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

乍看之下兩者毫無關連，

but I hope by the end of these 18 minutes,

但我希望在我演說的18分鐘結束後，

you'll see a little bit of a relation.

各位就可以看得出端倪。

It ties to origami. So let me start.

這一切都與摺紙有關，所以我們開始吧。

What is origami?

摺紙是什麼？

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

大部份的人都認為自己瞭解摺紙，

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

不過就是紙鶴、玩具、東南西北遊戲這類的東西。

And that is what origami used to be.

以前的摺紙確實就是這些，

But it's become something else.

但現在產生了新的變化。

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

摺紙現在變成一種藝術、一種雕塑，

The common theme -- what makes it origami --

摺紙的特點，也就是摺紙的精髓，

is folding is how we create the form.

在於摺的動作，在於成形的過程。

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

摺紙是一項非常老的技藝，這是一幅1797年的插圖，

It shows these women playing with these toys.

裡面的女士們正在玩一些東西，

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

仔細一看，原來是紙鶴。

Every Japanese kid

每個日本小孩

learns how to fold that crane.

都會摺紙鶴，

So this art has been around for hundreds of years,

這門藝術已經存在數百年之久，

and you would think something

我們很自然會認為，

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

存在了這麼久的技藝，就只有「摺」這個動作，

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

能玩的花樣老早就玩遍了。

And that might have been the case.

也許早期的確是這樣，

But in the twentieth century,

但是到了二十世紀，

a Japanese folder named Yoshizawa came along,

出現了一名叫做吉澤章的摺紙師傅，

and he created tens of thousands of new designs.

他發明上萬種新的摺紙設計。

But even more importantly, he created a language,

更重要的是，他發明了一種摺紙語言，

a way we could communicate,

一種摺紙的溝通方式，

a code of dots, dashes and arrows.

用點、虛線和箭頭所組成。

Harkening back to Susan Blackmore's talk,

如同蘇珊．布萊克摩爾在TED所發表的演說，

we now have a means of transmitting information

現在我們已經發展出一種資訊傳遞方式，

with heredity and selection,

經過不斷地傳承與改良，

and we know where that leads.

我們都知道最後會有什麼結果。

And where it has led in origami

而目前在摺紙的領域裡，

is to things like this.

就發展出這樣的成果。

This is an origami figure --

這是一個摺紙作品，

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

一張紙、沒有切割、只靠翻摺、有數百道摺痕。

This, too, is origami,

這也是摺紙，

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

看得出現代摺紙的發展趨勢，

Naturalism. Detail.

也就是注重自然主義與細節。

You can get horns, antlers --

你可以做出牛角、鹿角，

even, if you look close, cloven hooves.

再看仔細一點，還可以做出分趾蹄。

And it raises a question: what changed?

看了不禁讓人好奇，這和以前有何不同？

And what changed is something

不同的地方，

you might not have expected in an art,

是你從來不會與藝術聯想在一起的，

which is math.

就是數學。

That is, people applied mathematical principles

也就是說，

to the art,

現代摺紙應用了數學方法，

to discover the underlying laws.

開發潛藏在其中的規則。

And that leads to a very powerful tool.

在此不得不提到一項非常有用的工具，

The secret to productivity in so many fields --

同時也是其他許多領域提升生產力的祕訣，

and in origami --

摺紙也不例外，

is letting dead people do your work for you.

就是讓死去的人幫你做事。

(Laughter)

（笑聲）

Because what you can do is

你要做的，

take your problem,

就是把你的問題

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

和以前別人所遇到的問題做比對，

and use their solutions.

再利用他們已經想出的辦法來解決。

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

我來告訴各位我們在摺紙時是怎麼解決問題的。

Origami revolves around crease patterns.

摺紙就是在有摺痕圖案的紙上作業，

The crease pattern shown here is the underlying blueprint

現在各位所看到的，

for an origami figure.

是某個摺紙作品的草稿底圖。

And you can't just draw them arbitrarily.

當然，你不可能隨意畫出這些線，

They have to obey four simple laws.

至少要遵循四個簡單的法則，

And they're very simple, easy to understand.

四個簡單又容易記住的法則。

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

第一個法則是雙色運用，在任何一張草圖上，

with just two colors without ever having

都可以運用二種顏色來上色，

the same color meeting.

但相同顏色不得相鄰。

The directions of the folds at any vertex --

在任一頂點要摺出線時，

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

山摺線和谷摺線的摺線次數，

always differs by two. Two more or two less.

永遠都差二次，不管是多二次還是少二次，

Nothing else.

就是這樣。

If you look at the angles around the fold,

看看摺線旁的角，

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

如果你將圓圈裡的角編號，

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

將偶數角摺疊起來就是直線，

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

而將奇數角摺疊起來也是一條直線。

And if you look at how the layers stack,

再看看各個層次的堆疊，

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

不管你怎麼堆疊各個摺痕與紙張，

a sheet can never

紙張永遠不能

penetrate a fold.

穿透摺痕。

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

摺紙就只需要這四個簡單的法則，

All of origami comes from that.

