Benoit Mandelbrot: Fractals and the art of roughness

譯者: Geoff Chen 審譯者: Wang-Ju Tsai

Thank you very much.

謝謝

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

請原諒我坐著說話，我老了

(Laughter)

（笑聲）

Well, the topic I'm going to discuss

嗯，我今天要談論的主題

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

是一個在某種程度上非常特殊的主題

because it's very old.

因為它非常古老

Roughness is part of human life

粗糙度，自古以來

forever and forever,

就是人類生命的一部份

and ancient authors have written about it.

古老的作家曾寫過它

It was very much uncontrollable,

它是非常難以掌握的概念

and in a certain sense,

而且，在某種意義上說來，

it seemed to be the extreme of complexity,

它看起來極度複雜，

just a mess, a mess and a mess.

亂無章法，

There are many different kinds of mess.

有著許多不同種類的混亂。

Now, in fact,

現在，事實上

by a complete fluke,

我幸運地

I got involved many years ago

在許多年前參與了一項

in a study of this form of complexity,

關於這種複雜圖形的研究

and to my utter amazement,

我驚異地發現

I found traces --

一些蛛絲馬跡——

very strong traces, I must say --

我必須說——有非常顯著的蛛絲馬跡顯示，

of order in that roughness.

粗糙度具有次序

And so today, I would like to present to you

因此今天，我要向各位呈現

a few examples

一些關於這項研究

of what this represents.

的例子。

I prefer the word roughness

比起不規則度（irregularity）

to the word irregularity

我更喜歡用粗糙度（roughness）這個詞

because irregularity --

因為，不規則度（irregularity）

to someone who had Latin

對學過拉丁文的人來說

in my long-past youth --

（也就是在我遙遠的青少年時）

means the contrary of regularity.

是規律（regularity）的反義詞，

But it is not so.

然而，在真實世界裏，

Regularity is the contrary of roughness

粗糙度才是規律的反義詞。

because the basic aspect of the world

因為世界的基本外觀

is very rough.

是極度粗糙、崎嶇的。

So let me show you a few objects.

我給各位看看一些物體

Some of them are artificial.

有些是人工的

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

有些，在某種程度上，是非常真實的

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

而現在這一個是真的。這是一朵花椰菜

Now why do I show a cauliflower,

爲什麽我要展示花椰菜？

a very ordinary and ancient vegetable?

爲什麽要展示這麼一個普通、古老的蔬菜呢？

Because old and ancient as it may be,

因為古老的事物，恰如其分地，

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

非常複雜、

both at the same time.

同時也非常簡單。

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

如果你試著掂掂它的重量，當然，我們很容易可以量出來

and when you eat it, the weight matters --

當你要吃它時，重量是個問題

but suppose you try to

但是，假如你試著

measure its surface.

測量它的表面

Well, it's very interesting.

這就非常有意思了

If you cut, with a sharp knife,

如果你用一把鋒利的刀子

one of the florets of a cauliflower

切下花椰菜中一個小花

and look at it separately,

分開來看它，

you think of a whole cauliflower, but smaller.

你會想，這是一整個花椰菜，只是小了些，

And then you cut again,

接著，你再切一刀，

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

一而再，再而三地反復切它，

and you still get small cauliflowers.

最後，你仍會得到一朵朵小花椰菜。

So the experience of humanity

所以人類的經驗

has always been that there are some shapes

總是存在著一些

which have this peculiar property,

擁有特殊屬性的形狀，

that each part is like the whole,

每個部分就如同它的整體，

but smaller.

只是稍微小了一些。

Now, what did humanity do with that?

那麼此刻，人類對它做了些什麽研究？

Very, very little.

非常、非常少

(Laughter)

（笑聲）

So what I did actually is to

所以實際上我所做的是

study this problem,

研究這個問題

and I found something quite surprising.

找出某些令人詫異的東西

That one can measure roughness

找出可以衡量粗糙度的東西

by a number, a number,

透過數字，一個數目

2.3, 1.2 and sometimes much more.

