
Mapping Reality
An artistmathematician illuminates a world of infinite beauty
artworks and text By LunYi Tsai, A92
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.




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 earlytwentiethcentury 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. 




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 twodimensional 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 threedimensional sphere as seen through a function
called the Hopf Fibration, which shows the connection between it and a twodimensional
sphere. An ordinary sphere—the surface of a ball—is a twodimensional
structure that sits in threedimensional space. A threedimensional sphere
is an object that sits in fourdimensional 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 threedimensional sphere is a very strange object indeed. 




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, LUNYI 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 LunYi Tsai, visit go.tufts.edu/lunyitsai. 

