Marcus du Sautoy: Symmetry, reality's riddle

131,456 views ・ 2009-10-29

TED


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翻译人员: Beibei Mu 校对人员: Qian YANG
00:18
On the 30th of May, 1832,
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在1832年5月30日,
00:22
a gunshot was heard
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人们听到一声枪响,
00:24
ringing out across the 13th arrondissement in Paris.
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枪声穿透了巴黎的第十三区
00:27
(Gunshot)
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(枪声)
00:28
A peasant, who was walking to market that morning,
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一个农民,那天早晨正去往市场
00:31
ran towards where the gunshot had come from,
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朝枪声传来的地方跑了过去,
00:33
and found a young man writhing in agony on the floor,
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并发现一名年轻男子正痛得在地上打滚,
00:37
clearly shot by a dueling wound.
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显然他在决斗中遭到了枪击。
00:40
The young man's name was Evariste Galois.
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这个年轻人名叫伊瓦利斯特•伽罗瓦。
00:43
He was a well-known revolutionary in Paris at the time.
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他当时在巴黎是一个著名的革命者。
00:47
Galois was taken to the local hospital
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伽罗瓦被送到了当地的医院,
00:50
where he died the next day in the arms of his brother.
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在医院第二天死在了他兄弟的怀中。
00:53
And the last words he said to his brother were,
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他最后对他兄弟说的话是,
00:55
"Don't cry for me, Alfred.
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“阿尔弗雷德不要为我哭泣。
00:57
I need all the courage I can muster
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我需要聚集我能聚集的所有勇气
00:59
to die at the age of 20."
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让我在20岁时死去。”
01:03
It wasn't, in fact, revolutionary politics
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实际上,革命政治并不是
01:05
for which Galois was famous.
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使伽罗瓦著名的原因。
01:07
But a few years earlier, while still at school,
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而是几年前,当他还在上学时,
01:10
he'd actually cracked one of the big mathematical
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他实际上已经破解了
01:12
problems at the time.
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当时重大数学问题之一。
01:14
And he wrote to the academicians in Paris,
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随后他写信给巴黎的院士,
01:16
trying to explain his theory.
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尝试解释他的理论。
01:18
But the academicians couldn't understand anything that he wrote.
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但院士们弄不懂他写的任何东西。
01:21
(Laughter)
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(众笑)
01:22
This is how he wrote most of his mathematics.
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这就是他怎么写大部分数学理论的。
01:25
So, the night before that duel, he realized
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因此,在决斗的前一天晚上,他意识到
01:27
this possibly is his last chance
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这可能是他最后一次机会
01:30
to try and explain his great breakthrough.
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来尝试解释他的重大突破了。
01:32
So he stayed up the whole night, writing away,
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所以他彻夜未眠,不停地写东西,
01:35
trying to explain his ideas.
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试图解释他的想法。
01:37
And as the dawn came up and he went to meet his destiny,
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随着黎明的到来,他准备迎接自己的命运。
01:40
he left this pile of papers on the table for the next generation.
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他把桌子上的一堆文件留给了下一代。
01:44
Maybe the fact that he stayed up all night doing mathematics
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也许他彻夜研究数学
01:47
was the fact that he was such a bad shot that morning and got killed.
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是他那天早晨受到枪击且被杀的真正原因。
01:50
But contained inside those documents
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但包含在那些文件中的
01:52
was a new language, a language to understand
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是一种新的语言,这种语言能让人们理解
01:55
one of the most fundamental concepts
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科学的一个最基本的概念,
01:57
of science -- namely symmetry.
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即对称性。
02:00
Now, symmetry is almost nature's language.
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现今,对称性几乎是大自然的语言。
02:02
It helps us to understand so many
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它有助于我们了解许多
02:04
different bits of the scientific world.
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科学世界里不同的小东西。
02:06
For example, molecular structure.
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例如,分子结构。
02:08
What crystals are possible,
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什么晶体是能让
02:10
we can understand through the mathematics of symmetry.
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我们可以通过数学的对称性来了解的?
02:14
In microbiology you really don't want to get a symmetrical object,
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在微生物学中,你真的不想研究对称的东西。
02:16
because they are generally rather nasty.
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因为它们一般都比较令人讨厌。
02:18
The swine flu virus, at the moment, is a symmetrical object.
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目前的猪流感病毒就是一种结构对称的病毒。
02:21
And it uses the efficiency of symmetry
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而且它利用对称的功效
02:23
to be able to propagate itself so well.
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来使自己很好的增殖。
02:27
But on a larger scale of biology, actually symmetry is very important,
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但就生物学更大范围的而言,对称性事实上非常重要,
02:30
because it actually communicates genetic information.
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因为它能传递遗传信息。
02:32
I've taken two pictures here and I've made them artificially symmetrical.
