ABOUT THE SPEAKER
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.

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.

More profile about the speaker
Robert Lang | Speaker | TED.com
TED2008

Robert Lang: The math and magic of origami

Robert Lang全新型態的摺紙

Filmed:
2,647,209 views

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。
- 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

Double-click the English transcript below to play the video.

00:18
My talk is "Flapping Birds鳥類 and Space空間 Telescopes望遠鏡."
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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另一個,
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乍看之下兩者毫無關連,
00:23
but I hope希望 by the end結束 of these 18 minutes分鐘,
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但我希望在我演說的18分鐘結束後,
00:26
you'll你會 see a little bit of a relation關係.
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各位就可以看得出端倪。
00:29
It ties聯繫 to origami摺紙. So let me start開始.
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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摺紙?
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摺紙是什麼?
00:32
Most people think they know what origami摺紙 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鳥類, toys玩具, cootie catchers捕手, that sort分類 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摺紙 used to be.
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以前的摺紙確實就是這些,
00:40
But it's become成為 something else其他.
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但現在產生了新的變化。
00:42
It's become成為 an art藝術 form形成, a form形成 of sculpture雕塑.
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摺紙現在變成一種藝術、一種雕塑,
00:44
The common共同 theme主題 -- what makes品牌 it origami摺紙 --
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摺紙的特點,也就是摺紙的精髓,
00:46
is folding摺頁 is how we create創建 the form形成.
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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盤子 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節目 these women婦女 playing播放 with these toys玩具.
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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形狀, called a crane起重機.
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仔細一看,原來是紙鶴。
00:58
Every一切 Japanese日本 kid孩子
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每個日本小孩
01:00
learns獲悉 how to fold that crane起重機.
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都會摺紙鶴,
01:02
So this art藝術 has been around for hundreds數以百計 of years年份,
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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限制, folding摺頁 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 doneDONE has been doneDONE a long time ago.
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能玩的花樣老早就玩遍了。
01:12
And that might威力 have been the case案件.
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也許早期的確是這樣,
01:14
But in the twentieth第二十 century世紀,
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但是到了二十世紀,
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a Japanese日本 folder named命名 Yoshizawa吉澤 came來了 along沿,
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出現了一名叫做吉澤章的摺紙師傅,
01:19
and he created創建 tens of thousands數千 of new designs設計.
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他發明上萬種新的摺紙設計。
01:22
But even more importantly重要的, he created創建 a language語言,
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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通信,
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一種摺紙的溝通方式,
01:27
a code of dots, dashes破折號 and arrows箭頭.
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用點、虛線和箭頭所組成。
01:29
HarkeningHarkening back to Susan蘇珊 Blackmore's布萊克莫爾的 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手段 of transmitting發射 information信息
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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遺傳 and selection選擇,
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經過不斷地傳承與改良,
01:36
and we know where that leads引線.
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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摺紙
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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摺紙 figure數字 --
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這是一個摺紙作品,
01:44
one sheet, no cuts削減, folding摺頁 only, hundreds數以百計 of folds褶皺.
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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摺紙,
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這也是摺紙,
01:52
and this shows節目 where we've我們已經 gone走了 in the modern現代 world世界.
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看得出現代摺紙的發展趨勢,
01:55
Naturalism自然主義. Detail詳情.
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也就是注重自然主義與細節。
01:57
You can get horns牛角, antlers鹿角 --
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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加薪 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威力 not have expected預期 in an art藝術,
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是你從來不會與藝術聯想在一起的,
02:09
which哪一個 is math數學.
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就是數學。
02:11
That is, people applied應用的 mathematical數學的 principles原則
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也就是說,
02:13
to the art藝術,
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現代摺紙應用了數學方法,
02:16
to discover發現 the underlying底層 laws法律.
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開發潛藏在其中的規則。
02:18
And that leads引線 to a very powerful強大 tool工具.
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在此不得不提到一項非常有用的工具,
02:21
The secret秘密 to productivity生產率 in so many許多 fields領域 --
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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摺紙 --
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摺紙也不例外,
02:25
is letting出租 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笑聲)
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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問題,
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就是把你的問題
02:33
and turn it into a problem問題 that someone有人 else其他 has solved解決了,
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和以前別人所遇到的問題做比對,
02:36
and use their solutions解決方案.
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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摺紙.
