On the 30th of May, 1832, a gunshot was heard ringing out across the 13th arrondissement in Paris. (Gunshot) A peasant, who was walking to market that morning, ran towards where the gunshot had come from, and found a young man writhing in agony on the floor, clearly shot by a dueling wound. The young man's name was Evariste Galois. He was a well-known revolutionary in Paris at the time. Galois was taken to the local hospital where he died the next day in the arms of his brother. And the last words he said to his brother were, "Don't cry for me, Alfred. I need all the courage I can muster to die at the age of 20."
在1832年5月30日, 人們聽到一聲槍響, 槍聲穿透了巴黎的第十三區 (槍聲) 一個那天早晨正前往市集的農民 朝槍聲傳來的地方跑了過去, 發現一名年輕男子正痛得在地上打滾, 顯然他在決鬥中遭到了槍擊。 這個年輕人名叫 Evariste • Galois 是巴黎當時一個有有名的革命者 Galois 被送到了當地的醫院, 第二天死在他兄弟的懷中 他對他兄弟說的臨別遺言是 “不要為我哭泣, Alfred 我需要聚集我能聚集的所有勇氣 讓我在20歲時死去。 ”
It wasn't, in fact, revolutionary politics for which Galois was famous. But a few years earlier, while still at school, he'd actually cracked one of the big mathematical problems at the time. And he wrote to the academicians in Paris, trying to explain his theory. But the academicians couldn't understand anything that he wrote. (Laughter) This is how he wrote most of his mathematics.
實際上,革命政治並不是 使 Galois 著名的原因。 而是幾年前,當他還在上學時, 他破解了 當時重大數學問題之一 隨後他寫信給巴黎的院士 嘗試解釋他的理論 但院士們弄不懂他寫的任何東西。 (大笑) 這就是他怎麼寫大部分數學理論的
So, the night before that duel, he realized this possibly is his last chance to try and explain his great breakthrough. So he stayed up the whole night, writing away, trying to explain his ideas. And as the dawn came up and he went to meet his destiny, he left this pile of papers on the table for the next generation. Maybe the fact that he stayed up all night doing mathematics was the fact that he was such a bad shot that morning and got killed.
因此,在決鬥的前一天晚上,他意識到 這可能是他最後一次機會 來嘗試解釋他的重大突破 所以他徹夜未眠,不停地寫東西, 試圖解釋他的想法 隨著黎明的到來,他準備迎接自己的命運。 他把桌子上的一堆文件留給了下一代。 也許徹夜研究數學 是他那天早晨受到槍擊且被殺的真正原因
But contained inside those documents was a new language, a language to understand one of the most fundamental concepts of science -- namely symmetry. Now, symmetry is almost nature's language. It helps us to understand so many different bits of the scientific world. For example, molecular structure. What crystals are possible, we can understand through the mathematics of symmetry.
但包含在那些文件中的 是一種新的語言,這種語言能讓人們理解 科學的一個最基本的概念, 即對稱性。 現今,對稱性幾乎是大自然的語言。 它有助於我們了解許多 科學世界裡不同的小東西。 例如,分子結構。 什麼晶體是能讓 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, because they are generally rather nasty. The swine flu virus, at the moment, is a symmetrical object. And it uses the efficiency of symmetry to be able to propagate itself so well. But on a larger scale of biology, actually symmetry is very important, because it actually communicates genetic information.
在微生物學中,你真的不想研究對稱的東西。 因為它們一般都比較令人討厭。 目前的豬流感病毒就是一種結構對稱的病毒。 而且它利用對稱的功效 來增強自己繁殖的速度 但從大方向來說,對稱性事實上對生物學非常重要 因為它能傳遞遺傳信息
I've taken two pictures here and I've made them artificially symmetrical. And if I ask you which of these you find more beautiful, you're probably drawn to the lower two. Because it is hard to make symmetry. And if you can make yourself symmetrical, you're sending out a sign that you've got good genes, you've got a good upbringing and therefore you'll make a good mate. So symmetry is a language which can help to communicate genetic information.
我帶了兩張照片到這兒來,並人工的把他們做成了對稱的 如果我問你們覺得哪些更漂亮, 你們可能會被下面的兩張吸引住。 因為很難做到對稱, 所以如果你可以使自己對稱,那麼你在傳遞一種信號 它意味著你得到了好的遺傳基因,你有好的教養, 因而你會有一個好的伴侶。 所以,對稱性是一種語言,它能有助於傳遞 遺傳信息。
Symmetry can also help us to explain what's happening in the Large Hadron Collider in CERN. Or what's not happening in the Large Hadron Collider in CERN. To be able to make predictions about the fundamental particles we might see there, it seems that they are all facets of some strange symmetrical shape in a higher dimensional space.
