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MarcusduSautoy_2009G-_对称性——现实之谜_
On the 30th of May, 1832, a gunshot was heard ringing out across the 13th arrondissement in Paris. 在1832年5月30日, 人们听到一声枪响, 枪声穿透了巴黎的第十三区
gunshot:n.射击;射程;炮击;adj.射击的;枪炮射击所致的; arrondissement:n.郡,法国的行政区;
(Gunshot) （枪声）
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 . 一个农民,那天早晨正去往市场 朝枪声传来的地方跑了过去, 并发现一名年轻男子正痛得在地上打滚, 显然他在决斗中遭到了枪击。
writhing:v.(常指因剧痛不停地)扭动,翻滚;(writhe的现在分词) agony:n.苦恼;极大的痛苦;临死的挣扎; dueling:n.决斗;竞争;v.决斗;斗争(duel的ing形式); wound:n.创伤,伤口;v.使受伤;受伤,伤害;
The young man's name was Evariste Galois. 这个年轻人名叫伊瓦利斯特?伽罗瓦。
He was a well-known revolutionary in Paris at the time. 他当时在巴黎是一个著名的革命者。
well-known:adj.著名的;众所周知的;清楚明白的; revolutionary:adj.革命性的;革命的;彻底变革的;n.改革者;革命者;
Galois was taken to the local hospital where he died the next day in the arms of his brother. 伽罗瓦被送到了当地的医院, 在医院第二天死在了他兄弟的怀中。
And the last words he said to his brother were, "Don't cry for me Alfred. 他最后对他兄弟说的话是, “阿尔弗雷德不要为我哭泣。
I need all the courage I can muster to die at the age of 20." 我需要聚集我能聚集的所有勇气 让我在20岁时死去。”
muster:vt.召集;对…进行点名;使振作;n.集合;检阅;点名册;集合人员;vi.召集;聚集;
It wasn't in fact, revolutionary politics for which Galois was famous. 实际上,革命政治并不是 使伽罗瓦著名的原因。
politics:n.政治;钩心斗角;政治观点;v.(贬)从事政治活动;(politic的第三人称单数)
But a few years earlier, while still at school, he'd actually cracked one of the big mathematical problems at the time. 而是几年前,当他还在上学时, 他实际上已经破解了 当时重大数学问题之一。
cracked:adj.疯狂;破裂的;有裂纹的;v.破裂;裂开;断裂;(crack的过去分词和过去式) mathematical:adj.数学的,数学上的;精确的;
And he wrote to the academicians in Paris, trying to explain his theory. 随后他写信给巴黎的院士, 尝试解释他的理论。
But the academicians couldn't understand anything that he wrote. 但院士们弄不懂他写的任何东西。
(Laughter) （众笑）
This is how he wrote most of his mathematics . 这就是他怎么写大部分数学理论的。
mathematics:n.数学;数学运算;
So, the night before that duel, he realized this possibly is his last chance to try and explain his great breakthrough . 因此,在决斗的前一天晚上,他意识到 这可能是他最后一次机会 来尝试解释他的重大突破了。
breakthrough:n.突破;开始取得成功之时;adj.突破性的;
So he stayed up the whole night, writing away, trying to explain his ideas. 所以他彻夜未眠,不停地写东西, 试图解释他的想法。
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. 随着黎明的到来,他准备迎接自己的命运。 他把桌子上的一堆文件留给了下一代。
dawn:n.黎明;开端;vt.破晓;出现;被领悟; destiny:n.命运,定数,天命; pile:n.桩;堆;摞;桩柱;v.堆放;摞起;叠放;放置;
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. 也许他彻夜研究数学 是他那天早晨受到枪击且被杀的真正原因。
But contained inside those documents was a new language, a language to understand one of the most fundamental concepts of science -- namely symmetry . 但包含在那些文件中的 是一种新的语言,这种语言能让人们理解 科学的一个最基本的概念, 即对称性。
fundamental:n.基础; adj.十分重大的; namely:adv.也就是;即是;换句话说; symmetry:n.对称(性);整齐,匀称;
Now, symmetry is almost nature's language. 现今,对称性几乎是大自然的语言。
It helps us to understand so many different bits of the scientific world. 它有助于我们了解许多 科学世界里不同的小东西。
scientific:adj.科学的,系统的;
For example, molecular structure . 例如,分子结构。
molecular:adj.[化学]分子的;由分子组成的; structure:n.结构;构造;建筑物;vt.组织;构成;建造;
What crystals are possible we can understand through the mathematics of symmetry. 什么晶体是能让 我们可以通过数学的对称性来了解的？
crystals:n.[晶体]晶体(crystal的复数形式);石英晶体;
In microbiology you really don't want to get a symmetrical object. 在微生物学中,你真的不想研究对称的东西。
microbiology:n.微生物学; symmetrical:adj.匀称的,对称的;