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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摺紙 revolves圍繞 around crease patterns模式.
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摺紙就是在有摺痕圖案的紙上作業,
02:43
The crease pattern模式 shown顯示 here is the underlying底層 blueprint藍圖
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現在各位所看到的,
02:46
for an origami摺紙 figure數字.
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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任意.
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當然,你不可能隨意畫出這些線,
02:50
They have to obey遵守 four simple簡單 laws法律.
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至少要遵循四個簡單的法則,
02:53
And they're very simple簡單, easy簡單 to understand理解.
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四個簡單又容易記住的法則。
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The first law is two-colorability二著色. You can color顏色 any crease pattern模式
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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顏色 without ever having
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都可以運用二種顏色來上色,
03:00
the same相同 color顏色 meeting會議.
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但相同顏色不得相鄰。
03:03
The directions方向 of the folds褶皺 at any vertex頂點 --
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在任一頂點要摺出線時,
03:06
the number of mountain folds褶皺, the number of valley folds褶皺 --
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山摺線和谷摺線的摺線次數,
03:09
always differs不同 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其他.
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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偶數 angles add up to a straight直行 line,
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將偶數角摺疊起來就是直線,
03:20
all the odd-numbered奇數的 angles add up to a straight直行 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,
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you'll你會 find that no matter how you stack folds褶皺 and sheets床單,
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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穿透 a fold.
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穿透摺痕。
03:32
So that's four simple簡單 laws法律. That's all you need in origami摺紙.
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摺紙就只需要這四個簡單的法則,
03:35
All of origami摺紙 comes from that.
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所有的摺紙都是從這四個法則衍生出來,
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And you'd think, "Can four simple簡單 laws法律
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你會想:「用這四個法則,
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give rise上升 to that kind of complexity複雜?"
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就可以創造出那麼複雜的摺紙嗎?」
03:41
But indeed確實, the laws法律 of quantum量子 mechanics機械學
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看看量子力學的定律,
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can be written書面 down on a napkin餐巾,
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不也是可以寫在一張紙巾上嗎?
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and yet然而 they govern治理 all of chemistry化學,
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但他們卻可以統禦所有的化學、
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all of life, all of history歷史.
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生命科學和歷史啊!
03:49
If we obey遵守 these laws法律,
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如果我們遵循這些法則,
03:51
we can do amazing驚人 things.
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我們可以做出很棒的東西,
03:53
So in origami摺紙, to obey遵守 these laws法律,
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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簡單 patterns模式 --
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我們就可以將簡單的圖案,
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like this repeating重複 pattern模式 of folds褶皺, called textures紋理 --
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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本身 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跟隨 the laws法律 of origami摺紙,
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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模式 into another另一個 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本身 might威力 be something very, very simple簡單,
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呈現出一種很簡單的圖樣,
04:09
but when we put it together一起,
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再把它大量運用之後,
04:11
we get something a little different不同.
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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完整無缺 square廣場, only folding摺頁.
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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少數 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腳趾.
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做出烏龜的甲殼,或是腳趾。
04:27
Or you can ramp舷梯 up and go up to 50 stars明星
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也可以更進一步摺50顆星星,
04:30
on a flag, with 13 stripes條紋.
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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響尾蛇.
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可以做有一千個鱗片的響尾蛇,
04:38
And this guy's傢伙 on display顯示 downstairs樓下,
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這隻蛇在樓下展覽著,
04:40
so take a look if you get a chance機會.
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有機會可以去看看。
04:43
The most powerful強大 tools工具 in origami摺紙
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摺紙裡最有力的工具,
04:45
have related有關 to how we get parts部分 of creatures生物.
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就是解構物件的工具。
04:48
And I can put it in this simple簡單 equation方程.
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可以用一個簡單的方程式來解釋,
04:50
We take an idea理念,
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也就是先想出構想,
04:52
combine結合 it with a square廣場, and you get an origami摺紙 figure數字.
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再用一張紙,就可以摺出摺紙作品。
04:55
(Laughter笑聲)
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(笑聲)
04:59
What matters事項 is what we mean by those symbols符號.
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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威力 say, "Can you really be that specific具體?
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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甲蟲 -- 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天線. Can you be that specific具體 in the detail詳情?"
