TEDGlobal 2009
Marcus du Sautoy: Symmetry, reality's riddle
Marcus du Sautoy：對稱性——真實謎語
Filmed:
Readability: 3.6
1,158,477 views
世界變得對稱，從旋轉的亞原子粒子到阿拉貝司克舞姿令人眼花繚亂的美。但這些已經不能夠滿足人們的視覺享受。牛津的數學家 Marcus du Sautoy 讓大家簡單了解一下看不見的數字與對稱物體的結合。
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,
0
0
4000
在1832年5月30日，
00:22
a gunshot was heard
1
4000
2000
人們聽到一聲槍響，
00:24
ringing out across the 13th arrondissement in Paris.
2
6000
3000
槍聲穿透了巴黎的第十三區
00:27
(Gunshot)
3
9000
1000
（槍聲）
00:28
A peasant, who was walking to market that morning,
4
10000
3000
一個那天早晨正前往市集的農民
00:31
ran towards where the gunshot had come from,
5
13000
2000
朝槍聲傳來的地方跑了過去，
00:33
and found a young man writhing in agony on the floor,
6
15000
4000
發現一名年輕男子正痛得在地上打滾，
00:37
clearly shot by a dueling wound.
7
19000
3000
顯然他在決鬥中遭到了槍擊。
00:40
The young man's name was Evariste Galois.
8
22000
3000
這個年輕人名叫 Evariste • Galois
00:43
He was a well-known revolutionary in Paris at the time.
9
25000
4000
是巴黎當時一個有有名的革命者
00:47
Galois was taken to the local hospital
10
29000
3000
Galois 被送到了當地的醫院，
00:50
where he died the next day in the arms of his brother.
11
32000
3000
第二天死在他兄弟的懷中
00:53
And the last words he said to his brother were,
12
35000
2000
他對他兄弟說的臨別遺言是
00:55
"Don't cry for me, Alfred.
13
37000
2000
“不要為我哭泣, Alfred
00:57
I need all the courage I can muster
14
39000
2000
我需要聚集我能聚集的所有勇氣
00:59
to die at the age of 20."
15
41000
4000
讓我在20歲時死去。 ”
01:03
It wasn't, in fact, revolutionary politics
16
45000
2000
實際上，革命政治並不是
01:05
for which Galois was famous.
17
47000
2000
使 Galois 著名的原因。
01:07
But a few years earlier, while still at school,
18
49000
3000
而是幾年前，當他還在上學時，
01:10
he'd actually cracked one of the big mathematical
19
52000
2000
他破解了
01:12
problems at the time.
20
54000
2000
當時重大數學問題之一
01:14
And he wrote to the academicians in Paris,
21
56000
2000
隨後他寫信給巴黎的院士
01:16
trying to explain his theory.
22
58000
2000
嘗試解釋他的理論
01:18
But the academicians couldn't understand anything that he wrote.
23
60000
3000
但院士們弄不懂他寫的任何東西。
01:21
(Laughter)
24
63000
1000
（大笑）
01:22
This is how he wrote most of his mathematics.
25
64000
3000
這就是他怎麼寫大部分數學理論的
01:25
So, the night before that duel, he realized
26
67000
2000
因此，在決鬥的前一天晚上，他意識到
01:27
this possibly is his last chance
27
69000
3000
這可能是他最後一次機會
01:30
to try and explain his great breakthrough.
28
72000
2000
來嘗試解釋他的重大突破
01:32
So he stayed up the whole night, writing away,
29
74000
3000
所以他徹夜未眠，不停地寫東西，
01:35
trying to explain his ideas.
30
77000
2000
試圖解釋他的想法
01:37
And as the dawn came up and he went to meet his destiny,
31
79000
3000
隨著黎明的到來，他準備迎接自己的命運。
01:40
he left this pile of papers on the table for the next generation.
32
82000
4000
他把桌子上的一堆文件留給了下一代。
01:44
Maybe the fact that he stayed up all night doing mathematics
33
86000
3000
也許徹夜研究數學
01:47
was the fact that he was such a bad shot that morning and got killed.
34
89000
3000
是他那天早晨受到槍擊且被殺的真正原因
01:50
But contained inside those documents
35
92000
2000
但包含在那些文件中的
01:52
was a new language, a language to understand
36
94000
3000
是一種新的語言，這種語言能讓人們理解
01:55
one of the most fundamental concepts
37
97000
2000
科學的一個最基本的概念，
01:57
of science -- namely symmetry.
38
99000
3000
即對稱性。
02:00
Now, symmetry is almost nature's language.
39
102000
2000
現今，對稱性幾乎是大自然的語言。
02:02
It helps us to understand so many
40
104000
2000
它有助於我們了解許多
02:04
different bits of the scientific world.
41
106000
2000
科學世界裡不同的小東西。
02:06
For example, molecular structure.
42
108000
2000
例如，分子結構。
02:08
What crystals are possible,
43
110000
2000
什麼晶體是能讓
02:10
we can understand through the mathematics of symmetry.
44
112000
4000
我們可以通過數學的對稱性來了解的？
02:14
In microbiology you really don't want to get a symmetrical object,
45
116000
2000
在微生物學中，你真的不想研究對稱的東西。
02:16
because they are generally rather nasty.
46
118000
2000
因為它們一般都比較令人討厭。
02:18
The swine flu virus, at the moment, is a symmetrical object.
47
120000
3000
目前的豬流感病毒就是一種結構對稱的病毒。
02:21
And it uses the efficiency of symmetry
48
123000
2000
而且它利用對稱的功效
02:23
to be able to propagate itself so well.
49
125000
4000
來增強自己繁殖的速度
02:27
But on a larger scale of biology, actually symmetry is very important,
50
129000
3000
但從大方向來說，對稱性事實上對生物學非常重要
02:30
because it actually communicates genetic information.
