I don’t have much control over what my publishers put on my covers. If you read Edwin Abbott Abbott’s Flatland closely, you’ll notice that it’s set on December 31, 1999. So I thought I should write a onedimension-higher version of the book set in my present-day Silicon Valley. My character, Joe Cube, travels into a four-dimensional space called the All, and visits two lands there called Klupdom and Dronia. Of course there are a number of standard things one expects in a story of this type: getting past a wall by hopping over it in a higher dimension, reaching into a person’s body without crossing their skin, flipping over and becoming one’s own mirror image, unexpected attacks from unseen higher-dimensional beings, and so on. Abbott and the SF writers of the 1950s treated all of these. A fresh topic that intrigued me was what I would actually see if I were in four-dimensional space. Using analogies to Abbott’s hero A Square, I convinced myself that the only way to see properly in hyperspace is to be equipped with a four-dimensional eye. So I gave Joe Cube an eyestalk sticking out into hyperspace from the center of his brain. Q: Did you use any computer simulations to help you to visualize the All, four-dimensional space, and its three-dimensional crosssections as seen by Joe Cube? A: The first hyperspace simulation I used was in the 1970s. It was a set of eighty-one colored paper cubes which I made, following the instructions of the 19th century mathematician Charles Howard Hinton. These cubes were a kind of “Rubik’s” version of a hypercube (because 34=81). I edited a Dover collection of Hinton’s writings. He was quite a character. He was convicted of bigamy, fled to Japan with his two wives, then ended up on the faculty at Princeton, where he invented a baseball gun so the Princeton players could experience really fast pitches during their practices. He wrote some science fiction, too. In the 1980s I met Tom Banchoff of Brown University. He showed me the first computer simulations of four-dimensional space. They made a powerful impression on me. When I worked at the graphics company Autodesk in the 1990s, we were building a virtual reality platform, and I wrote some code so that I could look at tumbling solid hypercubes through the VR goggles. Here at San Jose State in the 2000s, I’ve had some computer science Master’s degree students do thesis projects involving creating programs to display four-dimensional polytopes. By now, I don’t find these programs all that useful. Certainly they’re suggestive, and they get the mental ball rolling. But they don’t show you a full four-dimensional world, which is what I was trying to visualize in Spaceland. They only show a few simple polytopes. I would very much like to see a good four-dimensional virtual reality simulation. It’s a problem that hasn’t been properly attacked. Most efforts in virtual reality are, quite reasonably, focused on building computer games, so I think what’s needed is a good four-dimensional computer game. My student Wyley Dai did create a good fourdimensional Space Invaders game. But what I want is a whole reality with naturalistic forms resembling hyperdimensional plants, animals, and geological formations. Q: You once said: “To take pictures, you need to have something you like taking pictures of. To learn how to write, you need to have something you want to write about. And to learn programming, you need something you want to program about”. What about applying this principle to learning mathematics? What does it sound like? A: You only learn mathematics by applying it to something that matters to you. Learning based on drill has a very short half-life. Each person has to find things that catch their fancy, say, squaring numbers on a calculator and looking at the digit patterns, or maybe trying out possible arrangements of regular tiles. I remember once I was riding in a car with a friend and he wondered how many ways there are to fold a map. And I told him there’s a little branch of mathematics devoted to that problem. He thought I was kidding. The good teachers come up with intriguing problems that students really want to know the answer to. Q: In one of your interviews you said that you want to be called a writer since “writing is far and away the most important thing that I do. Over the long run, only the written language matters”. Isn't mathematics a highly evolved language? A: Certainly my books are more important than whatever I’ve done in mathematics or computer science. Of course this says more about my relative abilities as a mathematician and as a writer than about the absolute significance of mathematics. It’s not a contest, anyway. One thing doesn’t have to be more important than another. I do believe that the language of mathematics is less widely applicable than, say, English. Certainly there are things you can say in English that are much clearer in mathematics. But mathematics doesn’t talk about how it feels to be alive. Yes, we can contrive clever chains of reasoning to try and quantify sensations and emotions --- but these models come far after the fact. Ordinarily language, on the other hand, can capture human experience on the fly. Q: Once you said: “My work with computers has very much affected the way I see the world”. Could you explain what you meant? Also, has your work in math had a comparable