# Mapping Reality

An artist-mathematician illuminates a world of infinite beauty

Math and art are two sides of the same coin. I like to think of math as a way of mapping reality, of abstracting certain essential features of the world we experience and specifying how they interconnect. We begin to learn math as infants. Crawling across the floor, we observe that some objects are close—reachable in a small number of movements—while others are distant, requiring many movements. The concept of closeness is fundamental to a branch of math called topology, and topology in turn provides the foundation for calculus, the discipline that studies change, models dynamic physical phenomena, and also allows us to calculate the lengths, areas, or volumes of all sorts of geometric objects. Thus math becomes part of the personal map of reality we all construct—usually unconsciously—as we move through life, collecting data and figuring out how it all fits together.

Art starts to happen when we project these personal maps back out on the world. Each of us has an impulse to express his or her experience of reality. At the most mundane level, we do so through things like conversation, writing, dress, cooking, or doodling on napkins. But when this projecting of personal maps becomes more deliberate, more conscious, we begin to produce what is ordinarily called art—painting, music, literature, and so on.

I have spent much of my life trying to become conscious of both processes—the mapping that gives rise to mathematics and the projecting that gives rise to art. As a college math instructor, I have to explain abstract concepts as vividly and concretely as possible. As a maker of mathematically inspired art—paintings, drawings, and, lately, sculptures—I attempt to translate my understanding of math into visual forms that mean something on an aesthetic level. As it happens, the two occupations reinforce each other.

My art usually begins with a mathematical idea I find intriguing. To make sure I understand it, I work out the steps of the construction or the proof of the proposition—these often appear as handwritten notes in the background of my paintings. As I pore over the mathematical details, a mental image emerges, and when it becomes clear, I’m ready to paint. Just as sketching a person gives me a deeper understanding of the sitter, painting a mathematical concept gives me a deeper knowledge of the math.

Math and art converged for me when I was an undergraduate at Tufts. For much of my teens, I hung out in New York galleries and museums, where I gravitated to abstract painters like Rothko and Pollock. I had a vague sense that I wanted to become such a painter myself, but I couldn’t escape the feeling that abstract art was missing something—depth of meaning, perhaps—behind those abstractions. It was while taking multidimensional calculus with Professor Montserrat Teixidor i Bigas that I realized the solution might lie in theoretical math, and immediately switched majors. In essence, I studied mathematics in order to make art.

Although some students have told me they chose to major in math after seeing my work, you don’t have to know math to appreciate my paintings and drawings. The point of my art isn’t to teach math, any more than the point of Gauguin’s Tahitian paintings was to teach people about Tahiti. Like Gauguin, I’m transfixed by the beauty of the world I find myself in, and can’t help but represent, in my own idiosyncratic way, my experience of it. If my work should inspire somebody to make the arduous voyage into the world of math, then I’m doubly happy.

