TED2008
Robert Lang: The math and magic of origami
Robert Lang全新型態的摺紙
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Robert Lang是新式摺紙的先驅，他運用數學及工程學上的原理，設計出令人驚艷的摺紙作品，不僅有藝術之美，還有實用性。
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. Full bio
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
Double-click the English transcript below to play the video.
00:18
My talk is "Flapping Birds and Space Telescopes."
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我的演講題目是「振翅的鳥與太空望遠鏡」，
00:21
And you would think that should have nothing to do with one another,
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乍看之下兩者毫無關連，
00:23
but I hope by the end of these 18 minutes,
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但我希望在我演說的18分鐘結束後，
00:26
you'll see a little bit of a relation.
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各位就可以看得出端倪。
00:29
It ties to origami. So let me start.
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這一切都與摺紙有關，所以我們開始吧。
00:30
What is origami?
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摺紙是什麼？
00:32
Most people think they know what origami is. It's this:
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大部份的人都認為自己瞭解摺紙，
00:35
flapping birds, toys, cootie catchers, that sort of thing.
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不過就是紙鶴、玩具、東南西北遊戲這類的東西。
00:38
And that is what origami used to be.
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以前的摺紙確實就是這些，
00:40
But it's become something else.
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但現在產生了新的變化。
00:42
It's become an art form, a form of sculpture.
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摺紙現在變成一種藝術、一種雕塑，
00:44
The common theme -- what makes it origami --
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摺紙的特點，也就是摺紙的精髓，
00:46
is folding is how we create the form.
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在於摺的動作，在於成形的過程。
00:50
You know, it's very old. This is a plate from 1797.
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摺紙是一項非常老的技藝，這是一幅1797年的插圖，
00:53
It shows these women playing with these toys.
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裡面的女士們正在玩一些東西，
00:55
If you look close, it's this shape, called a crane.
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仔細一看，原來是紙鶴。
00:58
Every Japanese kid
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每個日本小孩
01:00
learns how to fold that crane.
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都會摺紙鶴，
01:02
So this art has been around for hundreds of years,
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這門藝術已經存在數百年之久，
01:04
and you would think something
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我們很自然會認為，
01:06
that's been around that long -- so restrictive, folding only --
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存在了這麼久的技藝，就只有「摺」這個動作，
01:09
everything that could be done has been done a long time ago.
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能玩的花樣老早就玩遍了。
01:12
And that might have been the case.
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也許早期的確是這樣，
01:14
But in the twentieth century,
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但是到了二十世紀，
01:16
a Japanese folder named Yoshizawa came along,
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出現了一名叫做吉澤章的摺紙師傅，
01:19
and he created tens of thousands of new designs.
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他發明上萬種新的摺紙設計。
01:22
But even more importantly, he created a language,
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更重要的是，他發明了一種摺紙語言，
01:25
a way we could communicate,
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一種摺紙的溝通方式，
01:27
a code of dots, dashes and arrows.
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用點、虛線和箭頭所組成。
01:29
Harkening back to Susan Blackmore's talk,
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如同蘇珊．布萊克摩爾在TED所發表的演說，
01:31
we now have a means of transmitting information
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現在我們已經發展出一種資訊傳遞方式，
01:33
with heredity and selection,
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經過不斷地傳承與改良，
01:36
and we know where that leads.
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我們都知道最後會有什麼結果。
01:38
And where it has led in origami
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而目前在摺紙的領域裡，
01:40
is to things like this.
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就發展出這樣的成果。
01:42
This is an origami figure --
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這是一個摺紙作品，
01:44
one sheet, no cuts, folding only, hundreds of folds.
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一張紙、沒有切割、只靠翻摺、有數百道摺痕。
01:50
This, too, is origami,
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這也是摺紙，
01:52
and this shows where we've gone in the modern world.
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看得出現代摺紙的發展趨勢，
01:55
Naturalism. Detail.
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也就是注重自然主義與細節。
01:57
You can get horns, antlers --
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你可以做出牛角、鹿角，
01:59
even, if you look close, cloven hooves.
