ABOUT THE SPEAKER
Benoit Mandelbrot - Mathematician
Benoit Mandelbrot's work led the world to a deeper understanding of fractals, a broad and powerful tool in the study of roughness, both in nature and in humanity's works.

Why you should listen

Studying complex dynamics in the 1970s, Benoit Mandelbrot had a key insight about a particular set of mathematical objects: that these self-similar structures with infinitely repeating complexities were not just curiosities, as they'd been considered since the turn of the century, but were in fact a key to explaining non-smooth objects and complex data sets -- which make up, let's face it, quite a lot of the world. Mandelbrot coined the term "fractal" to describe these objects, and set about sharing his insight with the world.

The Mandelbrot set (expressed as z² + c) was named in Mandelbrot's honor by Adrien Douady and John H. Hubbard. Its boundary can be magnified infinitely and yet remain magnificently complicated, and its elegant shape made it a poster child for the popular understanding of fractals. Led by Mandelbrot's enthusiastic work, fractal math has brought new insight to the study of pretty much everything, from the behavior of stocks to the distribution of stars in the universe.

Benoit Mandelbrot appeared at the first TED in 1984, and returned in 2010 to give an overview of the study of fractals and the paradigm-flipping insights they've brought to many fields. He died in October 2010 at age 85. Read more about his life on NYBooks.com >>

More profile about the speaker
Benoit Mandelbrot | Speaker | TED.com
TED2010

Benoit Mandelbrot: Fractals and the art of roughness

Benoit Mandelbrot: 碎形與粗糙度的藝術

Filmed:
1,448,555 views

在 TED2010 大會上,數學界的傳奇人物 Benoit Mandelbrot 展開他在 1984 年首次於 TED 討論的主題:極度複雜的粗糙程度、以及碎型理論可以在看似複雜的圖形中找出秩序的方法。
- Mathematician
Benoit Mandelbrot's work led the world to a deeper understanding of fractals, a broad and powerful tool in the study of roughness, both in nature and in humanity's works. Full bio

Double-click the English transcript below to play the video.