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我带了两张照片到这儿来,并人工的把他们做成了对称的。
02:36
And if I ask you which of these you find more beautiful,
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如果我问你们觉得哪些更漂亮,
02:39
you're probably drawn to the lower two.
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你们可能会被下面的两张吸引住。
02:41
Because it is hard to make symmetry.
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因为很难做到对称,
02:44
And if you can make yourself symmetrical, you're sending out a sign
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所以如果你可以使自己对称,那么你在传递一种信号
02:46
that you've got good genes, you've got a good upbringing
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它意味着你得到了好的遗传基因,你有好的教养,
02:49
and therefore you'll make a good mate.
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因而你会有一个好的伴侣。
02:51
So symmetry is a language which can help to communicate
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所以,对称性是一种语言,它能有助于传递
02:54
genetic information.
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遗传信息。
02:56
Symmetry can also help us to explain
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对称性还可以帮助我们解释
02:58
what's happening in the Large Hadron Collider in CERN.
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欧洲粒子物理研究所大型强子对撞机正发生着什么事情。
03:01
Or what's not happening in the Large Hadron Collider in CERN.
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或者欧洲粒子物理研究所的大型强子对撞机没有发生什么事情。
03:04
To be able to make predictions about the fundamental particles
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为了能够对基本粒子作出预测,
03:06
we might see there,
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我们可能会在那儿看到的(基本粒子),
03:08
it seems that they are all facets of some strange symmetrical shape
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似乎所有的小平面都有某种奇怪的对称形状
03:12
in a higher dimensional space.
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当它们在更高维的空间中时。
03:14
And I think Galileo summed up, very nicely,
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我认为伽利略很好地概括了
03:16
the power of mathematics
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数学的力量:
03:18
to understand the scientific world around us.
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它让我们对周围的科学世界得以了解。
03:20
He wrote, "The universe cannot be read
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他写道:“我们无法阅读宇宙,
03:22
until we have learnt the language
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除非学会它的语言,
03:24
and become familiar with the characters in which it is written.
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且熟悉其写作特点。
03:27
It is written in mathematical language,
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它是用数学语言写的。
03:29
and the letters are triangles, circles and other geometric figures,
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字母是三角形、圆和其他的几何数字,
03:33
without which means it is humanly impossible
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没有这些字母就意味着在人力所能及的范围内是不可能
03:35
to comprehend a single word."
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理解任何一个字的。”
03:38
But it's not just scientists who are interested in symmetry.
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不只是科学家们对对称性感兴趣。
03:41
Artists too love to play around with symmetry.
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艺术家也喜欢摆弄对称性。
03:44
They also have a slightly more ambiguous relationship with it.
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他们与对称性有一些更模糊的关系。
03:47
Here is Thomas Mann talking about symmetry in "The Magic Mountain."
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这是托马斯•曼在《魔山》中谈到的对称性。
03:50
He has a character describing the snowflake,
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他对雪花有这样的描。
03:53
and he says he "shuddered at its perfect precision,
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他说,“他因其有完美的精确度而震撼,
03:56
found it deathly, the very marrow of death."
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发现它死亡的精髓让他想到死亡。”
03:59
But what artists like to do is to set up expectations
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但艺术家们想要做的是树立对对称性的期望,
04:01
of symmetry and then break them.
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然后打破它们。
04:03
And a beautiful example of this
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就这一点我找到了一个很好的例子,
04:05
I found, actually, when I visited a colleague of mine
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其实是当我拜访我的同事
04:07
in Japan, Professor Kurokawa.
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在日本的黑川纪章教授时发现的。
04:09
And he took me up to the temples in Nikko.
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他带我到日光市的寺庙去。
04:12
And just after this photo was taken we walked up the stairs.
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就在拍好这张照片后,我们走上楼梯,
04:15
And the gateway you see behind
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你们看到的这后面的大门
04:17
has eight columns, with beautiful symmetrical designs on them.
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有八根柱子,都有着漂亮的对称性设计。
04:20
Seven of them are exactly the same,
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其中七个是完全一样的,
04:22
and the eighth one is turned upside down.
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而第八个是颠倒过来的。
04:25
And I said to Professor Kurokawa,
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我就对黑川纪章教授说:
04:27
"Wow, the architects must have really been kicking themselves
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“哇,建筑师们肯定要踢自己了,
04:29
when they realized that they'd made a mistake and put this one upside down."
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要是他么发现犯了这么一个错误,把这根柱子弄倒了过来。”
04:32
And he said, "No, no, no. It was a very deliberate act."
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他说,“不,不,不。这是一个特意的设计。”
04:35
And he referred me to this lovely quote from the Japanese
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他还向我提到了这个可爱的引述,引自日本
04:37
"Essays in Idleness" from the 14th century,
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1 4世纪的《闲置的散文》。
04:40
in which the essayist wrote, "In everything,
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其中,散文家写道:“在一切事物中,
04:42
uniformity is undesirable.
