Mathematical performances: George Hart at TEDxWellesleyCollege


Translator: Yifat Adler
Reviewer: Denise RQ Thank you.
Thank you very much. I’m a mathematician, and I’m an artist,
and I also do visual art sculptures, but today I’m going to talk about
what’s current research. I’m trying to develop techniques
for doing mathematical performances, and I’ll explain what I mean
by that as I go along. But let me just start
to get us all on the same point by saying that to me
mathematics is an art, and you can read this quote
by Bertrand Russell in which he famously compares
mathematics to sculpture. That mathematics has this austere beauty. And if you are not a mathematician,
if you haven’t had advanced math courses, you might not realize, but what mathematicians do
all the time is to create a field. They are creating ideas
and structures that are beautiful. And if you are a math major you learn, and part of your training is
how to appreciate the beauty of math. That there is an aesthetic to what’s a beautiful proof
or what’s an elegant theorem. And that there is this art aspect
to mathematics. I will start with that. I can give you a whole lecture
on what that’s about. But what I want to talk about today
is math as a performing art. So, here is the real question: what’s the mathematical analog
to a performance that you go to – a symphony, an opera, a ballet – a theater performance in which
there is some sort of audience. A general audience comes in and sits down, and up on the stage, there are people
who are experts in their field or whatever that performance is,
and the experts do something. As a mathematician I want to convey some true, beautiful
mathematical experience in a real mathematical content
to this audience. But I want it to have
a kind of emotional experience. So, the thing about a performance,
as opposed to many visual arts, is that it has this drama,
has this development, it has conflicts, and climax,
and resolution and whatever. And it has the ability overtime to build
this deep emotional connection to you. People come out of a performance,
and they are moved, they are enriched in a way often
that they’ll go tell their friends: “What a great performance. You have
to see it. Come with me next time.” There’s a depth of human
connection with a performance that, I would say, is different than what you have
with static visual arts, sculptures. So, how can I get there? But I want to start by telling
what I mostly do, which is sculpture, just to give you a background. As a mathematician, I think of myself as an applied mathematician,
applying math to sculpture. Math can be applied to different fields, engineering, economics,
and social science, etc. I apply it to sculpture
and I create things. I’ll now just show you
a random selection of things. You can see lots more
on my website, GeorgeHart.com. Go there. (Laughter) These are things I would love
to tell you the whole story about, but that’s not today’s lecture. But you can see I enjoy working
with many different materials. A range of different engineering problems. How do I work with steel,
how do I work with CDs, how do I work with wood,
plastic, etc., to create things. And the things that I’m creating
are coming out of my head as mathematical ideas. So, I start with mathematical foundations that as a mathematician
I can see the beauty, but if you are not trained as one you don’t always know
what’s so beautiful about math. So I try and take something of that
and convey to you in some form that anyone can appreciate
something about what’s going on. And as you look at these,
you’ll see various aspects of patterns, and structures,
and relationships. And that’s what mathematics
is really about. The fundamental essence of mathematics is not about numbers, or arithmetic,
or geometry, or trigonometry. But in general mathematics
is the science or the study of patterns and understanding patterns
and how things relate to each other. And you’ll see there’s all kinds
of structures and patterns here. I’m obviously moved by things that relate to each other
in interesting, complex ways. I’m showing you sort of flavors
of some of that. Some people often talk about
the beauty of mathematical equations. Something that people will tell you: “Euler’s formula, e to the i pi,
is one of the most beautiful equations.” And it is in a way. But I want to emphasize
that to a mathematician, it’s not really the equation. When you see that equation
it brings to mind a whole structure. Mathematicians never just believe
something, we don’t believe in equations, they think of it in the context
of how it was derived from other facts and how it leads to other facts. So, it’s part of a structure. Mathematicians, when they look at anything – an equation or anything
in the real world – see structures, patterns,and think
of it at a more abstract level. And it’s that beauty of structure, which is what I’m trying to convey
in these sculptures and in different ways. Each has a different story
which I won’t explain. But the kinds of things that as a teacher,
the professor in me wants to also have a pedagogical back story
when you look at these. And you might begin to ask
mathematical questions and say: “When I look at that sculpture,
I first of all try and count. How many pieces are there?
How many corners are there?” This sculpture has 30 pieces in it, each is shaped like an X,
and each has some number. But could I do that with 31?
Could I do that with 29? There is something about the counting that might get you thinking
in a mathematical frame of mind. There are also things about patterns. So, you might look at this sculpture
and say: “How do those angles work? How do you calculate
that length and angles?” There are certain geometric questions that
might come to mind and motivate you all: “If I want to make cool things,
maybe I should learn some geometry.” And there is this [issue] of structure, that when you look at
a sculpture like this, you might think about
how do these things relate. When I look at a sculpture,
what I think about often is: can I visualize it in my mind,
can I close my eyes and see it? And I don’t really feel like
I know a sculpture until I can see it. I can go to a desert island
and reproduce it in some form or another. So, you might look at any these and say:
“Do I understand what’s going on? Can I see it from the other side?