00:15
Thank you very much.
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謝謝
00:17
Please excuse藉口 me for sitting坐在; I'm very old.
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請原諒我坐著說話,我老了
00:20
(Laughter笑聲)
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(笑聲)
00:22
Well, the topic話題 I'm going to discuss討論
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嗯,我今天要談論的主題
00:24
is one which哪一個 is, in a certain某些 sense, very peculiar奇特
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是一個在某種程度上非常特殊的主題
00:27
because it's very old.
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因為它非常古老
00:29
Roughness粗糙度 is part部分 of human人的 life
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粗糙度,自古以來
00:32
forever永遠 and forever永遠,
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就是人類生命的一部份
00:34
and ancient authors作者 have written書面 about it.
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古老的作家曾寫過它
00:37
It was very much uncontrollable不可控,
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它是非常難以掌握的概念
00:39
and in a certain某些 sense,
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而且,在某種意義上說來,
00:41
it seemed似乎 to be the extreme極端 of complexity複雜,
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它看起來極度複雜,
00:44
just a mess食堂, a mess食堂 and a mess食堂.
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亂無章法,
00:46
There are many許多 different不同 kinds of mess食堂.
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有著許多不同種類的混亂。
00:48
Now, in fact事實,
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現在,事實上
00:50
by a complete完成 fluke吸蟲,
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我幸運地
00:52
I got involved參與 many許多 years年份 ago
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在許多年前參與了一項
00:55
in a study研究 of this form形成 of complexity複雜,
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關於這種複雜圖形的研究
00:58
and to my utter說出 amazement驚愕,
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我驚異地發現
01:00
I found發現 traces痕跡 --
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一些蛛絲馬跡——
01:02
very strong強大 traces痕跡, I must必須 say --
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我必須說——有非常顯著的蛛絲馬跡顯示,
01:04
of order訂購 in that roughness粗糙度.
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粗糙度具有次序
01:07
And so today今天, I would like to present當下 to you
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因此今天,我要向各位呈現
01:09
a few少數 examples例子
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一些關於這項研究
01:11
of what this represents代表.
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的例子。
01:13
I prefer比較喜歡 the word roughness粗糙度
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比起不規則度(irregularity)
01:15
to the word irregularity不規則
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我更喜歡用粗糙度(roughness)這個詞
01:17
because irregularity不規則 --
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因為,不規則度(irregularity)
01:19
to someone有人 who had Latin拉丁
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對學過拉丁文的人來說
01:21
in my long-past長期以往 youth青年 --
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(也就是在我遙遠的青少年時)
01:23
means手段 the contrary相反 of regularity規律性.
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是規律(regularity)的反義詞,
01:25
But it is not so.
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然而,在真實世界裏,
01:27
Regularity規律 is the contrary相反 of roughness粗糙度
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粗糙度才是規律的反義詞。
01:30
because the basic基本 aspect方面 of the world世界
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因為世界的基本外觀
01:32
is very rough.
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是極度粗糙、崎嶇的。
01:34
So let me show顯示 you a few少數 objects對象.
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我給各位看看一些物體
01:37
Some of them are artificial人造.
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有些是人工的
01:39
Others其他 of them are very real真實, in a certain某些 sense.
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有些,在某種程度上,是非常真實的
01:42
Now this is the real真實. It's a cauliflower菜花.
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而現在這一個是真的。這是一朵花椰菜
01:45
Now why do I show顯示 a cauliflower菜花,
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爲什麽我要展示花椰菜?
01:48
a very ordinary普通 and ancient vegetable蔬菜?
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爲什麽要展示這麼一個普通、古老的蔬菜呢?
01:51
Because old and ancient as it may可能 be,
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因為古老的事物,恰如其分地,
01:54
it's very complicated複雜 and it's very simple簡單,
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非常複雜、
01:57
both at the same相同 time.
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同時也非常簡單。
01:59
If you try to weigh稱重 it -- of course課程 it's very easy簡單 to weigh稱重 it,
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如果你試著掂掂它的重量,當然,我們很容易可以量出來
02:02
and when you eat it, the weight重量 matters事項 --
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當你要吃它時,重量是個問題
02:05
but suppose假設 you try to
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但是,假如你試著
02:08
measure測量 its surface表面.
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測量它的表面
02:10
Well, it's very interesting有趣.
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這就非常有意思了
02:12
If you cut, with a sharp尖銳 knife,
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如果你用一把鋒利的刀子
02:15
one of the florets小花 of a cauliflower菜花
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切下花椰菜中一個小花
02:17
and look at it separately分別,
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分開來看它,
02:19
you think of a whole整個 cauliflower菜花, but smaller.
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你會想,這是一整個花椰菜,只是小了些,
02:22
And then you cut again,
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接著,你再切一刀,
02:24
again, again, again, again, again, again, again, again,
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一而再,再而三地反復切它,
02:27
and you still get small cauliflowers花椰菜.
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最後,你仍會得到一朵朵小花椰菜。
02:29
So the experience經驗 of humanity人性
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所以人類的經驗
02:31
has always been that there are some shapes形狀
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總是存在著一些
02:34
which哪一個 have this peculiar奇特 property屬性,
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擁有特殊屬性的形狀,
02:36
that each part部分 is like the whole整個,
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每個部分就如同它的整體,
02:39
but smaller.
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只是稍微小了一些。
02:41
Now, what did humanity人性 do with that?
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那麼此刻,人類對它做了些什麽研究?
02:44
Very, very little.
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非常、非常少
02:47
(Laughter笑聲)
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(笑聲)
02:50
So what I did actually其實 is to
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所以實際上我所做的是
02:53
study研究 this problem問題,
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研究這個問題
02:56
and I found發現 something quite相當 surprising奇怪.
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找出某些令人詫異的東西
02:59
That one can measure測量 roughness粗糙度
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找出可以衡量粗糙度的東西
03:02
by a number, a number,
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透過數字,一個數目
03:05
2.3, 1.2 and sometimes有時 much more.
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2.3、1.2,有時更多。
03:08
One day, a friend朋友 of mine,
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有一天,我的朋友
03:10
to bug竊聽器 me,
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試著激怒我,