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再看仔細一點，還可以做出分趾蹄。
02:01
And it raises a question: what changed?
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看了不禁讓人好奇，這和以前有何不同？
02:04
And what changed is something
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不同的地方，
02:06
you might not have expected in an art,
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是你從來不會與藝術聯想在一起的，
02:09
which is math.
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就是數學。
02:11
That is, people applied mathematical principles
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也就是說，
02:13
to the art,
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現代摺紙應用了數學方法，
02:16
to discover the underlying laws.
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開發潛藏在其中的規則。
02:18
And that leads to a very powerful tool.
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在此不得不提到一項非常有用的工具，
02:21
The secret to productivity in so many fields --
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同時也是其他許多領域提升生產力的祕訣，
02:23
and in origami --
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摺紙也不例外，
02:25
is letting dead people do your work for you.
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就是讓死去的人幫你做事。
02:28
(Laughter)
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（笑聲）
02:29
Because what you can do is
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你要做的，
02:31
take your problem,
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就是把你的問題
02:33
and turn it into a problem that someone else has solved,
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和以前別人所遇到的問題做比對，
02:36
and use their solutions.
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再利用他們已經想出的辦法來解決。
02:38
And I want to tell you how we did that in origami.
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我來告訴各位我們在摺紙時是怎麼解決問題的。
02:41
Origami revolves around crease patterns.
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摺紙就是在有摺痕圖案的紙上作業，
02:43
The crease pattern shown here is the underlying blueprint
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現在各位所看到的，
02:46
for an origami figure.
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是某個摺紙作品的草稿底圖。
02:48
And you can't just draw them arbitrarily.
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當然，你不可能隨意畫出這些線，
02:50
They have to obey four simple laws.
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至少要遵循四個簡單的法則，
02:53
And they're very simple, easy to understand.
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四個簡單又容易記住的法則。
02:55
The first law is two-colorability. You can color any crease pattern
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第一個法則是雙色運用，在任何一張草圖上，
02:58
with just two colors without ever having
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都可以運用二種顏色來上色，
03:00
the same color meeting.
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但相同顏色不得相鄰。
03:03
The directions of the folds at any vertex --
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在任一頂點要摺出線時，
03:06
the number of mountain folds, the number of valley folds --
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山摺線和谷摺線的摺線次數，
03:09
always differs by two. Two more or two less.
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永遠都差二次，不管是多二次還是少二次，
03:11
Nothing else.
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就是這樣。
03:13
If you look at the angles around the fold,
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看看摺線旁的角，
03:15
you find that if you number the angles in a circle,
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如果你將圓圈裡的角編號，
03:17
all the even-numbered angles add up to a straight line,
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將偶數角摺疊起來就是直線，
03:20
all the odd-numbered angles add up to a straight line.
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而將奇數角摺疊起來也是一條直線。
03:23
And if you look at how the layers stack,
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再看看各個層次的堆疊，
03:25
you'll find that no matter how you stack folds and sheets,
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不管你怎麼堆疊各個摺痕與紙張，
03:28
a sheet can never
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紙張永遠不能
03:30
penetrate a fold.
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穿透摺痕。
03:32
So that's four simple laws. That's all you need in origami.
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摺紙就只需要這四個簡單的法則，
03:35
All of origami comes from that.
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所有的摺紙都是從這四個法則衍生出來，
03:37
And you'd think, "Can four simple laws
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你會想：「用這四個法則，
03:39
give rise to that kind of complexity?"
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就可以創造出那麼複雜的摺紙嗎？」
03:41
But indeed, the laws of quantum mechanics
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看看量子力學的定律，
03:43
can be written down on a napkin,
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不也是可以寫在一張紙巾上嗎？
03:45
and yet they govern all of chemistry,
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但他們卻可以統禦所有的化學、
03:47
all of life, all of history.
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生命科學和歷史啊！
03:49
If we obey these laws,
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如果我們遵循這些法則，
03:51
we can do amazing things.
