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TEDGlobal 2009

Marcus du Sautoy: Symmetry, reality's riddle

マーカス・デュ・ソートイ: 対称性の秘密

July 24, 2009

素粒子のスピンから美しいアラベスクまで、世界に溢れる対称性。しかし目に見えるものが全てではありません。 オックスフォードの数学者マーカス・デュ・ソートイが、全ての対称性に共通する見えない数学を語ります。

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

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Double-click the English subtitles below to play the video.
On the 30th of May, 1832,
1832年5月30日
00:18
a gunshot was heard
一発の銃声が
00:22
ringing out across the 13th arrondissement in Paris.
パリ13区に響きました
00:24
(Gunshot)
(銃声)
00:27
A peasant, who was walking to market that morning,
朝市に来ていた農民が
00:28
ran towards where the gunshot had come from,
銃声のした方へ行くと
00:31
and found a young man writhing in agony on the floor,
青年がもがき苦しんでいました
00:33
clearly shot by a dueling wound.
決闘で撃たれたのです
00:37
The young man's name was Evariste Galois.
撃たれた青年の名はエヴァリスト・ガロア
00:40
He was a well-known revolutionary in Paris at the time.
当時は良く知られたパリの革命家でした
00:43
Galois was taken to the local hospital
ガロアは地元の病院で
00:47
where he died the next day in the arms of his brother.
弟アルフレッドに看取られ亡くなりました
00:50
And the last words he said to his brother were,
弟へ残した最後の言葉はこうです
00:53
"Don't cry for me, Alfred.
「アルフレッド 泣かないでくれ
00:55
I need all the courage I can muster
20歳で死ぬには
00:57
to die at the age of 20."
相当の勇気がいるのだから」
00:59
It wasn't, in fact, revolutionary politics
現代 ガロアが有名なのは
01:03
for which Galois was famous.
革命活動ではありません
01:05
But a few years earlier, while still at school,
その数年前 まだ学生のときに
01:07
he'd actually cracked one of the big mathematical
彼は当時の数学の最大の謎のひとつを
01:10
problems at the time.
解明したのです
01:12
And he wrote to the academicians in Paris,
この理論を説明した論文を
01:14
trying to explain his theory.
パリ科学アカデミーに提出しましたが
01:16
But the academicians couldn't understand anything that he wrote.
アカデミーのメンバーは 書かれた内容が
理解できませんでした
01:18
(Laughter)
(笑)
01:21
This is how he wrote most of his mathematics.
これが彼が数学を記述する流儀でした
01:22
So, the night before that duel, he realized
決闘の前夜に
01:25
this possibly is his last chance
これが最後のチャンスだと思い
01:27
to try and explain his great breakthrough.
彼の偉大な業績を書き残そうとしました
01:30
So he stayed up the whole night, writing away,
一晩中起きて書き続け 彼のアイディアを
01:32
trying to explain his ideas.
残そうとしました
01:35
And as the dawn came up and he went to meet his destiny,
そして夜が明けて 彼は運命の時を迎えたのです
01:37
he left this pile of papers on the table for the next generation.
机の上には 紙の山が残されていました
01:40
Maybe the fact that he stayed up all night doing mathematics
もしかしたら 徹夜して数学をしていたのが
01:44
was the fact that he was such a bad shot that morning and got killed.
決闘で負けてしまった理由かも知れません
01:47
But contained inside those documents
彼が書き残した書類には
01:50
was a new language, a language to understand
科学の根底にある概念を理解するための
01:52
one of the most fundamental concepts
新しい言語が記されていました
01:55
of science -- namely symmetry.
その概念とは 対称性です
01:57
Now, symmetry is almost nature's language.
対称性は 自然界の言語とも言えます
02:00
It helps us to understand so many
非常に多岐にわたる科学的なことを
02:02
different bits of the scientific world.
理解する助けとなります
02:04
For example, molecular structure.
たとえば分子構造では
02:06
What crystals are possible,
どんな結晶構造がありえるのか
02:08
we can understand through the mathematics of symmetry.
対称性の数学を通して理解できます
02:10
In microbiology you really don't want to get a symmetrical object,
微生物学では 対称的な物体は
02:14
because they are generally rather nasty.
大抵やっかいなものです
02:16
The swine flu virus, at the moment, is a symmetrical object.
豚インフルエンザのウイルスは
02:18
And it uses the efficiency of symmetry
対称性が効率的であることを利用して
02:21
to be able to propagate itself so well.