51
132000
2000
因為它能傳遞遺傳信息
02:32
I've taken two pictures here and I've made them artificially symmetrical.
52
134000
4000
我帶了兩張照片到這兒來，並人工的把他們做成了對稱的
02:36
And if I ask you which of these you find more beautiful,
53
138000
3000
如果我問你們覺得哪些更漂亮，
02:39
you're probably drawn to the lower two.
54
141000
2000
你們可能會被下面的兩張吸引住。
02:41
Because it is hard to make symmetry.
55
143000
3000
因為很難做到對稱，
02:44
And if you can make yourself symmetrical, you're sending out a sign
56
146000
2000
所以如果你可以使自己對稱，那麼你在傳遞一種信號
02:46
that you've got good genes, you've got a good upbringing
57
148000
3000
它意味著你得到了好的遺傳基因，你有好的教養，
02:49
and therefore you'll make a good mate.
58
151000
2000
因而你會有一個好的伴侶。
02:51
So symmetry is a language which can help to communicate
59
153000
3000
所以，對稱性是一種語言，它能有助於傳遞
02:54
genetic information.
60
156000
2000
遺傳信息。
02:56
Symmetry can also help us to explain
61
158000
2000
對稱性還可以幫助我們解釋
02:58
what's happening in the Large Hadron Collider in CERN.
62
160000
3000
歐洲粒子物理研究所大型強子對撞機正發生著什麼事情。
03:01
Or what's not happening in the Large Hadron Collider in CERN.
63
163000
3000
或者歐洲粒子物理研究所的大型強子對撞機沒有發生什麼事情。
03:04
To be able to make predictions about the fundamental particles
64
166000
2000
為了能夠對基本粒子作出預測，
03:06
we might see there,
65
168000
2000
我們可能會在那兒看到的（基本粒子），
03:08
it seems that they are all facets of some strange symmetrical shape
66
170000
4000
似乎所有的小平面都有某種奇怪的對稱形狀
03:12
in a higher dimensional space.
67
174000
2000
當它們在更高維的空間中時。
03:14
And I think Galileo summed up, very nicely,
68
176000
2000
我認為伽利略很好地概括了
03:16
the power of mathematics
69
178000
2000
數學的力量：
03:18
to understand the scientific world around us.
70
180000
2000
它讓我們得以了解周圍的科學世界
03:20
He wrote, "The universe cannot be read
71
182000
2000
他寫道：“我們無法閱讀宇宙，
03:22
until we have learnt the language
72
184000
2000
除非學會它的語言，
03:24
and become familiar with the characters in which it is written.
73
186000
3000
且熟悉其寫作特點。
03:27
It is written in mathematical language,
74
189000
2000
它是用數學語言寫的。
03:29
and the letters are triangles, circles and other geometric figures,
75
191000
4000
字母是三角形、圓和其他的幾何數字，
03:33
without which means it is humanly impossible
76
195000
2000
沒有這些字母就意味著在人力所能及的範圍內是不可能
03:35
to comprehend a single word."
77
197000
3000
理解任何一個字。 ”
03:38
But it's not just scientists who are interested in symmetry.
78
200000
3000
不只是科學家們對對稱性感興趣。
03:41
Artists too love to play around with symmetry.
79
203000
3000
藝術家也喜歡擺弄對稱性。
03:44
They also have a slightly more ambiguous relationship with it.
80
206000
3000
他們與對稱性有一些更模糊的關係。
03:47
Here is Thomas Mann talking about symmetry in "The Magic Mountain."
81
209000
3000
這是托馬斯•曼在《魔山》中談到的對稱性。
03:50
He has a character describing the snowflake,
82
212000
3000
他這樣描寫雪花
03:53
and he says he "shuddered at its perfect precision,
83
215000
3000
他說他，“因其有完美的精確度而震撼，
03:56
found it deathly, the very marrow of death."
84
218000
3000
發現它死亡的精髓讓他想到死亡。 ”
03:59
But what artists like to do is to set up expectations
85
221000
2000
但藝術家們想要做的是樹立對對稱性的期望，
04:01
of symmetry and then break them.
86
223000
2000
然後打破它們。
04:03
And a beautiful example of this
87
225000
2000
就這一點我找到了一個很好的例子，
04:05
I found, actually, when I visited a colleague of mine
88
227000
2000
其實是當我拜訪我的同事
04:07
in Japan, Professor Kurokawa.
89
229000
2000
在日本的黑川紀章教授時發現的
04:09
And he took me up to the temples in Nikko.
90
231000
3000
他帶我到日光市的寺廟去
04:12
And just after this photo was taken we walked up the stairs.
91
234000
3000
就在拍好這張照片後，我們走上樓梯，
04:15
And the gateway you see behind
92
237000
2000
你們看到的這後面的大門
04:17
has eight columns, with beautiful symmetrical designs on them.
93
239000
3000
有八根柱子，都有著漂亮的對稱性設計。
04:20
Seven of them are exactly the same,
94
242000
2000
其中七個是完全一樣的，
04:22
and the eighth one is turned upside down.
95
244000
3000
而第八個是顛倒過來的。
04:25
And I said to Professor Kurokawa,
96
247000
2000
我就對黑川紀章教授說：
04:27
"Wow, the architects must have really been kicking themselves
97
249000
2000
“哇，建築師們肯定自責的很
04:29
when they realized that they'd made a mistake and put this one upside down."
98
251000
3000
要是他麼發現犯了這麼一個錯誤，這根柱子竟然是相反的。 ”
04:32
And he said, "No, no, no. It was a very deliberate act."
