Answers to Questions from the people at Germany's Jahr der Mathematik 2008:
1. What was your first mathematical experience?
My first mathematical experience was that of crawling around as a baby. This is how each of us begins to understand the notion of closeness. That which can be reached in a finite sequence of movements is close, while something like the moon that cannot be attained in such a way is far. Henri Poincaré talks about this in “Science and Hypothesis.” This early mapping of reality continues in us to this very day. The reason why we don’t realize we are constantly mapping reality is that it has become completely automatic hence unconscious. My art attempts to make conscious this process of mapping, which is the process of abstraction.
But the first and most fundamental mathematical experience is the experience of time. Sequences arise naturally from this most basic experience. Each moment seems to be replaced by another moment; we have the moment that is passing away and the one that is arising. But this experience of time also has a continuous quality to it, which leads us to the fruitful notion of the limit.
But to answer the question in a more standard way. My first memory of being impressed by math happened in the class of Madame Chesnay at the United Nations School in New York City when I was 10 or 11. I remember learning about decomposition into primes, Pascal’s Triangle, and also compass and ruler constructions. Each of these things surprised me. And I really enjoyed drawing with compass and ruler. Perhaps that’s why to this day, I occasionally use compass and ruler when making my art.
2. What awakened your enthusiasm for math?
I really liked Euclidean geometry. I liked the step-by-step process by which we showed things to be true. It was my first experience of mathematical proof. I enjoyed the feeling of really knowing something with confidence. It also surprised me how one can prove the same thing in several different ways.
3. Tell us a little about your current fascination with mathematics?
Today, I focus more on the experience of coming to an understanding in math whether its from texts or from the people who create the math. I see the state of not understanding or misunderstanding as characteristic of human existence. Mathematics provides one with the framework to make sense of our chaotic world. I enjoy observing the movement of my mind as it attempts to piece together the mathematics. It is a very pleasant experience as one puts together pieces of what one knows only to be surprised by what is achieved. In a sense, that is what my art is about—the process of abstract thought.
Nowadays, I continue to read math and enjoy collaborating with mathematicians to create representations of their math. My training in math has made me privvy to their worlds.
4. To what extent is mathematics similar to art?
As an artist whose art is inspired by his mathematical training, I find that math and art are similar in that both are ways of interpreting the human experience. And both seek to reveal interesting, surprising, and unknown features and relationships in our world. And by world I mean the network of thought that we’ve constructed in our minds as a result of life experience both mental and physical.
5. What do you say to people who tell you that mathematics is a dry and unworldly?
When people tell me that in their experience math is dry and unworldly, I just show them my art work and tell them that these paintings were inspired by my study of math. One of the main reasons for making my art is to share the wonder of mathematics—to give others a glimpse of this amazing universe that would otherwise take years of disciplined study to access.
I have a painting of Newton’s Laws of Motion with ellipses and circles, which was inspired by a lecture that the American physicist Richard Feynman gave. I don’t think one can get closer to the reality of this world than the laws of physics, which are mathematical.
6. What famous mathematicians do you admire? Why?
I admire L.E.J. Brouwer because of his fierce independence of mind. I like the way he worked and his absolute rigor. If you read his early work “Life, Art, and Mysticism,” you can hardly believe it was written by one of the greatest mathematicians of the 20th century. Parts of it seem to have been written by some mystic seer of the east. Another mathematician that has this quality is Alexander Grothendieck. Both are certainly very controversial figures in the history of math, but undoubtably extremely interesting, creative, and human.
7. Do you like to work with computers?
I like to use paper and pencil to do math, which for me is coming up with proofs to propositions. I almost never use computer programs to understand math. If you look at all the great mathematicians, almost all of them including the ones living today do their work by hand. Andrew Wiles and Grigori Perelman are just two contemporary examples.
Similarly, my art is also made by hand. I use paper, pencil, and charcoal for my drawings; and canvas and brushes for my paintings. I enjoy the process of working with my hands. It is sort of meditative, because it makes you aware of each motion of the brush, which silences the mind. And it is in this stillness that creativity arises.
8. What advice can you give to young people regarding studying mathematics?
If you don’t know what you want to do in life, which is actually a good thing when you’re young, then study math. Not only will it teach you to think really clearly and precisely, which is invaluable in whatever you end up doing, but also it will teach you what real creativity is.
9. In what way do you bring mathematics to people in everyday life?
I think the most important way that I bring mathematics to people who otherwise never think of math is through my art. My drawings and paintings challenge people’s preconceptions of math. They look at my art and they see these very interesting paintings and drawings that capture their attention and imagination. Then when they find out that the art is actually inspired by math, it really surprises them, because until then they always thought of math as something dull and boring. This experience gives them an opportunity to revisit their idea of math allowing people to gain a new appreciation for the subject. On rare occasions, my work has inspired some people to even study math!
Another way that I bring mathematics to other people is by organizing math and art presentations and dialogs between mathematicians and artists. I like to have these during my art exhibitions, which serve as springboards to interesting topics. For example, for my Berlin exhibition this Saturday, August 9th, I’ve organized a dialog entitled “Is Math an Art? Is Art Mathematical?”