高い感染力をもっています
02:23
But on a larger scale of biology, actually symmetry is very important,
より大きなスケールの生物学では 遺伝子情報の伝達が
02:27
because it actually communicates genetic information.
対称性によって実現されています
02:30
I've taken two pictures here and I've made them artificially symmetrical.
二枚の写真と それを左右対称に
加工したものがあります
02:32
And if I ask you which of these you find more beautiful,
どちらが美しいと感じるでしょうか
02:36
you're probably drawn to the lower two.
おそらく 下の二枚でしょう
02:39
Because it is hard to make symmetry.
対称を作るのは困難なので
02:41
And if you can make yourself symmetrical, you're sending out a sign
あなたの体が左右対称ならば
02:44
that you've got good genes, you've got a good upbringing
よい遺伝子を持ち 健康に育った
02:46
and therefore you'll make a good mate.
良い結婚相手だという証になります
02:49
So symmetry is a language which can help to communicate
つまり対称性は 遺伝情報の伝達に役立つ
02:51
genetic information.
言語なのです
02:54
Symmetry can also help us to explain
CERNの大型ハドロン衝突型加速器の中で
02:56
what's happening in the Large Hadron Collider in CERN.
何が起きているのかを説明するにも役に立ちます
02:58
Or what's not happening in the Large Hadron Collider in CERN.
あるいは 何が起きていないのかを...
03:01
To be able to make predictions about the fundamental particles
見つかるかもしれない素粒子を
03:04
we might see there,
予測する助けにもなります
03:06
it seems that they are all facets of some strange symmetrical shape
素粒子は全て
高次元空間にある対称形の
03:08
in a higher dimensional space.
色々な「断面」である可能性があります
03:12
And I think Galileo summed up, very nicely,
身の回りの科学的な世界を理解するのに
03:14
the power of mathematics
数学がどれほど強力なのか
03:16
to understand the scientific world around us.
ガリレオがうまく表現しています
03:18
He wrote, "The universe cannot be read
「宇宙という壮大な書物は
03:20
until we have learnt the language
それを記述している言葉を学び
03:22
and become familiar with the characters in which it is written.
その文字に親しまなければ解読できない
03:24
It is written in mathematical language,
それは数学という言葉で書かれていて
03:27
and the letters are triangles, circles and other geometric figures,
その文字は三角、円、その他の幾何学的図形であり
03:29
without which means it is humanly impossible
それらを知らなければ
03:33
to comprehend a single word."
一言も理解できない」
03:35
But it's not just scientists who are interested in symmetry.
しかし 対称性に興味を持つのは
科学者だけではありません
03:38
Artists too love to play around with symmetry.
芸術家も対称性を愛しています
03:41
They also have a slightly more ambiguous relationship with it.
彼らは すこし違った視点を持っています
03:44
Here is Thomas Mann talking about symmetry in "The Magic Mountain."
小説「魔の山」の中でトーマス・マンは
03:47
He has a character describing the snowflake,
雪の結晶について登場人物にこう語らせています
03:50
and he says he "shuddered at its perfect precision,
「ぞっとするほどの完璧さで
03:53
found it deathly, the very marrow of death."
死の核心に思えた」
03:56
But what artists like to do is to set up expectations
しかし 芸術家は対称性を暗示して
03:59
of symmetry and then break them.
それをわざと壊したがるものです
04:01
And a beautiful example of this
日本を訪れたときに
04:03
I found, actually, when I visited a colleague of mine
研究仲間の黒川教授のところへ行ったとき
04:05
in Japan, Professor Kurokawa.
とても良い例に出会いました
04:07
And he took me up to the temples in Nikko.
彼は日光東照宮へ連れて行ってくれました
04:09
And just after this photo was taken we walked up the stairs.
これは階段を登ったすぐの場所で撮った写真で
04:12
And the gateway you see behind
背後に門が見えます
04:15
has eight columns, with beautiful symmetrical designs on them.
門には美しい対称形の柱が8本あります
04:17
Seven of them are exactly the same,
そのうち7本はまったく同じですが
04:20
and the eighth one is turned upside down.
一本だけが上下逆になっています
04:22
And I said to Professor Kurokawa,
「これをつくった建築家は
04:25
"Wow, the architects must have really been kicking themselves
こいつを間違えて逆さまにしたって気付いた時に
04:27
when they realized that they'd made a mistake and put this one upside down."
しまった! と思っただろうねぇ」と私が言うと
04:29
And he said, "No, no, no. It was a very deliberate act."