03:12
brought a picture圖片 and said,
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他帶一張照片給我,說:
03:14
"What is the roughness粗糙度 of this curve曲線?"
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「這個曲線的粗糙度為何?」
03:16
I said, "Well, just short of 1.5."
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我回答:「嗯,不到1.5」
03:19
It was 1.48.
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那粗糙度只有 1.48
03:21
Now, it didn't take me any time.
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不須花多少時間
03:23
I've been looking at these things for so long.
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這些東西我已經已經看了許久,
03:25
So these numbers數字 are the numbers數字
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這些數目是
03:27
which哪一個 denote表示 the roughness粗糙度 of these surfaces.
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用來表示表面的粗糙度
03:30
I hasten to say that these surfaces
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我必須事先聲明,這些表面外觀是
03:32
are completely全然 artificial人造.
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完全人工的
03:34
They were doneDONE on a computer電腦,
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它們由電腦做成
03:36
and the only input輸入 is a number,
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唯一要輸入,就是一個數字
03:38
and that number is roughness粗糙度.
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那數字就是粗糙度
03:41
So on the left,
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在那左邊
03:43
I took the roughness粗糙度 copied複製 from many許多 landscapes景觀.
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我複製許多景觀的表面粗糙度
03:46
To the right, I took a higher更高 roughness粗糙度.
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在右邊,我取較高的粗糙度
03:49
So the eye, after a while,
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所以,眼睛過了一會
03:51
can distinguish區分 these two very well.
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便可以容易地區分兩者了
03:54
Humanity人性 had to learn學習 about measuring測量 roughness粗糙度.
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人類必須學習如何衡量粗糙度
03:56
This is very rough, and this is sort分類 of smooth光滑, and this perfectly完美 smooth光滑.
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這非常粗糙、這有點平滑、而這又極度平滑
03:59
Very few少數 things are very smooth光滑.
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很少有東西是極度平滑的
04:03
So then if you try to ask questions問題:
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因此,假使你試著提出一個問題:
04:06
"What's the surface表面 of a cauliflower菜花?"
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花椰菜的表面積有多少?
04:08
Well, you measure測量 and measure測量 and measure測量.
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嗯,你會一量再量
04:11
Each time you're closer接近, it gets得到 bigger,
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每一次你靠近它,它就變得更大
04:14
down to very, very small distances距離.
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可無限遞迴到很小的距離
04:16
What's the length長度 of the coastline海岸線
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這些湖的沿岸
04:18
of these lakes湖泊?
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有多長?
04:20
The closer接近 you measure測量, the longer it is.
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當你越是測量它,它就越長
04:23
The concept概念 of length長度 of coastline海岸線,
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沿岸線的概念
04:25
which哪一個 seems似乎 to be so natural自然
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看起來是如此自然
04:27
because it's given特定 in many許多 cases,
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因為,它在許多情況下被給定了
04:29
is, in fact事實, complete完成 fallacy謬論; there's no such這樣 thing.
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但事實上,這完全謬誤。根本沒有這回事。
04:32
You must必須 do it differently不同.
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你必須採取不同的做法
04:35
What good is that, to know these things?
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要理解這些,該採取什麽樣的辦法呢?
04:37
Well, surprisingly出奇 enough足夠,
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令人驚訝的是,
04:39
it's good in many許多 ways方法.
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我們可以透過各種途徑
04:41
To begin開始 with, artificial人造 landscapes景觀,
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首先,我發明的
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which哪一個 I invented發明 sort分類 of,
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這些人造景觀
04:45
are used in cinema電影 all the time.
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都是用在電影上
04:48
We see mountains in the distance距離.
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我們看到遠處的山
04:50
They may可能 be mountains, but they may可能 be just formulae公式, just cranked手搖 on.
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也許真的是山,但也有可能是公式計算來的,
04:53
Now it's very easy簡單 to do.
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現在要做這個是很容易了
04:55
It used to be very time-consuming耗時的, but now it's nothing.
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以往,製作這些必須曠日費時,但現在根本沒什麼
04:58
Now look at that. That's a real真實 lung.
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現在,看那,那是一個真正的肺臟
05:01
Now a lung is something very strange奇怪.
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肺是一種非常古怪的東西
05:03
If you take this thing,
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如果你測量它
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you know very well it weighs very little.
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你知道它的重量極小
05:08
The volume of a lung is very small,
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肺的體積很小
05:10
but what about the area of the lung?
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但肺的面積又如何呢?
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Anatomists解剖學家 were arguing爭論 very much about that.
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針對這個問題,以前解剖學家常有激烈的爭論
05:16
Some say that a normal正常 male's男的 lung
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有些人說,普通男子的肺
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has an area of the inside
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面積有
05:21
of a basketball籃球 [court法庭].
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一個籃球場大
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And the others其他 say, no, five basketball籃球 [courts法院].
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另外有些人認為,不,它有五個籃球場大
05:27
Enormous巨大 disagreements分歧.
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大家所持的意見相當不同
05:29
Why so? Because, in fact事實, the area of the lung
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爲什麽呢?因為事實上,肺的面積
05:32
is something very ill-defined不明確.
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從沒有明確的定義。
05:35
The bronchi支氣管 branch, branch, branch
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支氣管不斷分出分支
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and they stop branching分枝,
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而在其末梢停止了分支
05:41
not because of any matter of principle原理,
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並不是和什麽原則有關
05:44
but because of physical物理 considerations注意事項:
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而是由於肺臟裡頭的物理因素
05:47
the mucus粘液, which哪一個 is in the lung.
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因為肺裏的粘液所致。
05:50
So what happens發生 is that in a way
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在某種情況之下
05:52
you have a much bigger lung,
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你會有較大的肺。
05:54
but it branches分支機構 and branches分支機構
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但假使它不斷地分支出來,
