# 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)

ViHart's father, right?