Benoit Mandelbrot: Fractals and the art of roughness

翻译人员: James Dang 校对人员: Xu Jiang

Thank you very much.

非常感谢。

Please excuse me for sitting; I'm very old.

请原谅我坐着讲； 我很老了。

(Laughter)

(笑声)

Well, the topic I'm going to discuss

我要讨论的主题

is one which is, in a certain sense, very peculiar

在某种意义上很古怪，

because it's very old.

因为它很古老。

Roughness is part of human life

粗糙永永远远是

forever and forever,

人类生活的一部分。

and ancient authors have written about it.

古代的作者描写过它。

It was very much uncontrollable,

它很不受控制。

and in a certain sense,

在某种意义上，

it seemed to be the extreme of complexity,

它似乎是极度的复杂，

just a mess, a mess and a mess.

一片混乱、 乱七八糟。

There are many different kinds of mess.

有许多不同类型的混乱。

Now, in fact,

那么，实际上

by a complete fluke,

完全是出于偶然，

I got involved many years ago

我在许多年前

in a study of this form of complexity,

涉足于这种复杂性的研究。

and to my utter amazement,

让我非常惊讶的是，

I found traces --

我发现了——

very strong traces, I must say --

很清晰的踪迹，我必须说——

of order in that roughness.

粗糙中秩序的踪迹

And so today, I would like to present to you

今天，我想向你们展示

a few examples

几个

of what this represents.

有代表性的例子

I prefer the word roughness

我喜欢“粗糙”这个词

to the word irregularity

而不是“不规则”

because irregularity --

因为“不规则”——

to someone who had Latin

对于象我这样

in my long-past youth --

年轻时学过拉丁文的人来讲——

means the contrary of regularity.

是“规则”的反义词

But it is not so.

其实并非如此。

Regularity is the contrary of roughness

“规则”是“粗糙”的反义词

because the basic aspect of the world

因为世界的基本面

is very rough.

是很粗糙的。

So let me show you a few objects.

那么让我给你们展示几个东西。

Some of them are artificial.

有些是人造的

Others of them are very real, in a certain sense.

另外一些在某种意义上讲是非常真实的。

Now this is the real. It's a cauliflower.

这个是真实的，这是一个菜花。

Now why do I show a cauliflower,

我为什么展示一个菜花，

a very ordinary and ancient vegetable?

一种非常普通和古老的蔬菜？

Because old and ancient as it may be,

因为尽管它很古老，

it's very complicated and it's very simple,

它却是非常复杂的，同时也是

both at the same time.

非常简单的。

If you try to weigh it -- of course it's very easy to weigh it,

如果您想称它的重量，当然称它是非常容易的。

and when you eat it, the weight matters --

当你吃它时，你关心的是重量。

but suppose you try to

但是假设您想

measure its surface.

测量它的表面积。

Well, it's very interesting.

那么，非常有意思。

If you cut, with a sharp knife,

如果您用一把锋利的刀，

one of the florets of a cauliflower

切下其中一朵花，

and look at it separately,

分别观察它，

you think of a whole cauliflower, but smaller.

您会看到一棵整菜花，只是小点儿。

And then you cut again,

您然后再切，

again, again, again, again, again, again, again, again,

再切，再切，再切，….

and you still get small cauliflowers.

您得到仍然是小菜花。

So the experience of humanity

在人类的经验中

has always been that there are some shapes

总是有一些形状

which have this peculiar property,

具有奇怪的特性

that each part is like the whole,

每个部分就象整体一样

but smaller.

只是更小

Now, what did humanity do with that?

现在人类是否对此做了些什么呢？

Very, very little.

非常非常少。

(Laughter)

(笑声)

So what I did actually is to

我实际上做的就是

study this problem,

研究这个问题，

and I found something quite surprising.

我发现了相当惊奇的事情。

That one can measure roughness

我们可以用数字来度量粗糙度

by a number, a number,

用一个数字

2.3, 1.2 and sometimes much more.

