The math and magic of origami | Robert Lang

翻译人员: Hua Liu 校对人员: Amy Zerotus

My talk is "Flapping Birds and Space Telescopes."

我演讲的题目是《展翅的鸟儿与太空望远镜》。

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

你会觉得他们相互之间没有联系，

but I hope by the end of these 18 minutes,

但我希望在18分钟以后，

you'll see a little bit of a relation.

你能看到一些关联。

It ties to origami. So let me start.

这与折纸有关。下面我就开始了。

What is origami?

什么是折纸？

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

很多人以为他们知道折纸是什么。它是这样的：

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

展翅的鸟儿、玩具、东西南北之类的东西。

And that is what origami used to be.

折纸术以前是这样的。

But it's become something else.

但它已经改变了。

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

它已经成为了一种艺术形式，一种雕塑形式。

The common theme -- what makes it origami --

共同的主题——折纸术的本质——

is folding is how we create the form.

是折叠，也是我们如何创造形态的。

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

你们知道，这非常古老。这是1797年的一幅画。

It shows these women playing with these toys.

上面是这些妇女们玩纸玩具的场景。

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

如果你靠近点看，它是这种形状的，叫做鹤。

Every Japanese kid

每个日本孩子

learns how to fold that crane.

都学折纸鹤。

So this art has been around for hundreds of years,

所以这种艺术已经存在了数百年，

and you would think something

你可能会想如果某种东西

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

已经存在了这么久——如此有限制性，只能折叠——

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

那么所有能做出的东西应该在很久以前就做出来了。

And that might have been the case.

实际情况也许会是如此。

But in the twentieth century,

但在20世纪，

a Japanese folder named Yoshizawa came along,

一位名为吉泽的日本折纸艺术家出现了，

and he created tens of thousands of new designs.

他创造出了数万种全新的设计。

But even more importantly, he created a language,

更重要的是，他创造了一种语言——

a way we could communicate,

一种我们可以交流的方式，

a code of dots, dashes and arrows.

一种由点、破折号和箭头构成的代码。

Harkening back to Susan Blackmore's talk,

联系到苏珊·布莱克摩尔的演讲，

we now have a means of transmitting information

我们现在有了一种通过传承与选择

with heredity and selection,

传递信息的方法，

and we know where that leads.

我们也知道它的走向。

And where it has led in origami

而它在折纸术中产生的

is to things like this.

是这样的东西。

This is an origami figure --

这是一个折纸作品：

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

一张纸，没有裁剪，只有折叠，数百次折叠。

This, too, is origami,

而这也是折纸，

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

它显示出我们在现代世界中的已经走到哪了。

Naturalism. Detail.

自然主义。细节。

You can get horns, antlers --

你可以做出犄角，鹿角——

even, if you look close, cloven hooves.

如果你靠近看，偶蹄。

And it raises a question: what changed?

这就引出一个问题：什么发生了改变？

And what changed is something

发生变化的是一种

you might not have expected in an art,

你在艺术中可能不曾期待的东西，

which is math.

那就是数学。

That is, people applied mathematical principles

也就是说，人们将数学原理应用

to the art,

到艺术中，

to discover the underlying laws.

来发现潜在的规律。

And that leads to a very powerful tool.

这就形成了一种强大的工具。

The secret to productivity in so many fields --

在众多领域提高生产力的秘密——

and in origami --

包括在折纸术中——

is letting dead people do your work for you.

是让死去的人为你工作。

(Laughter)

（笑声）

Because what you can do is

因为你所能做的

take your problem,

是将你的问题

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

转变成一个其他人已经解决的问题，

and use their solutions.

并运用他们的解决方法。

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

而我想要告诉你们，我们是如何在折纸术中做到这一点的。

Origami revolves around crease patterns.

折纸术是围绕折痕图进行的。

The crease pattern shown here is the underlying blueprint

这个折痕图就是一个折纸造型

for an origami figure.

的设计图

And you can't just draw them arbitrarily.