Can I answer questions about it? Could I reproduce it in clay or paper?” And there are questions going on
in many different fields. Some of these are sort of topological,
they involve issues of topologies. Some involve combinatorics
or group theory, graph theory, symmetry. There are many branches of mathematics
which are really beautiful which the public doesn’t see. The public goes through arithmetic, algebra, trigonometry,
maybe a little calculus. And that’s great stuff,
it’s really important. And it’s crucial if you want to work up towards being a scientist
and understanding differential equations. You can’t really understand
how complex systems evolve without a good intuitive sense
of how differential equations work. And all that stuff is designed
in the standard K to 12 college curriculum to get you to that point, and it’s great. But that’s the tiniest fraction
of what mathematics is. Real mathematics as practiced
by professional mathematicians includes so much more. My mission in life is to take this passion
which I have for all these things, and find some way
to show them to the public. This one is based on a uniform tessellation
in the hyperbolic plane, which is this beautiful alternate geometry
that I’ve mapped to the Poincaré disk into a form that you can see
based on a spherical envelope. It takes a long time to learn it,
but by seeing this, at least you can be exposed to the fact
that these subjects are out there, and that mathematicians
find them so beautiful, so lovely. And I hope to inspire different people
in different ways in these sculptures. And I would love to have lots of museums
or places that you can view these things and see that this mathematical
sculpture exists. And it’s not just sculptures. I also organize a conference
on mathematics and art every year. It’s called “The Bridges Conference”
– you make a bridge between math and art; and I do sculpture
and I’m showing you my work – but at such conference, you’ll find
people from many different fields. People are making oil paintings
or water colors, and that sort of art, but there’s people making quilts,
and people making bead work, and sundials, or origami. Many many different fields
and media can be used by people who are inspired by these
fundamental mathematical beauties. Some of these sculptures that I’m showing
are made on 3D printers. If you’re familiar with
this modern revolution and be able to think of something,
express it to the computer, hit the print button and then
you have it in your hand. It’s just fascinating and so empowering
that anything you can think of, these machines
can build for you robotically. So, it’s a great reason to learn about
mathematics, because it’s a language for expressing complex structures
of anything that you want to make. So, I said I’d like to have
museums of various sorts. And I should mention at some point that I’ve wanted that a long time,
and I spent five years on the project. I’m one of the co-founders of the Museum of Mathematics
in New York City so just let me give a commercial: If you happen to be in New York,
11 East 26th Street, go visit MoMath. Well, these are static works. The philosophy of the museum
is to give you hands-on things. So you can walk in, and like
a science museum you do this or that, and then you have
some kind of fun interaction. That’s designed to give you
this emotional connection. And so you should check that out. That emotional connection is important; and I’m trying to lead up here
to this topic of mathematical performance. This I won’t describe,
but I did some of it in the break, is how to cut a bagel
into two equal parts, and if you have never done that
you should go find out how that works. This next set of pictures here
that we’re moving to, involve things that I do with workshops
or things that I do with groups of people that takes us into this idea
of a performance. And while I’m devoting
my life in some sense to visual art, and sculpture,
and things of this sort, there’s a fundamental fact that if you go to a museum
and you see great stuff, [you] don’t cry at a museum in the way that you might go
to a performance and cry. There’s a much deeper emotional connection
when you have this sweep of time and this ability to develop an idea
and have this human involvement. The real question here is
how do we go from these things – which I’m not putting down,
but they have a limitation – to things that are more
of a performance sort of nature. And you might say:
“Well, let’s take some math experts, and put them on a stage,
and have them do something.” What would that thing be? Is there anything that math experts
can do on a stage that would give you
that emotional connection? And I think the answer there
is probably “No,” if the audience just has
to sit down and be passive. We’re beginning to see things here
where I involve a group of people. But the fundamental issue: math teachers
say that math is not a spectator sport, that you have to do math to get the math. And as a corollary, you have to do math in order to feel the math,
to have this emotional connection. So, what’s happening here is that I’ve designed various ways
in which people can be involved. So, imagine you go into
a concert hall, and you sit down, and the usher hands you a viola,
and you’re expected to play. You can’t really do that
because it doesn’t work with violas. People can’t play violas. I’m going to flip through these
and move to the video. If we can cue the video. [Is] the video rolling? So, with music it takes
certain skills to perform. But I’m going to show you here
just a montage of clips of people who, when you see them,
in the audience here they are the audience. But what I claim is that math
is different from music because everyone can do math. Math is the science of study of patterns,
and structures, and relationships. And you’ve been doing that
your whole life. You can do that. At these events, we have to think
of those people as being in the audience, but unlike an audience that sits,
it’s an audience that participates because you have to add that ingredient
if you are going to actually do any math. For me to convey any real math content,
I can’t just do that on a board. It doesn’t work.