Because they are generally rather nasty . 因为它们一般都比较令人讨厌。
generally:adv.通常;普遍地,一般地; nasty:adj.极差的:令人厌恶的:不友好的:n.令人不愉快的事物:
The swine flu virus, at the moment, is a symmetrical object. 目前的猪流感病毒就是一种结构对称的病毒。
swine:n.猪;卑贱的人;
And it uses the efficiency of symmetry to be able to propagate itself so well. 而且它利用对称的功效 来使自己很好的增殖。
efficiency:n.效率;效能;功效; propagate:vt.传播;传送;繁殖;宣传;vi.繁殖;增殖;
But on a larger scale of biology, actually symmetry is very important, because it actually communicates genetic information. 但就生物学更大范围的而言,对称性事实上非常重要, 因为它能传递遗传信息。
scale:n.规模;比例;鳞;刻度;天平;数值范围;v.衡量;攀登;剥落;生水垢; genetic:adj.基因的;遗传学的;
I've taken two pictures here and I've made them artificially symmetrical. 我带了两张照片到这儿来,并人工的把他们做成了对称的。
artificially:adv.人工地;人为地;不自然地;
And if I ask you which of these you find more beautiful, you're probably drawn to the lower two. 如果我问你们觉得哪些更漂亮, 你们可能会被下面的两张吸引住。
Because it is hard to make symmetry. 因为很难做到对称,
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 . 所以如果你可以使自己对称,那么你在传递一种信号 它意味着你得到了好的遗传基因,你有好的教养, 因而你会有一个好的伴侣。
genes:n.基因;(gene的复数) upbringing:n.教养;养育;抚育; mate:n.助手,大副; v.使配对;
So symmetry is a language which can help to communicate genetic information. 所以,对称性是一种语言,它能有助于传递 遗传信息。
Symmetry can also help us to explain what's happening the Large Hadron Collider in CERN. 对称性还可以帮助我们解释 欧洲粒子物理研究所大型强子对撞机正发生着什么事情。
Hadron:n.[高能]强子(参与强相互作用的基本粒子); Collider:n.对撞机;碰撞机;
Or what's not happening in the Large Hadron Collider in CERN. 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。
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. 为了能够对基本粒子作出预测, 我们可能会在那儿看到的（基本粒子）, 似乎所有的小平面都有某种奇怪的对称形状 当它们在更高维的空间中时。
predictions:n.预测,预言(prediction复数形式); particles:n.微粒,粒子;粒子系统;碎木料(particle的复数形式); facets:n.面状,[材]刻面;切面或瓣面(facet的复数形式); dimensional:adj.空间的;尺寸的;
And I think Galileo summed up, very nicely , the power of mathematics, to understand the scientific world around us. 我认为伽利略很好地概括了 数学的力量: 它让我们对周围的科学世界得以了解。
summed:v.归纳;总计;总结,概括;(sum的过去分词和过去式) nicely:adv.细致地;有吸引力;令人满意;令人愉快;adj.强健的;
He wrote, "The universe cannot be read until we have learnt the language and become familiar with the characters in which it is written. 他写道:“我们无法阅读宇宙, 除非学会它的语言, 且熟悉其写作特点。
familiar:adj.熟悉的;常见的;亲近的;n.常客;密友;
It is written in mathematical language. 它是用数学语言写的。
And the letters are triangles , circles, and other geometric figures, without which means that it's humanly impossible to comprehend a single word." 字母是三角形、圆和其他的几何数字, 没有这些字母就意味着在人力所能及的范围内是不可能 理解任何一个字的。”
triangles:n.[数]三角形,三角型态(triangle的复数形式); geometric:adj.几何学的;[数]几何学图形的; humanly:adv.以人力;像人地;在人力所能及的范围; comprehend:v.理解;包含;由…组成;
But it's not just scientists who are interested in symmetry. 不只是科学家们对对称性感兴趣。
Artists too love to play around with symmetry. 艺术家也喜欢摆弄对称性。
play around:玩耍;胡搞;轻率对待;
They also have a slightly more ambiguous relationship with it. 他们与对称性有一些更模糊的关系。
slightly:adv.些微地,轻微地;纤细地; ambiguous:adj.模糊不清的;引起歧义的;
Here is Thomas Mann talking about symmetry in "The Magic Mountain." 这是托马斯?曼在《魔山》中谈到的对称性。
He has a character describing the snowflake . 他对雪花有这样的描。
describing:v.描述;形容;把…称为;做…运动;(describe的现在分词) snowflake:n.雪花;雪片莲;雪巫;
And he says he " shuddered at its perfect precision , found it deathly , the very marrow of death." 他说,“他因其有完美的精确度而震撼, 发现它死亡的精髓让他想到死亡。”
