Thank you very much. Please excuse me for sitting; I'm very old. (Laughter) Well, the topic I'm going to discuss is one which is, in a certain sense, very peculiar because it's very old. Roughness is part of human life forever and forever, and ancient authors have written about it. It was very much uncontrollable, and in a certain sense, it seemed to be the extreme of complexity, just a mess, a mess and a mess. There are many different kinds of mess. Now, in fact, by a complete fluke, I got involved many years ago in a study of this form of complexity, and to my utter amazement, I found traces -- very strong traces, I must say -- of order in that roughness. And so today, I would like to present to you a few examples of what this represents. I prefer the word roughness to the word irregularity because irregularity -- to someone who had Latin in my long-past youth -- means the contrary of regularity. But it is not so. Regularity is the contrary of roughness because the basic aspect of the world is very rough.
非常感谢。 请原谅我坐着讲; 我很老了。 (笑声) 我要讨论的主题 在某种意义上很古怪, 因为它很古老。 粗糙永永远远是 人类生活的一部分。 古代的作者描写过它。 它很不受控制。 在某种意义上, 它似乎是极度的复杂, 一片混乱、 乱七八糟。 有许多不同类型的混乱。 那么,实际上 完全是出于偶然, 我在许多年前 涉足于这种复杂性的研究。 让我非常惊讶的是, 我发现了—— 很清晰的踪迹,我必须说—— 粗糙中秩序的踪迹 今天,我想向你们展示 几个 有代表性的例子 我喜欢“粗糙”这个词 而不是“不规则” 因为“不规则”—— 对于象我这样 年轻时学过拉丁文的人来讲—— 是“规则”的反义词 其实并非如此。 “规则”是“粗糙”的反义词 因为世界的基本面 是很粗糙的。
So let me show you a few objects. Some of them are artificial. Others of them are very real, in a certain sense. Now this is the real. It's a cauliflower. Now why do I show a cauliflower, a very ordinary and ancient vegetable? Because old and ancient as it may be, it's very complicated and it's very simple, both at the same time. If you try to weigh it -- of course it's very easy to weigh it, and when you eat it, the weight matters -- but suppose you try to measure its surface. Well, it's very interesting. If you cut, with a sharp knife, one of the florets of a cauliflower and look at it separately, you think of a whole cauliflower, but smaller. And then you cut again, again, again, again, again, again, again, again, again, and you still get small cauliflowers. So the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole, but smaller. Now, what did humanity do with that? Very, very little. (Laughter)
那么让我给你们展示几个东西。 有些是人造的 另外一些在某种意义上讲是非常真实的。 这个是真实的,这是一个菜花。 我为什么展示一个菜花, 一种非常普通和古老的蔬菜? 因为尽管它很古老, 它却是非常复杂的,同时也是 非常简单的。 如果您想称它的重量,当然称它是非常容易的。 当你吃它时,你关心的是重量。 但是假设您想 测量它的表面积。 那么,非常有意思。 如果您用一把锋利的刀, 切下其中一朵花, 分别观察它, 您会看到一棵整菜花,只是小点儿。 您然后再切, 再切,再切,再切,…. 您得到仍然是小菜花。 在人类的经验中 总是有一些形状 具有奇怪的特性 每个部分就象整体一样 只是更小 现在人类是否对此做了些什么呢? 非常非常少。 (笑声)
So what I did actually is to study this problem, and I found something quite surprising. That one can measure roughness by a number, a number, 2.3, 1.2 and sometimes much more. One day, a friend of mine, to bug me, brought a picture and said, "What is the roughness of this curve?" I said, "Well, just short of 1.5." It was 1.48. Now, it didn't take me any time. I've been looking at these things for so long. So these numbers are the numbers which denote the roughness of these surfaces. I hasten to say that these surfaces are completely artificial. They were done on a computer, and the only input is a number, and that number is roughness. So on the left, I took the roughness copied from many landscapes. To the right, I took a higher roughness. So the eye, after a while, can distinguish these two very well.