99
254000
3000
他說，“不，不，不。這是故意設計成這樣的。”
04:35
And he referred me to this lovely quote from the Japanese
100
257000
2000
他還向我提到了這個可愛的出處，來自日本
04:37
"Essays in Idleness" from the 14th century,
101
259000
3000
1 4世紀的《徒然草》
04:40
in which the essayist wrote, "In everything,
102
262000
2000
其中，散文家寫道：“在一切事物中，
04:42
uniformity is undesirable.
103
264000
3000
一致性是不可取的。
04:45
Leaving something incomplete makes it interesting,
104
267000
2000
留下一些不完整的東西會更有趣，
04:47
and gives one the feeling that there is room for growth."
105
269000
3000
而且一致性給人一種沒有發展空間的感覺。 ”
04:50
Even when building the Imperial Palace,
106
272000
2000
即使是建造皇宮時，
04:52
they always leave one place unfinished.
107
274000
4000
他們也總是留下一個未完工的地方。
04:56
But if I had to choose one building in the world
108
278000
3000
但如果我必須選擇這世界上的一個建築，
04:59
to be cast out on a desert island, to live the rest of my life,
109
281000
3000
將其扔到一個荒島上，且我要在那裡度過餘生，
05:02
being an addict of symmetry, I would probably choose the Alhambra in Granada.
110
284000
4000
作為一個對對稱性痴迷的人，我可能會選擇在格拉納達的阿爾罕布拉。
05:06
This is a palace celebrating symmetry.
111
288000
2000
這是一座歌頌對稱性的宮殿。
05:08
Recently I took my family --
112
290000
2000
最近，我帶我的家人——
05:10
we do these rather kind of nerdy mathematical trips, which my family love.
113
292000
3000
我們進行這種並沒有學術氣息的數學旅行，我的家人都很喜歡。
05:13
This is my son Tamer. You can see
114
295000
2000
這是我的兒子塔梅爾。你們可以看到
05:15
he's really enjoying our mathematical trip to the Alhambra.
115
297000
3000
他真的很喜歡我們在阿爾罕布拉的數學之旅。
05:18
But I wanted to try and enrich him.
116
300000
3000
但我想嘗試使他變得充實。
05:21
I think one of the problems about school mathematics
117
303000
2000
我認為學校教的數學存在的一個問題就是
05:23
is it doesn't look at how mathematics is embedded
118
305000
2000
它沒有關注數學是如何被運用於
05:25
in the world we live in.
119
307000
2000
我們所處的這個世界。
05:27
So, I wanted to open his eyes up to
120
309000
2000
所以，我想開拓他的眼界，讓他知道
05:29
how much symmetry is running through the Alhambra.
121
311000
3000
阿爾罕布拉運用著多少對稱性。
05:32
You see it already. Immediately you go in,
122
314000
2000
你們已經看到了。你一走進去，
05:34
the reflective symmetry in the water.
123
316000
2000
水中有反映出對稱。
05:36
But it's on the walls where all the exciting things are happening.
124
318000
3000
但是，所有令人興奮的事情發生在牆壁上。
05:39
The Moorish artists were denied the possibility
125
321000
2000
人們否認摩爾藝術家能夠
05:41
to draw things with souls.
126
323000
2000
用靈魂來繪畫。
05:43
So they explored a more geometric art.
127
325000
2000
因此，他們探索出一種更加幾何化的藝術。
05:45
And so what is symmetry?
128
327000
2000
那麼什麼是對稱性？
05:47
The Alhambra somehow asks all of these questions.
129
329000
3000
阿爾罕布拉以某種方式提出了所有這些問題。
05:50
What is symmetry? When [there] are two of these walls,
130
332000
2000
什麼是對稱性？當[那兒]有兩面牆時，
05:52
do they have the same symmetries?
131
334000
2000
他們有相同的對稱性嗎？
05:54
Can we say whether they discovered
132
336000
2000
我們可以說他們是否發現了
05:56
all of the symmetries in the Alhambra?
133
338000
3000
阿爾罕布拉所有的對稱性嗎？
05:59
And it was Galois who produced a language
134
341000
2000
是 Galois 研製出了一種語言
06:01
to be able to answer some of these questions.
135
343000
3000
來回答這樣的問題
06:04
For Galois, symmetry -- unlike for Thomas Mann,
136
346000
3000
對 Galois 來說，對稱性不是托馬斯曼所說的
06:07
which was something still and deathly --
137
349000
2000
靜態的和死一般的東西
06:09
for Galois, symmetry was all about motion.
138
351000
3000
對 Galois 來說，所有的對稱性都是關於運動的
06:12
What can you do to a symmetrical object,
139
354000
2000
你能對一個對稱性的物體做些什麼？
06:14
move it in some way, so it looks the same
140
356000
2000
用某種方法移動它，讓它看起來
06:16
as before you moved it?
141
358000
2000
跟你移動它之前一樣？
06:18
I like to describe it as the magic trick moves.
142
360000
2000
我喜歡把這形容為神奇的假動作。
06:20
What can you do to something? You close your eyes.
143
362000
2000
你對一些東西可以做些什麼？閉上你的眼睛。
06:22
I do something, put it back down again.
144
364000
2000
我移動它，再把它放回到原處。
06:24
It looks like it did before it started.
145
366000
2000
它看起來和動之前一樣。
06:26
So, for example, the walls in the Alhambra --
146
368000
2000
那麼，例如，阿爾罕布拉的牆壁。
06:28
I can take all of these tiles, and fix them at the yellow place,
147
370000
4000
我可以把所有的這些瓦片拿起來，把他們放在這個黃色的地方，
06:32
rotate them by 90 degrees,
148
374000
2000
並把它們旋轉九十度，
06:34
put them all back down again and they fit perfectly down there.
149
376000
3000
再把他們都放回去，它們非常吻合。
06:37
And if you open your eyes again, you wouldn't know that they'd moved.
150
379000
3000
如果你再睜開你的眼睛，你不會知道它們被移動過。
06:40
But it's the motion that really characterizes the symmetry
151
382000
3000
但正是運動才使對稱性
06:43
inside the Alhambra.