對稱性還可以幫助我們解釋 歐洲粒子物理研究所大型強子對撞機正發生著什麼事情。 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。 為了能夠對基本粒子作出預測, 我們可能會在那兒看到的(基本粒子), 似乎所有的小平面都有某種奇怪的對稱形狀 當它們在更高維的空間中時。
And I think Galileo summed up, very nicely, the power of mathematics to understand the scientific world around us. He wrote, "The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometric figures, without which means it is humanly impossible to comprehend a single word."
我認為伽利略很好地概括了 數學的力量: 它讓我們得以了解周圍的科學世界 他寫道:“我們無法閱讀宇宙, 除非學會它的語言, 且熟悉其寫作特點。 它是用數學語言寫的。 字母是三角形、圓和其他的幾何數字, 沒有這些字母就意味著在人力所能及的範圍內是不可能 理解任何一個字。 ”
But it's not just scientists who are interested in symmetry. Artists too love to play around with symmetry. They also have a slightly more ambiguous relationship with it. Here is Thomas Mann talking about symmetry in "The Magic Mountain." He has a character describing the snowflake, and he says he "shuddered at its perfect precision, found it deathly, the very marrow of death."
不只是科學家們對對稱性感興趣。 藝術家也喜歡擺弄對稱性。 他們與對稱性有一些更模糊的關係。 這是托馬斯•曼在《魔山》中談到的對稱性。 他這樣描寫雪花 他說他,“因其有完美的精確度而震撼, 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 of symmetry and then break them. And a beautiful example of this I found, actually, when I visited a colleague of mine in Japan, Professor Kurokawa. And he took me up to the temples in Nikko. And just after this photo was taken we walked up the stairs. And the gateway you see behind has eight columns, with beautiful symmetrical designs on them. Seven of them are exactly the same, and the eighth one is turned upside down.
但藝術家們想要做的是樹立對對稱性的期望, 然後打破它們。 就這一點我找到了一個很好的例子, 其實是當我拜訪我的同事 在日本的黑川紀章教授時發現的 他帶我到日光市的寺廟去 就在拍好這張照片後,我們走上樓梯, 你們看到的這後面的大門 有八根柱子,都有著漂亮的對稱性設計。 其中七個是完全一樣的, 而第八個是顛倒過來的。
And I said to Professor Kurokawa, "Wow, the architects must have really been kicking themselves when they realized that they'd made a mistake and put this one upside down." And he said, "No, no, no. It was a very deliberate act." And he referred me to this lovely quote from the Japanese "Essays in Idleness" from the 14th century, in which the essayist wrote, "In everything, uniformity is undesirable. Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." Even when building the Imperial Palace, they always leave one place unfinished.
我就對黑川紀章教授說: “哇,建築師們肯定自責的很 要是他麼發現犯了這麼一個錯誤,這根柱子竟然是相反的。 ” 他說,“不,不,不。這是故意設計成這樣的。” 他還向我提到了這個可愛的出處,來自日本 1 4世紀的《徒然草》 其中,散文家寫道:“在一切事物中, 一致性是不可取的。 留下一些不完整的東西會更有趣, 而且一致性給人一種沒有發展空間的感覺。 ” 即使是建造皇宮時, 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 to be cast out on a desert island, to live the rest of my life, being an addict of symmetry, I would probably choose the Alhambra in Granada. This is a palace celebrating symmetry. Recently I took my family -- we do these rather kind of nerdy mathematical trips, which my family love. This is my son Tamer. You can see he's really enjoying our mathematical trip to the Alhambra. But I wanted to try and enrich him. I think one of the problems about school mathematics is it doesn't look at how mathematics is embedded in the world we live in. So, I wanted to open his eyes up to how much symmetry is running through the Alhambra.