设计图可不能随便画。

They have to obey four simple laws.

它们必须遵循4个简单的规则。

And they're very simple, easy to understand.

它们非常简单，并且很好理解。

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

第一个规则是双可着色性。你可以用两种颜色

with just two colors without ever having

填充你想画的的折痕图而

the same color meeting.

相同的颜色不会相邻。

The directions of the folds at any vertex --

在任何顶点的折叠方向--

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

凸折法的数量，凹折法的数量--

always differs by two. Two more or two less.

之间总是相差两下。多折或少折两下。

Nothing else.

就这么简单。

If you look at the angles around the fold,

如果观察折痕周围的角，

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

你会发现在数围成一圈的角时，

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

所有列为偶数的角加起来是一条直线。

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

所有列为奇数的角加起来是一个直线。

And if you look at how the layers stack,

接下来，如果观察这些纸是怎么叠加起来的，

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

你会发现不论怎样叠加褶层和纸片，

a sheet can never

纸片永远不能

penetrate a fold.

穿透褶层。

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

这就是四则简单的规则。在折纸艺术中这就是全部。

All of origami comes from that.

所有的折纸都源于这些。

And you'd think, "Can four simple laws

现在你觉得：“那些复杂的工艺

give rise to that kind of complexity?"

能是从四则简单的规则中衍生出来的吗？”

But indeed, the laws of quantum mechanics

但是，事实上，量子力学的法则

can be written down on a napkin,

可以在一张餐巾纸上写出来。

and yet they govern all of chemistry,

而它们可以支配所有的化学，

all of life, all of history.

甚至生活和历史的全部。

If we obey these laws,

如果遵循这些规则，

we can do amazing things.

我们能做出令人吃惊的事。

So in origami, to obey these laws,

所以折纸时，在遵循这些规则的情况下，

we can take simple patterns --

我们可以做出简单的样式--

like this repeating pattern of folds, called textures --

比如这个重复的折叠样式，叫做纹理--

and by itself it's nothing.

虽然这样单独看起来很普通。

But if we follow the laws of origami,

但如果我们遵守折纸的规则，

we can put these patterns into another fold

我们能把这些样式加入另一种折法，

that itself might be something very, very simple,

这种折法本身非常非常的简单。

but when we put it together,

但当我们把它加进来，

we get something a little different.

会得到很不一样的东西。

This fish, 400 scales --

这条鱼有400片鱼鳞，

again, it is one uncut square, only folding.

同样，它是一张没被剪过的正方形纸张。

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

如果你不想折400片鱼鳞，

you can back off and just do a few things,

你可以退而求其次，做些简单的折叠

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

得到一只乌龟的背壳或脚趾。

Or you can ramp up and go up to 50 stars

或者可以提升成为一面拥有

on a flag, with 13 stripes.

50颗星星和13条横条的旗子（美国国旗）。

And if you want to go really crazy,

如果你想做些疯狂的事情，

1,000 scales on a rattlesnake.

这有一条有1000片鳞片的响尾蛇。

And this guy's on display downstairs,

这个作品展示在楼下，

so take a look if you get a chance.

所以你们有机会可以看看。

The most powerful tools in origami

在折纸艺术中，最有用的方法

have related to how we get parts of creatures.

和我们怎样构造生物的一部分有关。

And I can put it in this simple equation.

我可以用一个简单的等式来解释。

We take an idea,

我们产生了一个想法，

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

把它与张纸片结合，就能得到一个折纸作品。

(Laughter)

（笑声）

What matters is what we mean by those symbols.

重要的是这些符号代表什么。

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

你们可能会问：“真的能做到那么具体吗？

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

我是说一只鹿角虫有两个点状的嘴，

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

和触角。你真的能做到具体到细节吗？”

And yeah, you really can.

是的，真的可以。

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

那该怎么做呢？我们把它分成

into a few smaller steps.

几个小步骤。

So let me stretch out that equation.

为此，让我来展开这个等式。

I start with my idea. I abstract it.