152
385000
2000
在阿爾罕布拉具有特色。
06:45
But it's also about producing a language to describe this.
153
387000
2000
但也要創造一種語言來描繪它。
06:47
And the power of mathematics is often
154
389000
3000
數學的力量往往
06:50
to change one thing into another, to change geometry into language.
155
392000
4000
把一樣東西變成另一樣，把幾何變成語言。
06:54
So I'm going to take you through, perhaps push you a little bit mathematically --
156
396000
3000
因此，我將帶你經歷，可能強加一些數學的東西給你們，
06:57
so brace yourselves --
157
399000
2000
所以撐住自己，
06:59
push you a little bit to understand how this language works,
158
401000
3000
強加一些數學的知識讓你們了解這種語言是怎麼運作的，
07:02
which enables us to capture what is symmetry.
159
404000
2000
這讓我們能夠捕捉到什麼是對稱性。
07:04
So, let's take these two symmetrical objects here.
160
406000
3000
那讓我們把這兩個對稱物放到這兒。
07:07
Let's take the twisted six-pointed starfish.
161
409000
2000
拿這個扭曲了的六角海星來說。
07:09
What can I do to the starfish which makes it look the same?
162
411000
3000
我怎麼做能讓這個海星看起來和原來一樣呢？
07:12
Well, there I rotated it by a sixth of a turn,
163
414000
3000
嗯，我把它旋轉了六分之一圈，
07:15
and still it looks like it did before I started.
164
417000
2000
它看起來仍然跟我動過之前一樣。
07:17
I could rotate it by a third of a turn,
165
419000
3000
我可以把它旋轉三分之一圈，
07:20
or a half a turn,
166
422000
2000
或者半圈，
07:22
or put it back down on its image, or two thirds of a turn.
167
424000
3000
或將它恢復到原圖，或旋轉三分之二圈。
07:25
And a fifth symmetry, I can rotate it by five sixths of a turn.
168
427000
4000
第五種對稱，我可以把它旋轉六分之五圈。
07:29
And those are things that I can do to the symmetrical object
169
431000
3000
這些就是我能對對稱物所做的，
07:32
that make it look like it did before I started.
170
434000
3000
可以讓它看起來跟我動它們之前一樣。
07:35
Now, for Galois, there was actually a sixth symmetry.
171
437000
3000
對 Galois 來說，實際上還有第六種對稱。
07:38
Can anybody think what else I could do to this
172
440000
2000
大家能想到其它什麼辦法
07:40
which would leave it like I did before I started?
173
442000
3000
可以讓它跟我動它之前一樣？
07:43
I can't flip it because I've put a little twist on it, haven't I?
174
445000
3000
我不能翻轉它，因為我已經把它扭曲了一些，是吧？
07:46
It's got no reflective symmetry.
175
448000
2000
這樣它沒有反射對稱了。
07:48
But what I could do is just leave it where it is,
176
450000
3000
但我可以做的就是把它放在原處，
07:51
pick it up, and put it down again.
177
453000
2000
把它拿起來再把它放下。
07:53
And for Galois this was like the zeroth symmetry.
178
455000
3000
對 Galois 來說，這就像是第零個對稱。
07:56
Actually, the invention of the number zero
179
458000
3000
其實，數字零的發明
07:59
was a very modern concept, seventh century A.D., by the Indians.
180
461000
3000
是一個非常現代化的概念，它是公元七世紀印度人發明的。
08:02
It seems mad to talk about nothing.
181
464000
3000
談論無有感覺很瘋狂。
08:05
And this is the same idea. This is a symmetrical --
182
467000
2000
這是同樣的概念。這是對稱的——
08:07
so everything has symmetry, where you just leave it where it is.
183
469000
2000
所以一切事物都有對稱性，把它放在拿起它的地方。
08:09
So, this object has six symmetries.
184
471000
3000
所以這個物體有六種對稱。
08:12
And what about the triangle?
185
474000
2000
那三角形呢？
08:14
Well, I can rotate by a third of a turn clockwise
186
476000
4000
嗯，我可以把它順時針旋轉三分之一圈
08:18
or a third of a turn anticlockwise.
187
480000
2000
或逆時針旋轉三分之一圈。
08:20
But now this has some reflectional symmetry.
188
482000
2000
但現在有反射對稱
08:22
I can reflect it in the line through X,
189
484000
2000
我可以在X軸上翻轉它，
08:24
or the line through Y,
190
486000
2000
或在Y軸上，
08:26
or the line through Z.
191
488000
2000
或在Z軸上。
08:28
Five symmetries and then of course the zeroth symmetry
192
490000
3000
五種對稱，當然還有第零個對稱，
08:31
where I just pick it up and leave it where it is.
193
493000
3000
我把它拿起來，放回原處。
08:34
So both of these objects have six symmetries.
194
496000
3000
因此，這些物體都有六種對稱。
08:37
Now, I'm a great believer that mathematics is not a spectator sport,
195
499000
3000
現在，我十分相信數學不是旁觀者的運動，
08:40
and you have to do some mathematics
196
502000
2000
你必須做一些數學運算
08:42
in order to really understand it.
197
504000
2000
才能真正理解它。
08:44
So here is a little question for you.
198
506000
2000
這兒有個小問題問問你們。
08:46
And I'm going to give a prize at the end of my talk
199
508000
2000
我將在講座結束後給一個獎品
08:48
for the person who gets closest to the answer.
200
510000
2000
給那個給出的答案最接近的人。
08:50
The Rubik's Cube.
201
512000
2000
魔術方塊
08:52
How many symmetries does a Rubik's Cube have?
202
514000
3000
一個魔術方塊有多少種對稱？
08:55
How many things can I do to this object
203
517000
2000
有多少種方法可以在動了這個物體，
08:57
and put it down so it still looks like a cube?
