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