但如果我必須選擇這世界上的一個建築, 將其扔到一個荒島上,且我要在那裡度過餘生, 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。 這是一座歌頌對稱性的宮殿。 最近,我帶我的家人—— 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。 這是我的兒子塔梅爾。你們可以看到 他真的很喜歡我們在阿爾罕布拉的數學之旅。 但我想嘗試使他變得充實。 我認為學校教的數學存在的一個問題就是 它沒有關注數學是如何被運用於 我們所處的這個世界。 所以,我想開拓他的眼界,讓他知道 阿爾罕布拉運用著多少對稱性。
You see it already. Immediately you go in, the reflective symmetry in the water. But it's on the walls where all the exciting things are happening. The Moorish artists were denied the possibility to draw things with souls. So they explored a more geometric art. And so what is symmetry? The Alhambra somehow asks all of these questions. What is symmetry? When [there] are two of these walls, do they have the same symmetries? Can we say whether they discovered all of the symmetries in the Alhambra?
你們已經看到了。你一走進去, 水中有反映出對稱。 但是,所有令人興奮的事情發生在牆壁上。 人們否認摩爾藝術家能夠 用靈魂來繪畫。 因此,他們探索出一種更加幾何化的藝術。 那麼什麼是對稱性? 阿爾罕布拉以某種方式提出了所有這些問題。 什麼是對稱性?當[那兒]有兩面牆時, 他們有相同的對稱性嗎? 我們可以說他們是否發現了 阿爾罕布拉所有的對稱性嗎?
And it was Galois who produced a language to be able to answer some of these questions. For Galois, symmetry -- unlike for Thomas Mann, which was something still and deathly -- for Galois, symmetry was all about motion. What can you do to a symmetrical object, move it in some way, so it looks the same as before you moved it? I like to describe it as the magic trick moves. What can you do to something? You close your eyes. I do something, put it back down again. It looks like it did before it started.
是 Galois 研製出了一種語言 來回答這樣的問題 對 Galois 來說,對稱性不是托馬斯曼所說的 靜態的和死一般的東西 對 Galois 來說,所有的對稱性都是關於運動的 你能對一個對稱性的物體做些什麼? 用某種方法移動它,讓它看起來 跟你移動它之前一樣? 我喜歡把這形容為神奇的假動作。 你對一些東西可以做些什麼?閉上你的眼睛。 我移動它,再把它放回到原處。 它看起來和動之前一樣。
So, for example, the walls in the Alhambra -- I can take all of these tiles, and fix them at the yellow place, rotate them by 90 degrees, put them all back down again and they fit perfectly down there. And if you open your eyes again, you wouldn't know that they'd moved. But it's the motion that really characterizes the symmetry inside the Alhambra. But it's also about producing a language to describe this. And the power of mathematics is often to change one thing into another, to change geometry into language.
那麼,例如,阿爾罕布拉的牆壁。 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, 並把它們旋轉九十度, 再把他們都放回去,它們非常吻合。 如果你再睜開你的眼睛,你不會知道它們被移動過。 但正是運動才使對稱性 在阿爾罕布拉具有特色。 但也要創造一種語言來描繪它。 數學的力量往往 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 -- so brace yourselves -- push you a little bit to understand how this language works, which enables us to capture what is symmetry. So, let's take these two symmetrical objects here. Let's take the twisted six-pointed starfish. What can I do to the starfish which makes it look the same? Well, there I rotated it by a sixth of a turn, and still it looks like it did before I started. I could rotate it by a third of a turn, or a half a turn, or put it back down on its image, or two thirds of a turn. And a fifth symmetry, I can rotate it by five sixths of a turn. And those are things that I can do to the symmetrical object that make it look like it did before I started.
因此,我將帶你經歷,可能強加一些數學的東西給你們, 所以撐住自己, 強加一些數學的知識讓你們了解這種語言是怎麼運作的, 這讓我們能夠捕捉到什麼是對稱性。 那讓我們把這兩個對稱物放到這兒。 拿這個扭曲了的六角海星來說。 我怎麼做能讓這個海星看起來和原來一樣呢? 嗯,我把它旋轉了六分之一圈, 它看起來仍然跟我動過之前一樣。 我可以把它旋轉三分之一圈, 或者半圈, 或將它恢復到原圖,或旋轉三分之二圈。 第五種對稱,我可以把它旋轉六分之五圈。 這些就是我能對對稱物所做的, 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. Can anybody think what else I could do to this which would leave it like I did before I started? I can't flip it because I've put a little twist on it, haven't I? It's got no reflective symmetry. But what I could do is just leave it where it is, pick it up, and put it down again. And for Galois this was like the zeroth symmetry. Actually, the invention of the number zero was a very modern concept, seventh century A.D., by the Indians. It seems mad to talk about nothing. And this is the same idea. This is a symmetrical -- so everything has symmetry, where you just leave it where it is.