2.3，1.2. 有时需要多个数字

One day, a friend of mine,

一天，我一个朋友

to bug me,

来烦我，

brought a picture and said,

他带来了一张图片，说：

"What is the roughness of this curve?"

“这条曲线的粗糙度是多少？”

I said, "Well, just short of 1.5."

我说，“好的，小与1.5。”

It was 1.48.

是1.48。

Now, it didn't take me any time.

这一点也不费事。

I've been looking at these things for so long.

我观察这些事物很长时间了。

So these numbers are the numbers

这些数字表示

which denote the roughness of these surfaces.

这些表面的粗糙度。

I hasten to say that these surfaces

我急切地说这些表面

are completely artificial.

完全是人造的，

They were done on a computer,

是用计算机产生的。

and the only input is a number,

唯一的输入是一个数字.

and that number is roughness.

那个数字就是粗糙度.

So on the left,

在左边

I took the roughness copied from many landscapes.

我取的是从许多风景中复制的粗糙度

To the right, I took a higher roughness.

在右边，我采取了更高的粗糙度

So the eye, after a while,

过一会儿

can distinguish these two very well.

眼睛就可以很好地区分这两个.

Humanity had to learn about measuring roughness.

人类必须了解粗糙度的测量.

This is very rough, and this is sort of smooth, and this perfectly smooth.

这个非常粗糙，这个有点光滑，这个非常光滑。

Very few things are very smooth.

很少东西是很光滑的。

So then if you try to ask questions:

所以你如果要问：

"What's the surface of a cauliflower?"

一个菜花的表面积是多少？

Well, you measure and measure and measure.

那么，你反复地测量。

Each time you're closer, it gets bigger,

测量得越精确，得到的数值就会越大，

down to very, very small distances.

直到非常、 非常小的差距。

What's the length of the coastline

这些湖泊的湖岸线

of these lakes?

长度是多少？

The closer you measure, the longer it is.

你测量得越精确，结果越长。

The concept of length of coastline,

海岸线长度的概念

which seems to be so natural

似乎是那么自然，

because it's given in many cases,

在许多情况下都会用到它，

is, in fact, complete fallacy; there's no such thing.

但实际上，是完全错误的。根本没有这种东西。

You must do it differently.

你必须换种方式对待它。

What good is that, to know these things?

知道这些事情有什么好处呢？

Well, surprisingly enough,

足以让人吃惊的是，

it's good in many ways.

它的好处是多方面的。

To begin with, artificial landscapes,

首先，人工景观——

which I invented sort of,

我发明的名词——

are used in cinema all the time.

在电影中经常使用。

We see mountains in the distance.

我们看远处的群山。

They may be mountains, but they may be just formulae, just cranked on.

他们可能是山，也可能只是个公式，是手摇出来的。

Now it's very easy to do.

现在很容易做。

It used to be very time-consuming, but now it's nothing.

它曾经是非常耗时的，但现在没有什么。

Now look at that. That's a real lung.

现在看看这个，这是一个真正的肺。

Now a lung is something very strange.

肺是很奇怪的东西。

If you take this thing,

如果你把它拿在手里，

you know very well it weighs very little.

你就会知道它的重量很小。

The volume of a lung is very small,

肺的体积也很小。

but what about the area of the lung?

但肺的面积呢？

Anatomists were arguing very much about that.

解剖学家们对此争论很大。

Some say that a normal male's lung

有人说一个正常男性的肺

has an area of the inside

其面积相当于一个篮球

of a basketball [court].

内部的面积。

And the others say, no, five basketball [courts].

有人说，不对，是五个篮球。

Enormous disagreements.

分歧很大。

Why so? Because, in fact, the area of the lung

为何如此？因为实际上肺的面积的定义

is something very ill-defined.

非常含糊不清。

The bronchi branch, branch, branch

支气管分枝，分枝，分枝。

and they stop branching,

它们停止产生分枝

not because of any matter of principle,

不是因为规则的缘故，

but because of physical considerations:

而是因为物理的考虑——

the mucus, which is in the lung.