2.3、1.2，有時更多。

One day, a friend of mine,

有一天，我的朋友

to bug me,

試著激怒我，

brought a picture and said,

他帶一張照片給我，說：

"What is the roughness of this curve?"

「這個曲線的粗糙度為何？」

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

我回答：「嗯，不到1.5」

It was 1.48.

那粗糙度只有 1.48

Now, it didn't take me any time.

不須花多少時間

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

這些東西我已經已經看了許久，

So these numbers are the numbers

這些數目是

which denote the roughness of these surfaces.

用來表示表面的粗糙度

I hasten to say that these surfaces

我必須事先聲明，這些表面外觀是

are completely artificial.

完全人工的

They were done on a computer,

它們由電腦做成

and the only input is a number,

唯一要輸入，就是一個數字

and that number is roughness.

那數字就是粗糙度

So on the left,

在那左邊

I took the roughness copied from many landscapes.

我複製許多景觀的表面粗糙度

To the right, I took a higher roughness.

在右邊，我取較高的粗糙度

So the eye, after a while,

所以，眼睛過了一會

can distinguish these two very well.

便可以容易地區分兩者了

Humanity had to learn about measuring roughness.

人類必須學習如何衡量粗糙度

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

這非常粗糙、這有點平滑、而這又極度平滑

Very few things are very smooth.

很少有東西是極度平滑的

So then if you try to ask questions:

因此，假使你試著提出一個問題：

"What's the surface of a cauliflower?"

花椰菜的表面積有多少？

Well, you measure and measure and measure.

嗯，你會一量再量

Each time you're closer, it gets bigger,

每一次你靠近它，它就變得更大

down to very, very small distances.

可無限遞迴到很小的距離

What's the length of the coastline

這些湖的沿岸

of these lakes?

有多長？

The closer you measure, the longer it is.

當你越是測量它，它就越長

The concept of length of coastline,

沿岸線的概念

which seems to be so natural

看起來是如此自然

because it's given in many cases,

因為，它在許多情況下被給定了

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

但事實上，這完全謬誤。根本沒有這回事。

You must do it differently.

你必須採取不同的做法

What good is that, to know these things?

要理解這些，該採取什麽樣的辦法呢？

Well, surprisingly enough,

令人驚訝的是，

it's good in many ways.

我們可以透過各種途徑

To begin with, artificial landscapes,

首先，我發明的

which I invented sort of,

這些人造景觀

are used in cinema all the time.

都是用在電影上

We see mountains in the distance.

我們看到遠處的山

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

也許真的是山，但也有可能是公式計算來的，

Now it's very easy to do.

現在要做這個是很容易了

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

以往，製作這些必須曠日費時，但現在根本沒什麼

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

現在，看那，那是一個真正的肺臟

Now a lung is something very strange.

肺是一種非常古怪的東西

If you take this thing,

如果你測量它

you know very well it weighs very little.

你知道它的重量極小

The volume of a lung is very small,

肺的體積很小

but what about the area of the lung?

但肺的面積又如何呢？

Anatomists were arguing very much about that.

針對這個問題，以前解剖學家常有激烈的爭論

Some say that a normal male's lung

有些人說，普通男子的肺

has an area of the inside

面積有

of a basketball [court].

一個籃球場大

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

另外有些人認為，不，它有五個籃球場大

Enormous disagreements.

大家所持的意見相當不同

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

爲什麽呢？因為事實上，肺的面積

is something very ill-defined.

從沒有明確的定義。

The bronchi branch, branch, branch

支氣管不斷分出分支

and they stop branching,

而在其末梢停止了分支

not because of any matter of principle,

並不是和什麽原則有關

but because of physical considerations:

而是由於肺臟裡頭的物理因素

the mucus, which is in the lung.

因為肺裏的粘液所致。

So what happens is that in a way

在某種情況之下

you have a much bigger lung,

你會有較大的肺。

but it branches and branches

但假使它不斷地分支出來，

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

在很微觀的情形下，

and for a little rodent.