我先从我的构思开始，使它抽象化。

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

什么是最抽象的形式呢？线条画。

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

然后从这个线条画，我得用某种方式得到折叠的式样，

that has a part for every bit of the subject,

并且包括想要表现对象的所有部分。

a flap for every leg.

一片三角形折叠对应一条腿。

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

然后，我们称这个折叠的式样为基础。

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

你可以使它的腿变细，使其弯曲，

you can turn it into the finished shape.

你可以把它做成成品。

Now the first step, pretty easy.

第一步：很简单。

Take an idea, draw a stick figure.

做出一个构思，画一幅线条图。

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

最后一步也不是很难，但中间的一步--

going from the abstract description to the folded shape --

把抽象的描绘变为折叠的式样--

that's hard.

这很难。

But that's the place where the mathematical ideas

但就是在这，数学理论让我们

can get us over the hump.

翻越难关。

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

我要向你们展示怎样做，

so you can go out of here and fold something.

这样离开这里后，你们可以叠出些东西。

But we're going to start small.

但我们要先从小的开始。

This base has a lot of flaps in it.

这个基础有很多片状物。

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

我们要学习怎样做一个片状物。

How would you make a single flap?

你会怎样叠一个片状物呢？

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

拿一张正方形的纸，把它对折再对折，

until it gets long and narrow,

直到它变得又长又细，

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

然后这个的尾部就是一个片状物。

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

我能用它做一条腿，一只手臂，和所有相似的东西。

What paper went into that flap?

在片状物中是什么样的纸呢？

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

如果把它展开去看它的折痕图，

you can see that the upper left corner of that shape

你们可以看到在纸片的左上角的形状

is the paper that went into the flap.

就是构成片状物的纸。

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

所以那就是一个片状物，和所有剩下的纸。

I can use it for something else.

我可以用剩下的部分做点别的。

Well, there are other ways of making a flap.

也有另外的做片状物的方法。

There are other dimensions for flaps.

也有不同形状的片状物。

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

如果把片状物叠得更细一些，所用的纸会更少。

If I make the flap as skinny as possible,

如果把片状物尽可能的叠细，

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

就能只用片状物所需的最少的纸。

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

就像你们所看到的，只需要纸上四分之一个圆就可以作出一个片状物。

There's other ways of making flaps.

还有别的做片状物的方法。

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

如果把片状物放在纸片边上，就需要一个半圆的纸。

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

如果把片状物放在纸片的中心，就需要一整圆。

So, no matter how I make a flap,

就是说不论怎样叠，

it needs some part

片状物是由

of a circular region of paper.

纸上圆形区域的一部分做成的。

So now we're ready to scale up.

现在让我们来提升到新的水平。

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

如果要叠一个有很多片状物的东西该怎么办呢？

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

我需要什么？我需要很多的圆。

And in the 1990s,

在二十世纪九十年代，

origami artists discovered these principles

折纸艺术家发现了这些规则，

and realized we could make arbitrarily complicated figures

并了解到我们可以通过组合圆形

just by packing circles.

来叠出任意复杂的形状。

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

这就是那些死去的人能帮到我们的地方。

because lots of people have studied

因为很多人都研究过

the problem of packing circles.

组合圆形的问题。

I can rely on that vast history of mathematicians and artists

我可以依赖那些有关圆的组合和排列的

looking at disc packings and arrangements.

大量的数学与艺术的历史。

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

然后我可以用这些式样来制造折纸的形状。

So we figured out these rules whereby you pack circles,

我们可以依据这些规则来组合圆形，

you decorate the patterns of circles with lines

依据更多的规矩我们可以

according to more rules. That gives you the folds.

用线条来装饰圆。这就有了折叠线。

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

沿这些线折叠就可以得到大体形状。你们就做出了大体的形状。

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

你们得到一个折叠的形状，在这里，是一只蟑螂。

And it's so simple.