shuddered:n.发抖;战栗;震动;vi.发抖;战栗; precision:n.精度,[数]精密度;精确;adj.精密的,精确的; deathly:adj.死一般的;致命的;adv.死了一样地;非常; marrow:n.髓,骨髓;精华;活力;
But what artists like to do is to set up expectations of symmetry and then break them. 但艺术家们想要做的是树立对对称性的期望, 然后打破它们。
expectations:n.预料;预期;期待;希望;指望;(expectation的复数)
And a beautiful example of this 就这一点我找到了一个很好的例子,
I found, actually, when I visited a colleague of mine in Japan, Professor Kurokawa. 其实是当我拜访我的同事 在日本的黑川纪章教授时发现的。
colleague:n.同事,同僚;
And he took me up to the temples in Nikko. 他带我到日光市的寺庙去。
And just after this photo was taken we walked up the stairs. 就在拍好这张照片后,我们走上楼梯,
And the gateway you see behind has eight columns , with beautiful symmetrical designs on them. 你们看到的这后面的大门 有八根柱子,都有着漂亮的对称性设计。
columns:n.柱:(通常为)圆形石柱:(书,报纸印刷页上的)栏(column的复数)
Seven of them are exactly the same, and the eighth one is turned upside down . 其中七个是完全一样的, 而第八个是颠倒过来的。
upside down:adj.颠倒的;乱七八糟的;
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." 我就对黑川纪章教授说: “哇,建筑师们肯定要踢自己了, 要是他么发现犯了这么一个错误,把这根柱子弄倒了过来。”
architects:n.建筑师;设计师;创造者;(architect的复数);
And he said, "No no no. It was a very deliberate act." 他说,“不,不,不。这是一个特意的设计。”
deliberate:v.深思熟虑;仔细考虑;反复思考;adj.故意的;蓄意的;存心的;不慌不忙的;
And he referred me to this lovely quote from the Japanese "Essays In Idleness " from the 14th century. 他还向我提到了这个可爱的引述,引自日本 1 4世纪的《闲置的散文》。
referred:v.提到;引用;认为;指示;涉及;(refer的过去式和过去分词) quote:v.引用;报价;举例说明;开价;为(企业的股份)上市;n.引用; Idleness:n.懒惰;闲散;失业;
In which, the essayist wrote, "In everything, uniformity is undesirable . 其中,散文家写道:“在一切事物中, 一致性是不可取的。
essayist:n.随笔作家,散文家;评论家; uniformity:n.均匀性;一致;同样; undesirable:adj.不良的;不受欢迎的;不合需要的;n.不良分子;不受欢迎的人;
Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." 留下一些不完整的东西会更有趣, 而且一致性给人一种没有发展空间的感觉。”
incomplete:n.未完成;adj.不完整的;不完全的;不完善的;
Even when building the Imperial Palace, they always leave one place unfinished. 即使是建造皇宫时, 他们也总是留下一个未完工的地方。
Imperial:adj.帝国的;皇帝的;至高无上的;威严的;n.纸张尺寸;特等品;
But if I had to chose 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 chose the Alhambra in Granada. 但如果我必须选择这世界上的一个建筑, 将其扔到一个荒岛上,且我要在那里度过余生, 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。
desert island:n.热带荒岛; addict:n.有瘾的人;入迷的人;v.使沉溺;使上瘾;
This is a palace celebrating symmetry. 这是一座歌颂对称性的宫殿。
Recently I took my family -- we do this kind of rather nerdy mathematical trips, which my family love. 最近,我带我的家人—— 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。
Recently:adv.最近;新近; nerdy:adj.书呆子式的;n.令人厌烦的人;
This is my son Tamer . You can see he's really enjoying our mathematical trip to the Alhambra. 这是我的儿子塔梅尔。你们可以看到 他真的很喜欢我们在阿尔罕布拉的数学之旅。
Tamer:驯兽师
But I wanted to try and enrich him. 但我想尝试使他变得充实。
enrich:v.充实;使丰富;使富有;使富裕;
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. 我认为学校教的数学存在的一个问题就是 它没有关注数学是如何被运用于 我们所处的这个世界。
embedded:adj.嵌入式的;植入的;内含的;v.嵌入(embed的过去式和过去分词形式);
So, I wanted to open his eyes up to how much symmetry is running through the Alhambra. 所以,我想开拓他的眼界,让他知道 阿尔罕布拉运用着多少对称性。
You see it already. Immediately you go in, the reflective symmetry in the water. 你们已经看到了。你一走进去, 水中有反射的对称。
reflective:adj.反射的;反映的;沉思的;
But it's on the walls where all the exciting things are happening. 但是,所有令人兴奋的事情发生在墙壁上。
