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.
我演讲的题目是《展翅的鸟儿与太空望远镜》。 你会觉得他们相互之间没有联系, 但我希望在18分钟以后, 你能看到一些关联。 这与折纸有关。下面我就开始了。 什么是折纸? 很多人以为他们知道折纸是什么。它是这样的: 展翅的鸟儿、玩具、东西南北之类的东西。 折纸术以前是这样的。 但它已经改变了。 它已经成为了一种艺术形式,一种雕塑形式。
The common theme -- what makes it origami -- 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.
共同的主题——折纸术的本质—— 是折叠,也是我们如何创造形态的。 你们知道,这非常古老。这是1797年的一幅画。 上面是这些妇女们玩纸玩具的场景。 如果你靠近点看,它是这种形状的,叫做鹤。 每个日本孩子 都学折纸鹤。 所以这种艺术已经存在了数百年, 你可能会想如果某种东西 已经存在了这么久——如此有限制性,只能折叠—— 那么所有能做出的东西应该在很久以前就做出来了。 实际情况也许会是如此。
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.
但在20世纪, 一位名为吉泽的日本折纸艺术家出现了, 他创造出了数万种全新的设计。 更重要的是,他创造了一种语言—— 一种我们可以交流的方式, 一种由点、破折号和箭头构成的代码。 联系到苏珊·布莱克摩尔的演讲, 我们现在有了一种通过传承与选择 传递信息的方法, 我们也知道它的走向。 而它在折纸术中产生的 是这样的东西。 这是一个折纸作品: 一张纸,没有裁剪,只有折叠,数百次折叠。 而这也是折纸, 它显示出我们在现代世界中的已经走到哪了。 自然主义。细节。 你可以做出犄角,鹿角—— 如果你靠近看,偶蹄。
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.
这就引出一个问题:什么发生了改变? 发生变化的是一种 你在艺术中可能不曾期待的东西, 那就是数学。 也就是说,人们将数学原理应用 到艺术中, 来发现潜在的规律。 这就形成了一种强大的工具。 在众多领域提高生产力的秘密—— 包括在折纸术中—— 是让死去的人为你工作。
(Laughter)
(笑声)
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.
因为你所能做的 是将你的问题 转变成一个其他人已经解决的问题, 并运用他们的解决方法。 而我想要告诉你们,我们是如何在折纸术中做到这一点的。 折纸术是围绕折痕图进行的。 这个折痕图就是一个折纸造型 的设计图 设计图可不能随便画。 它们必须遵循4个简单的规则。 它们非常简单,并且很好理解。 第一个规则是双可着色性。你可以用两种颜色 填充你想画的的折痕图而 相同的颜色不会相邻。 在任何顶点的折叠方向-- 凸折法的数量,凹折法的数量-- 之间总是相差两下。多折或少折两下。 就这么简单。 如果观察折痕周围的角, 你会发现在数围成一圈的角时, 所有列为偶数的角加起来是一条直线。 所有列为奇数的角加起来是一个直线。 接下来,如果观察这些纸是怎么叠加起来的, 你会发现不论怎样叠加褶层和纸片, 纸片永远不能 穿透褶层。 这就是四则简单的规则。在折纸艺术中这就是全部。 所有的折纸都源于这些。
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.
现在你觉得:“那些复杂的工艺 能是从四则简单的规则中衍生出来的吗?” 但是,事实上,量子力学的法则 可以在一张餐巾纸上写出来。 而它们可以支配所有的化学, 甚至生活和历史的全部。 如果遵循这些规则, 我们能做出令人吃惊的事。 所以折纸时,在遵循这些规则的情况下, 我们可以做出简单的样式-- 比如这个重复的折叠样式,叫做纹理-- 虽然这样单独看起来很普通。 但如果我们遵守折纸的规则, 我们能把这些样式加入另一种折法, 这种折法本身非常非常的简单。 但当我们把它加进来, 会得到很不一样的东西。 这条鱼有400片鱼鳞, 同样,它是一张没被剪过的正方形纸张。 如果你不想折400片鱼鳞, 你可以退而求其次,做些简单的折叠 得到一只乌龟的背壳或脚趾。 或者可以提升成为一面拥有 50颗星星和13条横条的旗子(美国国旗)。 如果你想做些疯狂的事情, 这有一条有1000片鳞片的响尾蛇。 这个作品展示在楼下, 所以你们有机会可以看看。
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.
在折纸艺术中,最有用的方法 和我们怎样构造生物的一部分有关。 我可以用一个简单的等式来解释。 我们产生了一个想法, 把它与张纸片结合,就能得到一个折纸作品。
(Laughter)
(笑声)
What matters is what we mean by those symbols. 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.