我实际上做的就是 研究这个问题, 我发现了相当惊奇的事情。 我们可以用数字来度量粗糙度 用一个数字 2.3,1.2. 有时需要多个数字 一天,我一个朋友 来烦我, 他带来了一张图片,说: “这条曲线的粗糙度是多少?” 我说,“好的,小与1.5。” 是1.48。 这一点也不费事。 我观察这些事物很长时间了。 这些数字表示 这些表面的粗糙度。 我急切地说这些表面 完全是人造的, 是用计算机产生的。 唯一的输入是一个数字. 那个数字就是粗糙度. 在左边 我取的是从许多风景中复制的粗糙度 在右边,我采取了更高的粗糙度 过一会儿 眼睛就可以很好地区分这两个.
Humanity had to learn about measuring roughness. This is very rough, and this is sort of smooth, and this perfectly smooth. Very few things are very smooth. So then if you try to ask questions: "What's the surface of a cauliflower?" Well, you measure and measure and measure. Each time you're closer, it gets bigger, down to very, very small distances. What's the length of the coastline of these lakes? The closer you measure, the longer it is. The concept of length of coastline, which seems to be so natural because it's given in many cases, is, in fact, complete fallacy; there's no such thing. You must do it differently.
人类必须了解粗糙度的测量. 这个非常粗糙,这个有点光滑,这个非常光滑。 很少东西是很光滑的。 所以你如果要问: 一个菜花的表面积是多少? 那么,你反复地测量。 测量得越精确,得到的数值就会越大, 直到非常、 非常小的差距。 这些湖泊的湖岸线 长度是多少? 你测量得越精确,结果越长。 海岸线长度的概念 似乎是那么自然, 在许多情况下都会用到它, 但实际上,是完全错误的。根本没有这种东西。 你必须换种方式对待它。
What good is that, to know these things? Well, surprisingly enough, it's good in many ways. To begin with, artificial landscapes, which I invented sort of, are used in cinema all the time. We see mountains in the distance. They may be mountains, but they may be just formulae, just cranked on. Now it's very easy to do. It used to be very time-consuming, but now it's nothing. Now look at that. That's a real lung. Now a lung is something very strange. If you take this thing, you know very well it weighs very little. The volume of a lung is very small, but what about the area of the lung? Anatomists were arguing very much about that. Some say that a normal male's lung has an area of the inside of a basketball [court]. And the others say, no, five basketball [courts]. Enormous disagreements. Why so? Because, in fact, the area of the lung is something very ill-defined. The bronchi branch, branch, branch and they stop branching, not because of any matter of principle, but because of physical considerations: the mucus, which is in the lung. So what happens is that in a way you have a much bigger lung, but it branches and branches down to distances about the same for a whale, for a man and for a little rodent.
知道这些事情有什么好处呢? 足以让人吃惊的是, 它的好处是多方面的。 首先,人工景观—— 我发明的名词—— 在电影中经常使用。 我们看远处的群山。 他们可能是山,也可能只是个公式,是手摇出来的。 现在很容易做。 它曾经是非常耗时的,但现在没有什么。 现在看看这个,这是一个真正的肺。 肺是很奇怪的东西。 如果你把它拿在手里, 你就会知道它的重量很小。 肺的体积也很小。 但肺的面积呢? 解剖学家们对此争论很大。 有人说一个正常男性的肺 其面积相当于一个篮球 内部的面积。 有人说,不对,是五个篮球。 分歧很大。 为何如此?因为实际上肺的面积的定义 非常含糊不清。 支气管分枝,分枝,分枝。 它们停止产生分枝 不是因为规则的缘故, 而是因为物理的考虑—— 肺内的粘液。 假如您有一个很大的肺 它的分支产生分支, 那将会怎样呢? 对于鲸鱼、人和小的啮齿目动物来说 没有两个距离大致相同。
Now, what good is it to have that? Well, surprisingly enough, amazingly enough, the anatomists had a very poor idea of the structure of the lung until very recently. And I think that my mathematics, surprisingly enough, has been of great help to the surgeons studying lung illnesses and also kidney illnesses, all these branching systems, for which there was no geometry. So I found myself, in other words, constructing a geometry, a geometry of things which had no geometry. And a surprising aspect of it is that very often, the rules of this geometry are extremely short. You have formulas that long. And you crank it several times. Sometimes repeatedly: again, again, again, the same repetition. And at the end, you get things like that.