The Moorish artists were denied the possibility to draw things with souls. 人们否认摩尔艺术家能够 用灵魂来绘画。
Moorish:adj.摩尔人的,摩尔文化的;摩尔式的建筑风格; denied:v.否认;拒绝;否定;不承认;(deny的过去分词和过去式)
So they explored a more geometric art. 因此,他们探索出一种更加几何化的艺术。
explored:v.探索;勘探;考察;探险;(explore的过去式和过去分词)
And so what is symmetry? 那么什么是对称性？
The Alhambra somehow asks all of these questions. 阿尔罕布拉以某种方式提出了所有这些问题。
somehow:adv.以某种方法;莫名其妙地;
What is symmetry? When [there] are two of these walls, do they have the same symmetries ? 什么是对称性？当[那儿]有两面墙时, 他们有相同的对称性吗？
symmetries:匀称;
Can we say whether they discovered all of the symmetries in the Alhambra? 我们可以说他们是否发现了 阿尔罕布拉所有的对称性吗？
And it was Galois who produced a language to be able to answer some of these questions. 是伽罗瓦研制出了一种语言 能够回答一些这样的问题。
For Galois, symmetry -- unlike for Thomas Mann, which was something still and deathly -- for Galois, symmetry was all about motion . 对伽罗瓦来说,对称性,不同于托马斯曼所说的 对称性是一些静态的和死一般的东西。 对伽罗瓦来说,所有的对称性都是关于运动的。
motion:n.动作;移动;手势;请求;意向;议案;v.运动;打手势;
What can you do to a symmetrical object, move it in some way, so it looks the same as before you moved it? 你能对一个对称性的物体做些什么？ 用某种方法移动它,让它看起来 跟你移动它之前一样？
I like to describe it as the magic trick moves. 我喜欢把这形容为神奇的假动作。
describe:v.描述;形容;把…称为;画出…图形;
What can you do to something? You close your eyes. 你对一些东西可以做些什么？闭上你的眼睛。
I do something, put it back down again. 我动下它,再把它放回到原处。
It looks like it did before it started. 它看起来和动之前一样。
So, for example, the walls in the Alhambra, 那么,例如,阿尔罕布拉的墙壁。
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. 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, 并把它们旋转九十度, 再把他们都放回去,它们非常吻合。
tiles:n.[建]瓷砖(tile的复数形式);v.铺砖(tile的第三人称单数形式); rotate:v.旋转;转动;轮换;使…轮流;
And if you open your eyes again you wouldn't know that they'd moved. 如果你再睁开你的眼睛,你不会知道它们被移动过。
But it's the motion that really characterizes the symmetry inside the Alhambra. 但正是运动才使对称性 在阿尔罕布拉具有特色。
characterizes:vt.描绘…的特性;具有…的特征;vi.塑造人物;
But it's also about producing a language to describe this. 但也要创造一种语言来描绘它。
And the power of mathematics is often to change one thing into another, to change geometry into language. 数学的力量往往 把一样东西变成另一样,把几何变成语言。
geometry:n.几何学;几何结构;
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. 因此,我将带你经历,可能强加一些数学的东西给你们, 所以撑住自己, 强加一些数学的知识让你们了解这种语言是怎么运作的, 这让我们能够捕捉到什么是对称性。
mathematically:adv.算术地,数学上地; brace:n.箍子;夹子;支架;牙箍;背带;大括弧;一对;v.使作准备;使防备;顶住,抵住 enables:v.使得; (enable的第三人称单数) capture:v.俘虏;捕获;攻占;夺得;刻画,描述;n.(被)捕获;(被)俘获
So, let's take these two symmetrical objects here. 那让我们把这两个对称物放到这儿。
Let's take the twisted six-pointed starfish . 拿这个扭曲了的六角海星来说。
twisted:adj.扭曲的; v.使弯曲,使扭曲; (twist的过去分词和过去式) starfish:n.海星;
What can I do to the starfish which makes it look the same? 我怎么做能让这个海星看起来和原来一样呢？
Well, there I rotated it by a sixth of a turn, and still it looks like it did before I started. 嗯,我把它旋转了六分之一圈, 它看起来仍然跟我动过之前一样。
rotated:adj.旋转的;v.旋转;使…转动;循环(rotate的过去分词);
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. 我可以把它旋转三分之一圈, 或者半圈, 或将它恢复到原图,或旋转三分之二圈。
And a fifth symmetry, I can rotate it by five sixths of a turn. 第五种对称,我可以把它旋转六分之五圈。
And those are things that I can do to the symmetrical object that make it look like it did before I started. 这些就是我能对对称物所做的, 可以让它看起来跟我动它们之前一样。