肺内的粘液。

So what happens is that in a way

假如您有一个很大的肺

you have a much bigger lung,

它的分支产生分支，

but it branches and branches

那将会怎样呢？

down to distances about the same for a whale, for a man

对于鲸鱼、人和小的啮齿目动物来说

and for a little rodent.

没有两个距离大致相同。

Now, what good is it to have that?

那么，这有什么好处呢？

Well, surprisingly enough, amazingly enough,

足以令人吃惊、足以让人称奇的是，

the anatomists had a very poor idea

解剖学家直到最近才对肺的结构

of the structure of the lung until very recently.

有了一些正确的认识

And I think that my mathematics,

我认为我的数学，

surprisingly enough,

令人吃惊地

has been of great help

为研究肺病

to the surgeons

的外科医生

studying lung illnesses

帮了大忙。

and also kidney illnesses,

还有肾病.

all these branching systems,

这些器官都具有分枝系统，

for which there was no geometry.

但没有几何结构。

So I found myself, in other words,

因此我发现我自己，换句话说,

constructing a geometry,

为这种没有几何结构的事物

a geometry of things which had no geometry.

构造了几何规则。

And a surprising aspect of it

并且，一个惊奇的方面是，

is that very often, the rules of this geometry

这几何规则经常是

are extremely short.

极其简练的。

You have formulas that long.

你的公式只有这么长。

And you crank it several times.

你把它迭代多次。

Sometimes repeatedly: again, again, again,

有时需要一次一次地重复，

the same repetition.

重复同样的运算。

And at the end, you get things like that.

最后，你将得到这样的东西。

This cloud is completely,

这朵云彩是完全地，

100 percent artificial.

100%地人造的。

Well, 99.9.

好吧，99.9%。

And the only part which is natural

其中唯一自然的部分

is a number, the roughness of the cloud,

是一个数字，云的粗糙度，

which is taken from nature.

这是取自于自然的。

Something so complicated like a cloud,

象云这种团状的复杂东西，

so unstable, so varying,

如此不稳定，如此易变，

should have a simple rule behind it.

背后应该有一个简单规则。

Now this simple rule

这个简单规则

is not an explanation of clouds.

不是对云的一个解释。

The seer of clouds had to

云的观察者必须

take account of it.

把它考虑在内。

I don't know how much advanced

我不知道这些图片有多先进，

these pictures are. They're old.

他们是旧的。

I was very much involved in it,

我曾经很投入地研究它们，

but then turned my attention to other phenomena.

但后来我的注意力转向了其他现象。

Now, here is another thing

这是另一件

which is rather interesting.

相当有趣的事情

One of the shattering events

数学史上的

in the history of mathematics,

一次粉碎性事件，

which is not appreciated by many people,

没有多少人赞赏它，

occurred about 130 years ago,

发生于大约130年前，

145 years ago.

145年前。

Mathematicians began to create

数学家开始创造

shapes that didn't exist.

不存在的形状

Mathematicians got into self-praise

数学家们有点沾沾自喜，

to an extent which was absolutely amazing,

甚至在某种程度上喜不自胜，

that man can invent things

因为人类能发明出

that nature did not know.

大自然不知道的事物。

In particular, it could invent

具体来说，人类可以发明

things like a curve which fills the plane.

填装飞机的曲线。

A curve's a curve, a plane's a plane,

曲线是曲线，飞机是飞机，

and the two won't mix.

二者不会混淆

Well, they do mix.

哦，他们还真混淆了。

A man named Peano

一个名叫皮诺的人

did define such curves,

定义了这种曲线

and it became an object of extraordinary interest.

它成为了非常有意思的对象。

It was very important, but mostly interesting

它非常重要，但更有趣的是

because a kind of break,

因为它导致了数学的分裂，

a separation between

来自现实的数学

the mathematics coming from reality, on the one hand,

和纯粹来自人的头脑的新数学

and new mathematics coming from pure man's mind.

之间的分离。

Well, I was very sorry to point out

那么，我非常抱歉地指出，

that the pure man's mind

纯粹的人脑

has, in fact,

实际上

seen at long last

终于看见了

what had been seen for a long time.