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我們可以做出很棒的東西，
03:53
So in origami, to obey these laws,
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在摺紙這門學問裡，只要遵循這些法則，
03:55
we can take simple patterns --
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我們就可以將簡單的圖案，
03:57
like this repeating pattern of folds, called textures --
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像是這類重覆對摺的圖案，我們稱之為結構，
04:00
and by itself it's nothing.
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單一的結構做不出什麼東西，
04:02
But if we follow the laws of origami,
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但如果我們運用摺紙的四個法則，
04:04
we can put these patterns into another fold
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我們就可以將這種圖案放進另一種摺法裡，
04:07
that itself might be something very, very simple,
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呈現出一種很簡單的圖樣，
04:09
but when we put it together,
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再把它大量運用之後，
04:11
we get something a little different.
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我們就可以得出一些不一樣的圖形。
04:13
This fish, 400 scales --
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看看這條魚，有400個鱗片，
04:16
again, it is one uncut square, only folding.
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再次強調，這是用一張紙摺出來的，完全沒有剪裁。
04:20
And if you don't want to fold 400 scales,
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如果你不想摺400個鱗片，
04:22
you can back off and just do a few things,
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那就摺少一點，再加點別的，
04:24
and add plates to the back of a turtle, or toes.
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做出烏龜的甲殼，或是腳趾。
04:27
Or you can ramp up and go up to 50 stars
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也可以更進一步摺50顆星星，
04:30
on a flag, with 13 stripes.
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加13條線就是星條旗了。
04:33
And if you want to go really crazy,
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如果你真的想不開，
04:36
1,000 scales on a rattlesnake.
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可以做有一千個鱗片的響尾蛇，
04:38
And this guy's on display downstairs,
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這隻蛇在樓下展覽著，
04:40
so take a look if you get a chance.
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有機會可以去看看。
04:43
The most powerful tools in origami
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摺紙裡最有力的工具，
04:45
have related to how we get parts of creatures.
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就是解構物件的工具。
04:48
And I can put it in this simple equation.
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可以用一個簡單的方程式來解釋，
04:50
We take an idea,
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也就是先想出構想，
04:52
combine it with a square, and you get an origami figure.
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再用一張紙，就可以摺出摺紙作品。
04:55
(Laughter)
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（笑聲）
04:59
What matters is what we mean by those symbols.
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在這個程式裡，真正重要的是運算符號。
05:01
And you might say, "Can you really be that specific?
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你會說：「可不可以再說清楚一點啊？
05:04
I mean, a stag beetle -- it's got two points for jaws,
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你看，一隻鍬形蟲有二個顎，
05:06
it's got antennae. Can you be that specific in the detail?"
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還有觸角，可不可以把細節再講清楚一點？」
05:10
And yeah, you really can.
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當然可以啊...
05:13
So how do we do that? Well, we break it down
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該怎做呢？我們把作法
05:16
into a few smaller steps.
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再拆解成更小的步驟，
05:18
So let me stretch out that equation.
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把這個方程式再展開來，
05:20
I start with my idea. I abstract it.
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先想出構想，畫出個輪廓，
05:23
What's the most abstract form? It's a stick figure.
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輪廓要怎麼畫？用線條描繪出軀幹就行了，
05:26
And from that stick figure, I somehow have to get to a folded shape
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有了這個輪廓，就可以創造出摺紙作品，
05:29
that has a part for every bit of the subject,
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並生動摺出物件的各個部分，
05:32
a flap for every leg.
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包括每一隻腳。
05:34
And then once I have that folded shape that we call the base,
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一旦我們以這個做基礎，
05:37
you can make the legs narrower, you can bend them,
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你就可以做出更細的腳，還可以彎折腳的角度，
05:40
you can turn it into the finished shape.
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把成品做出來。
05:42
Now the first step, pretty easy.
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第一個步驟：很簡單，
05:44
Take an idea, draw a stick figure.
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只要有構想，再畫出輪廓就行了。
05:46
The last step is not so hard, but that middle step --
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最後一個步驟也沒有那麼難，但是中間這個步驟，
05:49
going from the abstract description to the folded shape --
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是要從輪廓做出物件，
05:52
that's hard.