05:56
down to distances距離 about the same相同 for a whale, for a man
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在很微觀的情形下,
05:59
and for a little rodent囓齒動物.
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鯨魚、人和齧齒目動物會有相等面積的肺。
06:02
Now, what good is it to have that?
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這有什麼好處嗎?
06:05
Well, surprisingly出奇 enough足夠, amazingly令人驚訝 enough足夠,
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嗯,令人訝異地
06:07
the anatomists解剖學家 had a very poor較差的 idea理念
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直到近日以來,解剖學家都不太理解
06:10
of the structure結構體 of the lung until直到 very recently最近.
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肺臟的構造,
06:13
And I think that my mathematics數學,
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我想我的數學
06:15
surprisingly出奇 enough足夠,
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令人驚訝地
06:17
has been of great help
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可以帶來許多幫助
06:19
to the surgeons外科醫生
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給外科醫生
06:21
studying研究 lung illnesses疾病
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幫助他們研究肺臟
06:23
and also kidney illnesses疾病,
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和腎臟
06:25
all these branching分枝 systems系統,
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這些分叉管的系統的疾病
06:27
for which哪一個 there was no geometry幾何.
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因爲在這些系統中沒有幾何學。
06:30
So I found發現 myself, in other words,
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所以,換句話說,我發現我自己,
06:32
constructing建設 a geometry幾何,
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正在建立一種幾何學
06:34
a geometry幾何 of things which哪一個 had no geometry幾何.
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一種沒有幾何圖形的東西的的幾何學
06:37
And a surprising奇怪 aspect方面 of it
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而且,令人訝異的是
06:39
is that very often經常, the rules規則 of this geometry幾何
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這個幾何學的規則
06:42
are extremely非常 short.
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經常是極短的,
06:44
You have formulas公式 that long.
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你有這麼長的公式,
06:46
And you crank曲柄 it several一些 times.
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曲折了好幾次
06:48
Sometimes有時 repeatedly反复: again, again, again,
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有時候就只是一味地重復
06:50
the same相同 repetition重複.
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再重複,循著同樣方式反複循環
06:52
And at the end結束, you get things like that.
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最後,你會得到像這樣的東西
06:54
This cloud is completely全然,
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這片雲是100%
06:56
100 percent百分 artificial人造.
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完全人工的
06:59
Well, 99.9.
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嗯,99.9。
07:01
And the only part部分 which哪一個 is natural自然
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唯一自然的地方
07:03
is a number, the roughness粗糙度 of the cloud,
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是數字,也就是這片雲的粗糙度,
07:05
which哪一個 is taken採取 from nature性質.
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是取自自然的
07:07
Something so complicated複雜 like a cloud,
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有時,像雲這麼複雜的東西,
07:09
so unstable不穩定, so varying不同,
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是這麼不穩定、變化多端
07:11
should have a simple簡單 rule規則 behind背後 it.
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在它背後,應該有一個簡單的規則才是
07:14
Now this simple簡單 rule規則
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現在,這個簡單規則
07:17
is not an explanation說明 of clouds.
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並不是解釋雲層
07:20
The seer先見者 of clouds had to
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看這片雲的人必須
07:22
take account帳戶 of it.
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有這個認知。
07:24
I don't know how much advanced高級
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我不認為這些照片有多先進,
07:27
these pictures圖片 are. They're old.
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它們很舊了
07:29
I was very much involved參與 in it,
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我以前涉獵極深,
07:31
but then turned轉身 my attention注意 to other phenomena現象.
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但後來,我轉而研究其他現象了
07:34
Now, here is another另一個 thing
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現在,這裡有另一個
07:36
which哪一個 is rather interesting有趣.
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更有趣的東西
07:39
One of the shattering驚天動地 events事件
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這是在數學史上一件
07:41
in the history歷史 of mathematics數學,
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令人震驚的事件,
07:43
which哪一個 is not appreciated讚賞 by many許多 people,
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當時沒多少人理解,
07:46
occurred發生 about 130 years年份 ago,
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發生在大約 130 年前、
07:48
145 years年份 ago.
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或 145 年前。
07:50
Mathematicians數學家 began開始 to create創建
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當時,數學家開始創造
07:52
shapes形狀 that didn't exist存在.
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不存在的形狀
07:54
Mathematicians數學家 got into self-praise自我表揚
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數學家陷入一種自我耽溺的地步
07:57
to an extent程度 which哪一個 was absolutely絕對 amazing驚人,
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他們完全沉浸於
07:59
that man can invent發明 things
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人類發明的喜悅之中
08:01
that nature性質 did not know.
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而這些發明是自然所不知曉的事物
08:03
In particular特定, it could invent發明
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特別是,發明一種
08:05
things like a curve曲線 which哪一個 fills填充 the plane平面.
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可以填補平面的曲線
08:08
A curve's曲線的 a curve曲線, a plane's飛機的 a plane平面,
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曲線是曲線,平面是平面,
08:10
and the two won't慣於 mix混合.
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兩者無法混合
08:12
Well, they do mix混合.
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但事實上,他們是可以混在一起的
08:14
A man named命名 Peano皮亞諾
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有一個叫 Peano 的先生
08:16
did define確定 such這樣 curves曲線,
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真的確立了這些曲線,
08:18
and it became成為 an object目的 of extraordinary非凡 interest利益.
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於是,這形成一個當時多數人極感興趣的研究對象
08:21
It was very important重要, but mostly大多 interesting有趣
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它在當時非常重要,但也相當有趣
08:24
because a kind of break打破,
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因為,一種突破
08:26
a separation分割 between之間
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必須是一種區隔,
08:28
the mathematics數學 coming未來 from reality現實, on the one hand,
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它區隔來自描述現實現象的數學
08:31
and new mathematics數學 coming未來 from pure man's男人的 mind心神.
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與來自人類純粹心智的新數學
08:34
Well, I was very sorry to point out
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嗯,我必須很遺憾地指出
08:37
that the pure man's男人的 mind心神
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純粹的人類心智
08:39
has, in fact事實,
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事實上
08:41
seen看到 at long last