一直是随处可见的东西

And so here I introduce something,

那么在这里我要介绍一下

the set of rivers of a plane-filling curve.

一套飞机填装曲线。

And well,

那么，

it's a story unto itself.

它本身就是一个故事。

So it was in 1875 to 1925,

那是在1875年至1925年，

an extraordinary period

一个数学本身

in which mathematics prepared itself to break out from the world.

准备在世界上爆发的非凡时期。

And the objects which were used

那些在数学与

as examples, when I was

可见现实分裂时，

a child and a student, as examples

那时我还是个孩子和学生，

of the break between mathematics

被用作例子

and visible reality --

的事物-

those objects,

那些对象，

I turned them completely around.

我完全地拿它们另作他用。

I used them for describing

我用它们来描述

some of the aspects of the complexity of nature.

自然复杂性的某些方面。

Well, a man named Hausdorff in 1919

那么，1919年，一个名叫豪斯多夫的人

introduced a number which was just a mathematical joke,

介绍了一个数字，这个数字简直是一个数学笑话。

and I found that this number

我发现这个数字

was a good measurement of roughness.

是一个很好的测量粗糙度的值。

When I first told it to my friends in mathematics

当我首先把它告诉我的数学朋友时

they said, "Don't be silly. It's just something [silly]."

他们说： “别傻了。 那只是一个数。”

Well actually, I was not silly.

事实上我不傻。

The great painter Hokusai knew it very well.

大画家葛饰北斋很了解它。

The things on the ground are algae.

地面上长的是海藻。

He did not know the mathematics; it didn't yet exist.

他不懂数学;那时还没有数学。

And he was Japanese who had no contact with the West.

他是日本人，没有接触过西方文化。

But painting for a long time had a fractal side.

但是他的绘画长期以来就有分数维的一面。

I could speak of that for a long time.

我讲这个可以将很长时间。

The Eiffel Tower has a fractal aspect.

埃佛尔铁塔也有分数维的方面。

I read the book that Mr. Eiffel wrote about his tower,

我读了埃菲尔先生写的关于他这座塔的书。

and indeed it was astonishing how much he understood.

他了解的程度的确使我吃惊。

This is a mess, mess, mess, Brownian loop.

这是一个乱糟糟的布朗环。

One day I decided --

一天，我决定

halfway through my career,

在我职业生涯的半途中，

I was held by so many things in my work --

我被工作中太多的事情所缠绕，

I decided to test myself.

我决定考验一下自己。

Could I just look at something

我能否在

which everybody had been looking at for a long time

每个人都很熟悉的事物中

and find something dramatically new?

找到一些戏剧性的新发现呢？

Well, so I looked at these

于是我观察这些

things called Brownian motion -- just goes around.

被称作布朗运动的现象——只是来回转圈.

I played with it for a while,

我玩了一会儿之后，

and I made it return to the origin.

又把它放回到原处。

Then I was telling my assistant,

然后我对我的助手说:

"I don't see anything. Can you paint it?"

“我没有看到任何东西。你能画出它来吗？”

So he painted it, which means

于是他画将它画了出来，这意味着

he put inside everything. He said:

他把一切都装进心里了。他说：

"Well, this thing came out ..." And I said, "Stop! Stop! Stop!

“那么，事情是...” 我说：“停！停！停！

I see; it's an island."

我看到了，这是一个岛。”

And amazing.

太神奇了。

So Brownian motion, which happens to have

所以布朗运动，

a roughness number of two, goes around.

碰巧粗糙度为2，就是转圈圈。

I measured it, 1.33.

我测量了它，1.33

Again, again, again.

一次又一次

Long measurements, big Brownian motions,

长的测量，大型的布朗运动

1.33.

1.33。

Mathematical problem: how to prove it?

数学问题：怎样证明它？

It took my friends 20 years.

这花了我朋友20年的时间。

Three of them were having incomplete proofs.

其中三个人得到了一个不完整的证明。

They got together, and together they had the proof.