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真的很難。
05:54
But that's the place where the mathematical ideas
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這裡就要靠一些數學頭腦
05:56
can get us over the hump.
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才能幫我們解決問題了。
05:58
And I'm going to show you all how to do that
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我會告訴各位每一個細節，
06:00
so you can go out of here and fold something.
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所以在散場之後，各位就會摺出些東西來了。
06:02
But we're going to start small.
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我們先從小的東西開始，
06:04
This base has a lot of flaps in it.
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這個基本形有很多分岔的肢體，
06:06
We're going to learn how to make one flap.
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我們先來學怎麼製作出各個肢體來，
06:09
How would you make a single flap?
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肢體要怎麼做？
06:11
Take a square. Fold it in half, fold it in half, fold it again,
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拿一張正方形的紙，對摺、對摺、再對摺，
06:14
until it gets long and narrow,
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讓它變得又長又細，
06:16
and then we'll say at the end of that, that's a flap.
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最後就變成了一個肢體。
06:18
I could use that for a leg, an arm, anything like that.
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這可以運用在腳、手臂或其他類似的肢體。
06:21
What paper went into that flap?
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這個肢體是用正方形的哪一個部分做成的呢？
06:23
Well, if I unfold it and go back to the crease pattern,
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把成品打開來，看看那些摺痕，
06:25
you can see that the upper left corner of that shape
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可以看到正方形的左上角
06:28
is the paper that went into the flap.
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就是摺出這個肢體的部分。
06:30
So that's the flap, and all the rest of the paper's left over.
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我們完成了肢體，還有其他部分的紙剩下來，
06:33
I can use it for something else.
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我可以用來做些別的。
06:35
Well, there are other ways of making a flap.
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要做出肢體還有別的方法，
06:37
There are other dimensions for flaps.
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還有別種形式的肢體，
06:39
If I make the flaps skinnier, I can use a bit less paper.
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如果我可以把肢體做得瘦一點，就可以用少一點的紙，
06:42
If I make the flap as skinny as possible,
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如果我做得夠瘦，
06:45
I get to the limit of the minimum amount of paper needed.
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就可以把紙的用量減到最少。
06:48
And you can see there, it needs a quarter-circle of paper to make a flap.
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看看那裡，我用四分之一圓就可以做出一個肢體，
06:52
There's other ways of making flaps.
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當然還有其他做肢體的方法。
06:54
If I put the flap on the edge, it uses a half circle of paper.
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如果我把肢體放在邊緣，就要用到二分之一圓，
06:57
And if I make the flap from the middle, it uses a full circle.
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但如果把肢體做在中間，就要用掉一整個圓。
07:00
So, no matter how I make a flap,
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所以不管怎麼摺出一個肢體，
07:02
it needs some part
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至少都會用去
07:04
of a circular region of paper.
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某個部分的圓才能摺出來。
07:06
So now we're ready to scale up.
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接下來我們就可以往下做，
07:08
What if I want to make something that has a lot of flaps?
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如果我想要做一個有很多肢體的東西呢？
07:11
What do I need? I need a lot of circles.
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需要的是什麼？就是很多個圓圈。
07:15
And in the 1990s,
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在1990年代，
07:17
origami artists discovered these principles
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摺紙師傅發現這些原理，
07:19
and realized we could make arbitrarily complicated figures
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只要把圓圈組合起來，
07:22
just by packing circles.
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就可以隨意做出複雜的作品。
07:25
And here's where the dead people start to help us out,
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這時候，那些死去的人就幫得上忙了。
07:28
because lots of people have studied
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因為很多人研究過
07:31
the problem of packing circles.
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圓圈堆疊這個題目，
07:33
I can rely on that vast history of mathematicians and artists
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我可以參考歷代數學家和藝術家的成果，
07:36
looking at disc packings and arrangements.
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看看圓圈要怎麼堆疊和組合，
07:39
And I can use those patterns now to create origami shapes.