那么,这有什么好处呢? 足以令人吃惊、足以让人称奇的是, 解剖学家直到最近才对肺的结构 有了一些正确的认识 我认为我的数学, 令人吃惊地 为研究肺病 的外科医生 帮了大忙。 还有肾病. 这些器官都具有分枝系统, 但没有几何结构。 因此我发现我自己,换句话说, 为这种没有几何结构的事物 构造了几何规则。 并且,一个惊奇的方面是, 这几何规则经常是 极其简练的。 你的公式只有这么长。 你把它迭代多次。 有时需要一次一次地重复, 重复同样的运算。 最后,你将得到这样的东西。
This cloud is completely, 100 percent artificial. Well, 99.9. And the only part which is natural is a number, the roughness of the cloud, which is taken from nature. Something so complicated like a cloud, so unstable, so varying, should have a simple rule behind it. Now this simple rule is not an explanation of clouds. The seer of clouds had to take account of it. I don't know how much advanced these pictures are. They're old. I was very much involved in it, but then turned my attention to other phenomena.
这朵云彩是完全地, 100%地人造的。 好吧,99.9%。 其中唯一自然的部分 是一个数字,云的粗糙度, 这是取自于自然的。 象云这种团状的复杂东西, 如此不稳定,如此易变, 背后应该有一个简单规则。 这个简单规则 不是对云的一个解释。 云的观察者必须 把它考虑在内。 我不知道这些图片有多先进, 他们是旧的。 我曾经很投入地研究它们, 但后来我的注意力转向了其他现象。
Now, here is another thing which is rather interesting. One of the shattering events in the history of mathematics, which is not appreciated by many people, occurred about 130 years ago, 145 years ago. Mathematicians began to create shapes that didn't exist. Mathematicians got into self-praise to an extent which was absolutely amazing, that man can invent things that nature did not know. In particular, it could invent things like a curve which fills the plane. A curve's a curve, a plane's a plane, and the two won't mix. Well, they do mix. A man named Peano did define such curves, and it became an object of extraordinary interest. It was very important, but mostly interesting because a kind of break, a separation between the mathematics coming from reality, on the one hand, and new mathematics coming from pure man's mind. Well, I was very sorry to point out that the pure man's mind has, in fact, seen at long last what had been seen for a long time. And so here I introduce something, the set of rivers of a plane-filling curve. And well, it's a story unto itself. So it was in 1875 to 1925, an extraordinary period in which mathematics prepared itself to break out from the world. And the objects which were used as examples, when I was a child and a student, as examples of the break between mathematics and visible reality -- those objects, I turned them completely around. I used them for describing some of the aspects of the complexity of nature.