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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打破 it down
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該怎做呢?我們把作法
05:16
into a few少數 smaller steps腳步.
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再拆解成更小的步驟,
05:18
So let me stretch伸展 out that equation方程.
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把這個方程式再展開來,
05:20
I start開始 with my idea理念. I abstract抽象 it.
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先想出構想,畫出個輪廓,
05:23
What's the most abstract抽象 form形成? It's a stick figure數字.
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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數字, I somehow不知何故 have to get to a folded折疊 shape形狀
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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部分 for every一切 bit of the subject學科,
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並生動摺出物件的各個部分,
05:32
a flap拍打 for every一切 leg.
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包括每一隻腳。
05:34
And then once一旦 I have that folded折疊 shape形狀 that we call the base基礎,
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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彎曲 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形狀.
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把成品做出來。
05:42
Now the first step, pretty漂亮 easy簡單.
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第一個步驟:很簡單,
05:44
Take an idea理念, draw a stick figure數字.
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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中間 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抽象 description描述 to the folded折疊 shape形狀 --
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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地點 where the mathematical數學的 ideas思路
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這裡就要靠一些數學頭腦
05:56
can get us over the hump駝峰.
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才能幫我們解決問題了。
05:58
And I'm going to show顯示 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開始 small.
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我們先從小的東西開始,
06:04
This base基礎 has a lot of flaps襟翼 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學習 how to make one flap拍打.
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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拍打?
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肢體要怎麼做?
06:11
Take a square廣場. 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直到 it gets得到 long and narrow狹窄,
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讓它變得又長又細,
06:16
and then we'll say at the end結束 of that, that's a flap拍打.
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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拍打?
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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展開 it and go back to the crease pattern模式,
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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形狀
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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拍打.
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就是摺出這個肢體的部分。
06:30
So that's the flap拍打, and all the rest休息 of the paper's文件 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其他.
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我可以用來做些別的。
06:35
Well, there are other ways方法 of making製造 a flap拍打.
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要做出肢體還有別的方法,
06:37
There are other dimensions尺寸 for flaps襟翼.
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還有別種形式的肢體,
06:39
If I make the flaps襟翼 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拍打 as skinny枯瘦 as possible可能,
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如果我做得夠瘦,
06:45
I get to the limit限制 of the minimum最低限度 amount of paper needed需要.
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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需求 a quarter-circle四分之一圓 of paper to make a flap拍打.
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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方法 of making製造 flaps襟翼.
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當然還有其他做肢體的方法。
06:54
If I put the flap拍打 on the edge邊緣, it uses使用 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拍打 from the middle中間, it uses使用 a full充分 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拍打,
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所以不管怎麼摺出一個肢體,
07:02
it needs需求 some part部分
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至少都會用去
07:04
of a circular region地區 of paper.
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某個部分的圓才能摺出來。
07:06
So now we're ready準備 to scale規模 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襟翼?
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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摺紙 artists藝術家 discovered發現 these principles原則
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摺紙師傅發現這些原理,
07:19
and realized實現 we could make arbitrarily任意 complicated複雜 figures人物
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只要把圓圈組合起來,
07:22
just by packing填料 circles.
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就可以隨意做出複雜的作品。
07:25
And here's這裡的 where the dead people start開始 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研究
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因為很多人研究過
07:31
the problem問題 of packing填料 circles.
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圓圈堆疊這個題目,
07:33
I can rely依靠 on that vast廣大 history歷史 of mathematicians數學家 and artists藝術家
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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圓盤 packings填料 and arrangements安排.
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看看圓圈要怎麼堆疊和組合,
07:39
And I can use those patterns模式 now to create創建 origami摺紙 shapes形狀.
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再運用到我的摺紙作品上。
07:43
So we figured想通 out these rules規則 whereby因此 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裝飾 the patterns模式 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根據 to more rules規則. That gives you the folds褶皺.
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這樣就可以畫出摺痕了,
07:50
Those folds褶皺 fold into a base基礎. You shape形狀 the base基礎.
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這些摺痕可以摺出一個大概輪廓,
07:53
You get a folded折疊 shape形狀 -- in this case案件, a cockroach蟑螂.