Now, for Galois, there was actually a sixth symmetry. 对伽罗瓦来说,实际上还有第六种对称。
Can anybody think what else I could do to this which would leave it like I did before I started? 大家能想到其它什么办法 可以让它跟我动它之前一样？
I can't flip it because I've put a little twist on it, haven't I? 我不能翻转它,因为我已经把它扭曲了一些,是吧？
flip:n.浏览;空翻;轻抛;捻掷;v.迅速翻动;按(开关);按(按钮);开(或关)(机器等);
It's got no reflective symmetry. 这样它没有反射对称了。
But what I could do is just leave it where it is, pick it up, and put it down again. 但我可以做的就是把它放在原处, 把它拿起来再把它放下。
And for Galois this was like the zeroth symmetry. 对伽罗瓦来说,这就像是第零个对称。
zeroth:adj.[数]零的;
Actually the invention of the number zero was a very modern concept, seventh century A.D., by the Indians. 其实,数字零的发明 是一个非常现代化的概念,它是公元七世纪印度人发明的。
It seems mad to talk about nothing. 什么都没有谈论看起来很疯狂。
And this is the same idea. This is a symmetrical -- 这是同样的概念。这是对称的——
So everything has symmetry, where you just leave it where it is. 所以一切事物都有对称性,把它放在拿起它的地方。
So, this object has six symmetries. 所以这个物体有六种对称。
And what about the triangle? 那三角形呢？
Well, I can rotate by a third of a turn clockwise or a third of a turn anticlockwise . 嗯,我可以把它顺时针旋转三分之一圈 或逆时针旋转三分之一圈。
anticlockwise:adj.逆时针的;反时针方向的;adv.逆时针地;
But now this has some reflectional symmetry. 但现在有反射对称。
I can reflect it in the line through X, or the line through Y, or the line through Z. 。我可以在X轴上翻转它, 或在Y轴上, 或在Z轴上。
Five symmetries and then of course the zeroth symmetry where I just pick it up and leave it where it is. 五种对称,当然还有第零个对称, 我把它拿起来,放回原处。
So both of these objects have six symmetries. 因此,这些物体都有六种对称。
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. 现在,我十分相信数学不是旁观者的运动, 你必须做一些数学运算 才能真正理解它。
believer:n.信徒;相信的人; spectator sport:n.群众爱看的体育运动;观赏性体育运动;
So here is a little question for you. 这儿有个小问题问问你们。
And I'm going to give a prize at the end of my talk for the person who gets closest to the answer. 我将在讲座结束后给一个奖品 给那个给出的答案最接近的人。
The Rubik's Cube. 魔方。
How many symmetries does a Rubik's Cube have? 一个魔方有多少种对称？
How many things can I do to this object and put it down so it still looks like a cube? 有多少种方法可以在动了这个物体, 且把它放下后它仍然看起来像一个立方体？
Okay? So I want you to think about that problem as we go on, and count how many symmetries there are. 好吗？我希望随着讲座的继续,你们可以考虑下这个问题, 数数它有多少种对称。
And there will be a prize for the person who gets closest at the end. 会有一个奖品在讲座结束后那个给出的答案最接近的人。
But let's go back down to symmetries that I got for these two objects. 让我们回到这两个物体的对称性上。
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. 伽罗瓦意识到这不仅仅是个体的对称性, 而是个体之间如何相互作用 才真正赋予了一个物体具有对称性的特点。
individual:n.个人;有个性的人;adj.单独的;个别的; interact:v.互相影响;互相作用;n.幕间剧;幕间休息;
If I do one magic trick move, followed by another, the combination is a third magic trick move. 如果我做一个神奇的假动作,然后再做一个, 两个合并起来就是第三个神奇的假动作。
combination:n.结合;组合;联合;[化学]化合;
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. 这里我们了解到伽罗瓦开始开发 一种语言来研究 看不见的东西所具有的内在含义,以及 物理物体中存在的对称性的抽象的概念。
substance:n.物质;实质;主旨;物品; unseen:adj.看不见的,未看见的;未经预习的;n.(事前未看过原文的)即席翻译; abstract:n.摘要; adj.抽象的; vt.摘要; vi.做摘要; underlying:adj.根本的; v.构成…的基础; (underlie的现在分词) physical:adj.[物]物理的;身体的;物质的;符合自然法则的;n.体格检查;
For example, what if I turn the starfish by a sixth of a turn, and then a third of a turn? 例如,如果我把海星旋转 六分之一圈, 然后再转三分之一圈会,结果会怎样？
what if:如果…怎么办?