對 Galois 來說,實際上還有第六種對稱。 大家能想到其它什麼辦法 可以讓它跟我動它之前一樣? 我不能翻轉它,因為我已經把它扭曲了一些,是吧? 這樣它沒有反射對稱了。 但我可以做的就是把它放在原處, 把它拿起來再把它放下。 對 Galois 來說,這就像是第零個對稱。 其實,數字零的發明 是一個非常現代化的概念,它是公元七世紀印度人發明的。 談論無有感覺很瘋狂。 這是同樣的概念。這是對稱的—— 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. And what about the triangle? Well, I can rotate by a third of a turn clockwise or a third of a turn anticlockwise. But now this has some reflectional symmetry. I can reflect it in the line through X, or the line through Y, or the line through Z. Five symmetries and then of course the zeroth symmetry where I just pick it up and leave it where it is. So both of these objects have six symmetries. Now, I'm a great believer that mathematics is not a spectator sport, and you have to do some mathematics in order to really understand it.
所以這個物體有六種對稱。 那三角形呢? 嗯,我可以把它順時針旋轉三分之一圈 或逆時針旋轉三分之一圈。 但現在有反射對稱 我可以在X軸上翻轉它, 或在Y軸上, 或在Z軸上。 五種對稱,當然還有第零個對稱, 我把它拿起來,放回原處。 因此,這些物體都有六種對稱。 現在,我十分相信數學不是旁觀者的運動, 你必須做一些數學運算 才能真正理解它。
So here is a little question for you. And I'm going to give a prize at the end of my talk for the person who gets closest to the answer. The Rubik's Cube. How many symmetries does a Rubik's Cube have? How many things can I do to this object and put it down so it still looks like a cube? Okay? So I want you to think about that problem as we go on, and count how many symmetries there are. And there will be a prize for the person who gets closest at the end.
這兒有個小問題問問你們。 我將在講座結束後給一個獎品 給那個給出的答案最接近的人。 魔術方塊 一個魔術方塊有多少種對稱? 有多少種方法可以在動了這個物體, 且把它放下後它仍然看起來像一個立方體? 好嗎?我希望隨著講座的繼續,你們可以想想這個問題, 數數它有多少種對稱。 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. What Galois realized: it isn't just the individual symmetries, but how they interact with each other which really characterizes the symmetry of an object. If I do one magic trick move followed by another, the combination is a third magic trick move. And here we see Galois starting to develop a language to see the substance of the things unseen, the sort of abstract idea of the symmetry underlying this physical object. For example, what if I turn the starfish by a sixth of a turn, and then a third of a turn?
讓我們回到這兩個物體的對稱性上。 Galois 意識到這不僅僅是個體的對稱性, 而是個體之間如何相互作用 才真正賦予了一個物體具有對稱性的特點。 如果我做一個神奇的假動作,然後再做一個, 兩個合併起來就是第三個神奇的假動作。 這裡我們了解到 Glaois 開始開發 一種語言來研究 看不見的東西所具有的內在含義,以及 物理物體中存在的對稱性的抽象的概念。 例如,如果我把海星旋轉 六分之一圈, 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, are the names for the rotations. B, for example, rotates the little yellow dot to the B on the starfish. And so on. So what if I do B, which is a sixth of a turn, followed by C, which is a third of a turn? Well let's do that. A sixth of a turn, followed by a third of a turn, the combined effect is as if I had just rotated it by half a turn in one go. So the little table here records how the algebra of these symmetries work. I do one followed by another, the answer is it's rotation D, half a turn. What I if I did it in the other order? Would it make any difference? Let's see. Let's do the third of the turn first, and then the sixth of a turn. Of course, it doesn't make any difference. It still ends up at half a turn.