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一致性是不可取的。
04:45
Leaving something incomplete makes it interesting,
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留下一些不完整的东西会更有趣,
04:47
and gives one the feeling that there is room for growth."
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而且一致性给人一种没有发展空间的感觉。”
04:50
Even when building the Imperial Palace,
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即使是建造皇宫时,
04:52
they always leave one place unfinished.
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他们也总是留下一个未完工的地方。
04:56
But if I had to choose one building in the world
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但如果我必须选择这世界上的一个建筑,
04:59
to be cast out on a desert island, to live the rest of my life,
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将其扔到一个荒岛上,且我要在那里度过余生,
05:02
being an addict of symmetry, I would probably choose the Alhambra in Granada.
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作为一个对对称性痴迷的人,我可能会选择在格拉纳达的阿尔罕布拉。
05:06
This is a palace celebrating symmetry.
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这是一座歌颂对称性的宫殿。
05:08
Recently I took my family --
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最近,我带我的家人——
05:10
we do these rather kind of nerdy mathematical trips, which my family love.
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我们进行这种并没有学术气息的数学旅行,我的家人都很喜欢。
05:13
This is my son Tamer. You can see
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这是我的儿子塔梅尔。你们可以看到
05:15
he's really enjoying our mathematical trip to the Alhambra.
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他真的很喜欢我们在阿尔罕布拉的数学之旅。
05:18
But I wanted to try and enrich him.
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但我想尝试使他变得充实。
05:21
I think one of the problems about school mathematics
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我认为学校教的数学存在的一个问题就是
05:23
is it doesn't look at how mathematics is embedded
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它没有关注数学是如何被运用于
05:25
in the world we live in.
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我们所处的这个世界。
05:27
So, I wanted to open his eyes up to
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所以,我想开拓他的眼界,让他知道
05:29
how much symmetry is running through the Alhambra.
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阿尔罕布拉运用着多少对称性。
05:32
You see it already. Immediately you go in,
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你们已经看到了。你一走进去,
05:34
the reflective symmetry in the water.
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水中有反射的对称。
05:36
But it's on the walls where all the exciting things are happening.
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但是,所有令人兴奋的事情发生在墙壁上。
05:39
The Moorish artists were denied the possibility
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人们否认摩尔艺术家能够
05:41
to draw things with souls.
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用灵魂来绘画。
05:43
So they explored a more geometric art.
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因此,他们探索出一种更加几何化的艺术。
05:45
And so what is symmetry?
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那么什么是对称性?
05:47
The Alhambra somehow asks all of these questions.
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阿尔罕布拉以某种方式提出了所有这些问题。
05:50
What is symmetry? When [there] are two of these walls,
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什么是对称性?当[那儿]有两面墙时,
05:52
do they have the same symmetries?
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他们有相同的对称性吗?
05:54
Can we say whether they discovered
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我们可以说他们是否发现了
05:56
all of the symmetries in the Alhambra?
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阿尔罕布拉所有的对称性吗?
05:59
And it was Galois who produced a language
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是伽罗瓦研制出了一种语言
06:01
to be able to answer some of these questions.
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能够回答一些这样的问题。
06:04
For Galois, symmetry -- unlike for Thomas Mann,
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对伽罗瓦来说,对称性,不同于托马斯曼所说的
06:07
which was something still and deathly --
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对称性是一些静态的和死一般的东西。
06:09
for Galois, symmetry was all about motion.
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对伽罗瓦来说,所有的对称性都是关于运动的。
06:12
What can you do to a symmetrical object,
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你能对一个对称性的物体做些什么?
06:14
move it in some way, so it looks the same
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用某种方法移动它,让它看起来
06:16
as before you moved it?
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跟你移动它之前一样?
06:18
I like to describe it as the magic trick moves.
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我喜欢把这形容为神奇的假动作。
06:20
What can you do to something? You close your eyes.
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你对一些东西可以做些什么?闭上你的眼睛。
06:22
I do something, put it back down again.
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我动下它,再把它放回到原处。
06:24
It looks like it did before it started.
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它看起来和动之前一样。
06:26
So, for example, the walls in the Alhambra --
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那么,例如,阿尔罕布拉的墙壁。
06:28
I can take all of these tiles, and fix them at the yellow place,
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我可以把所有的这些瓦片拿起来,把他们放在这个黄色的地方,
06:32
rotate them by 90 degrees,
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并把它们旋转九十度,
06:34
put them all back down again and they fit perfectly down there.
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再把他们都放回去,它们非常吻合。
06:37
And if you open your eyes again, you wouldn't know that they'd moved.
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如果你再睁开你的眼睛,你不会知道它们被移动过。
06:40
But it's the motion that really characterizes the symmetry
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但正是运动才使对称性
06:43
inside the Alhambra.