 Proof by Picture [1–3] Each of these paintings embodies the basic ideas or steps in the proof of a major mathematical statement: 1) the uncountability of real numbers, 2) Urysohn’s Lemma, and 3) Euclid’s proof of the Pythagorean Theorem. Depending on your knowledge of math, the proof will either reveal itself or remain to be taken on faith.
 1. Surprised Again on the Diagonal 2002, oil on canvas, 72 x 66 in Permanent Collection, Butler Institute of American Art Here I refer to a famous theorem known as the uncountability of the real numbers, proved by the German mathematician Georg Cantor in 1891. Cantor showed that the set of real numbers, which include all the numbers we use in daily life, cannot be counted even given infinite time. The theorem marked a turning point in math and logic: there were now at least two infinities—the countable and the uncountable. Cantor went on to show that there are an infinity of infinities. His proof introduced a powerful tool known as Cantor’s Diagonal Argument. He started with a list of all real numbers. Then, following a diagonal line down the list and changing each digit along it, he produced a number that couldn’t possibly be in the list, proving the real numbers’ uncountability. The orange squares represent those diagonal digits.
 2. Michelle’s Math Lesson 2004, mixed media on canvas, 48 x 48 in For several months, this canvas served as a “blackboard” for my wife, Michelle. I would give her a math lesson each time she came to my studio, and then I would cover her lesson with a slightly transparent layer of white paint. There are at least a dozen lessons hidden beneath the surface, making this one of my heaviest canvases for its size. The lesson at the surface is about a theorem from the area of math known as topology, which studies the notions of closeness and continuity. The proof of the theorem, using a tool called Urysohn’s Lemma (its discoverer, Pavel Urysohn, was an early-twentieth-century Russian mathematician who drowned at the age of 26), is, to me, visually exciting. The math is complicated, but its graphic depiction isn’t. You begin with two disjoint curves and then draw loops around them in a prescribed manner, with the idea that the process goes on to infinity. I decided to float this picture over Michelle’s notes about the proof.
 3. Change 2008, encaustic on plywood, 41 x 38 in This is Euclid’s proof of Pythagoras’s theorem (in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides). I painted it just before the 2008 presidential elections—hence the face of Barack Obama. I also embedded newspaper clippings I had found from 1954, which made me wonder what people back then would have thought if they were told that in fifty-four years a woman and a black man would be running for the nation’s highest office. I thought it would be interesting to include the clippings in a painting representing an eternal mathematical truth.
 Concepts [4–6] These works explore facets of a single profound concept, parameterization, which has to do with specifying a quantity that, when changed, gives rise to different kinds of things. When my Tufts math professor Loring Tu saw my painting Some Quadric Surfaces (6), he gave me the idea of showing how the different surfaces could be created by varying a single parameter of a single equation. Several series of artworks about parameterization have followed.
 4. Whitney Family 2008, charcoal and graphite on paper, 39 x 39 in The Whitney Umbrella, a topological surface named for the American mathematician Hassler Whitney, can be thought of as a plane that is cut along a ray and glued back in such a way that it intersects with itself in three dimensions. This parameterization creates a stack of umbrellas sitting on their sides.
 5. Quadric Parameterization I 2006, charcoal and graphite on paper, 42 x 71 in When I look at this piece, I remember the parameterization, but more strongly I remember the late afternoon light in the New York City neighborhood of my childhood. In the summertime, I would sit on the fire escape outside my window and watch the light fade on the New Museum of Contemporary Art. I took a lot of black-and-white photos of buildings and empty streets. When I began to work in charcoal again several years ago, I was surprised by the nostalgic force of the drawings.
 6. Some Quadric Surfaces 2004, mixed media on board, 48 x 24 in At the age of three or four, I had disturbing dreams where I was floating in dark space and a curved white surface would appear, a different shape each time. It would begin small and grow until it was huge enough to crush me. Then I would cry out in my sleep. This painting recalls a wonderful semester with Professor Teixidor when I conquered my fear of smooth surfaces, realizing I could study them with mathematical tools (the gradient and multiple integrals, for example). In the process, I discovered the beauty and excitement of doing math. Although each surface comes from the same kind of equation, I painted each as a separate entity. Professor Tu’s suggestion that I look for the parameterization was the flash of light that made me see all these surfaces as a connected whole.
 Objects [7–9] Certain mathematical objects have remarkable qualities that separate them from an infinity of other objects: in one way or another, they differentiate subtle concepts, delineate what is or is not possible, and provide a means to see things both literally and figuratively.
 7. The Hopf Fibration 2007, charcoal and graphite on paper 42 x 60 in When I learned about the imaginary number i = √-1 in high school (it’s imaginary because there is no real number that, multiplied by itself, equals -1), I thought it sounded rather mysterious. Later, when I learned more about imaginary numbers, I realized how fascinating they are. Just as you can think of real numbers as sitting on a line, you can think of complex numbers as sitting on a two-dimensional plane. When you multiply real numbers together, they move along the line, but when you multiply complex numbers, they spin around the origin (the intersection of the x- and y- axes). This drawing shows a three-dimensional sphere as seen through a function called the Hopf Fibration, which shows the connection between it and a two-dimensional sphere. An ordinary sphere—the surface of a ball—is a two-dimensional structure that sits in three-dimensional space. A three-dimensional sphere is an object that sits in four-dimensional space. Since most of us can’t see in four dimensions, the best we can do is see its intersection with three dimensions, similar to looking at flat slices of a 3D object in a CT scan. Obviously, a three-dimensional sphere is a very strange object indeed.
 8. Newton by the Sea or The Topologist’s Sine Curve 2004, mixed media on board, 24 x 48 in Working on this painting, I thought of the beach in summertime—the sky, the sea, the sand, the children picking up shells and pebbles as they examine the universe, and of course the rolling waves. The natural, periodic behavior of ocean waves is modeled by the sine curve. The topologist’s sine curve is quite different. Near the origin—the intersection of the x and y axes—the wave wiggles around like crazy, whether approaching from the left or the right.
 9. Riemann’s Integral 2004, mixed media on canvas, 39 x 39 in One of the great mathematical achievements of ancient times was Archimedes’ discovery of the volume of a sphere, which anticipated calculus by almost two thousand years. With calculus, we can figure out the areas and volumes of all sorts of geometric objects. We do this by finding a certain limit called the integral, first rigorously formulated by Bernhard Riemann. (The limit is the fundamental tool that allows us to do calculations with infinitely small quantities.) Essentially, the integral is the sum of the areas of infinitely many, infinitely thin vertical rectangles under a curve. (The Greek letter sigma, ∑, which can be seen at the center under the surface, represents “sum.”) One can find the area between two curves, as I’ve shown in this painting, by subtracting the integral of the lower curve from that of the upper curve. The blue and green rectangles show an approximation of this area between the two curves. Born in Cambridge, Massachusetts, LUN-YI TSAI, A92, grew up in Paris, where his father, a kinetic sculptor, had a studio, and in New York City’s SoHo. After Tufts, he received a master’s in mathematics from the University of Pittsburgh and spent six years living, working, and making art in China. In 2008, he was a Karl Hofer Gesellschaft artist in residence in Berlin. To view slide shows narrated by Lun-Yi Tsai, visit go.tufts.edu/lunyitsai.