而且它非常的简单。

(Laughter)

（笑声）

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

因为它很简单，电脑就可以把它做出来。

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

你们可能问“那能有多简单呢？”

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

但是要用电脑，你们需要用最基本的方法

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

来描述一件事物。而这里我们可以做到。

So I wrote a computer program a bunch of years ago

所以我在很多年前写了一个电脑程序，

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

叫做TreeMaker(造树者），你们可以在我的网页上下载它。

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

它是免费的。它可以在大部分的操作系统里面运行，甚至在Windows里。

(Laughter)

（笑声）

And you just draw a stick figure,

然后你们就可以自己画一个线条图，

and it calculates the crease pattern.

这个程序会根据线条图计算折痕。

It does the circle packing, calculates the crease pattern,

这个程序可以排列圆形，计算折痕，

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

还有如果你们用刚才我展示的线条图，

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

你们可以看出它是一只有角的鹿，

you'll get this crease pattern.

你们就可以得到这个折痕图。

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

用这个折痕图，折叠有虚线的地方，

you'll get a base that you can then shape

你们就能得到一个基础，然后再用

into a deer,

你们想用的方法

with exactly the crease pattern that you wanted.

叠出一只鹿。

And if you want a different deer,

如果你们想要一只不同种的鹿，

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

而不是白尾鹿，

you change the packing,

你们可以改变圆形的排列，

and you can do an elk.

然后得到一只麋鹿。

Or you could do a moose.

或是一只驼鹿。

Or, really, any other kind of deer.

或是其它任何一种鹿。

These techniques revolutionized this art.

这些技术改革了这门艺术。

We found we could do insects,

我们发现我们可以叠出昆虫，

spiders, which are close,

或是相近的蜘蛛，

things with legs, things with legs and wings,

有脚的东西，有脚和翅膀的东西，

things with legs and antennae.

和有脚和触角的东西。

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

如果用一张没剪过的正方形纸叠一只螳螂

wasn't interesting enough,

还不够有趣的话，

then you could do two praying mantises

你们可以用一张没剪过的正方形纸

from a single uncut square.

叠两只螳螂。

She's eating him.

她在吃他。

I call it "Snack Time."

我称之为“点心时间”。

And you can do more than just insects.

你们能做的不只是昆虫。

This -- you can put details,

你们可以把它做到有细节，

toes and claws. A grizzly bear has claws.

像指头和爪子。一只有爪子的北美洲灰熊。

This tree frog has toes.

和这只有脚趾的树蛙。

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

实际上，在折纸艺术中有很多人把指头加入到他们的模型中。

Toes have become an origami meme,

指头变成了折纸艺术的文化基因。

because everyone's doing it.

因为每个人都在做。

You can make multiple subjects.

你可以做出多种的物体。

So these are a couple of instrumentalists.

像这里有一些音乐家。

The guitar player from a single square,

一个正方形做出的吉他手。

the bass player from a single square.

一个正方形做出的贝斯手。

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

如果你说，“好吧，但吉他和贝斯

that's not so hot.

不够帅。

Do a little more complicated instrument."

做些更复杂的乐器吧。”

Well, then you could do an organ.

那你可以做一架风琴。

(Laughter)

（笑声）

And what this has allowed is the creation

所以在这个世界里我们能

of origami-on-demand.

做出所需要的创造。

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

如果现在有人说，我想要这个这个还有这个。

and you can go out and fold it.

你就可以精确的把它们叠出来。

And sometimes you create high art,

有时可以做纯艺术。

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

有时可以做些商品卖钱。

But I want to show you some examples.

但是我想给你们看一些例子。

Everything you'll see here,

除了车子，

except the car, is origami.

你们将看到的所有东西都是折纸。

(Video)

（影片）

(Applause)

（掌声）

Just to show you, this really was folded paper.

就是想展示给你们这些真实的折纸。

Computers made things move,

电脑使所有的东西动起来。

but these were all real, folded objects that we made.

但是这些折纸全都是货真价实的。

And we can use this not just for visuals,

我们不只可以在视觉上运用到折纸艺术，

but it turns out to be useful even in the real world.