So I've given names. The capital letters, A, B, C, D, E, F, are the names for the rotations . 所以我给它们取了名字。大写的字母A、B、C、D、E、F, 这些名字旋转的代号。
rotations:n.旋转;循环(rotation的复数);
B, for example, rotates the little yellow dot to the B on the starfish. And so on. 例如B,旋转小黄点, 它位于海星上的B处,诸如此类。
rotates:旋转;轮流;
So what if I do B, which is a sixth of a turn, followed by C, which is a third of a turn? 那么,如果我旋转B,转六分之一圈, 其次是C,转三分之一圈？
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. 嗯,让我们开始。六分之一圈, 接着是三分之一圈, 合并后的效果就像我刚刚把它一次旋转了半圈一样。
So the little table here records how the algebra of these symmetries work. 那这个小表格记载着 这些对称的代数是怎么运作的。
algebra:n.代数学;
I do one followed by another, the answer is it's rotation D, half a turn. 我将一个接一个的旋转,结果就是 D旋转了半圈。
What I if I did it in the other order? Would it make any difference? 如果我按其他顺序旋转呢？会有什么不同吗？
Let's see. Let's do the third of the turn first, and then the sixth of a turn. 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。
Of course, it doesn't make any difference. 当然,没有什么差别。
It still ends up at half a turn. 结果仍然是半圈。
And there is some symmetry here in the way the symmetries interact with each other. 某种对称方式是通过相互作用得到的。
in the way:妨碍;挡道;
But this is completely different to the symmetries of the triangle. 但这与三角形的对称性是完全不同的。
Let's see what happens if we do two symmetries with the triangle, one after the other . 让我们看看如果对三角形 一个接一个的进行两个对称旋转会怎样。
one after the other:相继地,一个接一个地;
Let's do a rotation by a third of a turn anticlockwise, and reflect in the line through X. 让我们逆时针旋转三分之一圈, 然后在X轴上翻转。
Well, the combined effect is as if I had just done the reflection in the line through Z to start with. 嗯,合并后的效果就像我刚刚以Z轴翻转 开始一样。
reflection:n.反映;沉思;映像;深思;
Now, let's do it in a different order. 现在,让我们按不同的顺序来一次。
Let's do the reflection in X first, followed by the rotation by a third of a turn anticlockwise. 我们先在X轴上翻转, 然后逆时针旋转三分之一圈。
The combined effect, the triangle ends up somewhere completely different. 合并后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轴上翻转了一样。
reflected:adj.反射的;得自他人的;v.反射;思考;(reflect的过去式和过去分词)
Now it matters what order you do the operations in. 现在看来这与你操作它的顺序有关。
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? 这使我们能够区分 为什么这些物体的对称性 都有六个。那么,为什么我们不能说 它们有相同的对称性呢？
distinguish:vt.区分;辨别;使杰出,使表现突出;vi.区别,区分;辨别;
But the way the symmetries interact enable us -- we've now got a language to distinguish why these symmetries are fundamentally different. 但对称相互作用的方式 使我们——我们现在已经有一种语言 来区分为什么这些对称在根本上是不同的。
fundamentally:adv.从根本上;基础地;重要地
And you can try this when you go down to the pub, later on. 你也可以尝试一下,当你去酒吧时,以后去的时候。
Take a beer mat, and rotate it by a quarter of a turn, then flip it. And then do it in the other order, 拿一个啤酒垫,把它旋转四分之一圈, 然后翻转它。然后再按其它顺序做,
And the picture will be facing in the opposite direction. 酒垫上的图将是朝反方向面对你的。
Now Galois produced some laws for how these tables how symmetries interact. 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。
It's almost like little Sudoku tables. 这像一个小数独表。
Sudoku:n.数独,九宫格游戏;
You don't see any symmetry twice in any row or column. 你看不到任何重复的对称 出现在任何一栏或一行中。
And, using those rules, he was able to say that there are in fact only two objects with six symmetries. 通过运用那些定律,他可以说 事实上只有两个物体 有六个对称。
And they'll be the same as the symmetries of the triangle, or the symmetries of the six-pointed starfish. 而且这六个对称将和三角形的对称, 或六角海星的对称是一样的。
I think this is an amazing development. 我觉得这是一个惊人的发展。
It's almost like the concept of number being developed for symmetry. 它几乎是为了对称而研制的数的概念。
In the front here, I've got one, two, three people sitting on one, two, three chairs. 在这前面,我请一、二、三个人 坐在一、二、三把椅子上。
The people on the chairs are very different, but the number, the abstract idea of the number, is the same. 坐在椅子上的人都不一样, 但是数字,数字的抽象的观念,都是一样的。
And we can see this now: we go back to the walls in the Alhambra. 现在我们可以看到这个:我们回到阿尔罕布拉的墙壁。
Here are two very different walls, very different geometric pictures. 这有两面很不一样的墙壁, 很不相同的几何图片。
But, using the language of Galois, we can understand that the underlying abstract symmetries of these things are actually the same. 但是,利用伽罗瓦的语言, 我们可以知道这些东西含有的抽象的对称 实际上是相同的。