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在阿尔罕布拉具有特色。
06:45
But it's also about producing a language to describe this.
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但也要创造一种语言来描绘它。
06:47
And the power of mathematics is often
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数学的力量往往
06:50
to change one thing into another, to change geometry into language.
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把一样东西变成另一样,把几何变成语言。
06:54
So I'm going to take you through, perhaps push you a little bit mathematically --
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因此,我将带你经历,可能强加一些数学的东西给你们,
06:57
so brace yourselves --
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所以撑住自己,
06:59
push you a little bit to understand how this language works,
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强加一些数学的知识让你们了解这种语言是怎么运作的,
07:02
which enables us to capture what is symmetry.
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这让我们能够捕捉到什么是对称性。
07:04
So, let's take these two symmetrical objects here.
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那让我们把这两个对称物放到这儿。
07:07
Let's take the twisted six-pointed starfish.
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拿这个扭曲了的六角海星来说。
07:09
What can I do to the starfish which makes it look the same?
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我怎么做能让这个海星看起来和原来一样呢?
07:12
Well, there I rotated it by a sixth of a turn,
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嗯,我把它旋转了六分之一圈,
07:15
and still it looks like it did before I started.
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它看起来仍然跟我动过之前一样。
07:17
I could rotate it by a third of a turn,
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我可以把它旋转三分之一圈,
07:20
or a half a turn,
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或者半圈,
07:22
or put it back down on its image, or two thirds of a turn.
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或将它恢复到原图,或旋转三分之二圈。
07:25
And a fifth symmetry, I can rotate it by five sixths of a turn.
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第五种对称,我可以把它旋转六分之五圈。
07:29
And those are things that I can do to the symmetrical object
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这些就是我能对对称物所做的,
07:32
that make it look like it did before I started.
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可以让它看起来跟我动它们之前一样。
07:35
Now, for Galois, there was actually a sixth symmetry.
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对伽罗瓦来说,实际上还有第六种对称。
07:38
Can anybody think what else I could do to this
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大家能想到其它什么办法
07:40
which would leave it like I did before I started?
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可以让它跟我动它之前一样?
07:43
I can't flip it because I've put a little twist on it, haven't I?
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我不能翻转它,因为我已经把它扭曲了一些,是吧?
07:46
It's got no reflective symmetry.
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这样它没有反射对称了。
07:48
But what I could do is just leave it where it is,
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但我可以做的就是把它放在原处,
07:51
pick it up, and put it down again.
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把它拿起来再把它放下。
07:53
And for Galois this was like the zeroth symmetry.
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对伽罗瓦来说,这就像是第零个对称。
07:56
Actually, the invention of the number zero
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其实,数字零的发明
07:59
was a very modern concept, seventh century A.D., by the Indians.
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是一个非常现代化的概念,它是公元七世纪印度人发明的。
08:02
It seems mad to talk about nothing.
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什么都没有谈论看起来很疯狂。
08:05
And this is the same idea. This is a symmetrical --
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这是同样的概念。这是对称的——
08:07
so everything has symmetry, where you just leave it where it is.
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所以一切事物都有对称性,把它放在拿起它的地方。
08:09
So, this object has six symmetries.
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所以这个物体有六种对称。
08:12
And what about the triangle?
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那三角形呢?
08:14
Well, I can rotate by a third of a turn clockwise
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嗯,我可以把它顺时针旋转三分之一圈
08:18
or a third of a turn anticlockwise.
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或逆时针旋转三分之一圈。
08:20
But now this has some reflectional symmetry.
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但现在有反射对称。
08:22
I can reflect it in the line through X,
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。我可以在X轴上翻转它,
08:24
or the line through Y,
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或在Y轴上,
08:26
or the line through Z.
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或在Z轴上。
08:28
Five symmetries and then of course the zeroth symmetry
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五种对称,当然还有第零个对称,
08:31
where I just pick it up and leave it where it is.
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我把它拿起来,放回原处。
08:34
So both of these objects have six symmetries.
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因此,这些物体都有六种对称。
08:37
Now, I'm a great believer that mathematics is not a spectator sport,
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现在,我十分相信数学不是旁观者的运动,
08:40
and you have to do some mathematics
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你必须做一些数学运算
08:42
in order to really understand it.
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才能真正理解它。
08:44
So here is a little question for you.
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这儿有个小问题问问你们。
08:46
And I'm going to give a prize at the end of my talk
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我将在讲座结束后给一个奖品
08:48
for the person who gets closest to the answer.
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给那个给出的答案最接近的人。
08:50
The Rubik's Cube.
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魔方。
08:52
How many symmetries does a Rubik's Cube have?
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一个魔方有多少种对称?
08:55
How many things can I do to this object
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有多少种方法可以在动了这个物体,
08:57
and put it down so it still looks like a cube?
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且把它放下后它仍然看起来像一个立方体?