204
519000
2000
且把它放下後它仍然看起來像一個立方體？
08:59
Okay? So I want you to think about that problem as we go on,
205
521000
3000
好嗎？我希望隨著講座的繼續，你們可以想想這個問題，
09:02
and count how many symmetries there are.
206
524000
2000
數數它有多少種對稱。
09:04
And there will be a prize for the person who gets closest at the end.
207
526000
4000
講座結束後獎品會給答案最接近的人
09:08
But let's go back down to symmetries that I got for these two objects.
208
530000
4000
讓我們回到這兩個物體的對稱性上。
09:12
What Galois realized: it isn't just the individual symmetries,
209
534000
3000
Galois 意識到這不僅僅是個體的對稱性，
09:15
but how they interact with each other
210
537000
2000
而是個體之間如何相互作用
09:17
which really characterizes the symmetry of an object.
211
539000
4000
才真正賦予了一個物體具有對稱性的特點。
09:21
If I do one magic trick move followed by another,
212
543000
3000
如果我做一個神奇的假動作，然後再做一個，
09:24
the combination is a third magic trick move.
213
546000
2000
兩個合併起來就是第三個神奇的假動作。
09:26
And here we see Galois starting to develop
214
548000
2000
這裡我們了解到 Glaois 開始開發
09:28
a language to see the substance
215
550000
3000
一種語言來研究
09:31
of the things unseen, the sort of abstract idea
216
553000
2000
看不見的東西所具有的內在含義，以及
09:33
of the symmetry underlying this physical object.
217
555000
3000
物理物體中存在的對稱性的抽象的概念。
09:36
For example, what if I turn the starfish
218
558000
3000
例如，如果我把海星旋轉
09:39
by a sixth of a turn,
219
561000
2000
六分之一圈，
09:41
and then a third of a turn?
220
563000
2000
然後再轉三分之一圈會，結果會怎樣？
09:43
So I've given names. The capital letters, A, B, C, D, E, F,
221
565000
3000
所以我給它們取了名字。大寫的字母A、B、C、D、E、F，
09:46
are the names for the rotations.
222
568000
2000
這些名字旋轉的代號。
09:48
B, for example, rotates the little yellow dot
223
570000
3000
例如B，旋轉小黃點，
09:51
to the B on the starfish. And so on.
224
573000
3000
它位於海星上的B處，諸如此類。
09:54
So what if I do B, which is a sixth of a turn,
225
576000
2000
那麼，如果我旋轉B，轉六分之一圈，
09:56
followed by C, which is a third of a turn?
226
578000
3000
其次是C，轉三分之一圈？
09:59
Well let's do that. A sixth of a turn,
227
581000
2000
嗯，讓我們開始。六分之一圈，
10:01
followed by a third of a turn,
228
583000
2000
接著是三分之一圈，
10:03
the combined effect is as if I had just rotated it by half a turn in one go.
229
585000
5000
合併後的效果就像我剛剛把它一次旋轉了半圈一樣。
10:08
So the little table here records
230
590000
2000
那這個小表格記載著
10:10
how the algebra of these symmetries work.
231
592000
3000
這些對稱的代數是怎麼運作的。
10:13
I do one followed by another, the answer is
232
595000
2000
我將一個接一個的旋轉，結果就是
10:15
it's rotation D, half a turn.
233
597000
2000
D旋轉了半圈。
10:17
What I if I did it in the other order? Would it make any difference?
234
599000
3000
如果我按其他順序旋轉呢？會有什麼不同嗎？
10:20
Let's see. Let's do the third of the turn first, and then the sixth of a turn.
235
602000
4000
讓我們來看看。讓我們先旋轉三分之一圈，然後旋轉六分之一圈。
10:24
Of course, it doesn't make any difference.
236
606000
2000
當然，沒有什麼差別。
10:26
It still ends up at half a turn.
237
608000
2000
結果仍然是半圈。
10:28
And there is some symmetry here in the way the symmetries interact with each other.
238
610000
5000
某種對稱方式是通過相互作用得到的。
10:33
But this is completely different to the symmetries of the triangle.
239
615000
3000
但這與三角形的對稱性是完全不同的。
10:36
Let's see what happens if we do two symmetries
240
618000
2000
讓我們看看如果對三角形
10:38
with the triangle, one after the other.
241
620000
2000
一個接一個的進行兩個對稱旋轉會怎樣。
10:40
Let's do a rotation by a third of a turn anticlockwise,
242
622000
3000
讓我們逆時針旋轉三分之一圈，
10:43
and reflect in the line through X.
243
625000
2000
然後在X軸上翻轉。
10:45
Well, the combined effect is as if I had just done the reflection in the line through Z
244
627000
4000
嗯，合併後的效果就像我剛剛以Z軸翻轉
10:49
to start with.
245
631000
2000
開始一樣。
10:51
Now, let's do it in a different order.
246
633000
2000
現在，讓我們按不同的順序來一次。
10:53
Let's do the reflection in X first,
247
635000
2000
我們先在X軸上翻轉，
10:55
followed by the rotation by a third of a turn anticlockwise.
248
637000
4000
然後逆時針旋轉三分之一圈。
10:59
The combined effect, the triangle ends up somewhere completely different.
249
641000
3000
合併後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.
250
644000
3000
就像是在Y軸上翻轉了一樣。
11:05
Now it matters what order you do the operations in.
251
647000
3000
現在看來這與你操作它的順序有關。
11:08
And this enables us to distinguish
252
650000
2000
這使我們能夠區分
11:10
why the symmetries of these objects --
253
652000
2000
為什麼這些物體的對稱性
11:12
they both have six symmetries. So why shouldn't we say
254
654000
2000
都有六個。那麼，為什麼我們不能說
11:14
they have the same symmetries?
