TEDGlobal 2009
Marcus du Sautoy: Symmetry, reality's riddle
马库斯•杜•沙托：对称性——现实之谜
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世界变得对称，从旋转的亚原子粒子到阿拉贝司克舞姿令人眼花缭乱的美。但这些已经不能够满足人们视觉享受。在这里，牛津的数学家马库斯•杜•沙托会让大家简单了解一下看不见的数字与对称物体的结合。
Marcus du Sautoy - Mathematician
Oxford's newest science ambassador Marcus du Sautoy is also author of The Times' Sexy Maths column. He'll take you footballing with prime numbers, whopping symmetry groups, higher dimensions and other brow-furrowers. Full bio
Oxford's newest science ambassador Marcus du Sautoy is also author of The Times' Sexy Maths column. He'll take you footballing with prime numbers, whopping symmetry groups, higher dimensions and other brow-furrowers. Full bio
Double-click the English transcript below to play the video.
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）
ABOUT THE SPEAKER
Marcus du Sautoy - MathematicianOxford's newest science ambassador Marcus du Sautoy is also author of The Times' Sexy Maths column. He'll take you footballing with prime numbers, whopping symmetry groups, higher dimensions and other brow-furrowers.
Why you should listen
Marcus du Sautoy only permits prime numbers on the uniforms of his football team, but that idiosyncrasy isn't (entirely) driven by superstition -- just pure love. (His number is 17.) You might say primes, "the atoms of mathematics," as he calls them, are du Sautoy's intellectual spouse, the passion that has driven him from humble-enough academic beginnings to a spectacular and awarded career in maths, including a Royal Society fellowship and, of course, his recent election to the Simonyi Professorship for the Public Understanding of Science, the post previously held by Richard Dawkins.
A gifted science communicator -- interesting fashion sense aside -- du Sautoy has most recently been host of the BBC miniseries "The Story of Maths," which explores fascinating mathematical theories and techniques from throughout history and across cultures. Before that, he hosted The Num8er My5teries, a lecture series on history's stubbornest math problems -- the sorts of conundrums that get your head griddle-hot with thinking. He's also author, perhaps most famously, of The Music of the Primes, an engaging look at the often Pyrrhic attempts at cracking the Riemann Hypothesis. His 2008 book, Symmetry: A Journey into the Patterns of Nature, looks at various kinds of mathematical and aesthetic symmetry, including a massive, mysterious object called "the Monster" that exists in 196,883 dimensions.
Marcus du Sautoy | Speaker | TED.com