鯨魚、人和齧齒目動物會有相等面積的肺。

Now, what good is it to have that?

這有什麼好處嗎？

Well, surprisingly enough, amazingly enough,

嗯，令人訝異地

the anatomists had a very poor idea

直到近日以來，解剖學家都不太理解

of the structure of the lung until very recently.

肺臟的構造，

And I think that my mathematics,

我想我的數學

surprisingly enough,

令人驚訝地

has been of great help

可以帶來許多幫助

to the surgeons

給外科醫生

studying lung illnesses

幫助他們研究肺臟

and also kidney illnesses,

和腎臟

all these branching systems,

這些分叉管的系統的疾病

for which there was no geometry.

因爲在這些系統中沒有幾何學。

So I found myself, in other words,

所以，換句話說，我發現我自己，

constructing a geometry,

正在建立一種幾何學

a geometry of things which had no geometry.

一種沒有幾何圖形的東西的的幾何學

And a surprising aspect of it

而且，令人訝異的是

is that very often, the rules of this geometry

這個幾何學的規則

are extremely short.

經常是極短的，

You have formulas that long.

你有這麼長的公式，

And you crank it several times.

曲折了好幾次

Sometimes repeatedly: again, again, again,

有時候就只是一味地重復

the same repetition.

再重複，循著同樣方式反複循環

And at the end, you get things like that.

最後，你會得到像這樣的東西

This cloud is completely,

這片雲是100%

100 percent artificial.

完全人工的

Well, 99.9.

嗯，99.9。

And the only part which is natural

唯一自然的地方

is a number, the roughness of the cloud,

是數字，也就是這片雲的粗糙度，

which is taken from nature.

是取自自然的

Something so complicated like a cloud,

有時，像雲這麼複雜的東西，

so unstable, so varying,

是這麼不穩定、變化多端

should have a simple rule behind it.

在它背後，應該有一個簡單的規則才是

Now this simple rule

現在，這個簡單規則

is not an explanation of clouds.

並不是解釋雲層

The seer of clouds had to

看這片雲的人必須

take account of it.

有這個認知。

I don't know how much advanced

我不認為這些照片有多先進，

these pictures are. They're old.

它們很舊了

I was very much involved in it,

我以前涉獵極深，

but then turned my attention to other phenomena.

但後來，我轉而研究其他現象了

Now, here is another thing

現在，這裡有另一個

which is rather interesting.

更有趣的東西

One of the shattering events

這是在數學史上一件

in the history of mathematics,

令人震驚的事件，

which is not appreciated by many people,

當時沒多少人理解，

occurred about 130 years ago,

發生在大約 130 年前、

145 years ago.

或 145 年前。

Mathematicians began to create

當時，數學家開始創造

shapes that didn't exist.

不存在的形狀

Mathematicians got into self-praise

數學家陷入一種自我耽溺的地步

to an extent which was absolutely amazing,

他們完全沉浸於

that man can invent things

人類發明的喜悅之中

that nature did not know.

而這些發明是自然所不知曉的事物

In particular, it could invent

特別是，發明一種

things like a curve which fills the plane.

可以填補平面的曲線

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

曲線是曲線，平面是平面，

and the two won't mix.

兩者無法混合

Well, they do mix.

但事實上，他們是可以混在一起的

A man named Peano

有一個叫 Peano 的先生

did define such curves,

真的確立了這些曲線，

and it became an object of extraordinary interest.

於是，這形成一個當時多數人極感興趣的研究對象

It was very important, but mostly interesting

它在當時非常重要，但也相當有趣

because a kind of break,

因為，一種突破

a separation between

必須是一種區隔，

the mathematics coming from reality, on the one hand,

它區隔來自描述現實現象的數學

and new mathematics coming from pure man's mind.

與來自人類純粹心智的新數學

Well, I was very sorry to point out

嗯，我必須很遺憾地指出

that the pure man's mind

純粹的人類心智

has, in fact,

事實上

seen at long last

最終見到了

what had been seen for a long time.