255
656000
2000
它們有相同的對稱性呢？
11:16
But the way the symmetries interact
256
658000
2000
但對稱相互作用的方式
11:18
enable us -- we've now got a language
257
660000
2000
使我們——我們現在已經有一種語言
11:20
to distinguish why these symmetries are fundamentally different.
258
662000
3000
來區分為什麼這些對稱在根本上是不同的。
11:23
And you can try this when you go down to the pub, later on.
259
665000
3000
你也可以嘗試一下，當你去酒吧時，以後去的時候。
11:26
Take a beer mat and rotate it by a quarter of a turn,
260
668000
3000
拿一個啤酒墊，把它旋轉四分之一圈，
11:29
then flip it. And then do it in the other order,
261
671000
2000
然後翻轉它。然後再按其它順序做，
11:31
and the picture will be facing in the opposite direction.
262
673000
4000
酒墊上的圖將是朝反方向面對你的。
11:35
Now, Galois produced some laws for how these tables -- how symmetries interact.
263
677000
4000
Galois 為這些表格以及對稱性如何相互作用研究出了一些定律。
11:39
It's almost like little Sudoku tables.
264
681000
2000
這像一個小數獨表。
11:41
You don't see any symmetry twice
265
683000
2000
你看不到任何重複的對稱
11:43
in any row or column.
266
685000
2000
出現在任何一欄或一行中。
11:45
And, using those rules, he was able to say
267
687000
4000
通過運用那些定律，他可以說
11:49
that there are in fact only two objects
268
691000
2000
事實上只有兩個物體
11:51
with six symmetries.
269
693000
2000
有六個對稱。
11:53
And they'll be the same as the symmetries of the triangle,
270
695000
3000
而且這六個對稱將和三角形的對稱,
11:56
or the symmetries of the six-pointed starfish.
271
698000
2000
或六角海星的對稱是一樣的。
11:58
I think this is an amazing development.
272
700000
2000
我覺得這是一個驚人的發展。
12:00
It's almost like the concept of number being developed for symmetry.
273
702000
4000
它幾乎是為了對稱而研製的數的概念。
12:04
In the front here, I've got one, two, three people
274
706000
2000
在這前面，我請一、二、三個人
12:06
sitting on one, two, three chairs.
275
708000
2000
坐在一、二、三把椅子上。
12:08
The people and the chairs are very different,
276
710000
3000
坐在椅子上的人都不一樣，
12:11
but the number, the abstract idea of the number, is the same.
277
713000
3000
但是數字，數字的抽象的觀念，都是一樣的。
12:14
And we can see this now: we go back to the walls in the Alhambra.
278
716000
3000
現在我們可以看到這個：我們回到阿爾罕布拉的牆壁。
12:17
Here are two very different walls,
279
719000
2000
這有兩面很不一樣的牆壁，
12:19
very different geometric pictures.
280
721000
2000
很不相同的幾何圖片。
12:21
But, using the language of Galois,
281
723000
2000
但是，利用 Galois 的語言
12:23
we can understand that the underlying abstract symmetries of these things
282
725000
3000
我們可以知道這些東西含有的抽象的對稱
12:26
are actually the same.
283
728000
2000
實際上是相同的。
12:28
For example, let's take this beautiful wall
284
730000
2000
例如，讓我們把這面漂亮的牆
12:30
with the triangles with a little twist on them.
285
732000
3000
和三角形稍微扭曲一下。
12:33
You can rotate them by a sixth of a turn
286
735000
2000
你可以把它們旋轉六分之一圈，
12:35
if you ignore the colors. We're not matching up the colors.
287
737000
2000
如果忽略他們的顏色。我們不是在做顏色配對
12:37
But the shapes match up if I rotate by a sixth of a turn
288
739000
3000
而是在形狀配對，如果我把他們旋轉六分之一圈，
12:40
around the point where all the triangles meet.
289
742000
3000
圍繞著所有三角形交彙的一點旋轉。
12:43
What about the center of a triangle? I can rotate
290
745000
2000
三角形的中心會怎麼樣？我可以
12:45
by a third of a turn around the center of the triangle,
291
747000
2000
圍繞著三角形的中心把他們旋轉三分之一圈，
12:47
and everything matches up.
292
749000
2000
那麼一切就都對上了。
12:49
And then there is an interesting place halfway along an edge,
293
751000
2000
這兒有個有趣的地方，沿著邊的一半
12:51
where I can rotate by 180 degrees.
294
753000
2000
我可以把它旋轉180度。
12:53
And all the tiles match up again.
295
755000
3000
那麼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.
296
758000
3000
所以，沿著邊的一半旋轉，那麼他們都能配合上。
12:59
Now, let's move to the very different-looking wall in the Alhambra.
297
761000
4000
現在，讓我們移動阿爾罕布拉的一面外觀非常不一樣的牆。
13:03
And we find the same symmetries here, and the same interaction.
298
765000
3000
我們在這兒發現同樣的對稱性和同樣的相互作用。
13:06
So, there was a sixth of a turn. A third of a turn where the Z pieces meet.
299
768000
5000
那麼是轉了六分之一轉。轉了三分之一圈時第Z片交會
13:11
And the half a turn is halfway between the six pointed stars.
300
773000
4000
旋轉半圈時離六角星交會還有一半。
13:15
And although these walls look very different,
301
777000
2000
儘管這些牆壁看起來非常不同，
13:17
Galois has produced a language to say
302
779000
3000
Galois 研製出一種語言說，
13:20
that in fact the symmetries underlying these are exactly the same.
303
782000
3000
其實這些東西所具有的對稱性是完全相同的。
13:23
And it's a symmetry we call 6-3-2.
304
785000
3000
這個對稱性我們稱之為6-3-2。
13:26
Here is another example in the Alhambra.