重要的是这些符号代表什么。 你们可能会问:“真的能做到那么具体吗? 我是说一只鹿角虫有两个点状的嘴, 和触角。你真的能做到具体到细节吗?” 是的,真的可以。 那该怎么做呢?我们把它分成 几个小步骤。 为此,让我来展开这个等式。 我先从我的构思开始,使它抽象化。 什么是最抽象的形式呢?线条画。 然后从这个线条画,我得用某种方式得到折叠的式样, 并且包括想要表现对象的所有部分。 一片三角形折叠对应一条腿。 然后,我们称这个折叠的式样为基础。 你可以使它的腿变细,使其弯曲, 你可以把它做成成品。
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.
第一步:很简单。 做出一个构思,画一幅线条图。 最后一步也不是很难,但中间的一步-- 把抽象的描绘变为折叠的式样-- 这很难。 但就是在这,数学理论让我们 翻越难关。 我要向你们展示怎样做, 这样离开这里后,你们可以叠出些东西。 但我们要先从小的开始。 这个基础有很多片状物。 我们要学习怎样做一个片状物。 你会怎样叠一个片状物呢? 拿一张正方形的纸,把它对折再对折, 直到它变得又长又细, 然后这个的尾部就是一个片状物。 我能用它做一条腿,一只手臂,和所有相似的东西。
What paper went into that flap? 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.
在片状物中是什么样的纸呢? 如果把它展开去看它的折痕图, 你们可以看到在纸片的左上角的形状 就是构成片状物的纸。 所以那就是一个片状物,和所有剩下的纸。 我可以用剩下的部分做点别的。 也有另外的做片状物的方法。 也有不同形状的片状物。 如果把片状物叠得更细一些,所用的纸会更少。 如果把片状物尽可能的叠细, 就能只用片状物所需的最少的纸。 就像你们所看到的,只需要纸上四分之一个圆就可以作出一个片状物。 还有别的做片状物的方法。 如果把片状物放在纸片边上,就需要一个半圆的纸。 如果把片状物放在纸片的中心,就需要一整圆。 就是说不论怎样叠, 片状物是由 纸上圆形区域的一部分做成的。 现在让我们来提升到新的水平。 如果要叠一个有很多片状物的东西该怎么办呢? 我需要什么?我需要很多的圆。
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.
在二十世纪九十年代, 折纸艺术家发现了这些规则, 并了解到我们可以通过组合圆形 来叠出任意复杂的形状。 这就是那些死去的人能帮到我们的地方。 因为很多人都研究过 组合圆形的问题。 我可以依赖那些有关圆的组合和排列的 大量的数学与艺术的历史。 然后我可以用这些式样来制造折纸的形状。 我们可以依据这些规则来组合圆形, 依据更多的规矩我们可以 用线条来装饰圆。这就有了折叠线。 沿这些线折叠就可以得到大体形状。你们就做出了大体的形状。 你们得到一个折叠的形状,在这里,是一只蟑螂。 而且它非常的简单。
(Laughter)
(笑声)
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.
因为它很简单,电脑就可以把它做出来。 你们可能问“那能有多简单呢?” 但是要用电脑,你们需要用最基本的方法 来描述一件事物。而这里我们可以做到。 所以我在很多年前写了一个电脑程序, 叫做TreeMaker(造树者),你们可以在我的网页上下载它。 它是免费的。它可以在大部分的操作系统里面运行,甚至在Windows里。
(Laughter)
(笑声)
And you just draw a stick figure, 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."
然后你们就可以自己画一个线条图, 这个程序会根据线条图计算折痕。 这个程序可以排列圆形,计算折痕, 还有如果你们用刚才我展示的线条图, 你们可以看出它是一只有角的鹿, 你们就可以得到这个折痕图。 用这个折痕图,折叠有虚线的地方, 你们就能得到一个基础,然后再用 你们想用的方法 叠出一只鹿。 如果你们想要一只不同种的鹿, 而不是白尾鹿, 你们可以改变圆形的排列, 然后得到一只麋鹿。 或是一只驼鹿。 或是其它任何一种鹿。 这些技术改革了这门艺术。 我们发现我们可以叠出昆虫, 或是相近的蜘蛛, 有脚的东西,有脚和翅膀的东西, 和有脚和触角的东西。 如果用一张没剪过的正方形纸叠一只螳螂 还不够有趣的话, 你们可以用一张没剪过的正方形纸 叠两只螳螂。 她在吃他。 我称之为“点心时间”。
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.
你们能做的不只是昆虫。 你们可以把它做到有细节, 像指头和爪子。一只有爪子的北美洲灰熊。 和这只有脚趾的树蛙。 实际上,在折纸艺术中有很多人把指头加入到他们的模型中。 指头变成了折纸艺术的文化基因。 因为每个人都在做。 你可以做出多种的物体。 像这里有一些音乐家。 一个正方形做出的吉他手。 一个正方形做出的贝斯手。 如果你说,“好吧,但吉他和贝斯 不够帅。 做些更复杂的乐器吧。” 那你可以做一架风琴。
(Laughter)
(笑声)
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.