08:59
Okay? So I want you to think about that problem as we go on,
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好吗?我希望随着讲座的继续,你们可以考虑下这个问题,
09:02
and count how many symmetries there are.
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数数它有多少种对称。
09:04
And there will be a prize for the person who gets closest at the end.
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会有一个奖品在讲座结束后那个给出的答案最接近的人。
09:08
But let's go back down to symmetries that I got for these two objects.
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让我们回到这两个物体的对称性上。
09:12
What Galois realized: it isn't just the individual symmetries,
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伽罗瓦意识到这不仅仅是个体的对称性,
09:15
but how they interact with each other
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而是个体之间如何相互作用
09:17
which really characterizes the symmetry of an object.
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才真正赋予了一个物体具有对称性的特点。
09:21
If I do one magic trick move followed by another,
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如果我做一个神奇的假动作,然后再做一个,
09:24
the combination is a third magic trick move.
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两个合并起来就是第三个神奇的假动作。
09:26
And here we see Galois starting to develop
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这里我们了解到伽罗瓦开始开发
09:28
a language to see the substance
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一种语言来研究
09:31
of the things unseen, the sort of abstract idea
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看不见的东西所具有的内在含义,以及
09:33
of the symmetry underlying this physical object.
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物理物体中存在的对称性的抽象的概念。
09:36
For example, what if I turn the starfish
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例如,如果我把海星旋转
09:39
by a sixth of a turn,
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六分之一圈,
09:41
and then a third of a turn?
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然后再转三分之一圈会,结果会怎样?
09:43
So I've given names. The capital letters, A, B, C, D, E, F,
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所以我给它们取了名字。大写的字母A、B、C、D、E、F,
09:46
are the names for the rotations.
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这些名字旋转的代号。
09:48
B, for example, rotates the little yellow dot
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例如B,旋转小黄点,
09:51
to the B on the starfish. And so on.
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它位于海星上的B处,诸如此类。
09:54
So what if I do B, which is a sixth of a turn,
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那么,如果我旋转B,转六分之一圈,
09:56
followed by C, which is a third of a turn?
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其次是C,转三分之一圈?
09:59
Well let's do that. A sixth of a turn,
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嗯,让我们开始。六分之一圈,
10:01
followed by a third of a turn,
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接着是三分之一圈,
10:03
the combined effect is as if I had just rotated it by half a turn in one go.
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合并后的效果就像我刚刚把它一次旋转了半圈一样。
10:08
So the little table here records
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那这个小表格记载着
10:10
how the algebra of these symmetries work.
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这些对称的代数是怎么运作的。
10:13
I do one followed by another, the answer is
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我将一个接一个的旋转,结果就是
10:15
it's rotation D, half a turn.
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D旋转了半圈。
10:17
What I if I did it in the other order? Would it make any difference?
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如果我按其他顺序旋转呢?会有什么不同吗?
10:20
Let's see. Let's do the third of the turn first, and then the sixth of a turn.
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让我们来看看。让我们先旋转三分之一圈,然后旋转六分之一圈。
10:24
Of course, it doesn't make any difference.
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当然,没有什么差别。
10:26
It still ends up at half a turn.
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结果仍然是半圈。
10:28
And there is some symmetry here in the way the symmetries interact with each other.
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某种对称方式是通过相互作用得到的。
10:33
But this is completely different to the symmetries of the triangle.
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但这与三角形的对称性是完全不同的。
10:36
Let's see what happens if we do two symmetries
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让我们看看如果对三角形
10:38
with the triangle, one after the other.
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一个接一个的进行两个对称旋转会怎样。
10:40
Let's do a rotation by a third of a turn anticlockwise,
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让我们逆时针旋转三分之一圈,
10:43
and reflect in the line through X.
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然后在X轴上翻转。
10:45
Well, the combined effect is as if I had just done the reflection in the line through Z
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嗯,合并后的效果就像我刚刚以Z轴翻转
10:49
to start with.
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开始一样。
10:51
Now, let's do it in a different order.
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现在,让我们按不同的顺序来一次。
10:53
Let's do the reflection in X first,
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我们先在X轴上翻转,
10:55
followed by the rotation by a third of a turn anticlockwise.
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然后逆时针旋转三分之一圈。
10:59
The combined effect, the triangle ends up somewhere completely different.
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合并后的效果是三角形停的地方完全不同。
11:02
It's as if it was reflected in the line through Y.
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就像是在Y轴上翻转了一样。
11:05
Now it matters what order you do the operations in.
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现在看来这与你操作它的顺序有关。
11:08
And this enables us to distinguish
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这使我们能够区分
11:10
why the symmetries of these objects --
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为什么这些物体的对称性
11:12
they both have six symmetries. So why shouldn't we say
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都有六个。那么,为什么我们不能说
11:14
they have the same symmetries?
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它们有相同的对称性呢?