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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簡單.
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就是這麼簡單。
07:59
(Laughter笑聲)
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(笑聲)
08:02
It's so simple簡單 that a computer電腦 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簡單 is that?"
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你可能會問:「這真的很簡單嗎?」
08:07
But computers電腦 -- you need to be able能夠 to describe描述 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基本 terms條款, 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電腦 program程序 a bunch of years年份 ago
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所以我在幾年前撰寫了一個電腦程式,
08:14
called TreeMakerTreeMaker, and you can download下載 it from my website網站.
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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自由. It runs運行 on all the major重大的 platforms平台 -- even Windows視窗.
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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笑聲)
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(笑聲)
08:21
And you just draw a stick figure數字,
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你只要用線條畫出輪廓,
08:23
and it calculates計算 the crease pattern模式.
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電腦就會幫你畫出摺痕圖案,
08:25
It does the circle packing填料, calculates計算 the crease pattern模式,
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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數字 that I just showed顯示 --
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如果以我剛才畫的線條輪廓為例,
08:30
which哪一個 you can kind of tell, it's a deer鹿, it's got antlers鹿角 --
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你可以看出它是一隻鹿,有角,
08:33
you'll你會 get this crease pattern模式.
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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模式, you fold on the dotted lines,
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照著圖案上的虛線摺,
08:37
you'll你會 get a base基礎 that you can then shape形狀
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就可以摺出大概的形狀,
08:40
into a deer鹿,
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再細修就會摺成一隻鹿,
08:42
with exactly究竟 the crease pattern模式 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不同 deer鹿,
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如果你想摺一隻不同品種的鹿,
08:46
not a white-tailed白尾 deer鹿, but you want a mule馬騾 deer鹿, or an elk麋鹿,
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而不要這隻白尾鹿,
08:49
you change更改 the packing填料,
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你只要改變圓圈堆疊的方式,
08:51
and you can do an elk麋鹿.
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你就可以做出一隻麋鹿,
08:53
Or you could do a moose駝鹿.
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或是一隻北美麋鹿,
08:55
Or, really, any other kind of deer鹿.
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或是任何一隻其他品種的鹿。
08:57
These techniques技術 revolutionized革命性 this art藝術.
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這些技術完全改造了這門技藝,
09:00
We found發現 we could do insects昆蟲,
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我們現在可以摺出昆蟲、
09:02
spiders蜘蛛, which哪一個 are close,
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蜘蛛,這二種很接近--
09:04
things with legs, things with legs and wings翅膀,
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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天線.
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或是有腳和有觸角的生物。
09:10
And if folding摺頁 a single praying祈禱 mantis from a single uncut完整無缺 square廣場
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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有趣 enough足夠,
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做出一隻螳螂還不夠好玩,
09:15
then you could do two praying祈禱 mantises螳螂
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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完整無缺 square廣場.
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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小吃 Time."
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這幅作品叫「點心時間」。
09:24
And you can do more than just insects昆蟲.
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摺紙不只可以做出昆蟲,
09:26
This -- you can put details細節,
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你還可以在細節上多所描繪,
09:28
toes腳趾 and claws. A grizzly灰熊 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青蛙 has toes腳趾.
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而這隻樹蛙則有腳趾。
09:33
Actually其實, lots of people in origami摺紙 now put toes腳趾 into their models楷模.
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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腳趾 have become成為 an origami摺紙 meme米姆,
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摺腳趾變成了一種流行,
09:38
because everyone's大家的 doing it.
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大家都在摺腳趾。
09:41
You can make multiple subjects主題.
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你還可以摺出多個物件,
09:43
So these are a couple一對 of instrumentalists器樂.
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像是這兩個音樂家,
09:45
The guitar吉他 player播放機 from a single square廣場,
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吉他手是用一張正方形的紙摺出來的,
09:48
the bass低音 player播放機 from a single square廣場.
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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吉他, bass低音 --
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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複雜 instrument儀器."
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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器官.
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那你可以做個風琴。
09:58
(Laughter笑聲)
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(笑聲)
10:01
And what this has allowed允許 is the creation創建
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摺紙就是這種
10:03
of origami-on-demand摺紙點播.
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隨心所欲的創作藝術,
10:05
So now people can say, "I want exactly究竟 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有時 you create創建 high art藝術,
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有時我做的是藝術品,
10:13
and sometimes有時 you pay工資 the bills票據 by doing some commercial廣告 work.
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有時則接受一些商業邀約。
10:16
But I want to show顯示 you some examples例子.
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我想給大家看一些範例,
10:18
Everything you'll你會 see here,