305
788000
2000
另一個阿爾罕布拉的例子。
13:28
This is a wall, a ceiling, and a floor.
306
790000
3000
這是一面牆、天花板和地板。
13:31
They all look very different. But this language allows us to say
307
793000
3000
它們都看起來都非常不一樣。但是，這種語言讓我們可以說
13:34
that they are representations of the same symmetrical abstract object,
308
796000
4000
它們是相同的對稱的抽象物體，
13:38
which we call 4-4-2. Nothing to do with football,
309
800000
2000
我們稱之為4-4-2。這與足球毫無關係，
13:40
but because of the fact that there are two places where you can rotate
310
802000
3000
而是因為他們都有兩個你可以旋轉
13:43
by a quarter of a turn, and one by half a turn.
311
805000
4000
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,
312
809000
2000
現在，這種語言的力量更加強大，
13:49
because Galois can say,
313
811000
2000
因為 Galois 可能會說，
13:51
"Did the Moorish artists discover all of the possible symmetries
314
813000
3000
“摩爾藝術家發現了阿爾罕布拉牆上所有可能對稱的
13:54
on the walls in the Alhambra?"
315
816000
2000
地方了嗎？ ”
13:56
And it turns out they almost did.
316
818000
2000
結果是他們幾乎都發現了。
13:58
You can prove, using Galois' language,
317
820000
2000
你可以用伽羅瓦的語言來證明，
14:00
there are actually only 17
318
822000
2000
實際上只有17種
14:02
different symmetries that you can do in the walls in the Alhambra.
319
824000
4000
可以在阿爾罕布拉的牆上得到的不同的對稱。
14:06
And they, if you try to produce a different wall with this 18th one,
320
828000
3000
而且他們，如果你嘗試研製出第18面不同的牆壁，
14:09
it will have to have the same symmetries as one of these 17.
321
831000
5000
這面牆肯定與17種對稱中的一種對稱是相同的。
14:14
But these are things that we can see.
322
836000
2000
但這些都是我們可以看到的。
14:16
And the power of Galois' mathematical language
323
838000
2000
而 Galois 的數學語言的力量
14:18
is it also allows us to create
324
840000
2000
也讓我們能
14:20
symmetrical objects in the unseen world,
325
842000
3000
在看不見的世界裡創造對稱的物體，
14:23
beyond the two-dimensional, three-dimensional,
326
845000
2000
超越二維、三維，
14:25
all the way through to the four- or five- or infinite-dimensional space.
327
847000
3000
全都向四維或五維或無窮維空間發展。
14:28
And that's where I work. I create
328
850000
2000
而這正是我在研究的東西。我創建
14:30
mathematical objects, symmetrical objects,
329
852000
2000
數學對象和對稱物體，
14:32
using Galois' language,
330
854000
2000
通過運用 Galois 的語言
14:34
in very high dimensional spaces.
331
856000
2000
在非常高維的空間裡創建。
14:36
So I think it's a great example of things unseen,
332
858000
2000
因此，我認為這是個對於看不見的東西的很好的例子，
14:38
which the power of mathematical language allows you to create.
333
860000
4000
數學語言的力量讓你可以創建出來。
14:42
So, like Galois, I stayed up all last night
334
864000
2000
因此，像 Glaois 一樣，我昨晚徹夜未眠
14:44
creating a new mathematical symmetrical object for you,
335
866000
4000
為你們建立了一個新的數學對稱物。
14:48
and I've got a picture of it here.
336
870000
2000
我這兒有一張它的照片。
14:50
Well, unfortunately it isn't really a picture. If I could have my board
337
872000
3000
但是，可惜的是它不是一張真正的照片。我可以把我的圖板
14:53
at the side here, great, excellent.
338
875000
2000
放在這邊嗎？很好，非常好。
14:55
Here we are. Unfortunately, I can't show you
339
877000
2000
這兒。可惜的是我無法向你們展示
14:57
a picture of this symmetrical object.
340
879000
2000
這個對稱物的照片。
14:59
But here is the language which describes
341
881000
3000
但這兒有語言能描繪
15:02
how the symmetries interact.
342
884000
2000
其對稱性怎麼相互作用的。
15:04
Now, this new symmetrical object
343
886000
2000
現在這個新的對稱物
15:06
does not have a name yet.
344
888000
2000
還沒有名字。
15:08
Now, people like getting their names on things,
345
890000
2000
就像人們給東西命名一樣，
15:10
on craters on the moon
346
892000
2000
給月球上的隕石坑命名，
15:12
or new species of animals.
347
894000
2000
或給新動物品種命名一樣。
15:14
So I'm going to give you the chance to get your name on a new symmetrical object
348
896000
4000
所以我想給你們機會來給新的對稱物命名，
15:18
which hasn't been named before.
349
900000
2000
以前沒有給它取過名字。
15:20
And this thing -- species die away,
350
902000
2000
而且這個東西——物種會逐漸消失，
15:22
and moons kind of get hit by meteors and explode --
351
904000
3000
月球可能會被隕石撞擊並發生爆炸——
15:25
but this mathematical object will live forever.
352
907000
2000
但是這個數學物體將長存於世。
15:27
It will make you immortal.
353
909000
2000
它將使你不朽。
15:29
In order to win this symmetrical object,
354
911000
3000
為了贏得這個對稱物，
15:32
what you have to do is to answer the question I asked you at the beginning.
355
914000
3000
你所要做的就是回答我在一開始問的問題。
15:35
How many symmetries does a Rubik's Cube have?
356
917000
4000
魔術方塊有多少種對稱呢？
15:39
Okay, I'm going to sort you out.
357
921000
2000
好吧，我來給你們整理一下。
15:41
Rather than you all shouting out, I want you to count how many digits there are
358
923000
3000
而不是大家都喊出來，我想讓你們數數有多少位數字
15:44
in that number. Okay?