It’s not a spectator sport. But what you see here are these events
where I design something – I’m the composer of the symphony,
and I’m also the conductor because I’m there getting
these people to put things together – and they are doing real math. That may not be clear, but there are different kinds of math
that you can be doing. There are issues of counting and assembly.
There are geometry issues. But part of what they’re doing
is extending patterns. You get these mathematical
exercises when they say: “Three, five, seven. What’s
the next number in that sequence?” That skill that your teacher
was trying to convey to you is how to look at something, find a pattern in data,
and then extend that pattern. That’s a crucial idea and so essential
to mathematics in general. Patterns in math generally aren’t numbers – that’s a tiny bit of the subject
domain of mathematics – but the ability to find a pattern. They might see pieces over pieces and have
to figure out where should each piece go. Should it go there? That’s doing mathematics.
They are also dealing with structure. They’re trying to understand
how should these parts go together. As they begin to do this,
they are realizing that: “Oh, each of these openings
has five different colors around it.” So, when I do these events, depending on the audience,
I do them in different levels. So, as I said with the development, in some cases I’m bringing
just pieces of cardboard, and if we have enough time
we develop it from zero, we have a major long term opera; in other cases, I have to give them
more specific steps: “Do this, do this, do this.” Like a square dance teacher telling you:
“Put part A into part B.” This can be done at different levels.
You can see the experiments here. As a researcher my job is to try out different things, experiment,
and see what happens. But there’s real math content
in each of these events. And the beauty of it to me is that
as a professor, I don’t have to just say, “OK, I’m going to lecture
to you on symmetry,” but I’ll say, “Let’s make something,”
and we start building. and I have a plan, but it always works out
in some different way than you plan. You have to be willing
to improvise as you go along. People are going to ask questions
out of curiosity, they are going to say: “How many parts do we need? And why?
And how does this go together?” And at each time that a question comes up,
you have the opportunity of a teacher. It’s one of these teachable moments, to give an answer when the audience
isn’t just forced to sit but they are actually involved. The answer is relevant
to what they are doing at that moment. It’s an answer that they care about
in the context of where’s relevant. So, it’s a very different
teaching experience. I can have much greater depth. And as I said, this performance aspect,
or conflict, can come in hand, like: “I can’t get this piece in.” But they have to resolve it
one way or the other. And I don’t always know
how it will resolve, but I can say, in general, there is a happy ending
to most of these performances. This is a recent one at Duke University. I’m using laser cut steel here
that’s been powder coated. I used it to cut aluminium in orbs, and in this case, four, five,
six and seven feet in diameter over a period of several hours
to put this together. It’s an event that people
get deeply invested in. One of my measures
to understand that it’s successful is the smiles on people faces,
and the things they tell me, but another great measure
that I’ve noticed again and again is a kind of ownership. When people are involved in this, after they’re done,
they’ll bring their friends and say: “I helped make that,
and that’s my sculpture.” They put one nut and one bolt,
but it’s their sculpture. Or maybe they spent four hours
just in love with the process. Everyone has a different
amount of involvement. But there is this feel of ownership,
there is this strong emotional connection. Another thing that people are doing
is they’re solving puzzles. And solving puzzles is a great skill. You need to develop this ability
that someone says: “I don’t know how to solve this problem.” It’s not like a homework problem
that I know the answers to. It’s a real problem and you have
to be comfortable with that. So I try and present people with
as much of a challenge as I can. I don’t say to them, “Here’s what to do.” I say: “I think these things go together.
Can you figure it out?” And they’re working on it
to make that happen. So that skill to be comfortable
facing a new puzzle that you don’t know what to do,
and you try stuff and some doesn’t work, that’s the essential skill you need to be
a creative problem solver in real life. One other thing that happens is
that at some point, I can step back. We’re working for a while,
and things are coming together, and then someone asks a question, and someone else,
instead of me, will answer it. And I can just relax
and let them take over from there. That’s when I feel
I’ve actually taught something, that they begin to become
this problem solving community working together
to figure out how things go. And again, these are not
quantitative assessments factors that let me know
that this is definitely working, but I know this is working
one way or the other. But this is just one sort
of teaching performance. I’d like other people to explore this. I’m encouraging you
to go on and figure out how to bring aspects of performance
into mathematical thinking. This particular sculpture
right now is hanging in the Math Department
at Princeton University. But there are plans and ideas
for lots of these things on my web site and lots of places you can go. Thank you. (Applause)