11:16
But the way the symmetries interact
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但对称相互作用的方式
11:18
enable us -- we've now got a language
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使我们——我们现在已经有一种语言
11:20
to distinguish why these symmetries are fundamentally different.
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来区分为什么这些对称在根本上是不同的。
11:23
And you can try this when you go down to the pub, later on.
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你也可以尝试一下,当你去酒吧时,以后去的时候。
11:26
Take a beer mat and rotate it by a quarter of a turn,
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拿一个啤酒垫,把它旋转四分之一圈,
11:29
then flip it. And then do it in the other order,
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然后翻转它。然后再按其它顺序做,
11:31
and the picture will be facing in the opposite direction.
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酒垫上的图将是朝反方向面对你的。
11:35
Now, Galois produced some laws for how these tables -- how symmetries interact.
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伽罗瓦为这些表格以及对称性如何相互作用研究出了一些定律。
11:39
It's almost like little Sudoku tables.
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这像一个小数独表。
11:41
You don't see any symmetry twice
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你看不到任何重复的对称
11:43
in any row or column.
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出现在任何一栏或一行中。
11:45
And, using those rules, he was able to say
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通过运用那些定律,他可以说
11:49
that there are in fact only two objects
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事实上只有两个物体
11:51
with six symmetries.
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有六个对称。
11:53
And they'll be the same as the symmetries of the triangle,
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而且这六个对称将和三角形的对称,
11:56
or the symmetries of the six-pointed starfish.
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或六角海星的对称是一样的。
11:58
I think this is an amazing development.
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我觉得这是一个惊人的发展。
12:00
It's almost like the concept of number being developed for symmetry.
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它几乎是为了对称而研制的数的概念。
12:04
In the front here, I've got one, two, three people
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在这前面,我请一、二、三个人
12:06
sitting on one, two, three chairs.
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坐在一、二、三把椅子上。
12:08
The people and the chairs are very different,
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坐在椅子上的人都不一样,
12:11
but the number, the abstract idea of the number, is the same.
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但是数字,数字的抽象的观念,都是一样的。
12:14
And we can see this now: we go back to the walls in the Alhambra.
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现在我们可以看到这个:我们回到阿尔罕布拉的墙壁。
12:17
Here are two very different walls,
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这有两面很不一样的墙壁,
12:19
very different geometric pictures.
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很不相同的几何图片。
12:21
But, using the language of Galois,
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但是,利用伽罗瓦的语言,
12:23
we can understand that the underlying abstract symmetries of these things
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我们可以知道这些东西含有的抽象的对称
12:26
are actually the same.
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实际上是相同的。
12:28
For example, let's take this beautiful wall
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例如,让我们把这面漂亮的墙
12:30
with the triangles with a little twist on them.
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和三角形稍微扭曲一下。
12:33
You can rotate them by a sixth of a turn
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你可以把它们旋转六分之一圈,
12:35
if you ignore the colors. We're not matching up the colors.
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如果忽略他们的颜色。我们不是在匹配颜色。
12:37
But the shapes match up if I rotate by a sixth of a turn
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但在匹配形状,如果我把他们旋转六分之一圈,
12:40
around the point where all the triangles meet.
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围绕着所有三角形交汇的一点旋转。
12:43
What about the center of a triangle? I can rotate
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三角形的中心会怎么样?我可以
12:45
by a third of a turn around the center of the triangle,
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围绕着三角形的中心把他们旋转三分之一圈,
12:47
and everything matches up.
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那么一切就都匹配上了。
12:49
And then there is an interesting place halfway along an edge,
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这儿有个有趣的地方,沿着边的一半
12:51
where I can rotate by 180 degrees.
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我可以把它旋转180度。
12:53
And all the tiles match up again.
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那么所有的瓦片又重新匹配了。
12:56
So rotate along halfway along the edge, and they all match up.
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所以,沿着边的一半旋转,那么他们都能匹配上。
12:59
Now, let's move to the very different-looking wall in the Alhambra.
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现在,让我们移动阿尔罕布拉的一面外观非常不一样的墙。
13:03
And we find the same symmetries here, and the same interaction.
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我们在这儿发现同样的对称性和同样的相互作用。
13:06
So, there was a sixth of a turn. A third of a turn where the Z pieces meet.
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那么是转了六分之一转。转了三分之一圈时第Z片交汇。
13:11
And the half a turn is halfway between the six pointed stars.
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旋转半圈时离六角星交汇还有一半。
13:15
And although these walls look very different,
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尽管这些墙壁看起来非常不同,
13:17
Galois has produced a language to say
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伽罗瓦研制出一种语言说,
13:20
that in fact the symmetries underlying these are exactly the same.
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其实这些东西所具有的对称性是完全相同的。
13:23
And it's a symmetry we call 6-3-2.
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这个对称性我们称之为6-3-2。
13:26
Here is another example in the Alhambra.