所以我給它們取了名字。大寫的字母A、B、C、D、E、F, 這些名字旋轉的代號。 例如B,旋轉小黃點, 它位於海星上的B處,諸如此類。 那麼,如果我旋轉B,轉六分之一圈, 其次是C,轉三分之一圈? 嗯,讓我們開始。六分之一圈, 接著是三分之一圈, 合併後的效果就像我剛剛把它一次旋轉了半圈一樣。 那這個小表格記載著 這些對稱的代數是怎麼運作的。 我將一個接一個的旋轉,結果就是 D旋轉了半圈。 如果我按其他順序旋轉呢?會有什麼不同嗎? 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。 當然,沒有什麼差別。 結果仍然是半圈。
And there is some symmetry here in the way the symmetries interact with each other. But this is completely different to the symmetries of the triangle. Let's see what happens if we do two symmetries with the triangle, one after the other. Let's do a rotation by a third of a turn anticlockwise, and reflect in the line through X. Well, the combined effect is as if I had just done the reflection in the line through Z to start with. Now, let's do it in a different order. Let's do the reflection in X first, followed by the rotation by a third of a turn anticlockwise. The combined effect, the triangle ends up somewhere completely different. It's as if it was reflected in the line through Y.
某種對稱方式是通過相互作用得到的。 但這與三角形的對稱性是完全不同的。 讓我們看看如果對三角形 一個接一個的進行兩個對稱旋轉會怎樣。 讓我們逆時針旋轉三分之一圈, 然後在X軸上翻轉。 嗯,合併後的效果就像我剛剛以Z軸翻轉 開始一樣。 現在,讓我們按不同的順序來一次。 我們先在X軸上翻轉, 然後逆時針旋轉三分之一圈。 合併後的效果是三角形停的地方完全不同。 就像是在Y軸上翻轉了一樣。
Now it matters what order you do the operations in. And this enables us to distinguish why the symmetries of these objects -- they both have six symmetries. So why shouldn't we say they have the same symmetries? But the way the symmetries interact enable us -- we've now got a language to distinguish why these symmetries are fundamentally different. And you can try this when you go down to the pub, later on. Take a beer mat and rotate it by a quarter of a turn, then flip it. And then do it in the other order, and the picture will be facing in the opposite direction.
現在看來這與你操作它的順序有關。 這使我們能夠區分 為什麼這些物體的對稱性 都有六個。那麼,為什麼我們不能說 它們有相同的對稱性呢? 但對稱相互作用的方式 使我們——我們現在已經有一種語言 來區分為什麼這些對稱在根本上是不同的。 你也可以嘗試一下,當你去酒吧時,以後去的時候。 拿一個啤酒墊,把它旋轉四分之一圈, 然後翻轉它。然後再按其它順序做, 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. It's almost like little Sudoku tables. You don't see any symmetry twice in any row or column. And, using those rules, he was able to say that there are in fact only two objects with six symmetries. And they'll be the same as the symmetries of the triangle, or the symmetries of the six-pointed starfish. I think this is an amazing development. It's almost like the concept of number being developed for symmetry. In the front here, I've got one, two, three people sitting on one, two, three chairs. The people and the chairs are very different, but the number, the abstract idea of the number, is the same.
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。 這像一個小數獨表。 你看不到任何重複的對稱 出現在任何一欄或一行中。 通過運用那些定律,他可以說 事實上只有兩個物體 有六個對稱。 而且這六個對稱將和三角形的對稱, 或六角海星的對稱是一樣的。 我覺得這是一個驚人的發展。 它幾乎是為了對稱而研製的數的概念。 在這前面,我請一、二、三個人 坐在一、二、三把椅子上。 坐在椅子上的人都不一樣, 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. Here are two very different walls, very different geometric pictures. But, using the language of Galois, we can understand that the underlying abstract symmetries of these things are actually the same. For example, let's take this beautiful wall with the triangles with a little twist on them. You can rotate them by a sixth of a turn if you ignore the colors. We're not matching up the colors. But the shapes match up if I rotate by a sixth of a turn around the point where all the triangles meet. What about the center of a triangle? I can rotate by a third of a turn around the center of the triangle, and everything matches up. And then there is an interesting place halfway along an edge, where I can rotate by 180 degrees. And all the tiles match up again. So rotate along halfway along the edge, and they all match up.
現在我們可以看到這個:我們回到阿爾罕布拉的牆壁。 這有兩面很不一樣的牆壁, 很不相同的幾何圖片。 但是,利用 Galois 的語言 我們可以知道這些東西含有的抽象的對稱 實際上是相同的。 例如,讓我們把這面漂亮的牆 和三角形稍微扭曲一下。 你可以把它們旋轉六分之一圈, 如果忽略他們的顏色。我們不是在做顏色配對 而是在形狀配對,如果我把他們旋轉六分之一圈, 圍繞著所有三角形交彙的一點旋轉。 三角形的中心會怎麼樣?我可以 圍繞著三角形的中心把他們旋轉三分之一圈, 那麼一切就都對上了。 這兒有個有趣的地方,沿著邊的一半 我可以把它旋轉180度。 那麼所有的瓦片又重新匹配了。 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. And we find the same symmetries here, and the same interaction. So, there was a sixth of a turn. A third of a turn where the Z pieces meet. And the half a turn is halfway between the six pointed stars. And although these walls look very different, Galois has produced a language to say that in fact the symmetries underlying these are exactly the same. And it's a symmetry we call 6-3-2.