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你所看到的一切,
10:20
except the car汽車, is origami摺紙.
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除了車子以外,都是摺紙作品。
10:23
(Video視頻)
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(影片)
10:51
(Applause掌聲)
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(掌聲)
10:54
Just to show顯示 you, this really was folded折疊 paper.
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這些全都是摺紙作品,
10:57
Computers電腦 made製作 things move移動,
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電腦則負責為他們添加動畫,
10:59
but these were all real真實, folded折疊 objects對象 that we made製作.
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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視覺效果,
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摺紙作品不只是好看而已,
11:06
but it turns out to be useful有用 even in the real真實 world世界.
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在真實世界裡還有可以應用的範圍。
11:09
Surprisingly出奇, origami摺紙
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很難想像,
11:10
and the structures結構 that we've我們已經 developed發達 in origami摺紙
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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應用 in medicine醫學, in science科學,
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竟然可以應用在醫療、科學、
11:16
in space空間, in the body身體, consumer消費者 electronics電子產品 and more.
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太空、人體、家電等地方上。
11:19
And I want to show顯示 you some of these examples例子.
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現在給大家看一些例子。
11:22
One of the earliest最早 was this pattern模式,
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這是最早期的一個圖形:
11:24
this folded折疊 pattern模式,
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這個由日本工程師
11:26
studied研究 by Koryo高麗 Miura三浦, a Japanese日本 engineer工程師.
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三浦公亮所研究的摺痕圖案,
11:29
He studied研究 a folding摺頁 pattern模式, and realized實現
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他研究後發現,
11:31
this could fold down into an extremely非常 compact緊湊 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簡單 opening開盤 and closing關閉 structure結構體.
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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設計 this solar太陽能 array排列.
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他用這個來設計太陽能板。
11:40
It's an artist's藝術家 rendition翻譯, but it flew in a Japanese日本 telescope望遠鏡
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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其實 a little origami摺紙
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詹姆斯.韋伯太空望遠鏡裡面
11:47
in the James詹姆士 Webb韋伯 Space空間 Telescope望遠鏡, but it's very simple簡單.
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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望遠鏡, going up in space空間,
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太空望遠鏡上到太空後,
11:52
it unfolds展開 in two places地方.
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要在二個地方展開,
11:55
It folds褶皺 in thirds三分之二. It's a very simple簡單 pattern模式 --
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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不會 even call that origami摺紙.
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你甚至不會認為那是摺紙技術。
11:59
They certainly當然 didn't need to talk to origami摺紙 artists藝術家.
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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更高 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威力 need some origami摺紙.
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可能就需要一點摺紙技巧。
12:07
Engineers工程師 at Lawrence勞倫斯 Livermore利弗莫爾 National國民 Lab實驗室
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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理念 for a telescope望遠鏡 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眼鏡.
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他們稱它為「大眼鏡」。
12:14
The design設計 called for geosynchronous地球同步 orbit軌道
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這項設計需要同步軌道,
12:16
25,000 miles英里 up,
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設定在4萬1千600公尺高空,
12:18
100-meter-儀表 diameter直徑 lens鏡片.
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還需要一個直徑100公尺的鏡片,
12:21
So, imagine想像 a lens鏡片 the size尺寸 of a football足球 field領域.
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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有興趣 in this:
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對這個設計有興趣的人有兩種:
12:26
planetary行星 scientists科學家們, 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是否 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空間? You've got to get it up there in a rocket火箭.
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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火箭 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玻璃 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不知何故.
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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模型.
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這是縮小的模型。
12:51
Folded折疊 lens鏡片, you divide劃分 up the panels面板, you add flexures彎曲.
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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模式的 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少數 meters.
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直徑大的玻璃縮小到只有幾公尺。
12:59
So the Livermore利弗莫爾 engineers工程師,
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因此利弗摩爾的工程師
13:01
wanting希望 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也許 live生活 origamistsorigamists, said,
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於是他們來找摺紙專家說:
13:06
"Let's see if someone有人 else其他 is doing this sort分類 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看著 into the origami摺紙 community社區,
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於是他們向摺紙團體求救,
13:12
we got in touch觸摸 with them, and I started開始 working加工 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發達 a pattern模式 together一起
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我們一起開發了這種圖案,
13:16
that scales to arbitrarily任意 large size尺寸,
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可以隨意放大到任何尺寸,
13:18
but that allows允許 any flat平面 ring or disc圓盤
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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整齊, compact緊湊 cylinder圓筒.
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摺疊成非常整齊、緊實的圓柱體。
13:25
