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

謝謝

(Laughter)

請原諒我坐著說話，我老了

Well, the topic I'm going to discuss

（笑聲）

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

嗯，我今天要談論的主題

because it's very old.

是一個在某種程度上非常特殊的主題

Roughness is part of human life

因為它非常古老

forever and forever,

粗糙度，自古以來

and ancient authors have written about it.

就是人類生命的一部份

It was very much uncontrollable,

古老的作家曾寫過它

and in a certain sense,

它是非常難以掌握的概念

it seemed to be the extreme of complexity,

而且，在某種意義上說來，

just a mess, a mess and a mess.

它看起來極度複雜，

There are many different kinds of mess.

亂無章法，

Now, in fact,

有著許多不同種類的混亂。

by a complete fluke,

現在，事實上

I got involved many years ago

我幸運地

in a study of this form of complexity,

在許多年前參與了一項

and to my utter amazement,

關於這種複雜圖形的研究

I found traces --

我驚異地發現

very strong traces, I must say --

一些蛛絲馬跡——

of order in that roughness.

我必須說——有非常顯著的蛛絲馬跡顯示，

And so today, I would like to present to you

粗糙度具有次序

a few examples

因此今天，我要向各位呈現

of what this represents.

一些關於這項研究

I prefer the word roughness

的例子。

to the word irregularity

比起不規則度（irregularity）

because irregularity --

我更喜歡用粗糙度（roughness）這個詞

to someone who had Latin

因為，不規則度（irregularity）

in my long-past youth --

對學過拉丁文的人來說

means the contrary of regularity.

（也就是在我遙遠的青少年時）

But it is not so.

是規律（regularity）的反義詞，

Regularity is the contrary of roughness

然而，在真實世界裏，

because the basic aspect of the world

粗糙度才是規律的反義詞。

is very rough.

因為世界的基本外觀

So let me show you a few objects.

是極度粗糙、崎嶇的。

Some of them are artificial.

我給各位看看一些物體

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

有些是人工的

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

有些，在某種程度上，是非常真實的

Now why do I show a cauliflower,

而現在這一個是真的。這是一朵花椰菜

a very ordinary and ancient vegetable?

爲什麽我要展示花椰菜？

Because old and ancient as it may be,

爲什麽要展示這麼一個普通、古老的蔬菜呢？

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

因為古老的事物，恰如其分地，

both at the same time.

非常複雜、

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

同時也非常簡單。

and when you eat it, the weight matters --

如果你試著掂掂它的重量，當然，我們很容易可以量出來

but suppose you try to

當你要吃它時，重量是個問題

measure its surface.

但是，假如你試著

Well, it's very interesting.

測量它的表面

If you cut, with a sharp knife,

這就非常有意思了

one of the florets of a cauliflower

如果你用一把鋒利的刀子

and look at it separately,

切下花椰菜中一個小花

you think of a whole cauliflower, but smaller.

分開來看它，

And then you cut again,

你會想，這是一整個花椰菜，只是小了些，

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

接著，你再切一刀，

and you still get small cauliflowers.

一而再，再而三地反復切它，

So the experience of humanity

最後，你仍會得到一朵朵小花椰菜。

has always been that there are some shapes

所以人類的經驗

which have this peculiar property,

總是存在著一些

that each part is like the whole,

擁有特殊屬性的形狀，

but smaller.

每個部分就如同它的整體，

Now, what did humanity do with that?

只是稍微小了一些。

Very, very little.

那麼此刻，人類對它做了些什麽研究？

(Laughter)

非常、非常少

So what I did actually is to

（笑聲）

study this problem,

所以實際上我所做的是

and I found something quite surprising.

研究這個問題

That one can measure roughness

找出某些令人詫異的東西

by a number, a number,

找出可以衡量粗糙度的東西

2.3, 1.2 and sometimes much more.

透過數字，一個數目

One day, a friend of mine,

2.3、1.2，有時更多。

to bug me,

有一天，我的朋友

brought a picture and said,

試著激怒我，

"What is the roughness of this curve?"

他帶一張照片給我，說：

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

「這個曲線的粗糙度為何？」

It was 1.48.

我回答：「嗯，不到1.5」

Now, it didn't take me any time.

那粗糙度只有 1.48

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

不須花多少時間

So these numbers are the numbers

這些東西我已經已經看了許久，

which denote the roughness of these surfaces.

這些數目是

I hasten to say that these surfaces

用來表示表面的粗糙度

are completely artificial.

我必須事先聲明，這些表面外觀是

They were done on a computer,

完全人工的

and the only input is a number,

它們由電腦做成

and that number is roughness.

唯一要輸入，就是一個數字

So on the left,

那數字就是粗糙度

I took the roughness copied from many landscapes.

在那左邊

To the right, I took a higher roughness.

我複製許多景觀的表面粗糙度

So the eye, after a while,

在右邊，我取較高的粗糙度

can distinguish these two very well.

所以，眼睛過了一會

Humanity had to learn about measuring roughness.