它实际上在现实世界中也很有用。

Surprisingly, origami

令人惊奇的，折纸

and the structures that we've developed in origami

和从折纸中发展出来的结构

turn out to have applications in medicine, in science,

可以在医药学，科学，

in space, in the body, consumer electronics and more.

太空，身体和电子产品等等上得到应用。

And I want to show you some of these examples.

我想展示一些例子。

One of the earliest was this pattern,

在最早的应用中有这样一个样式，

this folded pattern,

折纸样式，

studied by Koryo Miura, a Japanese engineer.

由日本的工程师Koryo Miura发明的。

He studied a folding pattern, and realized

他研究这个折纸样式，然后发现

this could fold down into an extremely compact package

可以折出很紧凑的包装，

that had a very simple opening and closing structure.

有很简单的开口和闭合结构。

And he used it to design this solar array.

他应用这个技术设计了这个太阳能电池板。

It's an artist's rendition, but it flew in a Japanese telescope

这是一个艺术家的表演，但它在1995应用到了

in 1995.

一架日本望远镜。

Now, there is actually a little origami

现在在詹姆斯韦伯太空望远镜中只有

in the James Webb Space Telescope, but it's very simple.

一点点的折纸艺术，但它十分之简单。

The telescope, going up in space,

这架进入太空的望远镜

it unfolds in two places.

在两处展开。

It folds in thirds. It's a very simple pattern --

它在第三个处折叠。它是一个很简单的式样，

you wouldn't even call that origami.

你都不会把它称作折纸。

They certainly didn't need to talk to origami artists.

这些科学家的确不用跟折纸艺术家讨论。

But if you want to go higher and go larger than this,

但当你要更深入的研究时，

then you might need some origami.

折纸术是必需的。

Engineers at Lawrence Livermore National Lab

劳伦斯利物穆尔国家实验室的工程师们

had an idea for a telescope much larger.

有一个关于一个更大的望远镜的构想。

They called it the Eyeglass.

他们称之为“镜片”。

The design called for geosynchronous orbit

这个设计需要同步轨道，

25,000 miles up,

高于地面26000英里，

100-meter diameter lens.

和直径100米的镜片。

So, imagine a lens the size of a football field.

所以镜片有一个橄榄球场那么大。

There were two groups of people who were interested in this:

有两类人对这个望远镜有兴趣：

planetary scientists, who want to look up,

想要观察太空的行星学家，

and then other people, who wanted to look down.

和其他想要观察地球的人。

Whether you look up or look down,

不论你想观察什么，

how do you get it up in space? You've got to get it up there in a rocket.

该怎么上太空呢？你需要一个火箭。

And rockets are small. So you have to make it smaller.

而且火箭一般都很小。所以你需要把望远镜做的小一些。

How do you make a large sheet of glass smaller?

怎么把一大片玻璃变小呢？

Well, about the only way is to fold it up somehow.

唯一的办法就是折叠。

So you have to do something like this.

所以你要做这样的事，

This was a small model.

这一个小型的模型。

Folded lens, you divide up the panels, you add flexures.

对于镜片，你把板面分区然后加上弯曲。

But this pattern's not going to work

但是这个样式不能把100米的东西

to get something 100 meters down to a few meters.

变成几米。

So the Livermore engineers,

所以利物穆尔的工程师们，

wanting to make use of the work of dead people,

想要利用那些死去的人的成果，

or perhaps live origamists, said,

或是活着的折纸艺术家的成果。

"Let's see if someone else is doing this sort of thing."

工程师们说“看看有没有别人在做这类事。”

So they looked into the origami community,

所以他们研究折纸圈。

we got in touch with them, and I started working with them.

我们和折纸艺术家取得联系，而我开始和他们一起工作。

And we developed a pattern together

我们一起开发了一个

that scales to arbitrarily large size,

可以应用到任意大小，

but that allows any flat ring or disc

但可以允许所有的平面环或圆盘

to fold down into a very neat, compact cylinder.