所以在这个世界里我们能 做出所需要的创造。 如果现在有人说,我想要这个这个还有这个。 你就可以精确的把它们叠出来。 有时可以做纯艺术。 有时可以做些商品卖钱。 但是我想给你们看一些例子。 除了车子, 你们将看到的所有东西都是折纸。
(Video)
(影片)
(Applause)
(掌声)
Just to show you, this really was folded paper. 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.
就是想展示给你们这些真实的折纸。 电脑使所有的东西动起来。 但是这些折纸全都是货真价实的。 我们不只可以在视觉上运用到折纸艺术, 它实际上在现实世界中也很有用。 令人惊奇的,折纸 和从折纸中发展出来的结构 可以在医药学,科学, 太空,身体和电子产品等等上得到应用。
And I want to show you some of these examples. 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.
我想展示一些例子。 在最早的应用中有这样一个样式, 折纸样式, 由日本的工程师Koryo Miura发明的。 他研究这个折纸样式,然后发现 可以折出很紧凑的包装, 有很简单的开口和闭合结构。 他应用这个技术设计了这个太阳能电池板。 这是一个艺术家的表演,但它在1995应用到了 一架日本望远镜。 现在在詹姆斯韦伯太空望远镜中只有 一点点的折纸艺术,但它十分之简单。 这架进入太空的望远镜 在两处展开。 它在第三个处折叠。它是一个很简单的式样, 你都不会把它称作折纸。 这些科学家的确不用跟折纸艺术家讨论。
But if you want to go higher and go larger than this, 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.
但当你要更深入的研究时, 折纸术是必需的。 劳伦斯利物穆尔国家实验室的工程师们 有一个关于一个更大的望远镜的构想。 他们称之为“镜片”。 这个设计需要同步轨道, 高于地面26000英里, 和直径100米的镜片。 所以镜片有一个橄榄球场那么大。 有两类人对这个望远镜有兴趣: 想要观察太空的行星学家, 和其他想要观察地球的人。 不论你想观察什么, 该怎么上太空呢?你需要一个火箭。 而且火箭一般都很小。所以你需要把望远镜做的小一些。 怎么把一大片玻璃变小呢? 唯一的办法就是折叠。 所以你要做这样的事, 这一个小型的模型。
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.
对于镜片,你把板面分区然后加上弯曲。 但是这个样式不能把100米的东西 变成几米。 所以利物穆尔的工程师们, 想要利用那些死去的人的成果, 或是活着的折纸艺术家的成果。 工程师们说“看看有没有别人在做这类事。” 所以他们研究折纸圈。 我们和折纸艺术家取得联系,而我开始和他们一起工作。 我们一起开发了一个 可以应用到任意大小, 但可以允许所有的平面环或圆盘 折成一个整洁紧凑的圆柱体的样式。 他们在第一代的望远镜中采用了这个样式。 而第一代并不是100米而是5米。 但是这个5米的望远镜 有0.25英里的焦距。 而且在它的测试范围内效果很好。 它也的确被叠成了一小捆。
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.
现在,还有别的折纸术应用到太空中。 日本航空【探索者】部门发射了一个太阳光帆。 你们可以看到帆伸展开, 还有帆上的折叠线。 在这里所被解决的问题是 做出了一个在旅途中很小 但在目的地很大的薄片状的物体。 这个可以作用于当你想进入太空, 或是想进入人的身体时。 这个例子就是进入人身体的。 这是由牛津大学的钟游发明的 心脏手术支架。 它在到达目的地时会打开被堵塞的动脉血管。 但在旅途中它需要变得很小才能通过 你的血管。 这个支架运用一种折纸术被叠小。 我们称这个模型为水弹模型。
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.
安全气囊的设计师也遇到了同样的 把大薄片塞进小空间里的 问题。 而且他们都是通过仿真技术来做设计。 所以他们需要在电脑里研究出 怎样使安全气囊变平。 我们所开发出的叠昆虫的 算法在这里变成了 在仿真技术中解决安全气囊问题的 方法。 所以设计师可以做一个这个的模仿。 那些就是折纸的折痕, 现在你们所看到的就是正在放气的安全气囊 并且大家可以知道这方法管不管用。 这个例子实际上可以 推导出一个十分有趣的构想。
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.
你们知道,这些发明设计都是从哪来么? 这个心脏手术支架 是从大家小学就学到的 纸气球中衍生来的。 它们有着相同的构造,称之为“水弹模型”。 那个使安全气囊变平的算法是 从那些实际上只是 发明出来用来叠昆虫, 也就是有腿的东西, 的数学理论。 其实呢,这样的事经常 发生在数学和科学里面。 当你运用数学,解决 你纯粹为了美学价值 或是创造美而想解决的问题时, 实际上结果反过来 在现实世界中也可以应用。 而且即使听上去很奇怪, 折纸术有一天可能会救人一命。 谢谢。
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