教授は「いやいや、これはわざとなんだよ」と答えました
04:32
And he referred me to this lovely quote from the Japanese
そして 日本の古典「徒然草」からの
04:35
"Essays in Idleness" from the 14th century,
素敵な一節を教えてくれたのです
04:37
in which the essayist wrote, "In everything,
「何でも全部が
04:40
uniformity is undesirable.
完全に整っているのはよくない
04:42
Leaving something incomplete makes it interesting,
やり残したことを そのままにしておくのが面白く
04:45
and gives one the feeling that there is room for growth."
先に楽しみを残すことにもなる」
04:47
Even when building the Imperial Palace,
皇居を建てるときにすら
04:50
they always leave one place unfinished.
常に一箇所 未完成の場所を残します
04:52
But if I had to choose one building in the world
でも私がもし建物をひとつ選んで
04:56
to be cast out on a desert island, to live the rest of my life,
その中で一生を暮らすならば
04:59
being an addict of symmetry, I would probably choose the Alhambra in Granada.
「対称性中毒」の私なら
アルハンブラ宮殿を選びます
05:02
This is a palace celebrating symmetry.
この宮殿は対称性の極致です
05:06
Recently I took my family --
私はよく家族と
05:08
we do these rather kind of nerdy mathematical trips, which my family love.
数学オタク的な旅行をするのですが
05:10
This is my son Tamer. You can see
これは私の息子 タマーです
05:13
he's really enjoying our mathematical trip to the Alhambra.
「数学的」な旅行を
とても楽しんでいるようですね
05:15
But I wanted to try and enrich him.
しかし もっと息子に見せたいものがありました
05:18
I think one of the problems about school mathematics
学校の数学の授業では
05:21
is it doesn't look at how mathematics is embedded
現実の世界にどう数学が関わっているのか
05:23
in the world we live in.
教えてくれません
05:25
So, I wanted to open his eyes up to
ですから アルハンブラ宮殿の対称性を
05:27
how much symmetry is running through the Alhambra.
私は息子に見せたかったのです
05:29
You see it already. Immediately you go in,
宮殿に入ってすぐにわかるのは
05:32
the reflective symmetry in the water.
水に映し出された対称性です
05:34
But it's on the walls where all the exciting things are happening.
でも 特にすばらしいのは
ここの壁です
05:36
The Moorish artists were denied the possibility
ムーア人の芸術家たちは 偶像を描くことを
05:39
to draw things with souls.
禁止されていました
05:41
So they explored a more geometric art.
ですから彼らは
幾何的な芸術を追求しました
05:43
And so what is symmetry?
では 対称性とは何でしょう?
05:45
The Alhambra somehow asks all of these questions.
アルハンブラに行くと
次々と質問が頭に浮かびます
05:47
What is symmetry? When [there] are two of these walls,
対称性とは何か?
2つの模様の対称性が
05:50
do they have the same symmetries?
同じだと言えるのは
どんな時か?
05:52
Can we say whether they discovered
ムーア人は 可能な対称性の型を全て
05:54
all of the symmetries in the Alhambra?
アハンブラ宮殿に残したのでしょうか
05:56
And it was Galois who produced a language
ガロアは まさにこの疑問に答えるための
05:59
to be able to answer some of these questions.
言語を作り出したのです
06:01
For Galois, symmetry -- unlike for Thomas Mann,
トーマス・マンにとって対称とは
06:04
which was something still and deathly --
死であり 静止したものでしたが
06:07
for Galois, symmetry was all about motion.
ガロアにとって対称性は動きでした
06:09
What can you do to a symmetrical object,
対称的な図形を動かして
06:12
move it in some way, so it looks the same
元の状態と同じに見えるようにするには
06:14
as before you moved it?
何ができるでしょう
06:16
I like to describe it as the magic trick moves.
手品の動きに例えて説明しましょう
06:18
What can you do to something? You close your eyes.
みなさんが目を閉じている間に
06:20
I do something, put it back down again.
こっそりと動かして もとの場所に戻すと
06:22
It looks like it did before it started.
最初の状態と同じに見えるのは
どんな動かし方でしょう?
06:24
So, for example, the walls in the Alhambra --
たとえばアルハンブラの壁のタイルなら
06:26
I can take all of these tiles, and fix them at the yellow place,
黄色の点の位置を軸にして
06:28
rotate them by 90 degrees,
90°回転すると
06:32
put them all back down again and they fit perfectly down there.
完全にもとの模様と一致します
06:34
And if you open your eyes again, you wouldn't know that they'd moved.