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最終見到了
08:43
what had been seen看到 for a long time.
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他們長久以來視而不見的事物
08:45
And so here I introduce介紹 something,
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所以,在這裡,我要向大家介紹
08:47
the set of rivers河流 of a plane-filling平面填充 curve曲線.
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一組河流的平面填充曲線
08:50
And well,
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而且
08:52
it's a story故事 unto itself本身.
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它本身就是一個故事。
08:54
So it was in 1875 to 1925,
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1875 年到 1925 年
08:57
an extraordinary非凡 period
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是一段了不起的時期
08:59
in which哪一個 mathematics數學 prepared準備 itself本身 to break打破 out from the world世界.
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在這段期間,數學正準備突破自己的世界,
09:02
And the objects對象 which哪一個 were used
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當我還是個小孩、學生的時候
09:04
as examples例子, when I was
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當時作為範例的
09:06
a child兒童 and a student學生, as examples例子
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物體
09:08
of the break打破 between之間 mathematics數學
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區分了數學與
09:11
and visible可見 reality現實 --
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可見的現實——
09:13
those objects對象,
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我把那些物體
09:15
I turned轉身 them completely全然 around.
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完全顛倒過來
09:17
I used them for describing說明
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我把它們用來描述
09:19
some of the aspects方面 of the complexity複雜 of nature性質.
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自然的若干繁複面向
09:22
Well, a man named命名 Hausdorff豪斯多夫 in 1919
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1919 年,有一位叫做 Hausdorff 的先生
09:25
introduced介紹 a number which哪一個 was just a mathematical數學的 joke玩笑,
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引介了一個數字,這個數字在當時被看作數學玩笑
09:28
and I found發現 that this number
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但我發現這個數值
09:30
was a good measurement測量 of roughness粗糙度.
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卻是衡量粗糙度的好工具
09:32
When I first told it to my friends朋友 in mathematics數學
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當我第一次把這個發現告訴我數學界的朋友時,
09:34
they said, "Don't be silly愚蠢. It's just something [silly愚蠢]."
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他們說:「別傻了,那只不過是件無聊蠢事。」
09:37
Well actually其實, I was not silly愚蠢.
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然而事實上,我當時並不傻,
09:40
The great painter畫家 Hokusai北齋 knew知道 it very well.
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偉大的畫家葛飾北齋(Hokusai)深知這個道理
09:43
The things on the ground地面 are algae藻類.
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這些涉及複數的問題
09:45
He did not know the mathematics數學; it didn't yet然而 exist存在.
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他不懂數學,那時數學尚未存在
09:48
And he was Japanese日本 who had no contact聯繫 with the West西.
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他是個日本人,從未接觸過西方世界
09:51
But painting繪畫 for a long time had a fractal分形 side.
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但長久以來,他的畫作擁有碎形面
09:54
I could speak說話 of that for a long time.
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我可以花很多時間談論這個
09:56
The Eiffel艾菲爾 Tower has a fractal分形 aspect方面.
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艾菲爾鐵塔有個碎形的外觀
09:59
I read the book that Mr先生. Eiffel艾菲爾 wrote about his tower,
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我在書上讀到,埃菲爾先生寫過他的鐵塔
10:02
and indeed確實 it was astonishing驚人 how much he understood了解.
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確實,令人驚訝地,他非常瞭解碎型
10:05
This is a mess食堂, mess食堂, mess食堂, Brownian布朗 loop循環.
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這是一個混亂、混亂、混亂的布朗寧迴圈
10:08
One day I decided決定 --
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有一天,在我職業生涯的半途
10:10
halfway through通過 my career事業,
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我發現
10:12
I was held保持 by so many許多 things in my work --
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我的工作被許多事情絆住
10:15
I decided決定 to test測試 myself.
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我決定測試自己
10:18
Could I just look at something
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看我是否可以
10:20
which哪一個 everybody每個人 had been looking at for a long time
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從每個人看了許久的事物中
10:23
and find something dramatically顯著 new?
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發現什麽戲劇化的新東西?
10:26
Well, so I looked看著 at these
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嗯,於是我看到了這些
10:29
things called Brownian布朗 motion運動 -- just goes around.
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叫布朗寧運動的東西,只有一圈
10:32
I played發揮 with it for a while,
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我和它玩了一會,
10:34
and I made製作 it return返回 to the origin起源.
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我使它回到原點
10:37
Then I was telling告訴 my assistant助理,
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接著,我告訴我的助理:
10:39
"I don't see anything. Can you paint塗料 it?"
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「我看不到任何東西。你能把它畫出來嗎?」
10:41
So he painted it, which哪一個 means手段
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於是他畫了出來,這意謂著
10:43
he put inside everything. He said:
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他把所有都放了進去。他說:
10:45
"Well, this thing came來了 out ..." And I said, "Stop! Stop! Stop!
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「這東西出現了......」我說:「停下來! 停下來! 停下來!
10:48
I see; it's an island."
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我明白了,這是一座島嶼。」
10:51
And amazing驚人.
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多麼驚人
10:53
So Brownian布朗 motion運動, which哪一個 happens發生 to have
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因此,布朗寧運動剛好有
10:55
a roughness粗糙度 number of two, goes around.
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一個粗糙度數字2,它繞了一圈
10:58
I measured測量 it, 1.33.
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我測量它,是1.33
11:00
Again, again, again.
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一而再,再而三
11:02
Long measurements測量, big Brownian布朗 motions運動,
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長尺寸,大的布朗寧運動,
11:04
1.33.
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1.33
11:06
Mathematical數學的 problem問題: how to prove證明 it?
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一個數學問題來了:該如何證明它?
11:09
It took my friends朋友 20 years年份.
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我的朋友曾花 20 年的時間研究
11:12
Three of them were having incomplete殘缺 proofs樣張.
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他們三個人產出一個不完全的證明
11:15
They got together一起, and together一起 they had the proof證明.
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他們聚在一起,一起證明它
11:19
So they got the big [Fields字段] medal勳章 in mathematics數學,
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因此,他們獲得了這個領域的大獎
11:22
one of the three medals獎牌 that people have received收到
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這些獲獎人當中,有一面獎牌
11:24
for proving證明 things which哪一個 I've seen看到
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並不能合理地證明
11:27
without being存在 able能夠 to prove證明 them.
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我所見到的東西
11:30
Now everybody每個人 asks me at one point or another另一個,
