Marcus du Sautoy: Symmetry, reality's riddle

譯者: Coco Shen 審譯者: Geoff Chen

On the 30th of May, 1832,

在1832年5月30日，

a gunshot was heard

人們聽到一聲槍響，

ringing out across the 13th arrondissement in Paris.

槍聲穿透了巴黎的第十三區

(Gunshot)

（槍聲）

A peasant, who was walking to market that morning,

一個那天早晨正前往市集的農民

ran towards where the gunshot had come from,

朝槍聲傳來的地方跑了過去，

and found a young man writhing in agony on the floor,

發現一名年輕男子正痛得在地上打滾，

clearly shot by a dueling wound.

顯然他在決鬥中遭到了槍擊。

The young man's name was Evariste Galois.

這個年輕人名叫 Evariste • Galois

He was a well-known revolutionary in Paris at the time.

是巴黎當時一個有有名的革命者

Galois was taken to the local hospital

Galois 被送到了當地的醫院，

where he died the next day in the arms of his brother.

第二天死在他兄弟的懷中

And the last words he said to his brother were,

他對他兄弟說的臨別遺言是

"Don't cry for me, Alfred.

“不要為我哭泣, Alfred

I need all the courage I can muster

我需要聚集我能聚集的所有勇氣

to die at the age of 20."

讓我在20歲時死去。 ”

It wasn't, in fact, revolutionary politics

實際上，革命政治並不是

for which Galois was famous.

使 Galois 著名的原因。

But a few years earlier, while still at school,

而是幾年前，當他還在上學時，

he'd actually cracked one of the big mathematical

他破解了

problems at the time.

當時重大數學問題之一

And he wrote to the academicians in Paris,

隨後他寫信給巴黎的院士

trying to explain his theory.

嘗試解釋他的理論

But the academicians couldn't understand anything that he wrote.

但院士們弄不懂他寫的任何東西。

(Laughter)

（大笑）

This is how he wrote most of his mathematics.

這就是他怎麼寫大部分數學理論的

So, the night before that duel, he realized

因此，在決鬥的前一天晚上，他意識到

this possibly is his last chance

這可能是他最後一次機會

to try and explain his great breakthrough.

來嘗試解釋他的重大突破

So he stayed up the whole night, writing away,

所以他徹夜未眠，不停地寫東西，

trying to explain his ideas.

試圖解釋他的想法

And as the dawn came up and he went to meet his destiny,

隨著黎明的到來，他準備迎接自己的命運。

he left this pile of papers on the table for the next generation.

他把桌子上的一堆文件留給了下一代。

Maybe the fact that he stayed up all night doing mathematics

也許徹夜研究數學

was the fact that he was such a bad shot that morning and got killed.

是他那天早晨受到槍擊且被殺的真正原因

But contained inside those documents

但包含在那些文件中的

was a new language, a language to understand

是一種新的語言，這種語言能讓人們理解

one of the most fundamental concepts

科學的一個最基本的概念，

of science -- namely symmetry.

即對稱性。

Now, symmetry is almost nature's language.

現今，對稱性幾乎是大自然的語言。

It helps us to understand so many

它有助於我們了解許多

different bits of the scientific world.

科學世界裡不同的小東西。

For example, molecular structure.

例如，分子結構。

What crystals are possible,

什麼晶體是能讓

we can understand through the mathematics of symmetry.

我們可以通過數學的對稱性來了解的？

In microbiology you really don't want to get a symmetrical object,

在微生物學中，你真的不想研究對稱的東西。

because they are generally rather nasty.

因為它們一般都比較令人討厭。

The swine flu virus, at the moment, is a symmetrical object.

目前的豬流感病毒就是一種結構對稱的病毒。

And it uses the efficiency of symmetry

而且它利用對稱的功效

to be able to propagate itself so well.

來增強自己繁殖的速度

But on a larger scale of biology, actually symmetry is very important,

但從大方向來說，對稱性事實上對生物學非常重要

because it actually communicates genetic information.

因為它能傳遞遺傳信息

I've taken two pictures here and I've made them artificially symmetrical.

我帶了兩張照片到這兒來，並人工的把他們做成了對稱的

And if I ask you which of these you find more beautiful,

如果我問你們覺得哪些更漂亮，

you're probably drawn to the lower two.