这是另一件 相当有趣的事情 数学史上的 一次粉碎性事件, 没有多少人赞赏它, 发生于大约130年前, 145年前。 数学家开始创造 不存在的形状 数学家们有点沾沾自喜, 甚至在某种程度上喜不自胜, 因为人类能发明出 大自然不知道的事物。 具体来说,人类可以发明 填装飞机的曲线。 曲线是曲线,飞机是飞机, 二者不会混淆 哦,他们还真混淆了。 一个名叫皮诺的人 定义了这种曲线 它成为了非常有意思的对象。 它非常重要,但更有趣的是 因为它导致了数学的分裂, 来自现实的数学 和纯粹来自人的头脑的新数学 之间的分离。 那么,我非常抱歉地指出, 纯粹的人脑 实际上 终于看见了 一直是随处可见的东西 那么在这里我要介绍一下 一套飞机填装曲线。 那么, 它本身就是一个故事。 那是在1875年至1925年, 一个数学本身 准备在世界上爆发的非凡时期。 那些在数学与 可见现实分裂时, 那时我还是个孩子和学生, 被用作例子 的事物- 那些对象, 我完全地拿它们另作他用。 我用它们来描述 自然复杂性的某些方面。
Well, a man named Hausdorff in 1919 introduced a number which was just a mathematical joke, and I found that this number was a good measurement of roughness. When I first told it to my friends in mathematics they said, "Don't be silly. It's just something [silly]." Well actually, I was not silly. The great painter Hokusai knew it very well. The things on the ground are algae. He did not know the mathematics; it didn't yet exist. And he was Japanese who had no contact with the West. But painting for a long time had a fractal side. I could speak of that for a long time. The Eiffel Tower has a fractal aspect. I read the book that Mr. Eiffel wrote about his tower, and indeed it was astonishing how much he understood.
那么,1919年,一个名叫豪斯多夫的人 介绍了一个数字,这个数字简直是一个数学笑话。 我发现这个数字 是一个很好的测量粗糙度的值。 当我首先把它告诉我的数学朋友时 他们说: “别傻了。 那只是一个数。” 事实上我不傻。 大画家葛饰北斋很了解它。 地面上长的是海藻。 他不懂数学;那时还没有数学。 他是日本人,没有接触过西方文化。 但是他的绘画长期以来就有分数维的一面。 我讲这个可以将很长时间。 埃佛尔铁塔也有分数维的方面。 我读了埃菲尔先生写的关于他这座塔的书。 他了解的程度的确使我吃惊。
This is a mess, mess, mess, Brownian loop. One day I decided -- halfway through my career, I was held by so many things in my work -- I decided to test myself. Could I just look at something which everybody had been looking at for a long time and find something dramatically new? Well, so I looked at these things called Brownian motion -- just goes around. I played with it for a while, and I made it return to the origin. Then I was telling my assistant, "I don't see anything. Can you paint it?" So he painted it, which means he put inside everything. He said: "Well, this thing came out ..." And I said, "Stop! Stop! Stop! I see; it's an island." And amazing. So Brownian motion, which happens to have a roughness number of two, goes around. I measured it, 1.33. Again, again, again. Long measurements, big Brownian motions, 1.33. Mathematical problem: how to prove it? It took my friends 20 years. Three of them were having incomplete proofs. They got together, and together they had the proof. So they got the big [Fields] medal in mathematics, one of the three medals that people have received for proving things which I've seen without being able to prove them.