所有的摺紙都是從這四個法則衍生出來，

And you'd think, "Can four simple laws

你會想：「用這四個法則，

give rise to that kind of complexity?"

就可以創造出那麼複雜的摺紙嗎？」

But indeed, the laws of quantum mechanics

看看量子力學的定律，

can be written down on a napkin,

不也是可以寫在一張紙巾上嗎？

and yet they govern all of chemistry,

但他們卻可以統禦所有的化學、

all of life, all of history.

生命科學和歷史啊！

If we obey these laws,

如果我們遵循這些法則，

we can do amazing things.

我們可以做出很棒的東西，

So in origami, to obey these laws,

在摺紙這門學問裡，只要遵循這些法則，

we can take simple patterns --

我們就可以將簡單的圖案，

like this repeating pattern of folds, called textures --

像是這類重覆對摺的圖案，我們稱之為結構，

and by itself it's nothing.

單一的結構做不出什麼東西，

But if we follow the laws of origami,

但如果我們運用摺紙的四個法則，

we can put these patterns into another fold

我們就可以將這種圖案放進另一種摺法裡，

that itself might be something very, very simple,

呈現出一種很簡單的圖樣，

but when we put it together,

再把它大量運用之後，

we get something a little different.

我們就可以得出一些不一樣的圖形。

This fish, 400 scales --

看看這條魚，有400個鱗片，

again, it is one uncut square, only folding.

再次強調，這是用一張紙摺出來的，完全沒有剪裁。

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

如果你不想摺400個鱗片，

you can back off and just do a few things,

那就摺少一點，再加點別的，

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

做出烏龜的甲殼，或是腳趾。

Or you can ramp up and go up to 50 stars

也可以更進一步摺50顆星星，

on a flag, with 13 stripes.

加13條線就是星條旗了。

And if you want to go really crazy,

如果你真的想不開，

1,000 scales on a rattlesnake.

可以做有一千個鱗片的響尾蛇，

And this guy's on display downstairs,

這隻蛇在樓下展覽著，

so take a look if you get a chance.

有機會可以去看看。

The most powerful tools in origami

摺紙裡最有力的工具，

have related to how we get parts of creatures.

就是解構物件的工具。

And I can put it in this simple equation.

可以用一個簡單的方程式來解釋，

We take an idea,

也就是先想出構想，

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

再用一張紙，就可以摺出摺紙作品。

(Laughter)

（笑聲）

What matters is what we mean by those symbols.

在這個程式裡，真正重要的是運算符號。

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

你會說：「可不可以再說清楚一點啊？

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

你看，一隻鍬形蟲有二個顎，

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

還有觸角，可不可以把細節再講清楚一點？」

And yeah, you really can.

當然可以啊...

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

該怎做呢？我們把作法

into a few smaller steps.

再拆解成更小的步驟，

So let me stretch out that equation.

把這個方程式再展開來，

I start with my idea. I abstract it.

先想出構想，畫出個輪廓，

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

輪廓要怎麼畫？用線條描繪出軀幹就行了，

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

有了這個輪廓，就可以創造出摺紙作品，

that has a part for every bit of the subject,

並生動摺出物件的各個部分，

a flap for every leg.

包括每一隻腳。

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

一旦我們以這個做基礎，

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

你就可以做出更細的腳，還可以彎折腳的角度，

you can turn it into the finished shape.

把成品做出來。

Now the first step, pretty easy.

第一個步驟：很簡單，

Take an idea, draw a stick figure.

只要有構想，再畫出輪廓就行了。

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

最後一個步驟也沒有那麼難，但是中間這個步驟，

going from the abstract description to the folded shape --

是要從輪廓做出物件，

that's hard.

真的很難。

But that's the place where the mathematical ideas

這裡就要靠一些數學頭腦

can get us over the hump.

才能幫我們解決問題了。

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

我會告訴各位每一個細節，

so you can go out of here and fold something.

所以在散場之後，各位就會摺出些東西來了。

But we're going to start small.

我們先從小的東西開始，

This base has a lot of flaps in it.

這個基本形有很多分岔的肢體，

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

我們先來學怎麼製作出各個肢體來，

How would you make a single flap?

肢體要怎麼做？

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

拿一張正方形的紙，對摺、對摺、再對摺，

until it gets long and narrow,

讓它變得又長又細，

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

最後就變成了一個肢體。

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

這可以運用在腳、手臂或其他類似的肢體。

What paper went into that flap?

這個肢體是用正方形的哪一個部分做成的呢？

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

把成品打開來，看看那些摺痕，

you can see that the upper left corner of that shape

可以看到正方形的左上角

is the paper that went into the flap.

就是摺出這個肢體的部分。

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

我們完成了肢體，還有其他部分的紙剩下來，

I can use it for something else.

我可以用來做些別的。

Well, there are other ways of making a flap.

要做出肢體還有別的方法，

There are other dimensions for flaps.

還有別種形式的肢體，

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

如果我可以把肢體做得瘦一點，就可以用少一點的紙，

If I make the flap as skinny as possible,

如果我做得夠瘦，

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

就可以把紙的用量減到最少。

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

看看那裡，我用四分之一圓就可以做出一個肢體，

There's other ways of making flaps.