折成一个整洁紧凑的圆柱体的样式。

And they adopted that for their first generation,

他们在第一代的望远镜中采用了这个样式。

which was not 100 meters -- it was a five-meter.

而第一代并不是100米而是5米。

But this is a five-meter telescope --

但是这个5米的望远镜

has about a quarter-mile focal length.

有0.25英里的焦距。

And it works perfectly on its test range,

而且在它的测试范围内效果很好。

and it indeed folds up into a neat little bundle.

它也的确被叠成了一小捆。

Now, there is other origami in space.

现在，还有别的折纸术应用到太空中。

Japan Aerospace [Exploration] Agency flew a solar sail,

日本航空【探索者】部门发射了一个太阳光帆。

and you can see here that the sail expands out,

你们可以看到帆伸展开，

and you can still see the fold lines.

还有帆上的折叠线。

The problem that's being solved here is

在这里所被解决的问题是

something that needs to be big and sheet-like at its destination,

做出了一个在旅途中很小

but needs to be small for the journey.

但在目的地很大的薄片状的物体。

And that works whether you're going into space,

这个可以作用于当你想进入太空，

or whether you're just going into a body.

或是想进入人的身体时。

And this example is the latter.

这个例子就是进入人身体的。

This is a heart stent developed by Zhong You

这是由牛津大学的钟游发明的

at Oxford University.

心脏手术支架。

It holds open a blocked artery when it gets to its destination,

它在到达目的地时会打开被堵塞的动脉血管。

but it needs to be much smaller for the trip there,

但在旅途中它需要变得很小才能通过

through your blood vessels.

你的血管。

And this stent folds down using an origami pattern,

这个支架运用一种折纸术被叠小。

based on a model called the water bomb base.

我们称这个模型为水弹模型。

Airbag designers also have the problem

安全气囊的设计师也遇到了同样的

of getting flat sheets

把大薄片塞进小空间里的

into a small space.

问题。

And they want to do their design by simulation.

而且他们都是通过仿真技术来做设计。

So they need to figure out how, in a computer,

所以他们需要在电脑里研究出

to flatten an airbag.

怎样使安全气囊变平。

And the algorithms that we developed

我们所开发出的叠昆虫的

to do insects

算法在这里变成了

turned out to be the solution for airbags

在仿真技术中解决安全气囊问题的

to do their simulation.

方法。

And so they can do a simulation like this.

所以设计师可以做一个这个的模仿。

Those are the origami creases forming,

那些就是折纸的折痕，

and now you can see the airbag inflate

现在你们所看到的就是正在放气的安全气囊

and find out, does it work?

并且大家可以知道这方法管不管用。

And that leads

这个例子实际上可以

to a really interesting idea.

推导出一个十分有趣的构想。

You know, where did these things come from?

你们知道，这些发明设计都是从哪来么？

Well, the heart stent

这个心脏手术支架

came from that little blow-up box

是从大家小学就学到的

that you might have learned in elementary school.

纸气球中衍生来的。

It's the same pattern, called the water bomb base.

它们有着相同的构造，称之为“水弹模型”。

The airbag-flattening algorithm

那个使安全气囊变平的算法是

came from all the developments

从那些实际上只是

of circle packing and the mathematical theory

发明出来用来叠昆虫，

that was really developed

也就是有腿的东西，

just to create insects -- things with legs.

的数学理论。

The thing is, that this often happens

其实呢，这样的事经常

in math and science.

发生在数学和科学里面。

When you get math involved, problems that you solve

当你运用数学，解决

for aesthetic value only,

你纯粹为了美学价值

or to create something beautiful,

或是创造美而想解决的问题时，

turn around and turn out

实际上结果反过来

to have an application in the real world.

在现实世界中也可以应用。

And as weird and surprising as it may sound,

而且即使听上去很奇怪，

origami may someday even save a life.

折纸术有一天可能会救人一命。

Thanks.

谢谢。

(Applause)

（掌声）