Hi, I'm Robert J. Lang.

I'm a physicist and origami artist

and today I've been challenged to explain origami

in five levels.

If you know a little origami

you might think it's nothing more than simple toys,

like cranes or cootie catchers,

but origami is much more than that.

Out of the vast cloud of origami possibilities

I've chosen five different levels

that illustrate the diversity of this art.

[thoughtful music]

Do you know what origami is?

Is that where you fold paper

to make different animals, like those?

Yes, in fact it is.

Have you ever done any origami before?

Nope.

[Robert] Would you like to give it a try?

Sure. Okay, so we'll do some,

but I want to tell you a little bit about origami.

Most origami follows two, I'll call them customs,

almost like rules.

It's usually from a square

and the other is it's usually folded with no cuts.

So these guys are folded from an uncut square.

That's awesome.

So you ready?

Yep. Okay.

We're going to start with a model

that every Japanese person learns in kindergarten,

it's called a crane, traditional origami design,

it's over 400 years old.

So, people have been doing what we're about to do

for 400 years. Wow.

Let's fold it in half from corner to corner, unfold it

and then we'll fold it in half the other direction,

also corner to corner but we're going to lift it up

and we're going to hold the fold with both hands.

We're going to bring these corners together,

making a little pocket and then,

this is the trickiest part of this whole design,

so you're going to put your finger underneath the top layer

and we're going to try to make that layer

fold right along the edge.

Now you see how the sides kind of want to come in

as you're doing that? Yeah.

It's called a petal fold,

it's a part of a lot of origami designs

and it's key to the crane.

Now we're ready for the magic.

We're going to hold it in between thumb and forefinger,

reach inside,

grab the skinny point that's between the two layers,

which are the wings,

and I'm going to slide it out so it pokes out at an angle.

We'll take the two wings, we spread them out to the side

and you have made your first origami crane.

Wow.

Now, this is a traditional Japanese design

but there are origami designs that have been around so long

we're not entirely sure where they originated.

We're going to learn how to fold a cootie catcher.

Okay, good.

So we'll start with the white side up

and we're going to fold it in half from corner to corner,

in one fold and now we're going to fold all four corners

to the crossing point in the center.

We'll fold it in half like a book.

On the folded side we'll take one of the folded corners

and I'm going to fold it up through all layers.

There's a pocket in the middle.

We're going to spread the pocket

and bring all four corners together.

Where you have original corners of the square,

we're gonna just pop those out.

This is one of the most satisfying moments,

I think- Yeah.

because it suddenly changes shape.

I have seen these before, my friends use these.

Yeah,

but there's something else we can do with this model.

If we set it down and push on the middle

then pop it inside out

so that three flaps come up and one stays down

and then it's called talking crow

because here's a little crow's beak and mouth.

Wow.

There's thousands of other origami designs

but these are some of the first people learn

and this was, in fact,

one of the first origami designs I learned

some 50 years ago. Wow.

So, what do you think of that?

What do you think of origami?

I think that the people that make them are talented.

It's hard.

Seeing the stuff that we've made here,

I'd bet that they could do rocket ships.

Just so much that you can do with them.

Thanks for coming.

Thanks for having me.

[thoughtful music]

A lot of origami is animals, birds and things.

There's also a branch of origami that is,

it's more abstract or geometric, called tessellations.

Tessellations, like most origami,

are folded from a single sheet of paper

but they make patterns,

whether it's woven patterns like that,

or woven patterns like this.

If you hold them up to the light

you can see patterns. Wow.

The thing that makes them cool

is they're sort of like tilings,

it looks like you could put this together

by cutting little pieces of paper and sliding them together,

but they're still one sheet.

They weren't cut?

There's no cuts in these just folding.

We can build these up from smaller building blocks of folds,

learn how to fold little pieces and put them together

in the same way that a tiling like this

looks like it's built up of little pieces.