閉じていた目を開けても
動かされたと気付きません
06:37
But it's the motion that really characterizes the symmetry
しかし この動きこそがアルハンブラ宮殿の対称性を
06:40
inside the Alhambra.
特徴付けるのです
06:43
But it's also about producing a language to describe this.
同時に対称性を記述する言語にも繋がります
06:45
And the power of mathematics is often
数学は あるものを別のものに変換することで
06:47
to change one thing into another, to change geometry into language.
力を発揮します
ここでは「幾何学」を「言語」に変換します
06:50
So I'm going to take you through, perhaps push you a little bit mathematically --
ここから 数学的に少し踏み込んだお話をしましょう
06:54
so brace yourselves --
心の準備はいいですか?
06:57
push you a little bit to understand how this language works,
少し踏み込んで この言語がどうやって
06:59
which enables us to capture what is symmetry.
対称とは何かを捉えるか説明しましょう
07:02
So, let's take these two symmetrical objects here.
ふたつの対称的な図形があるとしましょう
07:04
Let's take the twisted six-pointed starfish.
少しねじれた ヒトデ型の図形です
07:07
What can I do to the starfish which makes it look the same?
どう動かすと 元と同じに見えるでしょうか
07:09
Well, there I rotated it by a sixth of a turn,
そう 1/6回転させれば
07:12
and still it looks like it did before I started.
元と同じように見えます
07:15
I could rotate it by a third of a turn,
1/3回転でもいいですし
07:17
or a half a turn,
半回転でも
07:20
or put it back down on its image, or two thirds of a turn.
もしくは2/3回転でもいいですし
07:22
And a fifth symmetry, I can rotate it by five sixths of a turn.
5番目の対称性として 5/6回転でも良いでしょう
07:25
And those are things that I can do to the symmetrical object
対称図形を動かして元と同じに見えるようにするために
07:29
that make it look like it did before I started.
このような操作をすることができます
07:32
Now, for Galois, there was actually a sixth symmetry.
実はガロアにとって 6番目の対称性がありました
07:35
Can anybody think what else I could do to this
図形が元と同じに見えるようになる
07:38
which would leave it like I did before I started?
操作方法は他にあるでしょうか?
07:40
I can't flip it because I've put a little twist on it, haven't I?
突起が少しひねれているので 裏返しにはできません
07:43
It's got no reflective symmetry.
反射(鏡映)対称性はないのです
07:46
But what I could do is just leave it where it is,
でも 動かさないでおくことはできます
07:48
pick it up, and put it down again.
持ち上げて そのまま戻す
07:51
And for Galois this was like the zeroth symmetry.
ガロアにとっては これがゼロ番目の対称性でした
07:53
Actually, the invention of the number zero
実際 ゼロという概念の発明はつい最近で
07:56
was a very modern concept, seventh century A.D., by the Indians.
紀元前7世紀のインド人によるものです
07:59
It seems mad to talk about nothing.
「何もない」ということを数えるのも変ですが
08:02
And this is the same idea. This is a symmetrical --
このゼロ番目の対称性も
同じアイデアなのです
08:05
so everything has symmetry, where you just leave it where it is.
どんなものにも「そのまま動かさない」
という対称性があります
08:07
So, this object has six symmetries.
この図形の場合は6種類の対称性があります
08:09
And what about the triangle?
三角形の場合はどうでしょう
08:12
Well, I can rotate by a third of a turn clockwise
時計周りに1/3回転させるか
08:14
or a third of a turn anticlockwise.
反時計周りに1/3回転させられます
08:18
But now this has some reflectional symmetry.
今度は反射(鏡映)対称なので
08:20
I can reflect it in the line through X,
Xを通る線を軸に反転させるか
08:22
or the line through Y,
Yを通る線か
08:24
or the line through Z.
Zを通る線でも
08:26
Five symmetries and then of course the zeroth symmetry
5種類の対称性と それに加えて 「そのまま」の
08:28
where I just pick it up and leave it where it is.
ゼロ番目の対称性があります
08:31
So both of these objects have six symmetries.
2つの図形は両方とも6つの対称性があります
08:34
Now, I'm a great believer that mathematics is not a spectator sport,
数学はスポーツ観戦とは違って
08:37
and you have to do some mathematics
理解するためには
08:40
in order to really understand it.
実際に計算をするべきです
08:42
So here is a little question for you.
そこで簡単な質問をしましょう
08:44
And I'm going to give a prize at the end of my talk
そして この講演の最後に
08:46
for the person who gets closest to the answer.