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現在,每個人都問我
11:33
"How did it all start開始?
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「這是怎麼開始的?
11:35
What got you in that strange奇怪 business商業?"
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是什麼原因使你進入這個陌生的領域?」
11:38
What got you to be,
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是什麼讓我
11:40
at the same相同 time, a mechanical機械 engineer工程師,
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同時成為一個機械工程師、
11:42
a geographer地理學
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又成為地理學家、
11:44
and a mathematician數學家 and so on, a physicist物理學家?
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數學家、或物理學家等等?
11:46
Well actually其實 I started開始, oddly奇怪 enough足夠,
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嗯,事實上,我是從一個非常怪異的地方開始的
11:49
studying研究 stock股票 market市場 prices價格.
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我研究股票市場價格
11:51
And so here
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在這
11:53
I had this theory理論,
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我提過理論
11:56
and I wrote books圖書 about it --
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我也寫了有關這方面的書籍
11:58
financial金融 prices價格 increments增量.
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金融價格增長量
12:00
To the left you see data數據 over a long period.
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在左邊,你們看到長期的數據
12:02
To the right, on top最佳,
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在右邊,上方
12:04
you see a theory理論 which哪一個 is very, very fashionable時髦.
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你們可以看到一個非常、非常流行的理論
12:07
It was very easy簡單, and you can write many許多 books圖書 very fast快速 about it.
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它非常簡單,你可以用極短的時間寫下許多關於它的書籍
12:10
(Laughter笑聲)
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(笑聲)
12:12
There are thousands數千 of books圖書 on that.
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坊間有上千本這方面的著作
12:15
Now compare比較 that with real真實 price價錢 increments增量.
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現在,比較真實的價格增加量,
12:18
Where are real真實 price價錢 increments增量?
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哪裡是實際的價格增加量呢?
12:20
Well, these other lines
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嗯,其他這些曲線
12:22
include包括 some real真實 price價錢 increments增量
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包涵了一些真正的價格利潤
12:24
and some forgery偽造品 which哪一個 I did.
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還有一些是我偽造的
12:26
So the idea理念 there was
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所以,這裡的觀點是
12:28
that one must必須 be able能夠 to -- how do you say? --
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人們必須能夠 --怎麼說呢? --
12:30
model模型 price價錢 variation變異.
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把價格變化模組化
12:33
And it went really well 50 years年份 ago.
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在五十年前,這觀點被認為相當有道理
12:36
For 50 years年份, people were sort分類 of pooh-poohing維尼poohing me
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五十年來,人們多少有點輕視我的看法
12:39
because they could do it much, much easier更輕鬆.
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因為他們可以用非常簡單的方式換算出來
12:41
But I tell you, at this point, people listened聽了 to me.
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但我告訴你,在這一點上,人們聽信我
12:44
(Laughter笑聲)
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(笑聲)
12:46
These two curves曲線 are averages均線:
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這兩條曲線是平均值
12:48
Standard標準 & Poor較差的, the blue藍色 one;
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藍色的那條是標準普爾(Standard & Poor)的曲線,
12:50
and the red one is Standard標準 & Poor's
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而紅色的那條是標準普爾
12:52
from which哪一個 the five biggest最大 discontinuities間斷
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根據其中 5 個最大的不連續性
12:55
are taken採取 out.
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所畫出來的曲線
12:57
Now discontinuities間斷 are a nuisance滋擾,
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現在,不連續造成了累贅
12:59
so in many許多 studies學習 of prices價格,
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所以,在許多關於價格的研究上
13:02
one puts看跌期權 them aside在旁邊.
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人們把它們擱在一旁,說:
13:04
"Well, acts行為 of God.
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「嗯,這些是神的旨意(不可抗力的因素)
13:06
And you have the little nonsense廢話 which哪一個 is left.
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於是留下了少許無意義的東西,
13:09
Acts行為 of God." In this picture圖片,
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在這幅包涵不可抗力因素的照片中
13:12
five acts行為 of God are as important重要 as everything else其他.
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五個不可抗力的現象就如同所有其他事物一樣重要
13:15
In other words,
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換句話說,
13:17
it is not acts行為 of God that we should put aside在旁邊.
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事實上,我們不應擱置那不可抗拒的現象不談
13:19
That is the meat, the problem問題.
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那才是牛肉,是問題所在
13:22
If you master these, you master price價錢,
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如果你熟悉價格和這些癥結
13:25
and if you don't master these, you can master
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而且,如果你不熟悉這些癥結,你也可以試著
13:27
the little noise噪聲 as well as you can,
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盡可能地了解小問題
13:29
but it's not important重要.
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但它不重要
13:31
Well, here are the curves曲線 for it.
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嗯,這裡有關於它的曲線
13:33
Now, I get to the final最後 thing, which哪一個 is the set
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我來到最後這個
13:35
of which哪一個 my name名稱 is attached.
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附有我名字的這組
13:37
In a way, it's the story故事 of my life.
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在某種程度上,它是我一生的故事
13:39
My adolescence青春期 was spent花費
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我青少年是在
13:41
during the German德語 occupation佔用 of France法國.
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德軍佔領法國的期間度過的
13:43
Since以來 I thought that I might威力
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我曾想,也許我可能會
13:46
vanish消失 within a day or a week,
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在一天或一個星期內憑空消失
13:49
I had very big dreams.
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所以,我有一些大夢想
13:52
And after the war戰爭,
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戰後
13:54
I saw an uncle叔叔 again.
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我和我叔叔相遇
13:56
My uncle叔叔 was a very prominent突出 mathematician數學家, and he told me,
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我叔叔是個非常重要的數學家,他告訴我
13:58
"Look, there's a problem問題
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「看,這裡有一個我二十五年來
14:00
which哪一個 I could not solve解決 25 years年份 ago,
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都無法解決的問題,
14:02
and which哪一個 nobody沒有人 can solve解決.
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沒有人可以解答
14:04
This is a construction施工 of a man named命名 [Gaston加斯頓] Julia朱莉婭
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這是一個叫 [Gaston] Julia 和 [Pierre] Fatou
14:06
and [Pierre皮埃爾] Fatou法圖.
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共同建構的問題
14:08
If you could
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如果你可以,
14:10
find something new, anything,