他們長久以來視而不見的事物

And so here I introduce something,

所以，在這裡，我要向大家介紹

the set of rivers of a plane-filling curve.

一組河流的平面填充曲線

And well,

而且

it's a story unto itself.

它本身就是一個故事。

So it was in 1875 to 1925,

1875 年到 1925 年

an extraordinary period

是一段了不起的時期

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

在這段期間，數學正準備突破自己的世界，

And the objects which were used

當我還是個小孩、學生的時候

as examples, when I was

當時作為範例的

a child and a student, as examples

物體

of the break between mathematics

區分了數學與

and visible reality --

可見的現實——

those objects,

我把那些物體

I turned them completely around.

完全顛倒過來

I used them for describing

我把它們用來描述

some of the aspects of the complexity of nature.

自然的若干繁複面向

Well, a man named Hausdorff in 1919

1919 年，有一位叫做 Hausdorff 的先生

introduced a number which was just a mathematical joke,

引介了一個數字，這個數字在當時被看作數學玩笑

and I found that this number

但我發現這個數值

was a good measurement of roughness.

卻是衡量粗糙度的好工具

When I first told it to my friends in mathematics

當我第一次把這個發現告訴我數學界的朋友時，

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

他們說：「別傻了，那只不過是件無聊蠢事。」

Well actually, I was not silly.

然而事實上，我當時並不傻，

The great painter Hokusai knew it very well.

偉大的畫家葛飾北齋（Hokusai）深知這個道理

The things on the ground are algae.

這些涉及複數的問題

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

他不懂數學，那時數學尚未存在

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

他是個日本人，從未接觸過西方世界

But painting for a long time had a fractal side.

但長久以來，他的畫作擁有碎形面

I could speak of that for a long time.

我可以花很多時間談論這個

The Eiffel Tower has a fractal aspect.

艾菲爾鐵塔有個碎形的外觀

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

我在書上讀到，埃菲爾先生寫過他的鐵塔

and indeed it was astonishing how much he understood.

確實，令人驚訝地，他非常瞭解碎型

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

這是一個混亂、混亂、混亂的布朗寧迴圈

One day I decided --

有一天，在我職業生涯的半途

halfway through my career,

我發現

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

我的工作被許多事情絆住

I decided to test myself.

我決定測試自己

Could I just look at something

看我是否可以

which everybody had been looking at for a long time

從每個人看了許久的事物中

and find something dramatically new?

發現什麽戲劇化的新東西？

Well, so I looked at these

嗯，於是我看到了這些

things called Brownian motion -- just goes around.

叫布朗寧運動的東西，只有一圈

I played with it for a while,

我和它玩了一會，

and I made it return to the origin.

我使它回到原點

Then I was telling my assistant,

接著，我告訴我的助理：

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

「我看不到任何東西。你能把它畫出來嗎？」

So he painted it, which means

於是他畫了出來，這意謂著

he put inside everything. He said:

他把所有都放了進去。他說：

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

「這東西出現了......」我說：「停下來! 停下來! 停下來!

I see; it's an island."

我明白了，這是一座島嶼。」

And amazing.

多麼驚人

So Brownian motion, which happens to have

因此，布朗寧運動剛好有

a roughness number of two, goes around.

一個粗糙度數字2，它繞了一圈

I measured it, 1.33.

我測量它，是1.33

Again, again, again.

一而再，再而三

Long measurements, big Brownian motions,

長尺寸，大的布朗寧運動，

1.33.

1.33

Mathematical problem: how to prove it?

一個數學問題來了：該如何證明它？

It took my friends 20 years.

我的朋友曾花 20 年的時間研究

Three of them were having incomplete proofs.

他們三個人產出一個不完全的證明

They got together, and together they had the proof.

他們聚在一起，一起證明它

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

因此，他們獲得了這個領域的大獎

one of the three medals that people have received

這些獲獎人當中，有一面獎牌

for proving things which I've seen

並不能合理地證明

without being able to prove them.