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另一个阿尔罕布拉的例子。
13:28
This is a wall, a ceiling, and a floor.
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这是一面墙、天花板和地板。
13:31
They all look very different. But this language allows us to say
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它们都看起来都非常不一样。但是,这种语言让我们可以说
13:34
that they are representations of the same symmetrical abstract object,
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它们是相同的对称的抽象物体,
13:38
which we call 4-4-2. Nothing to do with football,
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我们称之为4-4-2。这与足球毫无关系,
13:40
but because of the fact that there are two places where you can rotate
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而是因为他们都有两个你可以旋转
13:43
by a quarter of a turn, and one by half a turn.
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四分之一圈和二分之一圈的地方。
13:47
Now, this power of the language is even more,
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现在,这种语言的力量更加强大,
13:49
because Galois can say,
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因为伽罗瓦可能会说,
13:51
"Did the Moorish artists discover all of the possible symmetries
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“摩尔艺术家发现了阿尔罕布拉墙上所有可能对称的
13:54
on the walls in the Alhambra?"
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地方了吗?”
13:56
And it turns out they almost did.
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结果是他们几乎都发现了。
13:58
You can prove, using Galois' language,
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你可以用伽罗瓦的语言来证明,
14:00
there are actually only 17
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实际上只有17种
14:02
different symmetries that you can do in the walls in the Alhambra.
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可以在阿尔罕布拉的墙上得到的不同的对称。
14:06
And they, if you try to produce a different wall with this 18th one,
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而且他们,如果你尝试研制出第18面不同的墙壁,
14:09
it will have to have the same symmetries as one of these 17.
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这面墙肯定与17种对称中的一种对称是相同的。
14:14
But these are things that we can see.
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但这些都是我们可以看到的。
14:16
And the power of Galois' mathematical language
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而伽罗瓦的数学语言的力量
14:18
is it also allows us to create
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也让我们能
14:20
symmetrical objects in the unseen world,
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在看不见的世界里创造对称的物体,
14:23
beyond the two-dimensional, three-dimensional,
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超越二维、三维,
14:25
all the way through to the four- or five- or infinite-dimensional space.
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全都向四维或五维或无穷维空间发展。
14:28
And that's where I work. I create
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而这正是我在研究的东西。我创建
14:30
mathematical objects, symmetrical objects,
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数学对象和对称物体,
14:32
using Galois' language,
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通过运用伽罗瓦的语言
14:34
in very high dimensional spaces.
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在非常高维的空间里创建。
14:36
So I think it's a great example of things unseen,
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因此,我认为这是个对于看不见的东西的很好的例子,,
14:38
which the power of mathematical language allows you to create.
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数学语言的力量让你可以创建出来。
14:42
So, like Galois, I stayed up all last night
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因此,像伽罗瓦一样,我昨晚彻夜未眠
14:44
creating a new mathematical symmetrical object for you,
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为你们建立了一个新的数学对称物。
14:48
and I've got a picture of it here.
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我这儿有一张它的照片。
14:50
Well, unfortunately it isn't really a picture. If I could have my board
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但是,可惜的是它不是一张真正的照片。我可以把我的图板
14:53
at the side here, great, excellent.
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放在这边吗?很好,非常好。
14:55
Here we are. Unfortunately, I can't show you
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这儿。可惜的是我无法向你们展示
14:57
a picture of this symmetrical object.
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这个对称物的照片。
14:59
But here is the language which describes
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但这儿有语言能描绘
15:02
how the symmetries interact.
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其对称性怎么相互作用的。
15:04
Now, this new symmetrical object
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现在这个新的对称物
15:06
does not have a name yet.
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还没有名字。
15:08
Now, people like getting their names on things,
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就像人们给东西命名一样,
15:10
on craters on the moon
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给月球上的陨石坑命名,
15:12
or new species of animals.
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或给动物新品种命名一样。
15:14
So I'm going to give you the chance to get your name on a new symmetrical object
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所以我想给你们机会来给新的对称物命名,
15:18
which hasn't been named before.
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以前没有给它取过名字。
15:20
And this thing -- species die away,
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而且这个东西——物种会逐渐消失,
15:22
and moons kind of get hit by meteors and explode --
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月球可能会被陨石撞击并发生爆炸——
15:25
but this mathematical object will live forever.
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但是这个数学物体将长存于世。
15:27
It will make you immortal.
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它将使你不朽。
15:29
In order to win this symmetrical object,
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为了赢得这个对称物,
15:32
what you have to do is to answer the question I asked you at the beginning.
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你所要做的就是回答我在一开始问的问题。
15:35
How many symmetries does a Rubik's Cube have?
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一个魔方有多少种对称呢?
15:39
Okay, I'm going to sort you out.
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好吧,我来给你们整理一下。
15:41
Rather than you all shouting out, I want you to count how many digits there are
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而不是大家都喊出来,我想让你们数数有多少位数字
15:44
in that number. Okay?