現在,讓我們移動阿爾罕布拉的一面外觀非常不一樣的牆。 我們在這兒發現同樣的對稱性和同樣的相互作用。 那麼是轉了六分之一轉。轉了三分之一圈時第Z片交會 旋轉半圈時離六角星交會還有一半。 儘管這些牆壁看起來非常不同, Galois 研製出一種語言說, 其實這些東西所具有的對稱性是完全相同的。 這個對稱性我們稱之為6-3-2。
Here is another example in the Alhambra. This is a wall, a ceiling, and a floor. They all look very different. But this language allows us to say that they are representations of the same symmetrical abstract object, which we call 4-4-2. Nothing to do with football, but because of the fact that there are two places where you can rotate by a quarter of a turn, and one by half a turn.
另一個阿爾罕布拉的例子。 這是一面牆、天花板和地板。 它們都看起來都非常不一樣。但是,這種語言讓我們可以說 它們是相同的對稱的抽象物體, 我們稱之為4-4-2。這與足球毫無關係, 而是因為他們都有兩個你可以旋轉 四分之一圈和二分之一圈的地方。
Now, this power of the language is even more, because Galois can say, "Did the Moorish artists discover all of the possible symmetries on the walls in the Alhambra?" And it turns out they almost did. You can prove, using Galois' language, there are actually only 17 different symmetries that you can do in the walls in the Alhambra. And they, if you try to produce a different wall with this 18th one, it will have to have the same symmetries as one of these 17.
現在,這種語言的力量更加強大, 因為 Galois 可能會說, “摩爾藝術家發現了阿爾罕布拉牆上所有可能對稱的 地方了嗎? ” 結果是他們幾乎都發現了。 你可以用伽羅瓦的語言來證明, 實際上只有17種 可以在阿爾罕布拉的牆上得到的不同的對稱。 而且他們,如果你嘗試研製出第18面不同的牆壁, 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. And the power of Galois' mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two-dimensional, three-dimensional, all the way through to the four- or five- or infinite-dimensional space. And that's where I work. I create mathematical objects, symmetrical objects, using Galois' language, in very high dimensional spaces. So I think it's a great example of things unseen, which the power of mathematical language allows you to create.
但這些都是我們可以看到的。 而 Galois 的數學語言的力量 也讓我們能 在看不見的世界裡創造對稱的物體, 超越二維、三維, 全都向四維或五維或無窮維空間發展。 而這正是我在研究的東西。我創建 數學對象和對稱物體, 通過運用 Galois 的語言 在非常高維的空間裡創建。 因此,我認為這是個對於看不見的東西的很好的例子, 數學語言的力量讓你可以創建出來。
So, like Galois, I stayed up all last night creating a new mathematical symmetrical object for you, and I've got a picture of it here. Well, unfortunately it isn't really a picture. If I could have my board at the side here, great, excellent. Here we are. Unfortunately, I can't show you a picture of this symmetrical object. But here is the language which describes how the symmetries interact.
因此,像 Glaois 一樣,我昨晚徹夜未眠 為你們建立了一個新的數學對稱物。 我這兒有一張它的照片。 但是,可惜的是它不是一張真正的照片。我可以把我的圖板 放在這邊嗎?很好,非常好。 這兒。可惜的是我無法向你們展示 這個對稱物的照片。 但這兒有語言能描繪 其對稱性怎麼相互作用的。
Now, this new symmetrical object does not have a name yet. Now, people like getting their names on things, on craters on the moon or new species of animals. So I'm going to give you the chance to get your name on a new symmetrical object which hasn't been named before. And this thing -- species die away, and moons kind of get hit by meteors and explode -- but this mathematical object will live forever. It will make you immortal. In order to win this symmetrical object, what you have to do is to answer the question I asked you at the beginning. How many symmetries does a Rubik's Cube have?