Can you make a fold that starts at the dot

that doesn't run all the way across the paper?

How about like that? Mm-hmm.

Each of these folds is peaked like a mountain

and we call these mountain folds

but if I made it the other way, then it's shaped this way

and we call it a valley fold.

In all of origami there's just mountains and valleys.

So all the folds are reversible?

So they're all reversible and it turns out

that in every origami shape that folds flat,

it's going to be either three mountains and a valley

or, if we're looking at the backside,

three valleys and a mountain,

they always differ by two. Oh.

That's a rule of all flat origami

no matter how many folds come together at a point

and I'm going to show you a building block of tessellations,

it's called a twist

because that center square, as I unfold it,

it twists, it rotates. Twists?

If I had another twist in the same sheet of paper

I could make these folds connect with that,

and these folds connect with that.

And if I had another one up here, I could make all three.

And if I had a square array and all the folds lined up

I could make bigger and bigger arrays, like these,

because these are just very large twists.

In this case it's an octagon rather than a square,

but they're arranged in rows and columns.

And let's just try going along.

All right, there is our tessellation

with squares and hexagons.

So you have now designed and folded

your first origami tessellation

and perhaps you can see how just using this idea

of building up tiles and small building blocks

you could make tessellations as big and complex as you want.

That was cool. Yeah,

so what do you think now of origami and tessellations?

Origami, I think,

is the folding of paper to make anything in general,

from 3D things to flat things

and I think origami is about turning simple things

into complex things and it's all about patterns.

That is a great definition.

[upbeat music]

So here's a dragon fly and he's got six legs, four wings.

Wow. Here's a spider

with eight legs, ants with legs

and these, just like the crane,

are folded from a single uncut square.

What?

To figure out how to do that

we need to learn a little bit about what makes a point.

So, let's come back to the crane.

You can probably tell

that the corners of the square ended up as points,

right? Yes.

That's a corner, four corners of the square, four points.

How would you make one point out of this sheet of paper?

I'm thinking of, like, a paper airplane.

Yeah, exactly.

Actually you've discovered something pretty neat

because you made your point not from a corner

so you've already discovered one of the key insights.

Any flap, any point, leg of the ant,

takes up a circular region of paper.

Here's our boundary.

To make your point from an edge you use that much paper

and the shape, it's almost a circle.

If we take the crane

we'll see if the circles are visible in the crane pattern.

Here's the crane pattern, and here's a boundary of the wing,

and here's the other wing. Okay.

The crane has four circles

but, actually, there's a little bit of a surprise

because what about this?

There's a fifth circle, which is like that,

but does the crane have a fifth flap in it?

Let's refold it and put the wings up.

Well, yes, there is, there's another point

and that point is the fifth circle of our crane.

Okay. And to do that

we use a new technique called circle packing

in which all of the long features of the design

are represented by circles.

So, each leg becomes a circle, each wing becomes a circle

and things that can be big and thick,

like the head or the abdomen, can be points in the middle.

Now we have the basic idea of how to design the pattern,

we just count the number of legs we want.

We want a spider, if it's got let's say eight legs,

it's also got an abdomen, that's another point,

and it's got a head, so maybe that's 10 points.

If we find an arrangement of 10 circles

we should be able to fold that into the spider.

So in this book, Origami Insects II, it's one of my books

and has some patterns, and this is one of them

for a flying ladybug and, in fact,

it is exactly this flying ladybug.

We've got the crease pattern here in the circles

and you might now be able to see

which circles end up as which parts,

knowing that the largest features like the wings

are going to be the largest circles,

smaller points will be smaller circles.

So any thoughts which might be?

Well, the legs and the antenna

would probably have to be these smaller ones,

in the middle. Yeah, that's right.

[College Student] Oh, this looks like the back

'cause there's a bunch of circles all the way down,

like here. Mm-hmm, exactly.

And then the wings?