And they adopted採用 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哪一個 was not 100 meters -- it was a five-meter五米.
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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五米 telescope望遠鏡 --
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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四分之一英里 focal length長度.
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需要1.6公尺的焦距長度,
13:33
And it works作品 perfectly完美 on its test測試 range範圍,
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在測試階段表現得非常好,
13:35
and it indeed確實 folds褶皺 up into a neat整齊 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摺紙 in space空間.
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目前,太空上還應用了其他的摺紙技術,
13:41
Japan日本 Aerospace航天 [Exploration勘探] Agency機構 flew a solar太陽能 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展開 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問題 that's being存在 solved解決了 here is
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我們幫他們解決的問題是,
13:52
something that needs需求 to be big and sheet-like片狀 at its destination目的地,
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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需求 to be small for the journey旅程.
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在運送的時候將它縮小,
13:57
And that works作品 whether是否 you're going into space空間,
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不管你是要上太空
14:00
or whether是否 you're just going into a body身體.
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或是進入人體都一樣。
14:03
And this example is the latter後者.
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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支架 developed發達 by Zhong You
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這是牛津大學的游忠博士
14:08
at Oxford牛津 University大學.
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所發明的心臟血管支架。
14:10
It holds持有 open打開 a blocked受阻 artery動脈 when it gets得到 to its destination目的地,
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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需求 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通過 your blood血液 vessels船隻.
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才能通過血管。
14:18
And this stent支架 folds褶皺 down using運用 an origami摺紙 pattern模式,
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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基於 on a model模型 called the water bomb炸彈 base基礎.
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利用摺紙上所常用的水雷方式摺疊起來。
14:25
Airbag安全氣囊 designers設計師 also have the problem問題
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設計安全氣囊的工程師也有相同困擾,
14:27
of getting得到 flat平面 sheets床單
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他們需要將一個扁平的袋子
14:29
into a small space空間.
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壓縮擠進一個很小的空間。
14:32
And they want to do their design設計 by simulation模擬.
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他們希望以模擬的方式來看看,
14:34
So they need to figure數字 out how, in a computer電腦,
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要怎麼樣利用電腦來模擬出
14:36
to flatten弄平 an airbag安全氣囊.
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壓縮安全氣囊的最佳方式。
14:38
And the algorithms算法 that we developed發達
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我們在摺昆蟲時所開發出來的
14:40
to do insects昆蟲
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演算法,
14:42
turned轉身 out to be the solution for airbags安全氣囊
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後來變成了模擬壓縮安全氣囊的
14:45
to do their simulation模擬.
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最佳解法。
14:47
And so they can do a simulation模擬 like this.
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他們做出的模擬是像這樣,
14:50
Those are the origami摺紙 creases摺痕 forming成型,
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那都是摺紙的摺痕圖案,
14:52
and now you can see the airbag安全氣囊 inflate膨脹
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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引線
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這讓我想到
14:59
to a really interesting有趣 idea理念.
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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支架
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心臟血管支架
15:06
came來了 from that little blow-up爆炸 box
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是由你在小學時就學過的
15:08
that you might威力 have learned學到了 in elementary初級 school學校.
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那種會打開的小盒子所啟發,
15:11
It's the same相同 pattern模式, called the water bomb炸彈 base基礎.
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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安全氣囊變平 algorithm算法
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而將安全氣囊壓縮起來的演算法,
15:16
came來了 from all the developments發展
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則是受到我們摺昆蟲的摺法所影響,
15:18
of circle packing填料 and the mathematical數學的 theory理論
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為了要摺出昆蟲的腳,
15:21
that was really developed發達
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我們得把各個圓圈堆疊起來,
15:23
just to create創建 insects昆蟲 -- 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經常 happens發生
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事實上,這些都與
15:29
in math數學 and science科學.
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數學及科學相關,
15:31
When you get math數學 involved參與, problems問題 that you solve解決
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當我們在解決美學上的問題,
15:34
for aesthetic審美 value only,
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或是試圖創造某些藝術品時,
15:36
or to create創建 something beautiful美麗,
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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應用 in the real真實 world世界.
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應用到真實的世界裡。
15:43
And as weird奇怪的 and surprising奇怪 as it may可能 sound聲音,
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這乍聽之下或許難以置信,
15:46
origami摺紙 may可能 someday日後 even save保存 a life.
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但有一天摺紙或許能救人一命。
15:50
Thanks謝謝.
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謝謝。
15:52
(Applause掌聲)
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(掌聲)
Translated by Marie Wu
Reviewed by Shelley Krishna TSANG

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ABOUT THE SPEAKER
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.

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.

More profile about the speaker
Robert Lang | Speaker | TED.com