你們可能會被下面的兩張吸引住。

Because it is hard to make symmetry.

因為很難做到對稱，

And if you can make yourself symmetrical, you're sending out a sign

所以如果你可以使自己對稱，那麼你在傳遞一種信號

that you've got good genes, you've got a good upbringing

它意味著你得到了好的遺傳基因，你有好的教養，

and therefore you'll make a good mate.

因而你會有一個好的伴侶。

So symmetry is a language which can help to communicate

所以，對稱性是一種語言，它能有助於傳遞

genetic information.

遺傳信息。

Symmetry can also help us to explain

對稱性還可以幫助我們解釋

what's happening in the Large Hadron Collider in CERN.

歐洲粒子物理研究所大型強子對撞機正發生著什麼事情。

Or what's not happening in the Large Hadron Collider in CERN.

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。

To be able to make predictions about the fundamental particles

為了能夠對基本粒子作出預測，

we might see there,

我們可能會在那兒看到的（基本粒子），

it seems that they are all facets of some strange symmetrical shape

似乎所有的小平面都有某種奇怪的對稱形狀

in a higher dimensional space.

當它們在更高維的空間中時。

And I think Galileo summed up, very nicely,

我認為伽利略很好地概括了

the power of mathematics

數學的力量：

to understand the scientific world around us.

它讓我們得以了解周圍的科學世界

He wrote, "The universe cannot be read

他寫道：“我們無法閱讀宇宙，

until we have learnt the language

除非學會它的語言，

and become familiar with the characters in which it is written.

且熟悉其寫作特點。

It is written in mathematical language,

它是用數學語言寫的。

and the letters are triangles, circles and other geometric figures,

字母是三角形、圓和其他的幾何數字，

without which means it is humanly impossible

沒有這些字母就意味著在人力所能及的範圍內是不可能

to comprehend a single word."

理解任何一個字。 ”

But it's not just scientists who are interested in symmetry.

不只是科學家們對對稱性感興趣。

Artists too love to play around with symmetry.

藝術家也喜歡擺弄對稱性。

They also have a slightly more ambiguous relationship with it.

他們與對稱性有一些更模糊的關係。

Here is Thomas Mann talking about symmetry in "The Magic Mountain."

這是托馬斯•曼在《魔山》中談到的對稱性。

He has a character describing the snowflake,

他這樣描寫雪花

and he says he "shuddered at its perfect precision,

他說他，“因其有完美的精確度而震撼，

found it deathly, the very marrow of death."

發現它死亡的精髓讓他想到死亡。 ”

But what artists like to do is to set up expectations

但藝術家們想要做的是樹立對對稱性的期望，

of symmetry and then break them.

然後打破它們。

And a beautiful example of this

就這一點我找到了一個很好的例子，

I found, actually, when I visited a colleague of mine

其實是當我拜訪我的同事

in Japan, Professor Kurokawa.

在日本的黑川紀章教授時發現的

And he took me up to the temples in Nikko.

他帶我到日光市的寺廟去

And just after this photo was taken we walked up the stairs.

就在拍好這張照片後，我們走上樓梯，

And the gateway you see behind

你們看到的這後面的大門

has eight columns, with beautiful symmetrical designs on them.

有八根柱子，都有著漂亮的對稱性設計。

Seven of them are exactly the same,

其中七個是完全一樣的，

and the eighth one is turned upside down.

而第八個是顛倒過來的。

And I said to Professor Kurokawa,

我就對黑川紀章教授說：

"Wow, the architects must have really been kicking themselves

“哇，建築師們肯定自責的很

when they realized that they'd made a mistake and put this one upside down."

要是他麼發現犯了這麼一個錯誤，這根柱子竟然是相反的。 ”

And he said, "No, no, no. It was a very deliberate act."

他說，“不，不，不。這是故意設計成這樣的。”

And he referred me to this lovely quote from the Japanese

他還向我提到了這個可愛的出處，來自日本

"Essays in Idleness" from the 14th century,

1 4世紀的《徒然草》

in which the essayist wrote, "In everything,

其中，散文家寫道：“在一切事物中，

uniformity is undesirable.