You've got four big wings

which you could see on the ends there

and then, I guess, the head.

You've got it, so you are ready to design origami.

Awesome.

Origami artists all around the world

now use ideas like this to design, not just insects,

but animals, and birds, and all sorts of things

that are, I think, unbelievably complex and realistic

but most importantly, beautiful.

Wow, that's so impressive.

I think I learned how to make one of these paper cranes

when I was in third grade but I guess I never unfolded it

to actually see where it was coming from.

And so now that it's all broken up into circles

it makes these super complicated insects and animals

and everything seem so much simpler, so that's so cool.

I'm pretty excited about it. That's so cool.

Thank you so much for telling me about this.

[upbeat music]

Whenever there's a part of a spacecraft

that is shaped somewhat like paper,

meaning it's big and flat,

we can use folding mechanisms from origami

to make it smaller.

Right. Telescopes, solar arrays,

they need to be packed into a rocket, go up,

but then expand in a very controlled, deterministic way

when they get up into space. Okay.

These are the building blocks

of many, many origami deployable shapes,

it's called a degree-4 vertex.

It's the number of lines.

So in this case, we use solid lines for mountain,

we use dash lines for valley.

We're going to fold it and use these two to illustrate

some important properties of origami mechanisms.

It's important in the study of mechanisms

to take into account the rigidity.

So what we're going to do to help simulate rigidity

is to take these rectangles

and we're going to fold them over and over

so that they just become stiff and rigid.

[Grad Student] Okay.

So this is what's called

a single degree of freedom mechanism.

You have one degree of freedom, I can choose this fold,

and then if these are perfectly rigid

every other fold angle is fully determined.

One of the key behaviors here

is that with the smaller angles up here,

the two folds that are the same parity

and the folds that are of opposite parity

move at about the same rate

but with this, as we're getting closer to 90 degrees,

we find they move at very different rates

and then at the end of the motion, the opposite happens.

This one is almost folded

but this one goes through a much larger motion so

the relative speeds differ. Right.

So when we start sticking together vertices like this,

if they're individually single degree of freedom

then we can make very large mechanisms that open and close

but with just one degree of freedom.

So, these are examples of a pattern called the Miura-Ori.

When you stretch them out

they're pretty big. Okay.

And they fold flat and a pattern almost exactly like this

was used for a solar array for a Japanese mission

that flew in 1995.

So then you like fly it up compactly

and then once you get up there,

there's like some sort of like motorized mechanism,

but you only need it on one fold.

Yeah, so typically the mechanism

will run from corner to corner,

to diagonally to opposite corners

because then you can stretch it out that way.

Notice some differences between the one you have

and the one I have

in how this one sort of opens out almost evenly

but this one opens out more one way and then the other.

Yeah.

What sort of angle would you want

so that they open up the same rate?

Infinitesimally small. Okay.

So, sadly,

the only way to get them at exactly the same rate

is when these are microscopic slivers

and then that's not useful. For sure, right, right.

And it's exactly the difference

between the motions of these two vertices.

So these angles are closer to right angles

and the closer you get to a right angle

the more asymmetry there is

between the two directions of motion.

And then the other difference is how efficiently they pack,

so these started at about the same size

but when they're flat

notice that yours is much more compact.

So if I were you making a solar array,

I'd say, oh, I want that one.

But if I say, well, I want them to open at the same rate,

then I want this one.

So, it's kind of a trade-off?

There's an engineering trade-off to get them both to work.

And there's another place

that shows up in deployable structures

in a very cool structure.

This is a folded tube, it sort of pops out like this

but it has this neat property that if you twist it quickly,

it changes color.

There's a Mars Rover application

where they need a sleeve that protects a drill

and as the drill goes down, the sleeve is going to collapse

and they're using a pattern very much like this.

Interesting.

There are many open mathematical questions

and so room for mathematicians, like yourself,

to have a big impact on the world of origami and mechanisms.