便可以容易地區分兩者了

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

人類必須學習如何衡量粗糙度

Very few things are very smooth.

這非常粗糙、這有點平滑、而這又極度平滑

So then if you try to ask questions:

很少有東西是極度平滑的

"What's the surface of a cauliflower?"

因此，假使你試著提出一個問題：

Well, you measure and measure and measure.

花椰菜的表面積有多少？

Each time you're closer, it gets bigger,

嗯，你會一量再量

down to very, very small distances.

每一次你靠近它，它就變得更大

What's the length of the coastline

可無限遞迴到很小的距離

of these lakes?

這些湖的沿岸

The closer you measure, the longer it is.

有多長？

The concept of length of coastline,

當你越是測量它，它就越長

which seems to be so natural

沿岸線的概念

because it's given in many cases,

看起來是如此自然

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

因為，它在許多情況下被給定了

You must do it differently.

但事實上，這完全謬誤。根本沒有這回事。

What good is that, to know these things?

你必須採取不同的做法

Well, surprisingly enough,

要理解這些，該採取什麽樣的辦法呢？

it's good in many ways.

令人驚訝的是，

To begin with, artificial landscapes,

我們可以透過各種途徑

which I invented sort of,

首先，我發明的

are used in cinema all the time.

這些人造景觀

We see mountains in the distance.

都是用在電影上

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

我們看到遠處的山

Now it's very easy to do.

也許真的是山，但也有可能是公式計算來的，

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

現在要做這個是很容易了

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

以往，製作這些必須曠日費時，但現在根本沒什麼

Now a lung is something very strange.

現在，看那，那是一個真正的肺臟

If you take this thing,

肺是一種非常古怪的東西

you know very well it weighs very little.

如果你測量它

The volume of a lung is very small,

你知道它的重量極小

but what about the area of the lung?

肺的體積很小

Anatomists were arguing very much about that.

但肺的面積又如何呢？

Some say that a normal male's lung

針對這個問題，以前解剖學家常有激烈的爭論

has an area of the inside

有些人說，普通男子的肺

of a basketball [court].

面積有

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

一個籃球場大

Enormous disagreements.

另外有些人認為，不，它有五個籃球場大

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

大家所持的意見相當不同

is something very ill-defined.

爲什麽呢？因為事實上，肺的面積

The bronchi branch, branch, branch

從沒有明確的定義。

and they stop branching,

支氣管不斷分出分支

not because of any matter of principle,

而在其末梢停止了分支

but because of physical considerations:

並不是和什麽原則有關

the mucus, which is in the lung.

而是由於肺臟裡頭的物理因素

So what happens is that in a way

因為肺裏的粘液所致。

you have a much bigger lung,

在某種情況之下

but it branches and branches

你會有較大的肺。

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

但假使它不斷地分支出來，

and for a little rodent.

在很微觀的情形下，

Now, what good is it to have that?

鯨魚、人和齧齒目動物會有相等面積的肺。

Well, surprisingly enough, amazingly enough,

這有什麼好處嗎？

the anatomists had a very poor idea

嗯，令人訝異地

of the structure of the lung until very recently.

直到近日以來，解剖學家都不太理解

And I think that my mathematics,

肺臟的構造，

surprisingly enough,

我想我的數學

has been of great help

令人驚訝地

to the surgeons

可以帶來許多幫助

studying lung illnesses

給外科醫生

and also kidney illnesses,

幫助他們研究肺臟

all these branching systems,

和腎臟

for which there was no geometry.

這些分叉管的系統的疾病

So I found myself, in other words,

因爲在這些系統中沒有幾何學。

constructing a geometry,

所以，換句話說，我發現我自己，

a geometry of things which had no geometry.

正在建立一種幾何學

And a surprising aspect of it

一種沒有幾何圖形的東西的的幾何學

is that very often, the rules of this geometry

而且，令人訝異的是

are extremely short.

這個幾何學的規則

You have formulas that long.

經常是極短的，

And you crank it several times.

你有這麼長的公式，

Sometimes repeatedly: again, again, again,

曲折了好幾次

the same repetition.

有時候就只是一味地重復

And at the end, you get things like that.

再重複，循著同樣方式反複循環

This cloud is completely,

最後，你會得到像這樣的東西

100 percent artificial.

這片雲是100%

Well, 99.9.

完全人工的

And the only part which is natural

嗯，99.9。

is a number, the roughness of the cloud,

唯一自然的地方

which is taken from nature.

是數字，也就是這片雲的粗糙度，

Something so complicated like a cloud,

是取自自然的

so unstable, so varying,

有時，像雲這麼複雜的東西，

should have a simple rule behind it.

是這麼不穩定、變化多端

Now this simple rule

在它背後，應該有一個簡單的規則才是

is not an explanation of clouds.

現在，這個簡單規則

The seer of clouds had to

並不是解釋雲層

take account of it.

看這片雲的人必須

I don't know how much advanced

有這個認知。

these pictures are. They're old.

