18:04
TED2008

Robert Lang: The math and magic of origami

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Robert Lang is a pioneer of the newest kind of origami -- using math and engineering principles to fold mind-blowingly intricate designs that are beautiful and, sometimes, very useful.

- Origamist
Robert Lang merges mathematics with aesthetics to fold elegant modern origami. His scientific approach helps him make folds once thought impossible -- and has secured his place as one of the first great Western masters of the art. Full bio

My talk is "Flapping Birds and Space Telescopes."
00:18
And you would think that should have nothing to do with one another,
00:21
but I hope by the end of these 18 minutes,
00:23
you'll see a little bit of a relation.
00:26
It ties to origami. So let me start.
00:29
What is origami?
00:30
Most people think they know what origami is. It's this:
00:32
flapping birds, toys, cootie catchers, that sort of thing.
00:35
And that is what origami used to be.
00:38
But it's become something else.
00:40
It's become an art form, a form of sculpture.
00:42
The common theme -- what makes it origami --
00:44
is folding is how we create the form.
00:46
You know, it's very old. This is a plate from 1797.
00:50
It shows these women playing with these toys.
00:53
If you look close, it's this shape, called a crane.
00:55
Every Japanese kid
00:58
learns how to fold that crane.
01:00
So this art has been around for hundreds of years,
01:02
and you would think something
01:04
that's been around that long -- so restrictive, folding only --
01:06
everything that could be done has been done a long time ago.
01:09
And that might have been the case.
01:12
But in the twentieth century,
01:14
a Japanese folder named Yoshizawa came along,
01:16
and he created tens of thousands of new designs.
01:19
But even more importantly, he created a language,
01:22
a way we could communicate,
01:25
a code of dots, dashes and arrows.
01:27
Harkening back to Susan Blackmore's talk,
01:29
we now have a means of transmitting information
01:31
with heredity and selection,
01:33
and we know where that leads.
01:36
And where it has led in origami
01:38
is to things like this.
01:40
This is an origami figure --
01:42
one sheet, no cuts, folding only, hundreds of folds.
01:44
This, too, is origami,
01:50
and this shows where we've gone in the modern world.
01:52
Naturalism. Detail.
01:55
You can get horns, antlers --
01:57
even, if you look close, cloven hooves.
01:59
And it raises a question: what changed?
02:01
And what changed is something
02:04
you might not have expected in an art,
02:06
which is math.
02:09
That is, people applied mathematical principles
02:11
to the art,
02:13
to discover the underlying laws.
02:16
And that leads to a very powerful tool.
02:18
The secret to productivity in so many fields --
02:21
and in origami --
02:23
is letting dead people do your work for you.
02:25
(Laughter)
02:28
Because what you can do is
02:29
take your problem,
02:31
and turn it into a problem that someone else has solved,
02:33
and use their solutions.
02:36
And I want to tell you how we did that in origami.
02:38
Origami revolves around crease patterns.
02:41
The crease pattern shown here is the underlying blueprint
02:43
for an origami figure.
02:46
And you can't just draw them arbitrarily.
02:48
They have to obey four simple laws.
02:50
And they're very simple, easy to understand.
02:53
The first law is two-colorability. You can color any crease pattern
02:55
with just two colors without ever having
02:58
the same color meeting.
03:00
The directions of the folds at any vertex --
03:03
the number of mountain folds, the number of valley folds --
03:06
always differs by two. Two more or two less.
03:09
Nothing else.
03:11
If you look at the angles around the fold,
03:13
you find that if you number the angles in a circle,
03:15
all the even-numbered angles add up to a straight line,
03:17
all the odd-numbered angles add up to a straight line.
03:20
And if you look at how the layers stack,
03:23
you'll find that no matter how you stack folds and sheets,
03:25
a sheet can never
03:28
penetrate a fold.
03:30
So that's four simple laws. That's all you need in origami.
03:32
All of origami comes from that.
03:35
And you'd think, "Can four simple laws
03:37
give rise to that kind of complexity?"
03:39
But indeed, the laws of quantum mechanics
03:41
can be written down on a napkin,
03:43
and yet they govern all of chemistry,
03:45
all of life, all of history.
03:47
If we obey these laws,
03:49
we can do amazing things.
03:51
So in origami, to obey these laws,
03:53
we can take simple patterns --
03:55
like this repeating pattern of folds, called textures --
03:57
and by itself it's nothing.
04:00