For example, let's take this beautiful wall with the triangles with a little twist on them. 例如,让我们把这面漂亮的墙 和三角形稍微扭曲一下。
You can rotate them by a sixth of a turn if you ignore the colors. We're not matching up the colors. 你可以把它们旋转六分之一圈, 如果忽略他们的颜色。我们不是在匹配颜色。
ignore:v.驳回诉讼;忽视;不理睬;
But the shapes match up if I rotate by a sixth of a turn around the point where all the triangles meet. 但在匹配形状,如果我把他们旋转六分之一圈, 围绕着所有三角形交汇的一点旋转。
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. 三角形的中心会怎么样？我可以 围绕着三角形的中心把他们旋转三分之一圈, 那么一切就都匹配上了。
And then there is an interesting place halfway along an edge, where I can rotate by 180 degrees. 这儿有个有趣的地方,沿着边的一半 我可以把它旋转180度。
And all the tiles match up again. 那么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,沿着边的一半旋转,那么他们都能匹配上。
Now, let's move to the very different-looking wall in the Alhambra. 现在,让我们移动阿尔罕布拉的一面外观非常不一样的墙。
And we find the same symmetries here, and the same interaction . 我们在这儿发现同样的对称性和同样的相互作用。
interaction:n.[计]交互,相互作用;相互交流;干扰;
So, there was a sixth of a turn. A third of a turn where the Z pieces meet. 那么是转了六分之一转。转了三分之一圈时第Z片交汇。
And the half a turn is halfway between the six pointed stars. 旋转半圈时离六角星交汇还有一半。
And although these walls look very different, 尽管这些墙壁看起来非常不同,
Galois has produced a language to say that in fact the symmetries underlying these are exactly the same. 伽罗瓦研制出一种语言说, 其实这些东西所具有的对称性是完全相同的。
And it's a symmetry we call 6-3-2. 这个对称性我们称之为6-3-2。
Here is another example in the Alhambra. 另一个阿尔罕布拉的例子。
This is a wall, a ceiling, and a floor. 这是一面墙、天花板和地板。
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, 它们都看起来都非常不一样。但是,这种语言让我们可以说 它们是相同的对称的抽象物体, 我们称之为4-4-2。这与足球毫无关系,
representations:代表;表现(representation的复数);
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. 而是因为他们都有两个你可以旋转 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?" 现在,这种语言的力量更加强大, 因为伽罗瓦可能会说, “摩尔艺术家发现了阿尔罕布拉墙上所有可能对称的 地方了吗？”
And it turns out they almost did. 结果是他们几乎都发现了。
You can prove, using Galois' language, there are actually only 17 different symmetries that you can do in the walls in the Alhambra. 你可以用伽罗瓦的语言来证明, 实际上只有17种 可以在阿尔罕布拉的墙上得到的不同的对称。
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. 而且他们,如果你尝试研制出第18面不同的墙壁, 这面墙肯定与17种对称中的一种对称是相同的。
But these are things that we can see. 但这些都是我们可以看到的。
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. 而伽罗瓦的数学语言的力量 也让我们能 在看不见的世界里创造对称的物体, 超越二维、三维, 全都向四维或五维或无穷维空间发展。
two-dimensional:adj.二维的;缺乏深度的; three-dimensional:adj.三维的;立体的;真实的;
And that's where I work. I create mathematical objects, symmetrical objects, using Galois' language, in very high dimensional spaces. 而这正是我在研究的东西。我创建 数学对象和对称物体, 通过运用伽罗瓦的语言 在非常高维的空间里创建。
So I think it's a great example of things unseen, which the power of mathematical language allows you to create. 因此,我认为这是个对于看不见的东西的很好的例子,, 数学语言的力量让你可以创建出来。
So, like Galois, I stayed up all last night creating a new mathematical symmetrical object for you. 因此,像伽罗瓦一样,我昨晚彻夜未眠 为你们建立了一个新的数学对称物。
And I've got a picture of it here. 我这儿有一张它的照片。
Well, unfortunately it isn't really a picture. If I could have my board at the side here, great, excellent. 但是,可惜的是它不是一张真正的照片。我可以把我的图板 放在这边吗？很好,非常好。
unfortunately:adv.不幸地;
Here we are. Unfortunately I can't show you a picture of this symmetrical object. 这儿。可惜的是我无法向你们展示 这个对称物的照片。
But here is the language which describes how the symmetries interact. 但这儿有语言能描绘 其对称性怎么相互作用的。
describes:v.描述;形容;把…称为;画出…图形;(describe的第三人称单数)
Now this new symmetrical object does not have a name yet. 现在这个新的对称物 还没有名字。
Now, people like getting their names on things, on craters on the moon, or new species of animals. 就像人们给东西命名一样, 给月球上的陨石坑命名, 或给动物新品种命名一样。
craters:n.[地质]火山口; v.形成坑; species:n.[生物]物种;种类;
So I'm going to give you the chance to get your name on a new symmetrical object which hasn't been named before. 所以我想给你们机会来给新的对称物命名, 以前没有给它取过名字。
And this thing -- species die away, and moons kind of get hit by meteors and explode -- but this mathematical object will live forever. 而且这个东西——物种会逐渐消失, 月球可能会被陨石撞击并发生爆炸—— 但是这个数学物体将长存于世。