effect? A: My background in math and computer science has a tremendous influence on the way I see the world and on how I write. For instance I think about the writing process itself as a fractal. I have the big arc of plot, the short-story-like chapters, the scenes within the chapter, the actions that make up the scenes, the nicely formed sentences to describe the actions, the carefully chosen words in the sentence. And hidden beneath each word is another fractal, the entire language with all my ramifying mental associations. I see computer science as experimental mathematics. Of course people can use computers for other kinds of things, but what I’ve been doing for the last twenty years or so is exploring ways of bringing mathematics to life. Over the years, I’ve adopted a variety of mathematics-influenced views about the nature of reality. As a series of personal thought experiments, I’ve thought of the world as made of infinite sets, of curved space, of fractals, of cellular automata, and of computations. These days I just think reality is a whole lot of things at once, and that there aren't any simple answers. I'm accepting .and savoring .the fact that the world is rich and complicated. Mathematics and computer science have taught me something about the range of possibilities. The waving of the branches of a tree in the wind, for instance -- - it’s wonderful to think of them in terms of chaotic oscillations, and then you have the coupling of the branches to think about as well. Or the air around us --- it’s mind-boggling to think of the complexity of the flow fields. If we could see the air, we’d be amazed. Though, come to think of it, we can see clouds, which also happen to be, of course, fractals. If there were only one spot on Earth where clouds formed, people would be unbelievably excited about traveling there to see them. It would be like whale watching. Lying on your back looking at clouds is a deeply satisfying experience. Q: Are you familiar with two recent papers by John H. Conway and Juan Pablo Rossetti, “Describing the Platycosms,” and “Hearing the Platycosms”? Conway explains that they proposed the term platycosm for the 3-dimensional analogues of the torus and Klein bottle, and in these papers they discuss, in particular, what you'd see if you lived in one of these “flat 3-manifolds without boundary”. Sounds rather science fictional, doesn't it? A: This sounds like a nice mathematical SF idea, I’ll have to look into it. Platycosm is a wonderful word. John Conway is a great man. I’m proud to say that I’ve occasionally exchanged email with him. When I was working on my novel Freeware, I was interested in higherdimensional non-repeating tilings of hyperspace, similar to Penrose’s Perplexing Poultry. And Conway helped me a little. Freeware ended up including some devices I called “stunglasses.” You wear them for fun; they tessellate the images of your surroundings into three -dimensional Perplexing Poultry. Peck! Another contact I had with Conway was when, after carrying out some computer experiments, I formulated the notion that the stitch on a baseball matches the space curve defined by saying it has constant curvature and torsion that varies as the sine of its arc length. It’s a closed curve that at least looks like the baseball stitch. And Conway wrote me, “I have a principle that whenever someone thinks they’ve discovered the formula for the baseball stitch curve, they’re wrong.” Eventually my colleague Roger Alperin proved my curve doesn’t actually lie on a sphere. And then some further research revealed the actual baseball stitch curve to be based on some hand-made trial-and -error drawings! Q: Would you agree with the proposition that mathematics is to all other intellectual endeavors as poetry is to the rest of literature? A: I think poetry tries to capture emotional states by unexpected juxtapositions of words. There is nothing at all scientific about it. We don’t understand ourselves well enough to turn our poetry into science, and I don’t think we ever will. At the highest levels of human creativity, we’re doing something more complicated than anything that we can roll up into an algorithm. You can’t simulate yourself writing poetry. In the early stages of creation, a mathematician tries to capture some aspect of the world’s structure by an unexpected juxtaposition of concepts. Mathematics starts with images, and once the mathematician has formed some interesting sequence of associations, the images can be converted into compact mathematical notation. This process also transcends any humanly conceivable algorithm. You might say that poetry and mathematics resemble each other in their conciseness. I used to write poetry, and then I learned to write novels. If you’re writing well, a novel has poetic passages in it. There’s also novel-length mathematics; we have a lot of long math books. The poetic parts of a math book are the definitions and the surprising results. The story part is perhaps the applications. I could go on trying to make comparisons, but really I do think literature is very different from mathematics. I love them both. I’ve been fortunate to be able to work in both fields. I’m a Sunday painter, too, and that’s different from both math and from writing. The world is big and beautiful.Interview to Rudy Rucker for Math Horizons magazine. San Jose, 10/31/2003