一致性是不可取的。

Leaving something incomplete makes it interesting,

留下一些不完整的東西會更有趣，

and gives one the feeling that there is room for growth."

而且一致性給人一種沒有發展空間的感覺。 ”

Even when building the Imperial Palace,

即使是建造皇宮時，

they always leave one place unfinished.

他們也總是留下一個未完工的地方。

But if I had to choose one building in the world

但如果我必須選擇這世界上的一個建築，

to be cast out on a desert island, to live the rest of my life,

將其扔到一個荒島上，且我要在那裡度過餘生，

being an addict of symmetry, I would probably choose the Alhambra in Granada.

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。

This is a palace celebrating symmetry.

這是一座歌頌對稱性的宮殿。

Recently I took my family --

最近，我帶我的家人——

we do these rather kind of nerdy mathematical trips, which my family love.

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。

This is my son Tamer. You can see

這是我的兒子塔梅爾。你們可以看到

he's really enjoying our mathematical trip to the Alhambra.

他真的很喜歡我們在阿爾罕布拉的數學之旅。

But I wanted to try and enrich him.

但我想嘗試使他變得充實。

I think one of the problems about school mathematics

我認為學校教的數學存在的一個問題就是

is it doesn't look at how mathematics is embedded

它沒有關注數學是如何被運用於

in the world we live in.

我們所處的這個世界。

So, I wanted to open his eyes up to

所以，我想開拓他的眼界，讓他知道

how much symmetry is running through the Alhambra.

阿爾罕布拉運用著多少對稱性。

You see it already. Immediately you go in,

你們已經看到了。你一走進去，

the reflective symmetry in the water.

水中有反映出對稱。

But it's on the walls where all the exciting things are happening.

但是，所有令人興奮的事情發生在牆壁上。

The Moorish artists were denied the possibility

人們否認摩爾藝術家能夠

to draw things with souls.

用靈魂來繪畫。

So they explored a more geometric art.

因此，他們探索出一種更加幾何化的藝術。

And so what is symmetry?

那麼什麼是對稱性？

The Alhambra somehow asks all of these questions.

阿爾罕布拉以某種方式提出了所有這些問題。

What is symmetry? When [there] are two of these walls,

什麼是對稱性？當[那兒]有兩面牆時，

do they have the same symmetries?

他們有相同的對稱性嗎？

Can we say whether they discovered

我們可以說他們是否發現了

all of the symmetries in the Alhambra?

阿爾罕布拉所有的對稱性嗎？

And it was Galois who produced a language

是 Galois 研製出了一種語言

to be able to answer some of these questions.

來回答這樣的問題

For Galois, symmetry -- unlike for Thomas Mann,

對 Galois 來說，對稱性不是托馬斯曼所說的

which was something still and deathly --

靜態的和死一般的東西

for Galois, symmetry was all about motion.

對 Galois 來說，所有的對稱性都是關於運動的

What can you do to a symmetrical object,

你能對一個對稱性的物體做些什麼？

move it in some way, so it looks the same

用某種方法移動它，讓它看起來

as before you moved it?

跟你移動它之前一樣？

I like to describe it as the magic trick moves.

我喜歡把這形容為神奇的假動作。

What can you do to something? You close your eyes.

你對一些東西可以做些什麼？閉上你的眼睛。

I do something, put it back down again.

我移動它，再把它放回到原處。

It looks like it did before it started.

它看起來和動之前一樣。

So, for example, the walls in the Alhambra --

那麼，例如，阿爾罕布拉的牆壁。

I can take all of these tiles, and fix them at the yellow place,

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，

rotate them by 90 degrees,

並把它們旋轉九十度，

put them all back down again and they fit perfectly down there.

再把他們都放回去，它們非常吻合。

And if you open your eyes again, you wouldn't know that they'd moved.

如果你再睜開你的眼睛，你不會知道它們被移動過。

But it's the motion that really characterizes the symmetry

但正是運動才使對稱性

inside the Alhambra.

在阿爾罕布拉具有特色。

But it's also about producing a language to describe this.

但也要創造一種語言來描繪它。

And the power of mathematics is often

數學的力量往往

to change one thing into another, to change geometry into language.