我所見到的東西

Now everybody asks me at one point or another,

現在，每個人都問我

"How did it all start?

「這是怎麼開始的？

What got you in that strange business?"

是什麼原因使你進入這個陌生的領域？」

What got you to be,

是什麼讓我

at the same time, a mechanical engineer,

同時成為一個機械工程師、

a geographer

又成為地理學家、

and a mathematician and so on, a physicist?

數學家、或物理學家等等？

Well actually I started, oddly enough,

嗯，事實上，我是從一個非常怪異的地方開始的

studying stock market prices.

我研究股票市場價格

And so here

在這

I had this theory,

我提過理論

and I wrote books about it --

我也寫了有關這方面的書籍

financial prices increments.

金融價格增長量

To the left you see data over a long period.

在左邊，你們看到長期的數據

To the right, on top,

在右邊，上方

you see a theory which is very, very fashionable.

你們可以看到一個非常、非常流行的理論

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

它非常簡單，你可以用極短的時間寫下許多關於它的書籍

(Laughter)

（笑聲）

There are thousands of books on that.

坊間有上千本這方面的著作

Now compare that with real price increments.

現在，比較真實的價格增加量，

Where are real price increments?

哪裡是實際的價格增加量呢？

Well, these other lines

嗯，其他這些曲線

include some real price increments

包涵了一些真正的價格利潤

and some forgery which I did.

還有一些是我偽造的

So the idea there was

所以，這裡的觀點是

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

人們必須能夠 --怎麼說呢？ --

model price variation.

把價格變化模組化

And it went really well 50 years ago.

在五十年前，這觀點被認為相當有道理

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

五十年來，人們多少有點輕視我的看法

because they could do it much, much easier.

因為他們可以用非常簡單的方式換算出來

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

但我告訴你，在這一點上，人們聽信我

(Laughter)

（笑聲）

These two curves are averages:

這兩條曲線是平均值

Standard & Poor, the blue one;

藍色的那條是標準普爾（Standard & Poor）的曲線，

and the red one is Standard & Poor's

而紅色的那條是標準普爾

from which the five biggest discontinuities

根據其中 5 個最大的不連續性

are taken out.

所畫出來的曲線

Now discontinuities are a nuisance,

現在，不連續造成了累贅

so in many studies of prices,

所以，在許多關於價格的研究上

one puts them aside.

人們把它們擱在一旁，說：

"Well, acts of God.

「嗯，這些是神的旨意（不可抗力的因素）

And you have the little nonsense which is left.

於是留下了少許無意義的東西，

Acts of God." In this picture,

在這幅包涵不可抗力因素的照片中

five acts of God are as important as everything else.

五個不可抗力的現象就如同所有其他事物一樣重要

In other words,

換句話說，

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

事實上，我們不應擱置那不可抗拒的現象不談

That is the meat, the problem.

那才是牛肉，是問題所在

If you master these, you master price,

如果你熟悉價格和這些癥結

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

而且，如果你不熟悉這些癥結，你也可以試著

the little noise as well as you can,

盡可能地了解小問題

but it's not important.

但它不重要

Well, here are the curves for it.

嗯，這裡有關於它的曲線

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

我來到最後這個

of which my name is attached.

附有我名字的這組

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

在某種程度上，它是我一生的故事

My adolescence was spent

我青少年是在

during the German occupation of France.

德軍佔領法國的期間度過的

Since I thought that I might

我曾想，也許我可能會

vanish within a day or a week,

在一天或一個星期內憑空消失

I had very big dreams.

所以，我有一些大夢想

And after the war,

戰後

I saw an uncle again.

我和我叔叔相遇

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

我叔叔是個非常重要的數學家，他告訴我

"Look, there's a problem

「看，這裡有一個我二十五年來

which I could not solve 25 years ago,

都無法解決的問題，

and which nobody can solve.