10. What is your favorite number? Why?
I don’t actually have a favorite number. When people find out I do art and math, they guess that my favorite number must be Phi, the golden section. But I actually don’t like that number much, because I think it limits people’s concept of what math and art is about—there’s so much more to math and art than just the golden ratio! I’ve only made one painting of this number and I called it “Goodbye, Golden Ratio.”
11. Are there family members that share your passion for mathematics?
Everyone in my family is “good at math,” but I don’t think they share my passion for mathematics. Being engineers and scientists, they’re more practical about math and don’t need to see the rigorous proof a theorem before they use it. For me it’s always been really important that I understand why something is true in math.
12. Have you other interests besides mathematics? How are these related to math?
I practice meditation daily. It allows me to observe myself, especially my mind. So when I stated earlier that sequences arise from our experience of time, I wasn’t talking about something intellectual, I was talking about the experience of each moment as observed in meditation. Meditation stills the mind, makes it sharper and more sensitive, which curiously, makes it more open to create, which is the most important thing in both math and art.
My art is predominantly concerned with the process of abstraction, the primary means by which we make sense of our experience of reality. The study of mathematics serves as a model for this process: the usually automatic thought processes that take place everyday are made conscious in mathematics. The rigor of mathematical proof necessitates careful observation of each mental step. This slowing down of thought provides an opportunity for awareness of the very process of abstraction.
In order to understand my art, it is important to distinguish the notions of graphing and visualization from that of representation. Most all of what may be termed “mathematical art” falls into the category of graphing and visualization. The most prominent examples of this type of art are the colorful images of fractals, which are generated by inserting equations into computer programs. My work is not to be confused with mathematical art.
Although graphs and visualizations appear in the body of my work, the central thrust of my art is the representation of abstract concepts and the process of abstraction itself. Representation is a larger and more ambiguous concept. It is the attempt to illustrate ideas (notation) in a manner (often inaccurate and sometimes not completely correct) that is conducive to their further development. Representation is a creative step.
Algebraic geometer Aravind Asok explains:
“To many people, [the mathematical] process consists of the formal manipulation of algebraic equations or the writing of proofs in plane geometry. While mathematicians often do write proofs, study geometry and perform formal manipulations, they don’t usually study ideas that are explained in a few lines. On the other hand, mathematicians often do use elementary pictures to guide their investigations and motivate their proofs.”
These “elementary pictures” contain in them certain essential features of a given theorem or idea; as such they are much more meaningful than, say, graphing the projection of an n-dimensional hypercube onto a 2-dimensional plane. Curiously, the same images or diagrams are often used in very different contexts. This suggests a common underlying mathematical structure, which motivates the modern viewpoint that it is not the objects one studies that truly matter, but the relationships among them.
If this process of synthesizing what is grasped into a representative picture were relevant only in the domain of mathematics, my art would only be a portrayal of a very limited, albeit rich and profound, human experience. My conviction is that this process of abstraction (so clearly exemplified by mathematics) is at the heart of all descriptions of reality whether they be in literature, visual arts, music, mathematics, philosophy, natural or social sciences. Abstraction is the quintessential and definitive human process.
Any given painting or drawing of mine is the representation
of an idea or collection of ideas. As such, it is both summary and development.
Its main purpose is to open a space to allow for further understanding
and evolution on the basis of what has already been accomplished. Each
work captures in mid-flight the arrow of creativity, and is a metaphor
for the process of abstraction itself in whose stillness lies ceaseless
change and perpetual movement.
My work explores radical insights into the abstraction of reality. The primary practice of my art is the effort to understand space. The experience of space beginning in childhood with bodily sensations culminates in adulthood with mental constructs that are essentially mathematical.
As a child growing up in a kinetic sculptor’s family in Paris and New York in the 1970s and 80s, I was exposed to a lot of art especially abstract art. My early favorites were Kandinsky, Miro, Klee, and Mondrian; then there were the abstract expressionists Pollock, Rothko, and de Kooning. But as a teenager, I began to have a suspicion that something was lacking in the abstract art that I knew so well.
I decided to become an artist that would fill this void in abstract art. To do so meant that I had to really understand what abstraction is. I somehow knew that art school would not teach this to me, so I chose to study theoretical math, which is the most rigorous of abstract disciplines. Essentially, I went into mathematics to get my artistic education.
I see mathematics as a quintessentially human activity.
It is the ultimate extension of our human cognitive ability. The very
fabric of the reality in which we are caught is a fundamental abstraction
of which we are, for the most part, unaware. Mathematics allows me to
grasp this reality in a constructive and intuitive manner. My art is the
process by which I come to an understanding of my world. I have to find
my own way of looking at the world, which turns out to be my particular
understanding of the mathematics that is at the core of reality. The artwork
is a record of this struggle to find a cogent picture.
Lun-Yi London Tsai
TANGENT SPACE IV
Charcoal on paper
21 x 24 in