But if we follow the laws of origami,
04:02
we can put these patterns into another fold
04:04
that itself might be something very, very simple,
04:07
but when we put it together,
04:09
we get something a little different.
04:11
This fish, 400 scales --
04:13
again, it is one uncut square, only folding.
04:16
And if you don't want to fold 400 scales,
04:20
you can back off and just do a few things,
04:22
and add plates to the back of a turtle, or toes.
04:24
Or you can ramp up and go up to 50 stars
04:27
on a flag, with 13 stripes.
04:30
And if you want to go really crazy,
04:33
1,000 scales on a rattlesnake.
04:36
And this guy's on display downstairs,
04:38
so take a look if you get a chance.
04:40
The most powerful tools in origami
04:43
have related to how we get parts of creatures.
04:45
And I can put it in this simple equation.
04:48
We take an idea,
04:50
combine it with a square, and you get an origami figure.
04:52
(Laughter)
04:55
What matters is what we mean by those symbols.
04:59
And you might say, "Can you really be that specific?
05:01
I mean, a stag beetle -- it's got two points for jaws,
05:04
it's got antennae. Can you be that specific in the detail?"
05:06
And yeah, you really can.
05:10
So how do we do that? Well, we break it down
05:13
into a few smaller steps.
05:16
So let me stretch out that equation.
05:18
I start with my idea. I abstract it.
05:20
What's the most abstract form? It's a stick figure.
05:23
And from that stick figure, I somehow have to get to a folded shape
05:26
that has a part for every bit of the subject,
05:29
a flap for every leg.
05:32
And then once I have that folded shape that we call the base,
05:34
you can make the legs narrower, you can bend them,
05:37
you can turn it into the finished shape.
05:40
Now the first step, pretty easy.
05:42
Take an idea, draw a stick figure.
05:44
The last step is not so hard, but that middle step --
05:46
going from the abstract description to the folded shape --
05:49
that's hard.
05:52
But that's the place where the mathematical ideas
05:54
can get us over the hump.
05:56
And I'm going to show you all how to do that
05:58
so you can go out of here and fold something.
06:00
But we're going to start small.
06:02
This base has a lot of flaps in it.
06:04
We're going to learn how to make one flap.
06:06
How would you make a single flap?
06:09
Take a square. Fold it in half, fold it in half, fold it again,
06:11
until it gets long and narrow,
06:14
and then we'll say at the end of that, that's a flap.
06:16
I could use that for a leg, an arm, anything like that.
06:18
What paper went into that flap?
06:21
Well, if I unfold it and go back to the crease pattern,
06:23
you can see that the upper left corner of that shape
06:25
is the paper that went into the flap.
06:28
So that's the flap, and all the rest of the paper's left over.
06:30
I can use it for something else.
06:33
Well, there are other ways of making a flap.
06:35
There are other dimensions for flaps.
06:37
If I make the flaps skinnier, I can use a bit less paper.
06:39
If I make the flap as skinny as possible,
06:42
I get to the limit of the minimum amount of paper needed.
06:45
And you can see there, it needs a quarter-circle of paper to make a flap.
06:48
There's other ways of making flaps.
06:52
If I put the flap on the edge, it uses a half circle of paper.
06:54
And if I make the flap from the middle, it uses a full circle.
06:57
So, no matter how I make a flap,
07:00
it needs some part
07:02
of a circular region of paper.
07:04
So now we're ready to scale up.
07:06
What if I want to make something that has a lot of flaps?
07:08
What do I need? I need a lot of circles.
07:11
And in the 1990s,
07:15
origami artists discovered these principles
07:17
and realized we could make arbitrarily complicated figures
07:19
just by packing circles.
07:22
And here's where the dead people start to help us out,
07:25
because lots of people have studied
07:28
the problem of packing circles.
07:31
I can rely on that vast history of mathematicians and artists
07:33
looking at disc packings and arrangements.
07:36
And I can use those patterns now to create origami shapes.
07:39
So we figured out these rules whereby you pack circles,
07:43
you decorate the patterns of circles with lines
07:45
according to more rules. That gives you the folds.
07:48
Those folds fold into a base. You shape the base.
07:50
You get a folded shape -- in this case, a cockroach.
07:53
And it's so simple.
07:57
(Laughter)
07:59
It's so simple that a computer could do it.
08:02
And you say, "Well, you know, how simple is that?"
08:05
But computers -- you need to be able to describe things
08:07
in very basic terms, and with this, we could.
08:09
So I wrote a computer program a bunch of years ago