把一樣東西變成另一樣，把幾何變成語言。

So I'm going to take you through, perhaps push you a little bit mathematically --

因此，我將帶你經歷，可能強加一些數學的東西給你們，

so brace yourselves --

所以撐住自己，

push you a little bit to understand how this language works,

強加一些數學的知識讓你們了解這種語言是怎麼運作的，

which enables us to capture what is symmetry.

這讓我們能夠捕捉到什麼是對稱性。

So, let's take these two symmetrical objects here.

那讓我們把這兩個對稱物放到這兒。

Let's take the twisted six-pointed starfish.

拿這個扭曲了的六角海星來說。

What can I do to the starfish which makes it look the same?

我怎麼做能讓這個海星看起來和原來一樣呢？

Well, there I rotated it by a sixth of a turn,

嗯，我把它旋轉了六分之一圈，

and still it looks like it did before I started.

它看起來仍然跟我動過之前一樣。

I could rotate it by a third of a turn,

我可以把它旋轉三分之一圈，

or a half a turn,

或者半圈，

or put it back down on its image, or two thirds of a turn.

或將它恢復到原圖，或旋轉三分之二圈。

And a fifth symmetry, I can rotate it by five sixths of a turn.

第五種對稱，我可以把它旋轉六分之五圈。

And those are things that I can do to the symmetrical object

這些就是我能對對稱物所做的，

that make it look like it did before I started.

可以讓它看起來跟我動它們之前一樣。

Now, for Galois, there was actually a sixth symmetry.

對 Galois 來說，實際上還有第六種對稱。

Can anybody think what else I could do to this

大家能想到其它什麼辦法

which would leave it like I did before I started?

可以讓它跟我動它之前一樣？

I can't flip it because I've put a little twist on it, haven't I?

我不能翻轉它，因為我已​​經把它扭曲了一些，是吧？

It's got no reflective symmetry.

這樣它沒有反射對稱了。

But what I could do is just leave it where it is,

但我可以做的就是把它放在原處，

pick it up, and put it down again.

把它拿起來再把它放下。

And for Galois this was like the zeroth symmetry.

對 Galois 來說，這就像是第零個對稱。

Actually, the invention of the number zero

其實，數字零的發明

was a very modern concept, seventh century A.D., by the Indians.

是一個非常現代化的概念，它是公元七世紀印度人發明的。

It seems mad to talk about nothing.

談論無有感覺很瘋狂。

And this is the same idea. This is a symmetrical --

這是同樣的概念。這是對稱的——

so everything has symmetry, where you just leave it where it is.

所以一切事物都有對稱性，把它放在拿起它的地方。

So, this object has six symmetries.

所以這個物體有六種對稱。

And what about the triangle?

那三角形呢？

Well, I can rotate by a third of a turn clockwise

嗯，我可以把它順時針旋轉三分之一圈

or a third of a turn anticlockwise.

或逆時針旋轉三分之一圈。

But now this has some reflectional symmetry.

但現在有反射對稱

I can reflect it in the line through X,

我可以在X軸上翻轉它，

or the line through Y,

或在Y軸上，

or the line through Z.

或在Z軸上。

Five symmetries and then of course the zeroth symmetry

五種對稱，當然還有第零個對稱，

where I just pick it up and leave it where it is.

我把它拿起來，放回原處。

So both of these objects have six symmetries.

因此，這些物體都有六種對稱。

Now, I'm a great believer that mathematics is not a spectator sport,

現在，我十分相信數學不是旁觀者的運動，

and you have to do some mathematics

你必須做一些數學運算

in order to really understand it.

才能真正理解它。

So here is a little question for you.

這兒有個小問題問問你們。

And I'm going to give a prize at the end of my talk

我將在講座結束後給一個獎品

for the person who gets closest to the answer.

給那個給出的答案最接近的人。

The Rubik's Cube.

魔術方塊

How many symmetries does a Rubik's Cube have?

一個魔術方塊有多少種對稱？

How many things can I do to this object

有多少種方法可以在動了這個物體，

and put it down so it still looks like a cube?

且把它放下後它仍然看起來像一個立方體？

Okay? So I want you to think about that problem as we go on,

好嗎？我希望隨著講座的繼續，你們可以想想這個問題，

and count how many symmetries there are.