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再運用到我的摺紙作品上。
07:43
So we figured out these rules whereby you pack circles,
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我們在堆疊圓圈的過程裡發現了一些規則，
07:45
you decorate the patterns of circles with lines
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我們還運用其他的規則來畫出線條與圓圈，
07:48
according to more rules. That gives you the folds.
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這樣就可以畫出摺痕了，
07:50
Those folds fold into a base. You shape the base.
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這些摺痕可以摺出一個大概輪廓，
07:53
You get a folded shape -- in this case, a cockroach.
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細部修正就可以完成一個摺紙作品，像是這隻蟑螂，
07:57
And it's so simple.
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就是這麼簡單。
07:59
(Laughter)
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（笑聲）
08:02
It's so simple that a computer could do it.
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簡單到可以用電腦解決。
08:05
And you say, "Well, you know, how simple is that?"
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你可能會問：「這真的很簡單嗎？」
08:07
But computers -- you need to be able to describe things
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電腦只能用一些最基本的條件繪製出東西，
08:09
in very basic terms, and with this, we could.
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而摺紙正具備這些條件。
08:12
So I wrote a computer program a bunch of years ago
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所以我在幾年前撰寫了一個電腦程式，
08:14
called TreeMaker, and you can download it from my website.
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叫做TreeMaker，各位可以在我的網頁上下載這個程式，
08:16
It's free. It runs on all the major platforms -- even Windows.
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完全免費，可以在各主要作業系統上運作，連Windows也可以。
08:19
(Laughter)
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（笑聲）
08:21
And you just draw a stick figure,
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你只要用線條畫出輪廓，
08:23
and it calculates the crease pattern.
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電腦就會幫你畫出摺痕圖案，
08:25
It does the circle packing, calculates the crease pattern,
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它會幫你堆疊那些圓圈，計算出摺痕位置。
08:28
and if you use that stick figure that I just showed --
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如果以我剛才畫的線條輪廓為例，
08:30
which you can kind of tell, it's a deer, it's got antlers --
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你可以看出它是一隻鹿，有角，
08:33
you'll get this crease pattern.
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你可以用這個程式繪製出摺痕圖案。
08:35
And if you take this crease pattern, you fold on the dotted lines,
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照著圖案上的虛線摺，
08:37
you'll get a base that you can then shape
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就可以摺出大概的形狀，
08:40
into a deer,
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再細修就會摺成一隻鹿，
08:42
with exactly the crease pattern that you wanted.
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那就是用剛才那個圖案摺出來的成品。
08:44
And if you want a different deer,
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如果你想摺一隻不同品種的鹿，
08:46
not a white-tailed deer, but you want a mule deer, or an elk,
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而不要這隻白尾鹿，
08:49
you change the packing,
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你只要改變圓圈堆疊的方式，
08:51
and you can do an elk.
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你就可以做出一隻麋鹿，
08:53
Or you could do a moose.
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或是一隻北美麋鹿，
08:55
Or, really, any other kind of deer.
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或是任何一隻其他品種的鹿。
08:57
These techniques revolutionized this art.
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這些技術完全改造了這門技藝，
09:00
We found we could do insects,
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我們現在可以摺出昆蟲、
09:02
spiders, which are close,
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蜘蛛，這二種很接近--
09:04
things with legs, things with legs and wings,
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就是有腳的生物，或是有腳和有翅膀的生物，
09:08
things with legs and antennae.
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或是有腳和有觸角的生物。
09:10
And if folding a single praying mantis from a single uncut square
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如果你覺得用一張完全沒有裁切的紙，
09:13
wasn't interesting enough,
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做出一隻螳螂還不夠好玩，
09:15
then you could do two praying mantises
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你可以試試用一張完全沒有裁切的紙，
09:17
from a single uncut square.
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做出二隻螳螂看看。
09:19
She's eating him.
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母螳螂在吃公螳螂！
09:21
I call it "Snack Time."
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這幅作品叫「點心時間」。
09:24
And you can do more than just insects.
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摺紙不只可以做出昆蟲，
09:26
This -- you can put details,
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你還可以在細節上多所描繪，
09:28
toes and claws. A grizzly bear has claws.