08:12
called TreeMaker, and you can download it from my website.
08:14
It's free. It runs on all the major platforms -- even Windows.
08:16
(Laughter)
08:19
And you just draw a stick figure,
08:21
and it calculates the crease pattern.
08:23
It does the circle packing, calculates the crease pattern,
08:25
and if you use that stick figure that I just showed --
08:28
which you can kind of tell, it's a deer, it's got antlers --
08:30
you'll get this crease pattern.
08:33
And if you take this crease pattern, you fold on the dotted lines,
08:35
you'll get a base that you can then shape
08:37
into a deer,
08:40
with exactly the crease pattern that you wanted.
08:42
And if you want a different deer,
08:44
not a white-tailed deer, but you want a mule deer, or an elk,
08:46
you change the packing,
08:49
and you can do an elk.
08:51
Or you could do a moose.
08:53
Or, really, any other kind of deer.
08:55
These techniques revolutionized this art.
08:57
We found we could do insects,
09:00
spiders, which are close,
09:02
things with legs, things with legs and wings,
09:04
things with legs and antennae.
09:08
And if folding a single praying mantis from a single uncut square
09:10
wasn't interesting enough,
09:13
then you could do two praying mantises
09:15
from a single uncut square.
09:17
She's eating him.
09:19
I call it "Snack Time."
09:21
And you can do more than just insects.
09:24
This -- you can put details,
09:26
toes and claws. A grizzly bear has claws.
09:28
This tree frog has toes.
09:31
Actually, lots of people in origami now put toes into their models.
09:33
Toes have become an origami meme,
09:36
because everyone's doing it.
09:38
You can make multiple subjects.
09:41
So these are a couple of instrumentalists.
09:43
The guitar player from a single square,
09:45
the bass player from a single square.
09:48
And if you say, "Well, but the guitar, bass --
09:50
that's not so hot.
09:52
Do a little more complicated instrument."
09:54
Well, then you could do an organ.
09:56
(Laughter)
09:58
And what this has allowed is the creation
10:01
of origami-on-demand.
10:03
So now people can say, "I want exactly this and this and this,"
10:05
and you can go out and fold it.
10:08
And sometimes you create high art,
10:11
and sometimes you pay the bills by doing some commercial work.
10:13
But I want to show you some examples.
10:16
Everything you'll see here,
10:18
except the car, is origami.
10:20
(Video)
10:23
(Applause)
10:51
Just to show you, this really was folded paper.
10:54
Computers made things move,
10:57
but these were all real, folded objects that we made.
10:59
And we can use this not just for visuals,
11:03
but it turns out to be useful even in the real world.
11:06
Surprisingly, origami
11:09
and the structures that we've developed in origami
11:10
turn out to have applications in medicine, in science,
11:13
in space, in the body, consumer electronics and more.
11:16
And I want to show you some of these examples.
11:19
One of the earliest was this pattern,
11:22
this folded pattern,
11:24
studied by Koryo Miura, a Japanese engineer.
11:26
He studied a folding pattern, and realized
11:29
this could fold down into an extremely compact package
11:31
that had a very simple opening and closing structure.
11:34
And he used it to design this solar array.
11:37
It's an artist's rendition, but it flew in a Japanese telescope
11:40
in 1995.
11:43
Now, there is actually a little origami
11:45
in the James Webb Space Telescope, but it's very simple.
11:47
The telescope, going up in space,
11:50
it unfolds in two places.
11:52
It folds in thirds. It's a very simple pattern --
11:55
you wouldn't even call that origami.
11:57
They certainly didn't need to talk to origami artists.
11:59
But if you want to go higher and go larger than this,
12:02
then you might need some origami.
12:05
Engineers at Lawrence Livermore National Lab
12:07
had an idea for a telescope much larger.
12:09
They called it the Eyeglass.
12:12
The design called for geosynchronous orbit
12:14
25,000 miles up,
12:16
100-meter diameter lens.
12:18
So, imagine a lens the size of a football field.
12:21
There were two groups of people who were interested in this:
12:24
planetary scientists, who want to look up,
12:26
and then other people, who wanted to look down.
12:29
Whether you look up or look down,
12:33
how do you get it up in space? You've got to get it up there in a rocket.
12:35
And rockets are small. So you have to make it smaller.
12:38
How do you make a large sheet of glass smaller?
12:41
Well, about the only way is to fold it up somehow.
12:43
So you have to do something like this.
12:46
This was a small model.
12:48
Folded lens, you divide up the panels, you add flexures.
12:51
But this pattern's not going to work