meteors:n.[天]流星(meteor的复数); explode:v.爆炸;爆裂;勃然(大怒);突然发生(危险);
It will make you immortal . 它将使你不朽。
immortal:adj.不朽的;神仙的;长生的;n.神仙;不朽人物;
In order to win this symmetrical object, what you have to do is to answer the question I asked you at the beginning . 为了赢得这个对称物, 你所要做的就是回答我在一开始问的问题。
at the beginning:首先;从一开始;起初;从头开始;
How many symmetries does a Rubik's Cube have? 一个魔方有多少种对称呢？
Okay, I'm going to sort you out. 好吧,我来给你们整理一下。
Rather than you all shouting out, I want you to count how many digits there are in that number. Okay? 而不是大家都喊出来,我想让你们数数有多少位数字 在那个答案里。好吗？
If you've got it as a factorial, you've got to expand the factorials . 如果你得出的结果是一个阶乘,那么你要扩大它的阶乘。
expand:v.扩张;使膨胀;详述;发展;张开,展开; factorials:n.[数]阶乘;[数]析因(factorial的复数);
Okay, now if you want to play, 好了,现在如果你想参与,
I want you to stand up, okay? 我希望你能站起来,好吗？
If you think you've got an estimate for how many digits, right -- we've already got one competitor here -- 如果你认为你已经估计出了它有多少位数字, 好的——我们在这儿已经有了一位竞争者——
estimate:v.估计;估算;估价;n.估价;(对大小、数量、成本等的)估计;估计的成本;
If you all stay down he wins it automatically . 如果你们都继续坐着,那么他就自动赢了。
automatically:adv.自动地;机械地;无意识地;adj.不经思索的;
Okay. Excellent. So we've got four here, five, six. 好的。很好。我们已经有四位、五位、六位。
Great. Excellent. That should get us going. All right. 很好。太好了。让我们继续。好了。
Anybody with five or less digits, you've got to sit down. 你们中有人的答案是等于或少于五位数的,那你得坐下了。
Because you've underestimated . 因为你们估计少了。
underestimated:v.低估;对…估计不足;轻视(underestimate的过去分词和过去式)
Five or less digits. So, if you're in the tens of thousands you've got to sit down. 五位数或更少的。那么,如果你的答案是几万的话,你得坐下。
60 digits or more, you've got to sit down. 六十或六十多位数的,你必须坐下。
You've over estimated. 你估计得多了。
20 digits or less, sit down. 二十位数或二十位以下的,坐下。
How many digits are there in your number? 你的答案是几位数？
Two? So you should have sat down earlier. 两个？那你早就该坐下了。
(Laughter) （众笑）
Let's have the other ones, who sat down during the 20, up again. Okay? 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？
If I told you 20 or less, stand up. 如果我告诉你是二十位或二十位以下,请站起来。
Because this one. I think there were a few here. 因为这一个。我想应该有一些人。
The people who just last sat down. 谁是最后一个做下去的。
Okay, how many digits do you have in your number? 好的,你的答案是多少位数？
(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。那么是这位女士赢了。
21 is the closest. 21是最接近的。
It actually has -- the number of symmetries in the Rubik's cube has 25 digits. 实际上,魔方对称种数的答案 有25位数字。
So now I need to name this object. 那么现在我需要给这个物体命名。
So, what is your name? 嗯,你叫什么名字？
I need your surname . Symmetrical objects generally -- 我需要你的姓氏。对称的物体一般——
surname:n.姓,姓氏;绰号,别名;vt.给…起别名;给…姓氏;
Spell it for me. 为我拼写一下。
G-H-E-Z G-H-E-Z
No SO2 has already been used, actually, in the mathematical language. So you can't have that one. 不,SO2已经用过了,其实, 在数学语言里。你不能用那个名字。
So Ghez, there we go. That's your new symmetrical object. Ghez,就是这个名字啦。这是你的新的对称物。
You are now immortal. 你现在是不朽的了。
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And if you'd like your own symmetrical object, 而且,如果你想用你自己的对称物,
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. 我有一个项目,是为在瓜地马拉的慈善筹钱的, 我可以熬夜为你发明一个物体, 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。
charity:n.慈善;施舍;慈善团体;宽容;施舍物; Guatemala:n.危地马拉(拉丁美洲国家);危地马拉人; stay up all night:一夜未睡;一整夜不睡觉; devise:v.设计; n.[法]遗赠财产的遗嘱(或其中的条款); donation:n.捐赠;捐赠物;赠送;
And I think what drives me, as a mathematician , are those things which are not seen, the things that we haven't discovered. 我认为,作为一个数学家, 给我动力的是那些看不到的东西,是我们还未发现的东西。
mathematician:n.数学家;善作数字计算的人;
It's all the unanswered questions which make mathematics a living subject. 它们都是悬而未决的问题,这使数学成为一个鲜活的主题。
unanswered:adj.未答复的;无反应的;
And I will always come back to this quote from the Japanese "Essays in Idleness": "In everything, uniformity is undesirable. 我常常想起引自日本《闲置的散文》中的这句话: “在一切事物中,一致性是不可取的。
Leaving something incomplete makes it interesting, and gives one the feeling that there is room for growth." Thank you. 留下一些不完整的东西会更有趣, 一致性并且给人一种没有发展空间的感觉。”谢谢。
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