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發掘新的解決辦法,任何解決辦法,
14:12
you will get your career事業 made製作."
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你的事業必定有所成就。」
14:14
Very simple簡單.
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非常簡單
14:16
So I looked看著,
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於是,我試試看
14:18
and like the thousands數千 of people that had tried試著 before,
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就像許許多多前人試過的一樣
14:20
I found發現 nothing.
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我什麽也沒找到
14:23
But then the computer電腦 came來了,
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然而接著,電腦出現了
14:25
and I decided決定 to apply應用 the computer電腦,
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我決定使用電腦
14:27
not to new problems問題 in mathematics數學 --
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不是用在數學的新問題——
14:30
like this wiggle擺動 wiggle擺動, that's a new problem問題 --
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比如這條擺動的曲線,這是新問題——
14:32
but to old problems問題.
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而是,把電腦應用於舊的問題之上
14:34
And I went from what's called
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我從那稱為實數(real number)
14:36
real真實 numbers數字, which哪一個 are points on a line,
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的地方開始,這是一條線上的點
14:38
to imaginary假想, complex複雜 numbers數字,
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到虛數、複數
14:40
which哪一個 are points on a plane平面,
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這些是平面的數
14:42
which哪一個 is what one should do there,
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也是人們必須去研究的事
14:44
and this shape形狀 came來了 out.
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於是,這個圖形出現了
14:46
This shape形狀 is of an extraordinary非凡 complication並發症.
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形狀極其複雜
14:49
The equation方程 is hidden there,
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該方程式隱藏在那裡
14:51
z goes into z squared平方, plus c.
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z 進入z 平方,加上 c
14:54
It's so simple簡單, so dry.
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它是如此簡單、如此枯燥、
14:56
It's so uninteresting枯燥.
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如此無趣
14:58
Now you turn the crank曲柄 once一旦, twice兩次:
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現在,你轉動曲軸兩次
15:01
twice兩次,
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兩次
15:04
marvels奇蹟 come out.
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奇蹟就出現了。
15:06
I mean this comes out.
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我指的是,出現了這個
15:08
I don't want to explain說明 these things.
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我不想解釋這些東西
15:10
This comes out. This comes out.
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出現了這個,出現了這個
15:12
Shapes形狀 which哪一個 are of such這樣 complication並發症,
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出現了如此這般複雜的形狀
15:14
such這樣 harmony和諧 and such這樣 beauty美女.
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它們具有如此的和諧與美感
15:17
This comes out
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出現了這個
15:19
repeatedly反复, again, again, again.
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它們一而再,再而三地重複著
15:21
And that was one of my major重大的 discoveries發現,
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這就是過去我最主要的發現之一
15:23
to find that these islands島嶼 were the same相同
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我發現這些島嶼是相同的
15:25
as the whole整個 big thing, more or less.
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或多或少,就如同它較大的整體
15:27
And then you get these
380
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於是,你在所有地方得到這些
15:29
extraordinary非凡 baroque巴洛克 decorations all over the place地點.
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非凡的巴洛克式裝飾
15:32
All that from this little formula,
382
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它們全都來自這個小小的方程式
15:35
which哪一個 has whatever隨你, five symbols符號 in it.
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這方程式有五種符號
15:38
And then this one.
384
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接著是這個
15:40
The color顏色 was added添加 for two reasons原因.
385
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加上兩種顏色的原因是
15:42
First of all, because these shapes形狀
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首先,因為這些圖形
15:44
are so complicated複雜
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是如此複雜
15:47
that one couldn't不能 make any sense of the numbers數字.
388
932000
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以致於人們無法辨識任何數目
15:50
And if you plot情節 them, you must必須 choose選擇 some system系統.
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如果你要繪製它們,你必須選擇某些系統
15:53
And so my principle原理 has been
390
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所以,我的原則是
15:55
to always present當下 the shapes形狀
391
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永遠以不同的顏色
15:58
with different不同 colorings色素
392
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呈現這些圖形
16:00
because some colorings色素 emphasize注重 that,
393
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因為某些顏色強調某些部份
16:02
and others其他 it is that or that.
394
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而其他的強調這,或強調那
16:04
It's so complicated複雜.
395
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實在真的很複雜
16:06
(Laughter笑聲)
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(笑聲)
16:08
In 1990, I was in Cambridge劍橋, U.K.
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1990 年,我在英國劍橋
16:10
to receive接收 a prize from the university大學,
398
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獲得大學一個獎項
16:13
and three days later後來,
399
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三天後
16:15
a pilot飛行員 was flying飛行 over the landscape景觀 and found發現 this thing.
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有位駕駛飛越田野上空,發現了這東西
16:18
So where did this come from?
401
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這是來自哪裡呢?
16:20
Obviously明顯, from extraterrestrials外星人.
402
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很顯然,這來自外星人
16:22
(Laughter笑聲)
403
967000
3000
(笑聲)
16:25
Well, so the newspaper報紙 in Cambridge劍橋
404
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2000
嗯,所以劍橋的報紙
16:27
published發表 an article文章 about that "discovery發現"
405
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登載了關於那「發現」的文章
16:29
and received收到 the next下一個 day
406
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隔天後,他們收到了
16:31
5,000 letters from people saying,
407
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5000 封信,人們在信上說:
16:33
"But that's simply只是 a Mandelbrot曼德爾布羅 set very big."
408
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「這只不過是一個非常大的 Mandelbrot 圖組罷了。」
16:37
Well, let me finish.
409
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嗯,讓我這麼結束吧
16:39
This shape形狀 here just came來了
410
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這兒的圖形,只是來自
16:41
out of an exercise行使 in pure mathematics數學.
411
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純數學的演算
16:43
Bottomless萬丈 wonders奇蹟 spring彈簧 from simple簡單 rules規則,
412
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3000
深不可測的奇觀,源自簡單的規則
16:46
which哪一個 are repeated重複 without end結束.
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它們無止無盡地反復
16:49
Thank you very much.
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謝謝大家
16:51
(Applause掌聲)
415
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(掌聲)
Translated by Geoff Chen
Reviewed by Wang-Ju Tsai