And even though those studies

are mathematically interesting,

they're going to also have real-world applications in space,

solar arrays, drills, telescopes, and more.

Any questions or thoughts about this?

If you want to send something to space

it probably makes sense to do it compactly,

so if you have something that you can fold up

and then unfold, just one of the folds,

that's going to be probably the easiest way

to get something up there

and expand it to what it needs to be.

[upbeat music]

I'm Tom Hull, I'm a math professor, a mathematician.

I've been doing origami since I was eight years old

and studying the mathematics of origami

ever since grad school, at least.

The first thing I want to show you

is origami in the real world.

This is the origami lamp.

It comes shipped flat but it folds, clip holds it together.

The lamp has LEDs on the inside

so when we power it up we get light, we have a lampshade

and we get the base.

Why does origami lend itself

to, say, this type of application?

Origami applications have in common,

is that at some stage the thing is flat

and so whenever you need to either start from a flat state

and then take it to a 3D state,

or conversely, for deployables like space,

you want to have it in a fully folded flat state

but then take it to a 3D state,

or possibly an unfolded flat state.

Whenever a flat state is involved,

origami is a really effective way

of making the transition between those states.

Another aspect of origami and origami mechanisms

that has leant itself to many different uses

is the fact that it's scalable.

When you have an origami crease pattern

like the Miura-Ori used in solar panel deployment,

the type of motion that you see happening here

will happen whether this is on a piece of paper

that's small like this, or on a larger scale,

or even on a smaller, smaller, smaller, smaller scale.

Engineers, in particular robotics engineers,

are turning to origami

toward designing mechanisms that will either be really big

or really, really small.

This looks like the most promising way

of getting nano robotics to work.

This is another real-world application

but this particular implementation

is used to make a wheel for a Rover.

Cool, so this is something

that can actually get really, really tiny

but then get big and fat and roll.

New problems arise

when we try to make origami out of things other than paper,

but also new opportunities.

An example here

which is a kind of a variant of the Miura-Ori.

It's got a three-dimensional structure.

If I stretch it one way, it expands the other

but because it has these S-bends in the pattern,

if you squeeze it, it doesn't go all the way flat.

This is a epoxy impregnated aramid fiber

and so if I put this fold pattern into it

and then compress it

and then put a skin on the top and bottom,

this becomes incredibly lightweight but incredibly strong.

Yeah!

Another origami challenge

that comes up with these patterns

is if we're going to make an aircraft out of this thing

we're going to need hundreds of yards of folded origami.

We're not going to do it by hand

and this might be the new frontier in origami engineering,

which is the design of machines

that can fold patterns that have applications.

So you're talking about a machine

that is actually folding it into this,

not just making the creases but actually folding it.

Yeah, so what goes in as sheet

and what comes out is this, or something this wide.

That's cool, yeah.

What do you see as kind of like the next big breakthrough?

Is there anything out there on the horizon

that you're just like, oh wow, this is really exciting?

It's something we've talked about a little bit

that with all the richness of behavior

of origami from a flat sheet,

it seems like there ought to be an equally rich world

of things that don't start flat

but are still made from flat sheets of paper.

So like a cone? Bi-stable properties

and you can combine them together with copies of themselves

to make cellular structures.

They're astonishingly stiff and rigid, useful for mechanics.

The thing that I think I'm the most excited about

comes from math mainly.

When I look at origami,

when I look at all these applications

or just all these different origami folds, I see structure.

Math is really about patterns.

The patterns that we see in origami

are reflecting some kind of mathematical structure

and we don't quite know yet what all of that structure is

and if we can tie a mathematical structure

that's already well-studied

to something we see happening in origami,

then we can use the math tools right away

to help solve the engineering problems

and the origami problems.

And the fact that there's so many applications to this

is really making people excited who are working in the area.

I'm really excited to see what happens with that

in the next five years or so.

[encouraging music]