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摺出腳趾和利爪，大灰熊有利爪，
09:31
This tree frog has toes.
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而這隻樹蛙則有腳趾。
09:33
Actually, lots of people in origami now put toes into their models.
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現在很多人會在自己的摺紙物件裡加入腳趾，
09:36
Toes have become an origami meme,
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摺腳趾變成了一種流行，
09:38
because everyone's doing it.
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大家都在摺腳趾。
09:41
You can make multiple subjects.
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你還可以摺出多個物件，
09:43
So these are a couple of instrumentalists.
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像是這兩個音樂家，
09:45
The guitar player from a single square,
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吉他手是用一張正方形的紙摺出來的，
09:48
the bass player from a single square.
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貝斯手則是用另一張紙摺出來的。
09:50
And if you say, "Well, but the guitar, bass --
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如果你說：「吉他和貝斯，
09:52
that's not so hot.
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不是什麼熱門的題材，
09:54
Do a little more complicated instrument."
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做個複雜一點的樂器來看看。」
09:56
Well, then you could do an organ.
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那你可以做個風琴。
09:58
(Laughter)
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（笑聲）
10:01
And what this has allowed is the creation
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摺紙就是這種
10:03
of origami-on-demand.
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隨心所欲的創作藝術，
10:05
So now people can say, "I want exactly this and this and this,"
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如果有人說他要這個和這個，
10:08
and you can go out and fold it.
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我絕對做得出來。
10:11
And sometimes you create high art,
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有時我做的是藝術品，
10:13
and sometimes you pay the bills by doing some commercial work.
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有時則接受一些商業邀約。
10:16
But I want to show you some examples.
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我想給大家看一些範例，
10:18
Everything you'll see here,
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你所看到的一切，
10:20
except the car, is origami.
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除了車子以外，都是摺紙作品。
10:23
(Video)
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（影片）
10:51
(Applause)
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（掌聲）
10:54
Just to show you, this really was folded paper.
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這些全都是摺紙作品，
10:57
Computers made things move,
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電腦則負責為他們添加動畫，
10:59
but these were all real, folded objects that we made.
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我們為他們創作了這些摺紙作品。
11:03
And we can use this not just for visuals,
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摺紙作品不只是好看而已，
11:06
but it turns out to be useful even in the real world.
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在真實世界裡還有可以應用的範圍。
11:09
Surprisingly, origami
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很難想像，
11:10
and the structures that we've developed in origami
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摺紙和我們為摺紙發展出來的結構圖，
11:13
turn out to have applications in medicine, in science,
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竟然可以應用在醫療、科學、
11:16
in space, in the body, consumer electronics and more.
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太空、人體、家電等地方上。
11:19
And I want to show you some of these examples.
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現在給大家看一些例子。
11:22
One of the earliest was this pattern,
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這是最早期的一個圖形：
11:24
this folded pattern,
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這個由日本工程師
11:26
studied by Koryo Miura, a Japanese engineer.
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三浦公亮所研究的摺痕圖案，
11:29
He studied a folding pattern, and realized
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他研究後發現，
11:31
this could fold down into an extremely compact package
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這個圖案可以把東西摺疊成很小的體積，
11:34
that had a very simple opening and closing structure.
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結構就僅僅只是簡單的開闔而已，
11:37
And he used it to design this solar array.
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他用這個來設計太陽能板。
11:40
It's an artist's rendition, but it flew in a Japanese telescope
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現在看到的是描摩圖，但在1995年真的跟著日本的太空望遠鏡
11:43
in 1995.
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上到太空去。
11:45
Now, there is actually a little origami
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詹姆斯．韋伯太空望遠鏡裡面
11:47
in the James Webb Space Telescope, but it's very simple.
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也有點摺紙的技術，但其實是很簡單的形式。
11:50
The telescope, going up in space,
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太空望遠鏡上到太空後，
11:52
it unfolds in two places.
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要在二個地方展開，
11:55
It folds in thirds. It's a very simple pattern --
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然後在第三個地方摺疊起來，整個形式非常簡單，
11:57
you wouldn't even call that origami.