沒有人可以解答

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

這是一個叫 [Gaston] Julia 和 [Pierre] Fatou

and [Pierre] Fatou.

共同建構的問題

If you could

如果你可以，

find something new, anything,

發掘新的解決辦法，任何解決辦法，

you will get your career made."

你的事業必定有所成就。」

Very simple.

非常簡單

So I looked,

於是，我試試看

and like the thousands of people that had tried before,

就像許許多多前人試過的一樣

I found nothing.

我什麽也沒找到

But then the computer came,

然而接著，電腦出現了

and I decided to apply the computer,

我決定使用電腦

not to new problems in mathematics --

不是用在數學的新問題——

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

比如這條擺動的曲線，這是新問題——

but to old problems.

而是，把電腦應用於舊的問題之上

And I went from what's called

我從那稱為實數（real number）

real numbers, which are points on a line,

的地方開始，這是一條線上的點

to imaginary, complex numbers,

到虛數、複數

which are points on a plane,

這些是平面的數

which is what one should do there,

也是人們必須去研究的事

and this shape came out.

於是，這個圖形出現了

This shape is of an extraordinary complication.

形狀極其複雜

The equation is hidden there,

該方程式隱藏在那裡

z goes into z squared, plus c.

z 進入z 平方，加上 c

It's so simple, so dry.

它是如此簡單、如此枯燥、

It's so uninteresting.

如此無趣

Now you turn the crank once, twice:

現在，你轉動曲軸兩次

twice,

兩次

marvels come out.

奇蹟就出現了。

I mean this comes out.

我指的是，出現了這個

I don't want to explain these things.

我不想解釋這些東西

This comes out. This comes out.

出現了這個，出現了這個

Shapes which are of such complication,

出現了如此這般複雜的形狀

such harmony and such beauty.

它們具有如此的和諧與美感

This comes out

出現了這個

repeatedly, again, again, again.

它們一而再，再而三地重複著

And that was one of my major discoveries,

這就是過去我最主要的發現之一

to find that these islands were the same

我發現這些島嶼是相同的

as the whole big thing, more or less.

或多或少，就如同它較大的整體

And then you get these

於是，你在所有地方得到這些

extraordinary baroque decorations all over the place.

非凡的巴洛克式裝飾

All that from this little formula,

它們全都來自這個小小的方程式

which has whatever, five symbols in it.

這方程式有五種符號

And then this one.

接著是這個

The color was added for two reasons.

加上兩種顏色的原因是

First of all, because these shapes

首先，因為這些圖形

are so complicated

是如此複雜

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

以致於人們無法辨識任何數目

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

如果你要繪製它們，你必須選擇某些系統

And so my principle has been

所以，我的原則是

to always present the shapes

永遠以不同的顏色

with different colorings

呈現這些圖形

because some colorings emphasize that,

因為某些顏色強調某些部份

and others it is that or that.

而其他的強調這，或強調那

It's so complicated.

實在真的很複雜

(Laughter)

（笑聲）

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

1990 年，我在英國劍橋

to receive a prize from the university,

獲得大學一個獎項

and three days later,

三天後

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

有位駕駛飛越田野上空，發現了這東西

So where did this come from?

這是來自哪裡呢？

Obviously, from extraterrestrials.

很顯然，這來自外星人

(Laughter)

（笑聲）

Well, so the newspaper in Cambridge

嗯，所以劍橋的報紙

published an article about that "discovery"

登載了關於那「發現」的文章

and received the next day

隔天後，他們收到了

5,000 letters from people saying,

5000 封信，人們在信上說：

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

「這只不過是一個非常大的 Mandelbrot 圖組罷了。」

Well, let me finish.

嗯，讓我這麼結束吧

This shape here just came

這兒的圖形，只是來自

out of an exercise in pure mathematics.

純數學的演算

Bottomless wonders spring from simple rules,

深不可測的奇觀，源自簡單的規則

which are repeated without end.

它們無止無盡地反復

Thank you very much.

謝謝大家

(Applause)

（掌聲）