數數它有多少種對稱。

And there will be a prize for the person who gets closest at the end.

講座結束後獎品會給答案最接近的人

But let's go back down to symmetries that I got for these two objects.

讓我們回到這兩個物體的對稱性上。

What Galois realized: it isn't just the individual symmetries,

Galois 意識到這不僅僅是個體的對稱性，

but how they interact with each other

而是個體之間如何相互作用

which really characterizes the symmetry of an object.

才真正賦予了一個物體具有對稱性的特點。

If I do one magic trick move followed by another,

如果我做一個神奇的假動作，然後再做一個，

the combination is a third magic trick move.

兩個合併起來就是第三個神奇的假動作。

And here we see Galois starting to develop

這裡我們了解到 Glaois 開始開發

a language to see the substance

一種語言來研究

of the things unseen, the sort of abstract idea

看不見的東西所具有的內在含義，以及

of the symmetry underlying this physical object.

物理物體中存在的對稱性的抽象的概念。

For example, what if I turn the starfish

例如，如果我把海星旋轉

by a sixth of a turn,

六分之一圈，

and then a third of a turn?

然後再轉三分之一圈會，結果會怎樣？

So I've given names. The capital letters, A, B, C, D, E, F,

所以我給它們取了名字。大寫的字母A、B、C、D、E、F，

are the names for the rotations.

這些名字旋轉的代號。

B, for example, rotates the little yellow dot

例如B，旋轉小黃點，

to the B on the starfish. And so on.

它位於海星上的B處，諸如此類。

So what if I do B, which is a sixth of a turn,

那麼，如果我旋轉B，轉六分之一圈，

followed by C, which is a third of a turn?

其次是C，轉三分之一圈？

Well let's do that. A sixth of a turn,

嗯，讓我們開始。六分之一圈，

followed by a third of a turn,

接著是三分之一圈，

the combined effect is as if I had just rotated it by half a turn in one go.

合併後的效果就像我剛剛把它一次旋轉了半圈一樣。

So the little table here records

那這個小表格記載著

how the algebra of these symmetries work.

這些對稱的代數是怎麼運作的。

I do one followed by another, the answer is

我將一個接一個的旋轉，結果就是

it's rotation D, half a turn.

D旋轉了半圈。

What I if I did it in the other order? Would it make any difference?

如果我按其他順序旋轉呢？會有什麼不同嗎？

Let's see. Let's do the third of the turn first, and then the sixth of a turn.

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。

Of course, it doesn't make any difference.

當然，沒有什麼差別。

It still ends up at half a turn.

結果仍然是半圈。

And there is some symmetry here in the way the symmetries interact with each other.

某種對稱方式是通過相互作用得到的。

But this is completely different to the symmetries of the triangle.

但這與三角形的對稱性是完全不同的。

Let's see what happens if we do two symmetries

讓我們看看如果對三角形

with the triangle, one after the other.

一個接一個的進行兩個對稱旋轉會怎樣。

Let's do a rotation by a third of a turn anticlockwise,

讓我們逆時針旋轉三分之一圈，

and reflect in the line through X.

然後在X軸上翻轉。

Well, the combined effect is as if I had just done the reflection in the line through Z

嗯，合併後的效果就像我剛剛以Z軸翻轉

to start with.

開始一樣。

Now, let's do it in a different order.

現在，讓我們按不同的順序來一次。

Let's do the reflection in X first,

我們先在X軸上翻轉，

followed by the rotation by a third of a turn anticlockwise.

然後逆時針旋轉三分之一圈。

The combined effect, the triangle ends up somewhere completely different.

合併後的效果是三角形停的地方完全不同。

It's as if it was reflected in the line through Y.

就像是在Y軸上翻轉了一樣。

Now it matters what order you do the operations in.

現在看來這與你操作它的順序有關。

And this enables us to distinguish

這使我們能夠區分

why the symmetries of these objects --

為什麼這些物體的對稱性

they both have six symmetries. So why shouldn't we say

都有六個。那麼，為什麼我們不能說

they have the same symmetries?