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ABOUT THE SPEAKER
Benoit Mandelbrot - Mathematician
Benoit Mandelbrot's work led the world to a deeper understanding of fractals, a broad and powerful tool in the study of roughness, both in nature and in humanity's works.

Why you should listen

Studying complex dynamics in the 1970s, Benoit Mandelbrot had a key insight about a particular set of mathematical objects: that these self-similar structures with infinitely repeating complexities were not just curiosities, as they'd been considered since the turn of the century, but were in fact a key to explaining non-smooth objects and complex data sets -- which make up, let's face it, quite a lot of the world. Mandelbrot coined the term "fractal" to describe these objects, and set about sharing his insight with the world.

The Mandelbrot set (expressed as z² + c) was named in Mandelbrot's honor by Adrien Douady and John H. Hubbard. Its boundary can be magnified infinitely and yet remain magnificently complicated, and its elegant shape made it a poster child for the popular understanding of fractals. Led by Mandelbrot's enthusiastic work, fractal math has brought new insight to the study of pretty much everything, from the behavior of stocks to the distribution of stars in the universe.

Benoit Mandelbrot appeared at the first TED in 1984, and returned in 2010 to give an overview of the study of fractals and the paradigm-flipping insights they've brought to many fields. He died in October 2010 at age 85. Read more about his life on NYBooks.com >>

More profile about the speaker
Benoit Mandelbrot | Speaker | TED.com