359
926000
2000
在那個答案裡。好嗎？
15:46
If you've got it as a factorial, you've got to expand the factorials.
360
928000
3000
如果你得出的結果是一個階乘，那麼你要擴大它的階乘。
15:49
Okay, now if you want to play,
361
931000
2000
好了，現在如果你想參與，
15:51
I want you to stand up, okay?
362
933000
2000
我希望你能站起來，好嗎？
15:53
If you think you've got an estimate for how many digits,
363
935000
2000
如果你認為你已經估計出了它有多少位數字，
15:55
right -- we've already got one competitor here.
364
937000
3000
好的——我們在這兒已經有了一位參賽者——
15:58
If you all stay down he wins it automatically.
365
940000
2000
如果你們都繼續坐著，那麼他就自動贏了。
16:00
Okay. Excellent. So we've got four here, five, six.
366
942000
3000
好的。很好。我們已經有四位、五位、六位。
16:03
Great. Excellent. That should get us going. All right.
367
945000
5000
很好。太好了。讓我們繼續。好了。
16:08
Anybody with five or less digits, you've got to sit down,
368
950000
3000
你們中有人的答案是等於或少於五位數的，那你得坐下了。
16:11
because you've underestimated.
369
953000
2000
因為你們估計少了。
16:13
Five or less digits. So, if you're in the tens of thousands you've got to sit down.
370
955000
4000
五位數或更少的。那麼，如果你的答案是幾萬的話，你得坐下。
16:17
60 digits or more, you've got to sit down.
371
959000
3000
六十或六十多位數的，你必須坐下。
16:20
You've overestimated.
372
962000
2000
你估計得多了。
16:22
20 digits or less, sit down.
373
964000
4000
二十位數或二十位以下的，坐下。
16:26
How many digits are there in your number?
374
968000
5000
你的答案是幾位數？
16:31
Two? So you should have sat down earlier.
375
973000
2000
兩個？那你早就該坐下了。
16:33
(Laughter)
376
975000
1000
（大笑）
16:34
Let's have the other ones, who sat down during the 20, up again. Okay?
377
976000
4000
讓我們再來問問其他人，誰估計的是二十位的，請再次站起來，好嗎？
16:38
If I told you 20 or less, stand up.
378
980000
2000
如果我告訴你是二十位或二十位以下，請站起來。
16:40
Because this one. I think there were a few here.
379
982000
2000
因為這一個。我想應該有一些人。
16:42
The people who just last sat down.
380
984000
3000
誰是最後一個坐下去的。
16:45
Okay, how many digits do you have in your number?
381
987000
5000
好的，你的答案是多少位數？
16:50
(Laughs)
382
992000
3000
（笑）
16:53
21. Okay good. How many do you have in yours?
383
995000
2000
21。好的，很好。你的是多少位？
16:55
18. So it goes to this lady here.
384
997000
3000
18。那麼是這位女士贏了。
16:58
21 is the closest.
385
1000000
2000
21是最接近的。
17:00
It actually has -- the number of symmetries in the Rubik's cube
386
1002000
2000
實際上，魔術方塊對稱種數的答案
17:02
has 25 digits.
387
1004000
2000
有25位數字。
17:04
So now I need to name this object.
388
1006000
2000
那麼現在我需要給這個物體命名。
17:06
So, what is your name?
389
1008000
2000
嗯，你叫什麼名字？
17:08
I need your surname. Symmetrical objects generally --
390
1010000
3000
我需要你的姓氏。對稱的物體一般——
17:11
spell it for me.
391
1013000
2000
為我拼寫一下。
17:13
G-H-E-Z
392
1015000
7000
G-H-E-Z
17:20
No, SO2 has already been used, actually,
393
1022000
2000
不，SO2已經用過了，其實，
17:22
in the mathematical language. So you can't have that one.
394
1024000
2000
在數學語言裡。你不能用那個名字。
17:24
So Ghez, there we go. That's your new symmetrical object.
395
1026000
2000
Ghez，就是這個名字啦。這是你的新的對稱物。
17:26
You are now immortal.
396
1028000
2000
你現在是不朽的了。
17:28
(Applause)
397
1030000
6000
（鼓掌）
17:34
And if you'd like your own symmetrical object,
398
1036000
2000
而且，如果你想用你自己的對稱物，
17:36
I have a project raising money for a charity in Guatemala,
399
1038000
3000
我有一個項目，是為在瓜地馬拉的慈善籌錢的，
17:39
where I will stay up all night and devise an object for you,
400
1041000
3000
我可以熬夜為你發明一個物體，
17:42
for a donation to this charity to help kids get into education in Guatemala.
401
1044000
4000
讓你可以為慈善捐款來幫助瓜地馬拉的孩子們，讓他們能接受教育。
17:46
And I think what drives me, as a mathematician,
402
1048000
3000
我認為，作為一個數學家，
17:49
are those things which are not seen, the things that we haven't discovered.
403
1051000
4000
給我動力的是那些看不到的東西，是我們還未發現的東西。
17:53
It's all the unanswered questions which make mathematics a living subject.
404
1055000
4000
它們都是懸而未決的問題，這使數學繼續活著
17:57
And I will always come back to this quote from the Japanese "Essays in Idleness":
405
1059000
3000
我常常想起引自日本《徒然草》中的這句話：
18:00
"In everything, uniformity is undesirable.
406
1062000
3000
“在一切事物中，一致性是不可取的。
18:03
Leaving something incomplete makes it interesting,
407
1065000
3000
留下若干不完整，會更有趣，
18:06
and gives one the feeling that there is room for growth." Thank you.
408
1068000
3000
並給予一種仍有發展空間的感覺。 ”謝謝。
18:09
(Applause)
409
1071000
7000
（掌聲）
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