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在那个答案里。好吗?
15:46
If you've got it as a factorial, you've got to expand the factorials.
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如果你得出的结果是一个阶乘,那么你要扩大它的阶乘。
15:49
Okay, now if you want to play,
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好了,现在如果你想参与,
15:51
I want you to stand up, okay?
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我希望你能站起来,好吗?
15:53
If you think you've got an estimate for how many digits,
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如果你认为你已经估计出了它有多少位数字,
15:55
right -- we've already got one competitor here.
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好的——我们在这儿已经有了一位竞争者——
15:58
If you all stay down he wins it automatically.
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如果你们都继续坐着,那么他就自动赢了。
16:00
Okay. Excellent. So we've got four here, five, six.
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好的。很好。我们已经有四位、五位、六位。
16:03
Great. Excellent. That should get us going. All right.
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很好。太好了。让我们继续。好了。
16:08
Anybody with five or less digits, you've got to sit down,
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你们中有人的答案是等于或少于五位数的,那你得坐下了。
16:11
because you've underestimated.
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因为你们估计少了。
16:13
Five or less digits. So, if you're in the tens of thousands you've got to sit down.
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五位数或更少的。那么,如果你的答案是几万的话,你得坐下。
16:17
60 digits or more, you've got to sit down.
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六十或六十多位数的,你必须坐下。
16:20
You've overestimated.
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你估计得多了。
16:22
20 digits or less, sit down.
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二十位数或二十位以下的,坐下。
16:26
How many digits are there in your number?
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你的答案是几位数?
16:31
Two? So you should have sat down earlier.
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两个?那你早就该坐下了。
16:33
(Laughter)
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(众笑)
16:34
Let's have the other ones, who sat down during the 20, up again. Okay?
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让我们再来问问其他人,谁估计的是二十位的,请再次站起来,好吗?
16:38
If I told you 20 or less, stand up.
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如果我告诉你是二十位或二十位以下,请站起来。
16:40
Because this one. I think there were a few here.
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因为这一个。我想应该有一些人。
16:42
The people who just last sat down.
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谁是最后一个做下去的。
16:45
Okay, how many digits do you have in your number?
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好的,你的答案是多少位数?
16:50
(Laughs)
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(笑)
16:53
21. Okay good. How many do you have in yours?
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21。好的,很好。你的是多少位?
16:55
18. So it goes to this lady here.
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18。那么是这位女士赢了。
16:58
21 is the closest.
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21是最接近的。
17:00
It actually has -- the number of symmetries in the Rubik's cube
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实际上,魔方对称种数的答案
17:02
has 25 digits.
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有25位数字。
17:04
So now I need to name this object.
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那么现在我需要给这个物体命名。
17:06
So, what is your name?
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嗯,你叫什么名字?
17:08
I need your surname. Symmetrical objects generally --
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我需要你的姓氏。对称的物体一般——
17:11
spell it for me.
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为我拼写一下。
17:13
G-H-E-Z
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G-H-E-Z
17:20
No, SO2 has already been used, actually,
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不,SO2已经用过了,其实,
17:22
in the mathematical language. So you can't have that one.
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在数学语言里。你不能用那个名字。
17:24
So Ghez, there we go. That's your new symmetrical object.
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Ghez,就是这个名字啦。这是你的新的对称物。
17:26
You are now immortal.
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你现在是不朽的了。
17:28
(Applause)
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(鼓掌)
17:34
And if you'd like your own symmetrical object,
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而且,如果你想用你自己的对称物,
17:36
I have a project raising money for a charity in Guatemala,
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我有一个项目,是为在瓜地马拉的慈善筹钱的,
17:39
where I will stay up all night and devise an object for you,
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我可以熬夜为你发明一个物体,
17:42
for a donation to this charity to help kids get into education in Guatemala.
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让你可以为慈善捐款来帮助瓜地马拉的孩子们,让他们能接受教育。
17:46
And I think what drives me, as a mathematician,
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我认为,作为一个数学家,
17:49
are those things which are not seen, the things that we haven't discovered.
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给我动力的是那些看不到的东西,是我们还未发现的东西。
17:53
It's all the unanswered questions which make mathematics a living subject.
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它们都是悬而未决的问题,这使数学成为一个鲜活的主题。
17:57
And I will always come back to this quote from the Japanese "Essays in Idleness":
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我常常想起引自日本《闲置的散文》中的这句话:
18:00
"In everything, uniformity is undesirable.
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“在一切事物中,一致性是不可取的。
18:03
Leaving something incomplete makes it interesting,
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留下一些不完整的东西会更有趣,
18:06
and gives one the feeling that there is room for growth." Thank you.
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一致性并且给人一种没有发展空间的感觉。”谢谢。
18:09
(Applause)
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(鼓掌)
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