他们不断地聚在一起研究，得到了这个证明。

So they got the big [Fields] medal in mathematics,

所以他们获得到了一个数学大奖（菲尔茨奖），

one of the three medals that people have received

是三大数学奖项之一，

for proving things which I've seen

用来奖励那些证明了

without being able to prove them.

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,

大家经常问我，

"How did it all start?

“这一切是怎么开始的？

What got you in that strange business?"

是什么让你做起了这个奇怪的行当？”

What got you to be,

是什么使我

at the same time, a mechanical engineer,

同时成为一名机械工程师、

a geographer

一名地理学家

and a mathematician and so on, a physicist?

和一名数学家，等等，还有物理学家？

Well actually I started, oddly enough,

那么，很奇怪的是，我实际上是从

studying stock market prices.

研究股市价格开始的

And so here

于是

I had this theory,

我提出了这个理论

and I wrote books about it --

并且写了关于它的书，

financial prices increments.

金融价格增量。

To the left you see data over a long period.

在左边您看到的是长期数据。

To the right, on top,

在右上角，

you see a theory which is very, very fashionable.

您看到是一个非常非常时髦的理论。

It was very easy, and you can write many books very fast about it.

它非常容易，您可以很快地写出许多关于它的书。

(Laughter)

(笑声)

There are thousands of books on that.

有数以千计的写它的书。

Now compare that with real price increments.

现在把它与真实的价格增量比较一下。

Where are real price increments?

真实的价格增量在哪里呢？

Well, these other lines

这些曲线包括了

include some real price increments

真实的价格增量

and some forgery which I did.

和我的伪造。

So the idea there was

这里的想法是

that one must be able to -- how do you say? --

人必须能 --怎么说呢? –

model price variation.

模拟价格变化。

And it went really well 50 years ago.

50年前这方法运行的很好。

For 50 years, people were sort of pooh-poohing me

50年来，人们有点儿看不起我，

because they could do it much, much easier.

因为他们可以很容易地做到它。

But I tell you, at this point, people listened to me.

但是我告诉您，此时此刻，人们听我的。

(Laughter)

(笑声)

These two curves are averages:

这两条曲线是均线。

Standard & Poor, the blue one;

标准普尔，蓝色的那个

and the red one is Standard & Poor's

而红色的一个是

from which the five biggest discontinuities

去掉不连续性最大的五个股票后的

are taken out.

标准普尔。

Now discontinuities are a nuisance,

不连续性是有害的。

so in many studies of prices,

因此所有价格研究，

one puts them aside.

人们总是把它们放到一边。

"Well, acts of God.

“哦，不可抗力

And you have the little nonsense which is left.

您就没有什么好胡搅蛮缠的了。

Acts of God." In this picture,

不可抗力。”在这张图片中，

five acts of God are as important as everything else.

五个不可抗力同其它因素是同样重要的。

In other words,

换句话说，

it is not acts of God that we should put aside.

不可抗力是不应该被放到一边的。

That is the meat, the problem.

那才是肉，是问题的所在。

If you master these, you master price,

如果您掌握了这些，您就掌握了价格。

and if you don't master these, you can master

如果您掌握不了这些，

the little noise as well as you can,

您可以尽量掌握小噪音。

but it's not important.

但是这不重要。

Well, here are the curves for it.

那么，这是它的曲线。

Now, I get to the final thing, which is the set

现在，我讲最后一个事情，

of which my name is attached.

用我名字命名的一个集合。

In a way, it's the story of my life.

在某种意义上它是我生命的故事。

My adolescence was spent

我的青春期是在

during the German occupation of France.

德军占领下的法国度过的。

Since I thought that I might

因为我认为我也许会

vanish within a day or a week,

在一天或一个星期之内消失

I had very big dreams.

我过有大的梦想。

And after the war,

战争过后，

I saw an uncle again.

我又见到我的叔叔。

My uncle was a very prominent mathematician, and he told me,

我的叔叔是一位非常著名数学家，他告诉我，

"Look, there's a problem

“你看，有一道难题，

which I could not solve 25 years ago,

我花了25年也没有解决，

and which nobody can solve.

别人也没有解决。

This is a construction of a man named [Gaston] Julia

这是一个名叫（加斯顿）朱丽叶和

and [Pierre] Fatou.

