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  • My talk is "Flapping Birds and Space Telescopes."

  • And you would think that should have nothing to do with one another,

  • but I hope by the end of these 18 minutes,

  • you'll see a little bit of a relation.

  • It ties to origami. So let me start.

  • What is origami?

  • Most people think they know what origami is. It's this:

  • flapping birds, toys, cootie catchers, that sort of thing.

  • And that is what origami used to be.

  • But it's become something else.

  • It's become an art form, a form of sculpture.

  • The common theme -- what makes it origami --

  • is folding is how we create the form.

  • You know, it's very old. This is a plate from 1797.

  • It shows these women playing with these toys.

  • If you look close, it's this shape, called a crane.

  • Every Japanese kid

  • learns how to fold that crane.

  • So this art has been around for hundreds of years,

  • and you would think something

  • that's been around that long -- so restrictive, folding only --

  • everything that could be done has been done a long time ago.

  • And that might have been the case.

  • But in the twentieth century,

  • a Japanese folder named Yoshizawa came along,

  • and he created tens of thousands of new designs.

  • But even more importantly, he created a language,

  • a way we could communicate,

  • a code of dots, dashes and arrows.

  • Harkening back to Susan Blackmore's talk,

  • we now have a means of transmitting information

  • with heredity and selection,

  • and we know where that leads.

  • And where it has led in origami

  • is to things like this.

  • This is an origami figure --

  • one sheet, no cuts, folding only, hundreds of folds.

  • This, too, is origami,

  • and this shows where we've gone in the modern world.

  • Naturalism. Detail.

  • You can get horns, antlers --

  • even, if you look close, cloven hooves.

  • And it raises a question: what changed?

  • And what changed is something

  • you might not have expected in an art,

  • which is math.

  • That is, people applied mathematical principles

  • to the art,

  • to discover the underlying laws.

  • And that leads to a very powerful tool.

  • The secret to productivity in so many fields --

  • and in origami --

  • is letting dead people do your work for you.

  • (Laughter)

  • Because what you can do is

  • take your problem,

  • and turn it into a problem that someone else has solved,

  • and use their solutions.

  • And I want to tell you how we did that in origami.

  • Origami revolves around crease patterns.

  • The crease pattern shown here is the underlying blueprint

  • for an origami figure.

  • And you can't just draw them arbitrarily.

  • They have to obey four simple laws.

  • And they're very simple, easy to understand.

  • The first law is two-colorability. You can color any crease pattern

  • with just two colors without ever having

  • the same color meeting.

  • The directions of the folds at any vertex --

  • the number of mountain folds, the number of valley folds --

  • always differs by two. Two more or two less.

  • Nothing else.

  • If you look at the angles around the fold,

  • you find that if you number the angles in a circle,

  • all the even-numbered angles add up to a straight line,

  • all the odd-numbered angles add up to a straight line.

  • And if you look at how the layers stack,

  • you'll find that no matter how you stack folds and sheets,

  • a sheet can never

  • penetrate a fold.

  • So that's four simple laws. That's all you need in origami.

  • All of origami comes from that.

  • And you'd think, "Can four simple laws

  • give rise to that kind of complexity?"

  • But indeed, the laws of quantum mechanics

  • can be written down on a napkin,

  • and yet they govern all of chemistry,

  • all of life, all of history.

  • If we obey these laws,

  • we can do amazing things.

  • So in origami, to obey these laws,

  • we can take simple patterns --

  • like this repeating pattern of folds, called textures --

  • and by itself it's nothing.

  • But if we follow the laws of origami,

  • we can put these patterns into another fold

  • that itself might be something very, very simple,

  • but when we put it together,

  • we get something a little different.

  • This fish, 400 scales --

  • again, it is one uncut square, only folding.

  • And if you don't want to fold 400 scales,

  • you can back off and just do a few things,

  • and add plates to the back of a turtle, or toes.

  • Or you can ramp up and go up to 50 stars

  • on a flag, with 13 stripes.

  • And if you want to go really crazy,

  • 1,000 scales on a rattlesnake.

  • And this guy's on display downstairs,

  • so take a look if you get a chance.

  • The most powerful tools in origami

  • have related to how we get parts of creatures.

  • And I can put it in this simple equation.

  • We take an idea,

  • combine it with a square, and you get an origami figure.

  • (Laughter)

  • What matters is what we mean by those symbols.

  • And you might say, "Can you really be that specific?

  • I mean, a stag beetle -- it's got two points for jaws,

  • it's got antennae. Can you be that specific in the detail?"

  • And yeah, you really can.

  • So how do we do that? Well, we break it down

  • into a few smaller steps.

  • So let me stretch out that equation.

  • I start with my idea. I abstract it.

  • What's the most abstract form? It's a stick figure.

  • And from that stick figure, I somehow have to get to a folded shape

  • that has a part for every bit of the subject,

  • a flap for every leg.