它們有相同的對稱性呢？

But the way the symmetries interact

但對稱相互作用的方式

enable us -- we've now got a language

使我們——我們現在已經有一種語言

to distinguish why these symmetries are fundamentally different.

來區分為什麼這些對稱在根本上是不同的。

And you can try this when you go down to the pub, later on.

你也可以嘗試一下，當你去酒吧時，以後去的時候。

Take a beer mat and rotate it by a quarter of a turn,

拿一個啤酒墊，把它旋轉四分之一圈，

then flip it. And then do it in the other order,

然後翻轉它。然後再按其它順序做，

and the picture will be facing in the opposite direction.

酒墊上的圖將是朝反方向面對你的。

Now, Galois produced some laws for how these tables -- how symmetries interact.

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。

It's almost like little Sudoku tables.

這像一個小數獨表。

You don't see any symmetry twice

你看不到任何重複的對稱

in any row or column.

出現在任何一欄或一行中。

And, using those rules, he was able to say

通過運用那些定律，他可以說

that there are in fact only two objects

事實上只有兩個物體

with six symmetries.

有六個對稱。

And they'll be the same as the symmetries of the triangle,

而且這六個對稱將和三角形的對稱,

or the symmetries of the six-pointed starfish.

或六角海星的對稱是一樣的。

I think this is an amazing development.

我覺得這是一個驚人的發展。

It's almost like the concept of number being developed for symmetry.

它幾乎是為了對稱而研製的數的概念。

In the front here, I've got one, two, three people

在這前面，我請一、二、三個人

sitting on one, two, three chairs.

坐在一、二、三把椅子上。

The people and the chairs are very different,

坐在椅子上的人都不一樣，

but the number, the abstract idea of the number, is the same.

但是數字，數字的抽象的觀念，都是一樣的。

And we can see this now: we go back to the walls in the Alhambra.

現在我們可以看到這個：我們回到阿爾罕布拉的牆壁。

Here are two very different walls,

這有兩面很不一樣的牆壁，

very different geometric pictures.

很不相同的幾何圖片。

But, using the language of Galois,

但是，利用 Galois 的語言

we can understand that the underlying abstract symmetries of these things

我們可以知道這些東西含有的抽象的對稱

are actually the same.

實際上是相同的。

For example, let's take this beautiful wall

例如，讓我們把這面漂亮的牆

with the triangles with a little twist on them.

和三角形稍微扭曲一下。

You can rotate them by a sixth of a turn

你可以把它們旋轉六分之一圈，

if you ignore the colors. We're not matching up the colors.

如果忽略他們的顏色。我們不是在做顏色配對

But the shapes match up if I rotate by a sixth of a turn

而是在形狀配對，如果我把他們旋轉六分之一圈，

around the point where all the triangles meet.

圍繞著所有三角形交彙的一點旋轉。

What about the center of a triangle? I can rotate

三角形的中心會怎麼樣？我可以

by a third of a turn around the center of the triangle,

圍繞著三角形的中心把他們旋轉三分之一圈，

and everything matches up.

那麼一切就都對上了。

And then there is an interesting place halfway along an edge,

這兒有個有趣的地方，沿著邊的一半

where I can rotate by 180 degrees.

我可以把它旋轉180度。

And all the tiles match up again.

那麼所有的瓦片又重新匹配了。

So rotate along halfway along the edge, and they all match up.

所以，沿著邊的一半旋轉，那麼他們都能配合上。

Now, let's move to the very different-looking wall in the Alhambra.

現在，讓我們移動阿爾罕布拉的一面外觀非常不一樣的牆。

And we find the same symmetries here, and the same interaction.

我們在這兒發現同樣的對稱性和同樣的相互作用。

So, there was a sixth of a turn. A third of a turn where the Z pieces meet.

那麼是轉了六分之一轉。轉了三分之一圈時第Z片交會

And the half a turn is halfway between the six pointed stars.

旋轉半圈時離六角星交會還有一半。

And although these walls look very different,

儘管這些牆壁看起來非常不同，

Galois has produced a language to say

Galois 研製出一種語言說，

that in fact the symmetries underlying these are exactly the same.

其實這些東西所具有的對稱性是完全相同的。

And it's a symmetry we call 6-3-2.