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你甚至不會認為那是摺紙技術。
11:59
They certainly didn't need to talk to origami artists.
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這樣的設計當然沒必要諮詢摺紙專家，
12:02
But if you want to go higher and go larger than this,
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但如果你想要更大、更高階的東西，
12:05
then you might need some origami.
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可能就需要一點摺紙技巧。
12:07
Engineers at Lawrence Livermore National Lab
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勞倫斯．利弗摩爾國家實驗室裡的工程師，
12:09
had an idea for a telescope much larger.
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就希望能建造一個更大型的太空望遠鏡，
12:12
They called it the Eyeglass.
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他們稱它為「大眼鏡」。
12:14
The design called for geosynchronous orbit
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這項設計需要同步軌道，
12:16
25,000 miles up,
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設定在4萬1千600公尺高空，
12:18
100-meter diameter lens.
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還需要一個直徑100公尺的鏡片，
12:21
So, imagine a lens the size of a football field.
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那簡直就像一個足球場大小的望遠鏡鏡片。
12:24
There were two groups of people who were interested in this:
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對這個設計有興趣的人有兩種：
12:26
planetary scientists, who want to look up,
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一種是想往上看的太空科學家，
12:29
and then other people, who wanted to look down.
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另一種是想往下看的人。
12:33
Whether you look up or look down,
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不管往上看或往下看，
12:35
how do you get it up in space? You've got to get it up there in a rocket.
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要怎麼把望遠鏡送上太空？當然是用火箭。
12:38
And rockets are small. So you have to make it smaller.
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可是火箭不大，望遠鏡一定要比火箭小。
12:41
How do you make a large sheet of glass smaller?
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要怎麼讓一大片玻璃縮小？
12:43
Well, about the only way is to fold it up somehow.
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唯一的方法就是想辦法摺起來。
12:46
So you have to do something like this.
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所以必須這樣做，
12:48
This was a small model.
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這是縮小的模型。
12:51
Folded lens, you divide up the panels, you add flexures.
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針對玻璃，你只能把它切割成較小的玻璃，增加些曲度，
12:53
But this pattern's not going to work
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但還是沒有辦法把這100公尺
12:56
to get something 100 meters down to a few meters.
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直徑大的玻璃縮小到只有幾公尺。
12:59
So the Livermore engineers,
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因此利弗摩爾的工程師
13:01
wanting to make use of the work of dead people,
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也想參考死去的人的成果，
13:03
or perhaps live origamists, said,
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於是他們來找摺紙專家說：
13:06
"Let's see if someone else is doing this sort of thing."
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「我們想看看有沒有人在做這種事。」
13:09
So they looked into the origami community,
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於是他們向摺紙團體求救，
13:12
we got in touch with them, and I started working with them.
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他們找上了我們，請我們和他們一起工作。
13:14
And we developed a pattern together
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我們一起開發了這種圖案，
13:16
that scales to arbitrarily large size,
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可以隨意放大到任何尺寸，
13:18
but that allows any flat ring or disc
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也可以將任何平面的環或圓盤
13:22
to fold down into a very neat, compact cylinder.
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摺疊成非常整齊、緊實的圓柱體。
13:25
And they adopted that for their first generation,
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他們將這個圖案應用在第一代的設計中，
13:27
which was not 100 meters -- it was a five-meter.
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那還不是100公尺大的玻璃，只有5公尺而已。
13:29
But this is a five-meter telescope --
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但這個只有5公尺的太空望遠鏡，
13:31
has about a quarter-mile focal length.
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需要1.6公尺的焦距長度，
13:33
And it works perfectly on its test range,
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在測試階段表現得非常好，
13:35
and it indeed folds up into a neat little bundle.
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確實能摺疊成很整齊的一捆。
13:39
Now, there is other origami in space.
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目前，太空上還應用了其他的摺紙技術，
13:41
Japan Aerospace [Exploration] Agency flew a solar sail,
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日本太空總署發射過太陽風帆，
13:44
and you can see here that the sail expands out,
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這裡可以看到帆張開來，
13:47
and you can still see the fold lines.