當然還有其他做肢體的方法。

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

如果我把肢體放在邊緣，就要用到二分之一圓，

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

但如果把肢體做在中間，就要用掉一整個圓。

So, no matter how I make a flap,

所以不管怎麼摺出一個肢體，

it needs some part

至少都會用去

of a circular region of paper.

某個部分的圓才能摺出來。

So now we're ready to scale up.

接下來我們就可以往下做，

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

如果我想要做一個有很多肢體的東西呢？

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

需要的是什麼？就是很多個圓圈。

And in the 1990s,

在1990年代，

origami artists discovered these principles

摺紙師傅發現這些原理，

and realized we could make arbitrarily complicated figures

只要把圓圈組合起來，

just by packing circles.

就可以隨意做出複雜的作品。

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

這時候，那些死去的人就幫得上忙了。

because lots of people have studied

因為很多人研究過

the problem of packing circles.

圓圈堆疊這個題目，

I can rely on that vast history of mathematicians and artists

我可以參考歷代數學家和藝術家的成果，

looking at disc packings and arrangements.

看看圓圈要怎麼堆疊和組合，

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

再運用到我的摺紙作品上。

So we figured out these rules whereby you pack circles,

我們在堆疊圓圈的過程裡發現了一些規則，

you decorate the patterns of circles with lines

我們還運用其他的規則來畫出線條與圓圈，

according to more rules. That gives you the folds.

這樣就可以畫出摺痕了，

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

這些摺痕可以摺出一個大概輪廓，

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

細部修正就可以完成一個摺紙作品，像是這隻蟑螂，

And it's so simple.

就是這麼簡單。

(Laughter)

（笑聲）

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

簡單到可以用電腦解決。

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

你可能會問：「這真的很簡單嗎？」

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

電腦只能用一些最基本的條件繪製出東西，

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

而摺紙正具備這些條件。

So I wrote a computer program a bunch of years ago

所以我在幾年前撰寫了一個電腦程式，

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

叫做TreeMaker，各位可以在我的網頁上下載這個程式，

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

完全免費，可以在各主要作業系統上運作，連Windows也可以。

(Laughter)

（笑聲）

And you just draw a stick figure,

你只要用線條畫出輪廓，

and it calculates the crease pattern.

電腦就會幫你畫出摺痕圖案，

It does the circle packing, calculates the crease pattern,

它會幫你堆疊那些圓圈，計算出摺痕位置。

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

如果以我剛才畫的線條輪廓為例，

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

你可以看出它是一隻鹿，有角，

you'll get this crease pattern.

你可以用這個程式繪製出摺痕圖案。

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

照著圖案上的虛線摺，

you'll get a base that you can then shape

就可以摺出大概的形狀，

into a deer,

再細修就會摺成一隻鹿，

with exactly the crease pattern that you wanted.

那就是用剛才那個圖案摺出來的成品。

And if you want a different deer,

如果你想摺一隻不同品種的鹿，

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

而不要這隻白尾鹿，

you change the packing,

你只要改變圓圈堆疊的方式，

and you can do an elk.

你就可以做出一隻麋鹿，

Or you could do a moose.

或是一隻北美麋鹿，

Or, really, any other kind of deer.

或是任何一隻其他品種的鹿。

These techniques revolutionized this art.

這些技術完全改造了這門技藝，

We found we could do insects,

我們現在可以摺出昆蟲、

spiders, which are close,

蜘蛛，這二種很接近--

things with legs, things with legs and wings,

就是有腳的生物，或是有腳和有翅膀的生物，

things with legs and antennae.

或是有腳和有觸角的生物。

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

如果你覺得用一張完全沒有裁切的紙，

wasn't interesting enough,

做出一隻螳螂還不夠好玩，

then you could do two praying mantises

你可以試試用一張完全沒有裁切的紙，

from a single uncut square.

做出二隻螳螂看看。

She's eating him.

母螳螂在吃公螳螂！

I call it "Snack Time."

這幅作品叫「點心時間」。

And you can do more than just insects.

摺紙不只可以做出昆蟲，

This -- you can put details,

你還可以在細節上多所描繪，

toes and claws. A grizzly bear has claws.

摺出腳趾和利爪，大灰熊有利爪，

This tree frog has toes.

而這隻樹蛙則有腳趾。

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

現在很多人會在自己的摺紙物件裡加入腳趾，

Toes have become an origami meme,

摺腳趾變成了一種流行，

because everyone's doing it.

大家都在摺腳趾。

You can make multiple subjects.

你還可以摺出多個物件，

So these are a couple of instrumentalists.

像是這兩個音樂家，

The guitar player from a single square,

吉他手是用一張正方形的紙摺出來的，

the bass player from a single square.

貝斯手則是用另一張紙摺出來的。

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

如果你說：「吉他和貝斯，

that's not so hot.

不是什麼熱門的題材，

Do a little more complicated instrument."

做個複雜一點的樂器來看看。」

Well, then you could do an organ.

那你可以做個風琴。

(Laughter)

（笑聲）

And what this has allowed is the creation