這個對稱性我們稱之為6-3-2。

Here is another example in the Alhambra.

另一個阿爾罕布拉的例子。

This is a wall, a ceiling, and a floor.

這是一面牆、天花板和地板。

They all look very different. But this language allows us to say

它們都看起來都非常不一樣。但是，這種語言讓我們可以說

that they are representations of the same symmetrical abstract object,

它們是相同的對稱的抽象物體，

which we call 4-4-2. Nothing to do with football,

我們稱之為4-4-2。這與足球毫無關係，

but because of the fact that there are two places where you can rotate

而是因為他們都有兩個你可以旋轉

by a quarter of a turn, and one by half a turn.

四分之一圈和二分之一圈的地方。

Now, this power of the language is even more,

現在，這種語言的力量更加強大，

because Galois can say,

因為 Galois 可能會說，

"Did the Moorish artists discover all of the possible symmetries

“摩爾藝術家發現了阿爾罕布拉牆上所有可能對稱的

on the walls in the Alhambra?"

地方了嗎？ ”

And it turns out they almost did.

結果是他們幾乎都發現了。

You can prove, using Galois' language,

你可以用伽羅瓦的語言來證明，

there are actually only 17

實際上只有17種

different symmetries that you can do in the walls in the Alhambra.

可以在阿爾罕布拉的牆上得到的不同的對稱。

And they, if you try to produce a different wall with this 18th one,

而且他們，如果你嘗試研製出第18面不同的牆壁，

it will have to have the same symmetries as one of these 17.

這面牆肯定與17種對稱中的一種對稱是相同的。

But these are things that we can see.

但這些都是我們可以看到的。

And the power of Galois' mathematical language

而 Galois 的數學語言的力量

is it also allows us to create

也讓我們能

symmetrical objects in the unseen world,

在看不見的世界裡創造對稱的物體，

beyond the two-dimensional, three-dimensional,

超越二維、三維，

all the way through to the four- or five- or infinite-dimensional space.

全都向四維或五維或無窮維空間發展。

And that's where I work. I create

而這正是我在研究的東西。我創建

mathematical objects, symmetrical objects,

數學對象和對稱物體，

using Galois' language,

通過運用 Galois 的語言

in very high dimensional spaces.

在非常高維的空間裡創建。

So I think it's a great example of things unseen,

因此，我認為這是個對於看不見的東西的很好的例子，

which the power of mathematical language allows you to create.

數學語言的力量讓你可以創建出來。

So, like Galois, I stayed up all last night

因此，像 Glaois 一樣，我昨晚徹夜未眠

creating a new mathematical symmetrical object for you,

為你們建立了一個新的數學對稱物。

and I've got a picture of it here.

我這兒有一張它的照片。

Well, unfortunately it isn't really a picture. If I could have my board

但是，可惜的是它不是一張真正的照片。我可以把我的圖板

at the side here, great, excellent.

放在這邊嗎？很好，非常好。

Here we are. Unfortunately, I can't show you

這兒。可惜的是我無法向你們展示

a picture of this symmetrical object.

這個對稱物的照片。

But here is the language which describes

但這兒有語言能描繪

how the symmetries interact.

其對稱性怎麼相互作用的。

Now, this new symmetrical object

現在這個新的對稱物

does not have a name yet.

還沒有名字。

Now, people like getting their names on things,

就像人們給東西命名一樣，

on craters on the moon

給月球上的隕石坑命名，

or new species of animals.

或給新動物品種命名一樣。

So I'm going to give you the chance to get your name on a new symmetrical object

所以我想給你們機會來給新的對稱物命名，

which hasn't been named before.

以前沒有給它取過名字。

And this thing -- species die away,

而且這個東西——物種會逐漸消失，

and moons kind of get hit by meteors and explode --

月球可能會被隕石撞擊並發生爆炸——

but this mathematical object will live forever.

但是這個數學物體將長存於世。

It will make you immortal.

它將使你不朽。

In order to win this symmetrical object,

為了贏得這個對稱物，

what you have to do is to answer the question I asked you at the beginning.

你所要做的就是回答我在一開始問的問題。

How many symmetries does a Rubik's Cube have?

魔術方塊有多少種對稱呢？

Okay, I'm going to sort you out.