这是一个乱糟糟的布朗环。 一天,我决定 在我职业生涯的半途中, 我被工作中太多的事情所缠绕, 我决定考验一下自己。 我能否在 每个人都很熟悉的事物中 找到一些戏剧性的新发现呢? 于是我观察这些 被称作布朗运动的现象——只是来回转圈. 我玩了一会儿之后, 又把它放回到原处。 然后我对我的助手说: “我没有看到任何东西。你能画出它来吗?” 于是他画将它画了出来,这意味着 他把一切都装进心里了。他说: “那么,事情是...” 我说:“停!停!停! 我看到了,这是一个岛。” 太神奇了。 所以布朗运动, 碰巧粗糙度为2,就是转圈圈。 我测量了它,1.33 一次又一次 长的测量,大型的布朗运动 1.33。 数学问题:怎样证明它? 这花了我朋友20年的时间。 其中三个人得到了一个不完整的证明。 他们不断地聚在一起研究,得到了这个证明。 所以他们获得到了一个数学大奖(菲尔茨奖), 是三大数学奖项之一, 用来奖励那些证明了 别人看到了但无法证明的事情的人们。
Now everybody asks me at one point or another, "How did it all start? What got you in that strange business?" What got you to be, at the same time, a mechanical engineer, a geographer and a mathematician and so on, a physicist? Well actually I started, oddly enough, studying stock market prices. And so here I had this theory, and I wrote books about it -- financial prices increments. To the left you see data over a long period. To the right, on top, you see a theory which is very, very fashionable. It was very easy, and you can write many books very fast about it. (Laughter) There are thousands of books on that. Now compare that with real price increments. Where are real price increments? Well, these other lines include some real price increments and some forgery which I did. So the idea there was that one must be able to -- how do you say? -- model price variation. And it went really well 50 years ago. For 50 years, people were sort of pooh-poohing me because they could do it much, much easier. But I tell you, at this point, people listened to me. (Laughter) These two curves are averages: Standard & Poor, the blue one; and the red one is Standard & Poor's from which the five biggest discontinuities are taken out. Now discontinuities are a nuisance, so in many studies of prices, one puts them aside. "Well, acts of God. And you have the little nonsense which is left. Acts of God." In this picture, five acts of God are as important as everything else. In other words, it is not acts of God that we should put aside. That is the meat, the problem. If you master these, you master price, and if you don't master these, you can master the little noise as well as you can, but it's not important. Well, here are the curves for it.
大家经常问我, “这一切是怎么开始的? 是什么让你做起了这个奇怪的行当?” 是什么使我 同时成为一名机械工程师、 一名地理学家 和一名数学家,等等,还有物理学家? 那么,很奇怪的是,我实际上是从 研究股市价格开始的 于是 我提出了这个理论 并且写了关于它的书, 金融价格增量。 在左边您看到的是长期数据。 在右上角, 您看到是一个非常非常时髦的理论。 它非常容易,您可以很快地写出许多关于它的书。 (笑声) 有数以千计的写它的书。 现在把它与真实的价格增量比较一下。 真实的价格增量在哪里呢? 这些曲线包括了 真实的价格增量 和我的伪造。 这里的想法是 人必须能 --怎么说呢? – 模拟价格变化。 50年前这方法运行的很好。 50年来,人们有点儿看不起我, 因为他们可以很容易地做到它。 但是我告诉您,此时此刻,人们听我的。 (笑声) 这两条曲线是均线。 标准普尔,蓝色的那个 而红色的一个是 去掉不连续性最大的五个股票后的 标准普尔。 不连续性是有害的。 因此所有价格研究, 人们总是把它们放到一边。 “哦,不可抗力 您就没有什么好胡搅蛮缠的了。 不可抗力。”在这张图片中, 五个不可抗力同其它因素是同样重要的。 换句话说, 不可抗力是不应该被放到一边的。 那才是肉,是问题的所在。 如果您掌握了这些,您就掌握了价格。 如果您掌握不了这些, 您可以尽量掌握小噪音。 但是这不重要。 那么,这是它的曲线。
Now, I get to the final thing, which is the set of which my name is attached. In a way, it's the story of my life. My adolescence was spent during the German occupation of France. Since I thought that I might vanish within a day or a week, I had very big dreams. And after the war, I saw an uncle again. My uncle was a very prominent mathematician, and he told me, "Look, there's a problem which I could not solve 25 years ago, and which nobody can solve. This is a construction of a man named [Gaston] Julia and [Pierre] Fatou. If you could find something new, anything, you will get your career made." Very simple. So I looked, and like the thousands of people that had tried before, I found nothing.