現在這個新的對稱物 還沒有名字。 就像人們給東西命名一樣, 給月球上的隕石坑命名, 或給新動物品種命名一樣。 所以我想給你們機會來給新的對稱物命名, 以前沒有給它取過名字。 而且這個東西——物種會逐漸消失, 月球可能會被隕石撞擊並發生爆炸—— 但是這個數學物體將長存於世。 它將使你不朽。 為了贏得這個對稱物, 你所要做的就是回答我在一開始問的問題。 魔術方塊有多少種對稱呢?
Okay, I'm going to sort you out. Rather than you all shouting out, I want you to count how many digits there are in that number. Okay? If you've got it as a factorial, you've got to expand the factorials. Okay, now if you want to play, I want you to stand up, okay? If you think you've got an estimate for how many digits, right -- we've already got one competitor here. If you all stay down he wins it automatically. Okay. Excellent. So we've got four here, five, six. Great. Excellent. That should get us going. All right.
好吧,我來給你們整理一下。 而不是大家都喊出來,我想讓你們數數有多少位數字 在那個答案裡。好嗎? 如果你得出的結果是一個階乘,那麼你要擴大它的階乘。 好了,現在如果你想參與, 我希望你能站起來,好嗎? 如果你認為你已經估計出了它有多少位數字, 好的——我們在這兒已經有了一位參賽者—— 如果你們都繼續坐著,那麼他就自動贏了。 好的。很好。我們已經有四位、五位、六位。 很好。太好了。讓我們繼續。好了。
Anybody with five or less digits, you've got to sit down, because you've underestimated. Five or less digits. So, if you're in the tens of thousands you've got to sit down. 60 digits or more, you've got to sit down. You've overestimated. 20 digits or less, sit down. How many digits are there in your number? Two? So you should have sat down earlier. (Laughter) Let's have the other ones, who sat down during the 20, up again. Okay? If I told you 20 or less, stand up. Because this one. I think there were a few here. The people who just last sat 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。 因為你們估計少了。 五位數或更少的。那麼,如果你的答案是幾萬的話,你得坐下。 六十或六十多位數的,你必須坐下。 你估計得多了。 二十位數或二十位以下的,坐下。 你的答案是幾位數? 兩個?那你早就該坐下了。 (大笑) 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? 如果我告訴你是二十位或二十位以下,請站起來。 因為這一個。我想應該有一些人。 誰是最後一個坐下去的。
Okay, how many digits do you have in your number? (Laughs) 21. Okay good. How many do you have in yours? 18. So it goes to this lady here. 21 is the closest. It actually has -- the number of symmetries in the Rubik's cube has 25 digits. So now I need to name this object. So, what is your name? I need your surname. Symmetrical objects generally -- spell it for me. G-H-E-Z No, SO2 has already been used, actually, in the mathematical language. So you can't have that one. So Ghez, there we go. That's your new symmetrical object. You are now immortal. (Applause)
好的,你的答案是多少位數? (笑) 21。好的,很好。你的是多少位? 18。那麼是這位女士贏了。 21是最接近的。 實際上,魔術方塊對稱種數的答案 有25位數字。 那麼現在我需要給這個物體命名。 嗯,你叫什麼名字? 我需要你的姓氏。對稱的物體一般—— 為我拼寫一下。 G-H-E-Z 不,SO2已經用過了,其實, 在數學語言裡。你不能用那個名字。 Ghez,就是這個名字啦。這是你的新的對稱物。 你現在是不朽的了。 (鼓掌)
And if you'd like your own symmetrical object, I have a project raising money for a charity in Guatemala, where I will stay up all night and devise an object for you, for a donation to this charity to help kids get into education in Guatemala. And I think what drives me, as a mathematician, are those things which are not seen, the things that we haven't discovered. It's all the unanswered questions which make mathematics a living subject. And I will always come back to this quote from the Japanese "Essays in Idleness": "In everything, uniformity is undesirable. Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." Thank you. (Applause)
而且,如果你想用你自己的對稱物, 我有一個項目,是為在瓜地馬拉的慈善籌錢的, 我可以熬夜為你發明一個物體, 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。 我認為,作為一個數學家, 給我動力的是那些看不到的東西,是我們還未發現的東西。 它們都是懸而未決的問題,這使數學繼續活著 我常常想起引自日本《徒然草》中的這句話: “在一切事物中,一致性是不可取的。 留下若干不完整,會更有趣, 並給予一種仍有發展空間的感覺。 ”謝謝。 (掌聲)