好吧，我來給你們整理一下。

Rather than you all shouting out, I want you to count how many digits there are

而不是大家都喊出來，我想讓你們數數有多少位數字

in that number. Okay?

在那個答案裡。好嗎？

If you've got it as a factorial, you've got to expand the factorials.

如果你得出的結果是一個階乘，那麼你要擴大它的階乘。

Okay, now if you want to play,

好了，現在如果你想參與，

I want you to stand up, okay?

我希望你能站起來，好嗎？

If you think you've got an estimate for how many digits,

如果你認為你已經估計出了它有多少位數字，

right -- we've already got one competitor here.

好的——我們在這兒已經有了一位參賽者——

If you all stay down he wins it automatically.

如果你們都繼續坐著，那麼他就自動贏了。

Okay. Excellent. So we've got four here, five, six.

好的。很好。我們已經有四位、五位、六位。

Great. Excellent. That should get us going. All right.

很好。太好了。讓我們繼續。好了。

Anybody with five or less digits, you've got to sit down,

你們中有人的答案是等於或少於五位數的，那你得坐下了。

because you've underestimated.

因為你們估計少了。

Five or less digits. So, if you're in the tens of thousands you've got to sit down.

五位數或更少的。那麼，如果你的答案是幾萬的話，你得坐下。

60 digits or more, you've got to sit down.

六十或六十多位數的，你必須坐下。

You've overestimated.

你估計得多了。

20 digits or less, sit down.

二十位數或二十位以下的，坐下。

How many digits are there in your number?

你的答案是幾位數？

Two? So you should have sat down earlier.

兩個？那你早就該坐下了。

(Laughter)

（大笑）

Let's have the other ones, who sat down during the 20, up again. Okay?

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？

If I told you 20 or less, stand up.

如果我告訴你是二十位或二十位以下，請站起來。

Because this one. I think there were a few here.

因為這一個。我想應該有一些人。

The people who just last sat down.

誰是最後一個坐下去的。

Okay, how many digits do you have in your number?

好的，你的答案是多少位數？

(Laughs)

（笑）

21. Okay good. How many do you have in yours?

21。好的，很好。你的是多少位？

18. So it goes to this lady here.

18。那麼是這位女士贏了。

21 is the closest.

21是最接近的。

It actually has -- the number of symmetries in the Rubik's cube

實際上，魔術方塊對稱種數的答案

has 25 digits.

有25位數字。

So now I need to name this object.

那麼現在我需要給這個物體命名。

So, what is your name?

嗯，你叫什麼名字？

I need your surname. Symmetrical objects generally --

我需要你的姓氏。對稱的物體一般——

spell it for me.

為我拼寫一下。

G-H-E-Z

G-H-E-Z

No, SO2 has already been used, actually,

不，SO2已經用過了，其實，

in the mathematical language. So you can't have that one.

在數學語言裡。你不能用那個名字。

So Ghez, there we go. That's your new symmetrical object.

Ghez，就是這個名字啦。這是你的新的對稱物。

You are now immortal.

你現在是不朽的了。

(Applause)

（鼓掌）

And if you'd like your own symmetrical object,

而且，如果你想用你自己的對稱物，

I have a project raising money for a charity in Guatemala,

我有一個項目，是為在瓜地馬拉的慈善籌錢的，

where I will stay up all night and devise an object for you,

我可以熬夜為你發明一個物體，

for a donation to this charity to help kids get into education in Guatemala.

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。

And I think what drives me, as a mathematician,

我認為，作為一個數學家，

are those things which are not seen, the things that we haven't discovered.

給我動力的是那些看不到的東西，是我們還未發現的東西。

It's all the unanswered questions which make mathematics a living subject.

它們都是懸而未決的問題，這使數學繼續活著

And I will always come back to this quote from the Japanese "Essays in Idleness":

我常常想起引自日本《徒然草》中的這句話：

"In everything, uniformity is undesirable.

“在一切事物中，一致性是不可取的。

Leaving something incomplete makes it interesting,

留下若干不完整，會更有趣，

and gives one the feeling that there is room for growth." Thank you.

並給予一種仍有發展空間的感覺。 ”謝謝。

(Applause)

（掌聲）