现在,我讲最后一个事情, 用我名字命名的一个集合。 在某种意义上它是我生命的故事。 我的青春期是在 德军占领下的法国度过的。 因为我认为我也许会 在一天或一个星期之内消失 我过有大的梦想。 战争过后, 我又见到我的叔叔。 我的叔叔是一位非常著名数学家,他告诉我, “你看,有一道难题, 我花了25年也没有解决, 别人也没有解决。 这是一个名叫(加斯顿)朱丽叶和 一个名叫(皮埃尔)费托的人提出来的。 如果你能够 有任何新发现 你将成就你的事业。” 非常简单。 于是我就看这道题, 象之前做过尝试的成千上万的人一样, 我什么也没有发现。
But then the computer came, and I decided to apply the computer, not to new problems in mathematics -- like this wiggle wiggle, that's a new problem -- but to old problems. And I went from what's called real numbers, which are points on a line, to imaginary, complex numbers, which are points on a plane, which is what one should do there, and this shape came out. This shape is of an extraordinary complication. The equation is hidden there, z goes into z squared, plus c. It's so simple, so dry. It's so uninteresting. Now you turn the crank once, twice: twice, marvels come out. I mean this comes out. I don't want to explain these things. This comes out. This comes out. Shapes which are of such complication, such harmony and such beauty. This comes out repeatedly, again, again, again. And that was one of my major discoveries, to find that these islands were the same as the whole big thing, more or less. And then you get these extraordinary baroque decorations all over the place. All that from this little formula, which has whatever, five symbols in it. And then this one. The color was added for two reasons. First of all, because these shapes are so complicated that one couldn't make any sense of the numbers. And if you plot them, you must choose some system. And so my principle has been to always present the shapes with different colorings because some colorings emphasize that, and others it is that or that. It's so complicated.
然后出现了计算机。 我决定研究计算机, 而不是新的数学问题- 例如这个“摆动”的问题,这是新问题- 而是建立在旧问题上。 我由所谓“实数”开始, 也就是数轴上的点, 到虚的“复数”, 也就是平面上的点, 人们应该在平面上研究。 这个形状出来了。 这个形状异常复杂。 公式就隐藏在那里, z等于 z的 平方加c。 它是那么简单,相当简单。 一点意思也没有 现在你把它重复一次、 两次, 两次 奇迹出现了 我是说这个出现了 我不想解释这些东西。 这个出来了。这个出来了。 多么复杂、多么和谐、 多么美丽的形状啊。 这个出来了, 不断地,一而再,再而三地出来, 这就是我的一个主要发现 我发现这些小岛的形状 与整体的大形状相同,或多或少 于是你得到这些 随处可见的非凡的巴洛克式装饰。 所有这些来自这个 只有五个符号的小小的公式 然后这一个 加颜色是由于两个原因 首先,因为这些形状 是如此的复杂, 以至于人根本意识不到这些数字。 如果你想突出它们,您必须选择一些系统 所以我的原则是 总是在展示不同的形状时 涂上不同的颜色 因为有些颜色突出这个, 有些颜色突出那个。 非常复杂。
(Laughter)
(笑声)
In 1990, I was in Cambridge, U.K. to receive a prize from the university, and three days later, a pilot was flying over the landscape and found this thing. So where did this come from? Obviously, from extraterrestrials. (Laughter) Well, so the newspaper in Cambridge published an article about that "discovery" and received the next day 5,000 letters from people saying, "But that's simply a Mandelbrot set very big."
1990 年,我在英国的剑桥大学 接受了一个奖项。 三天后, 一个飞行员在飞行时发现了这个。 这是从哪里来的? 显然,从外星人那里来的。 (笑声) 于是剑桥的校报上 发表一篇有关这一“发现”的文章。 第二天, 收到了5000封来信,人们说: “那只是一个放得很大的曼德尔布罗特图形。”
Well, let me finish. This shape here just came out of an exercise in pure mathematics. Bottomless wonders spring from simple rules, which are repeated without end.
好吧,让我结束演讲。 这个形状仅仅出自 纯数学的一个练习 无边的奇迹源自简单规则的 无限重复。
Thank you very much.
非常感谢。
(Applause)
(掌声)