一个名叫（皮埃尔）费托的人提出来的。

If you could

如果你能够

find something new, anything,

有任何新发现

you will get your career made."

你将成就你的事业。”

Very simple.

非常简单。

So I looked,

于是我就看这道题，

and like the thousands of people that had tried before,

象之前做过尝试的成千上万的人一样，

I found nothing.

我什么也没有发现。

But then the computer came,

然后出现了计算机。

and I decided to apply the computer,

我决定研究计算机，

not to new problems in mathematics --

而不是新的数学问题-

like this wiggle wiggle, that's a new problem --

例如这个“摆动”的问题，这是新问题-

but to old problems.

而是建立在旧问题上。

And I went from what's called

我由所谓“实数”开始，

real numbers, which are points on a line,

也就是数轴上的点，

to imaginary, complex numbers,

到虚的“复数”，

which are points on a plane,

也就是平面上的点，

which is what one should do there,

人们应该在平面上研究。

and this shape came out.

这个形状出来了。

This shape is of an extraordinary complication.

这个形状异常复杂。

The equation is hidden there,

公式就隐藏在那里，

z goes into z squared, plus c.

z等于 z的 平方加c。

It's so simple, so dry.

它是那么简单，相当简单。

It's so uninteresting.

一点意思也没有

Now you turn the crank once, twice:

现在你把它重复一次、 两次，

twice,

两次

marvels come out.

奇迹出现了

I mean this comes out.

我是说这个出现了

I don't want to explain these things.

我不想解释这些东西。

This comes out. This comes out.

这个出来了。这个出来了。

Shapes which are of such complication,

多么复杂、多么和谐、

such harmony and such beauty.

多么美丽的形状啊。

This comes out

这个出来了，

repeatedly, again, again, again.

不断地，一而再，再而三地出来，

And that was one of my major discoveries,

这就是我的一个主要发现

to find that these islands were the same

我发现这些小岛的形状

as the whole big thing, more or less.

与整体的大形状相同，或多或少

And then you get these

于是你得到这些

extraordinary baroque decorations all over the place.

随处可见的非凡的巴洛克式装饰。

All that from this little formula,

所有这些来自这个

which has whatever, five symbols in it.

只有五个符号的小小的公式

And then this one.

然后这一个

The color was added for two reasons.

加颜色是由于两个原因

First of all, because these shapes

首先，因为这些形状

are so complicated

是如此的复杂，

that one couldn't make any sense of the numbers.

以至于人根本意识不到这些数字。

And if you plot them, you must choose some system.

如果你想突出它们，您必须选择一些系统

And so my principle has been

所以我的原则是

to always present the shapes

总是在展示不同的形状时

with different colorings

涂上不同的颜色

because some colorings emphasize that,

因为有些颜色突出这个，

and others it is that or that.

有些颜色突出那个。

It's so complicated.

非常复杂。

(Laughter)

（笑声）

In 1990, I was in Cambridge, U.K.

1990 年，我在英国的剑桥大学

to receive a prize from the university,

接受了一个奖项。

and three days later,

三天后，

a pilot was flying over the landscape and found this thing.

一个飞行员在飞行时发现了这个。

So where did this come from?

这是从哪里来的？

Obviously, from extraterrestrials.

显然，从外星人那里来的。

(Laughter)

(笑声)

Well, so the newspaper in Cambridge

于是剑桥的校报上

published an article about that "discovery"

发表一篇有关这一“发现”的文章。

and received the next day

第二天，

5,000 letters from people saying,

收到了5000封来信，人们说：

"But that's simply a Mandelbrot set very big."

“那只是一个放得很大的曼德尔布罗特图形。”

Well, let me finish.

好吧，让我结束演讲。

This shape here just came

这个形状仅仅出自

out of an exercise in pure mathematics.

纯数学的一个练习

Bottomless wonders spring from simple rules,

无边的奇迹源自简单规则的

which are repeated without end.

无限重复。

Thank you very much.

非常感谢。

(Applause)

（掌声）