  • And then once I have that folded shape that we call the base,

  • you can make the legs narrower, you can bend them,

  • you can turn it into the finished shape.

  • Now the first step, pretty easy.

  • Take an idea, draw a stick figure.

  • The last step is not so hard, but that middle step --

  • going from the abstract description to the folded shape --

  • that's hard.

  • But that's the place where the mathematical ideas

  • can get us over the hump.

  • And I'm going to show you all how to do that

  • so you can go out of here and fold something.

  • But we're going to start small.

  • This base has a lot of flaps in it.

  • We're going to learn how to make one flap.

  • How would you make a single flap?

  • Take a square. Fold it in half, fold it in half, fold it again,

  • until it gets long and narrow,

  • and then we'll say at the end of that, that's a flap.

  • I could use that for a leg, an arm, anything like that.

  • What paper went into that flap?

  • Well, if I unfold it and go back to the crease pattern,

  • you can see that the upper left corner of that shape

  • is the paper that went into the flap.

  • So that's the flap, and all the rest of the paper's left over.

  • I can use it for something else.

  • Well, there are other ways of making a flap.

  • There are other dimensions for flaps.

  • If I make the flaps skinnier, I can use a bit less paper.

  • If I make the flap as skinny as possible,

  • I get to the limit of the minimum amount of paper needed.

  • And you can see there, it needs a quarter-circle of paper to make a flap.

  • There's other ways of making flaps.

  • If I put the flap on the edge, it uses a half circle of paper.

  • And if I make the flap from the middle, it uses a full circle.

  • So, no matter how I make a flap,

  • it needs some part

  • of a circular region of paper.

  • So now we're ready to scale up.

  • What if I want to make something that has a lot of flaps?

  • What do I need? I need a lot of circles.

  • And in the 1990s,

  • origami artists discovered these principles

  • and realized we could make arbitrarily complicated figures

  • just by packing circles.

  • And here's where the dead people start to help us out,

  • because lots of people have studied

  • the problem of packing circles.

  • I can rely on that vast history of mathematicians and artists

  • looking at disc packings and arrangements.

  • And I can use those patterns now to create origami shapes.

  • So we figured out these rules whereby you pack circles,

  • you decorate the patterns of circles with lines

  • according to more rules. That gives you the folds.

  • Those folds fold into a base. You shape the base.

  • You get a folded shape -- in this case, a cockroach.

  • And it's so simple.

  • (Laughter)

  • It's so simple that a computer could do it.

  • And you say, "Well, you know, how simple is that?"

  • But computers -- you need to be able to describe things

  • in very basic terms, and with this, we could.

  • So I wrote a computer program a bunch of years ago

  • called TreeMaker, and you can download it from my website.

  • It's free. It runs on all the major platforms -- even Windows.

  • (Laughter)

  • And you just draw a stick figure,

  • and it calculates the crease pattern.

  • It does the circle packing, calculates the crease pattern,

  • and if you use that stick figure that I just showed --

  • which you can kind of tell, it's a deer, it's got antlers --

  • you'll get this crease pattern.

  • And if you take this crease pattern, you fold on the dotted lines,

  • you'll get a base that you can then shape

  • into a deer,

  • with exactly the crease pattern that you wanted.

  • And if you want a different deer,

  • not a white-tailed deer, but you want a mule deer, or an elk,

  • you change the packing,

  • and you can do an elk.

  • Or you could do a moose.

  • Or, really, any other kind of deer.

  • These techniques revolutionized this art.

  • We found we could do insects,

  • spiders, which are close,

  • things with legs, things with legs and wings,

  • things with legs and antennae.

  • And if folding a single praying mantis from a single uncut square

  • wasn't interesting enough,

  • then you could do two praying mantises

  • from a single uncut square.

  • She's eating him.

  • I call it "Snack Time."

  • And you can do more than just insects.

  • This -- you can put details,

  • toes and claws. A grizzly bear has claws.

  • This tree frog has toes.

  • Actually, lots of people in origami now put toes into their models.

  • Toes have become an origami meme,

  • because everyone's doing it.

  • You can make multiple subjects.

  • So these are a couple of instrumentalists.

  • The guitar player from a single square,

  • the bass player from a single square.

  • And if you say, "Well, but the guitar, bass --

  • that's not so hot.

  • Do a little more complicated instrument."

  • Well, then you could do an organ.

  • (Laughter)

  • And what this has allowed is the creation

  • of origami-on-demand.

  • So now people can say, "I want exactly this and this and this,"

  • and you can go out and fold it.

  • And sometimes you create high art,

  • and sometimes you pay the bills by doing some commercial work.

  • But I want to show you some examples.

  • Everything you'll see here,

  • except the car, is origami.

  • (Video)

  • (Applause)

  • Just to show you, this really was folded paper.

  • Computers made