12:53
to get something 100 meters down to a few meters.
12:56
So the Livermore engineers,
12:59
wanting to make use of the work of dead people,
13:01
or perhaps live origamists, said,
13:03
"Let's see if someone else is doing this sort of thing."
13:06
So they looked into the origami community,
13:09
we got in touch with them, and I started working with them.
13:12
And we developed a pattern together
13:14
that scales to arbitrarily large size,
13:16
but that allows any flat ring or disc
13:18
to fold down into a very neat, compact cylinder.
13:22
And they adopted that for their first generation,
13:25
which was not 100 meters -- it was a five-meter.
13:27
But this is a five-meter telescope --
13:29
has about a quarter-mile focal length.
13:31
And it works perfectly on its test range,
13:33
and it indeed folds up into a neat little bundle.
13:35
Now, there is other origami in space.
13:39
Japan Aerospace [Exploration] Agency flew a solar sail,
13:41
and you can see here that the sail expands out,
13:44
and you can still see the fold lines.
13:47
The problem that's being solved here is
13:49
something that needs to be big and sheet-like at its destination,
13:52
but needs to be small for the journey.
13:55
And that works whether you're going into space,
13:57
or whether you're just going into a body.
14:00
And this example is the latter.
14:03
This is a heart stent developed by Zhong You
14:05
at Oxford University.
14:08
It holds open a blocked artery when it gets to its destination,
14:10
but it needs to be much smaller for the trip there,
14:13
through your blood vessels.
14:16
And this stent folds down using an origami pattern,
14:18
based on a model called the water bomb base.
14:21
Airbag designers also have the problem
14:25
of getting flat sheets
14:27
into a small space.
14:29
And they want to do their design by simulation.
14:32
So they need to figure out how, in a computer,
14:34
to flatten an airbag.
14:36
And the algorithms that we developed
14:38
to do insects
14:40
turned out to be the solution for airbags
14:42
to do their simulation.
14:45
And so they can do a simulation like this.
14:47
Those are the origami creases forming,
14:50
and now you can see the airbag inflate
14:52
and find out, does it work?
14:54
And that leads
14:57
to a really interesting idea.
14:59
You know, where did these things come from?
15:01
Well, the heart stent
15:04
came from that little blow-up box
15:06
that you might have learned in elementary school.
15:08
It's the same pattern, called the water bomb base.
15:11
The airbag-flattening algorithm
15:14
came from all the developments
15:16
of circle packing and the mathematical theory
15:18
that was really developed
15:21
just to create insects -- things with legs.
15:23
The thing is, that this often happens
15:27
in math and science.
15:29
When you get math involved, problems that you solve
15:31
for aesthetic value only,
15:34
or to create something beautiful,
15:36
turn around and turn out
15:38
to have an application in the real world.
15:40
And as weird and surprising as it may sound,
15:43
origami may someday even save a life.
15:46
Thanks.
15:50
(Applause)
15:52

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About the Speaker:

Robert Lang - Origamist
Robert Lang merges mathematics with aesthetics to fold elegant modern origami. His scientific approach helps him make folds once thought impossible -- and has secured his place as one of the first great Western masters of the art.

Why you should listen

Origami, as Robert Lang describes it, is simple: "You take a creature, you combine it with a square, and you get an origami figure." But Lang's own description belies the technicality of his art; indeed, his creations inspire awe by sheer force of their intricacy. His repertoire includes a snake with one thousand scales, a two-foot-tall allosaurus skeleton, and a perfect replica of a Black Forest cuckoo clock. Each work is the result of software (which Lang himself pioneered) that manipulates thousands of mathematical calculations in the production of a "folding map" of a single creature.

The marriage of mathematics and origami harkens back to Lang's own childhood.  As a first-grader, Lang proved far too clever for elementary mathematics and quickly became bored, prompting his teacher to give him a book on origami. His acuity for mathematics would lead him to become a physicist at the California Institute of Technology, and the owner of nearly fifty patents on lasers and optoelectronics. Now a professional origami master, Lang practices his craft as both artist and engineer, one day folding the smallest of insects and the next the largest of space-bound telescope lenses.

More profile about the speaker
Robert Lang | Speaker | TED.com