我不認為這些照片有多先進，

I was very much involved in it,

它們很舊了

but then turned my attention to other phenomena.

我以前涉獵極深，

Now, here is another thing

但後來，我轉而研究其他現象了

which is rather interesting.

現在，這裡有另一個

One of the shattering events

更有趣的東西

in the history of mathematics,

這是在數學史上一件

which is not appreciated by many people,

令人震驚的事件，

occurred about 130 years ago,

當時沒多少人理解，

145 years ago.

發生在大約 130 年前、

Mathematicians began to create

或 145 年前。

shapes that didn't exist.

當時，數學家開始創造

Mathematicians got into self-praise

不存在的形狀

to an extent which was absolutely amazing,

數學家陷入一種自我耽溺的地步

that man can invent things

他們完全沉浸於

that nature did not know.

人類發明的喜悅之中

In particular, it could invent

而這些發明是自然所不知曉的事物

things like a curve which fills the plane.

特別是，發明一種

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

可以填補平面的曲線

and the two won't mix.

曲線是曲線，平面是平面，

Well, they do mix.

兩者無法混合

A man named Peano

但事實上，他們是可以混在一起的

did define such curves,

有一個叫 Peano 的先生

and it became an object of extraordinary interest.

真的確立了這些曲線，

It was very important, but mostly interesting

於是，這形成一個當時多數人極感興趣的研究對象

because a kind of break,

它在當時非常重要，但也相當有趣

a separation between

因為，一種突破

the mathematics coming from reality, on the one hand,

必須是一種區隔，

and new mathematics coming from pure man's mind.

它區隔來自描述現實現象的數學

Well, I was very sorry to point out

與來自人類純粹心智的新數學

that the pure man's mind

嗯，我必須很遺憾地指出

has, in fact,

純粹的人類心智

seen at long last

事實上

what had been seen for a long time.

最終見到了

And so here I introduce something,

他們長久以來視而不見的事物

the set of rivers of a plane-filling curve.

所以，在這裡，我要向大家介紹

And well,

一組河流的平面填充曲線

it's a story unto itself.

而且

So it was in 1875 to 1925,

它本身就是一個故事。

an extraordinary period

1875 年到 1925 年

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

是一段了不起的時期

And the objects which were used

在這段期間，數學正準備突破自己的世界，

as examples, when I was

當我還是個小孩、學生的時候

a child and a student, as examples

當時作為範例的

of the break between mathematics

物體

and visible reality --

區分了數學與

those objects,

可見的現實——

I turned them completely around.

我把那些物體

I used them for describing

完全顛倒過來

some of the aspects of the complexity of nature.

我把它們用來描述

Well, a man named Hausdorff in 1919

自然的若干繁複面向

introduced a number which was just a mathematical joke,

1919 年，有一位叫做 Hausdorff 的先生

and I found that this number

引介了一個數字，這個數字在當時被看作數學玩笑

was a good measurement of roughness.

但我發現這個數值

When I first told it to my friends in mathematics

卻是衡量粗糙度的好工具

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

當我第一次把這個發現告訴我數學界的朋友時，

Well actually, I was not silly.

他們說：「別傻了，那只不過是件無聊蠢事。」

The great painter Hokusai knew it very well.

然而事實上，我當時並不傻，

The things on the ground are algae.

偉大的畫家葛飾北齋（Hokusai）深知這個道理

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

這些涉及複數的問題

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

他不懂數學，那時數學尚未存在

But painting for a long time had a fractal side.

他是個日本人，從未接觸過西方世界

I could speak of that for a long time.

但長久以來，他的畫作擁有碎形面

The Eiffel Tower has a fractal aspect.

我可以花很多時間談論這個

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

艾菲爾鐵塔有個碎形的外觀

and indeed it was astonishing how much he understood.

我在書上讀到，埃菲爾先生寫過他的鐵塔

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

確實，令人驚訝地，他非常瞭解碎型

One day I decided --

這是一個混亂、混亂、混亂的布朗寧迴圈

halfway through my career,

有一天，在我職業生涯的半途

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

我發現

I decided to test myself.

我的工作被許多事情絆住

Could I just look at something

我決定測試自己

which everybody had been looking at for a long time

看我是否可以

and find something dramatically new?

從每個人看了許久的事物中

Well, so I looked at these

發現什麽戲劇化的新東西？

things called Brownian motion -- just goes around.

嗯，於是我看到了這些

I played with it for a while,

叫布朗寧運動的東西，只有一圈

and I made it return to the origin.

我和它玩了一會，

Then I was telling my assistant,

我使它回到原點

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

接著，我告訴我的助理：

So he painted it, which means

「我看不到任何東西。你能把它畫出來嗎？」

he put inside everything. He said:

於是他畫了出來，這意謂著

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

他把所有都放了進去。他說：

I see; it's an island."

「這東西出現了......」我說：「停下來! 停下來! 停下來!

And amazing.

我明白了，這是一座島嶼。」

So Brownian motion, which happens to have

多麼驚人