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還可以看到摺痕。
13:49
The problem that's being solved here is
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我們幫他們解決的問題是，
13:52
something that needs to be big and sheet-like at its destination,
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把一個在目的地必須呈現出很大一張的東西，
13:55
but needs to be small for the journey.
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在運送的時候將它縮小，
13:57
And that works whether you're going into space,
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不管你是要上太空
14:00
or whether you're just going into a body.
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或是進入人體都一樣。
14:03
And this example is the latter.
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現在所看到的例子是要進入人體，
14:05
This is a heart stent developed by Zhong You
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這是牛津大學的游忠博士
14:08
at Oxford University.
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所發明的心臟血管支架。
14:10
It holds open a blocked artery when it gets to its destination,
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當這個血管支架被送到目地的後，就會撐開被阻塞的血管，
14:13
but it needs to be much smaller for the trip there,
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但在運送的過程裡必須將它縮到很小，
14:16
through your blood vessels.
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才能通過血管。
14:18
And this stent folds down using an origami pattern,
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於是他利用摺紙原理，將這個支架摺疊起來，
14:21
based on a model called the water bomb base.
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利用摺紙上所常用的水雷方式摺疊起來。
14:25
Airbag designers also have the problem
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設計安全氣囊的工程師也有相同困擾，
14:27
of getting flat sheets
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他們需要將一個扁平的袋子
14:29
into a small space.
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壓縮擠進一個很小的空間。
14:32
And they want to do their design by simulation.
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他們希望以模擬的方式來看看，
14:34
So they need to figure out how, in a computer,
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要怎麼樣利用電腦來模擬出
14:36
to flatten an airbag.
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壓縮安全氣囊的最佳方式。
14:38
And the algorithms that we developed
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我們在摺昆蟲時所開發出來的
14:40
to do insects
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演算法，
14:42
turned out to be the solution for airbags
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後來變成了模擬壓縮安全氣囊的
14:45
to do their simulation.
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最佳解法。
14:47
And so they can do a simulation like this.
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他們做出的模擬是像這樣，
14:50
Those are the origami creases forming,
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那都是摺紙的摺痕圖案，
14:52
and now you can see the airbag inflate
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現在可以看到安全氣囊被充氣了，
14:54
and find out, does it work?
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看看是否能成功？
14:57
And that leads
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這讓我想到
14:59
to a really interesting idea.
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一個很有趣的想法，
15:01
You know, where did these things come from?
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這些東西究竟是從何而來？
15:04
Well, the heart stent
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心臟血管支架
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came from that little blow-up box
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是由你在小學時就學過的
15:08
that you might have learned in elementary school.
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那種會打開的小盒子所啟發，
15:11
It's the same pattern, called the water bomb base.
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也就是那種我們稱為水雷的基本摺法；
15:14
The airbag-flattening algorithm
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而將安全氣囊壓縮起來的演算法，
15:16
came from all the developments
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則是受到我們摺昆蟲的摺法所影響，
15:18
of circle packing and the mathematical theory
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為了要摺出昆蟲的腳，
15:21
that was really developed
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我們得把各個圓圈堆疊起來，
15:23
just to create insects -- things with legs.
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還得運用一些數學運算技巧。
15:27
The thing is, that this often happens
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事實上，這些都與
15:29
in math and science.
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數學及科學相關，
15:31
When you get math involved, problems that you solve
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當我們在解決美學上的問題，
15:34
for aesthetic value only,
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或是試圖創造某些藝術品時，
15:36
or to create something beautiful,
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只要運用一些數學運算，
15:38
turn around and turn out
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最終就有可能
15:40
to have an application in the real world.
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應用到真實的世界裡。
15:43
And as weird and surprising as it may sound,
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這乍聽之下或許難以置信，
15:46
origami may someday even save a life.
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但有一天摺紙或許能救人一命。
15:50
Thanks.
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謝謝。
15:52
(Applause)
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（掌聲）
ABOUT THE SPEAKER
Robert Lang - OrigamistRobert 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.
Robert Lang | Speaker | TED.com