正解に一番近かった人には 賞をあげます
08:48
The Rubik's Cube.
ルービックキューブです
08:50
How many symmetries does a Rubik's Cube have?
ルービックキューブには 対称性はいくつあるでしょう?
08:52
How many things can I do to this object
この様に動かして
08:55
and put it down so it still looks like a cube?
キューブの形を保つ操作はいくつあるでしょう
08:57
Okay? So I want you to think about that problem as we go on,
いいですか? これからしばらくの間
08:59
and count how many symmetries there are.
いくつ対称性があるか 考えてみてください
09:02
And there will be a prize for the person who gets closest at the end.
そして 正解に最も近かった人に賞を差し上げます
09:04
But let's go back down to symmetries that I got for these two objects.
それではまた 先程の2つの図形の話に戻りましょう
09:08
What Galois realized: it isn't just the individual symmetries,
物体の対称性を特徴付けるのは
09:12
but how they interact with each other
物体の持つ ひとつひとつの対称型ではなく
09:15
which really characterizes the symmetry of an object.
それらの関連性だと ガロアは気付きました
09:17
If I do one magic trick move followed by another,
もし複数の種類の操作を続けて行えば
09:21
the combination is a third magic trick move.
この組み合わせは別の操作に相当します
09:24
And here we see Galois starting to develop
これこそが ガロアが作り出した
09:26
a language to see the substance
この物体の対称性にひそむ
09:28
of the things unseen, the sort of abstract idea
抽象的なアイデアを理解するための
09:31
of the symmetry underlying this physical object.
言語だったのです
09:33
For example, what if I turn the starfish
例えば ヒトデ型を まず1/6回転させ
09:36
by a sixth of a turn,
次に1/3回転したら
09:39
and then a third of a turn?
どうなるでしょう?
09:41
So I've given names. The capital letters, A, B, C, D, E, F,
説明のために 回転操作に名前を付ましょう
09:43
are the names for the rotations.
A, B, C, D, E, Fです
09:46
B, for example, rotates the little yellow dot
例えば B は 黄色の点が
09:48
to the B on the starfish. And so on.
図形のbの点に合うよう回転させます
09:51
So what if I do B, which is a sixth of a turn,
B つまり1/6回転の次に
09:54
followed by C, which is a third of a turn?
C つまり1/3回転したら どうなりますか?
09:56
Well let's do that. A sixth of a turn,
やってみましょう まず 1/6回転
09:59
followed by a third of a turn,
続けて1/3回転
10:01
the combined effect is as if I had just rotated it by half a turn in one go.
これらを足し合わせると 半回転になります
10:03
So the little table here records
この表に記録されるのは
10:08
how the algebra of these symmetries work.
これらの対称性の計算の結果です
10:10
I do one followed by another, the answer is
1つ目の操作をして
続けて2番目の操作をすると
10:13
it's rotation D, half a turn.
結果はDの回転 つまり1/2回転です
10:15
What I if I did it in the other order? Would it make any difference?
ではもし この順番を変えたら違いはあるでしょうか
10:17
Let's see. Let's do the third of the turn first, and then the sixth of a turn.
まず1/3回転させて 次に1/6回転させます
10:20
Of course, it doesn't make any difference.
当然 同じ結果になります
10:24
It still ends up at half a turn.
つまり1/2回転します
10:26
And there is some symmetry here in the way the symmetries interact with each other.
この操作の組み合わせにも対称性が見られます
10:28
But this is completely different to the symmetries of the triangle.
三角形の場合は まったく違います
10:33
Let's see what happens if we do two symmetries
二種類の対称的操作を続けて行うと
10:36
with the triangle, one after the other.
三角形がどうなるか見てみましょう
10:38
Let's do a rotation by a third of a turn anticlockwise,
反時計回りに1/3回転してから
10:40
and reflect in the line through X.
Xを通る線で鏡像反転させます
10:43
Well, the combined effect is as if I had just done the reflection in the line through Z
組み合わせた結果はZを通る軸で
鏡像反転させた場合と
10:45
to start with.
同じです
10:49
Now, let's do it in a different order.
今度は違う順番にしてみましょう
10:51
Let's do the reflection in X first,
X軸での反転を先に行って
10:53
followed by the rotation by a third of a turn anticlockwise.
それから反時計回りに1/3回転させます
10:55
The combined effect, the triangle ends up somewhere completely different.
結果は まったく違ったものになります
10:59
It's as if it was reflected in the line through Y.
これはYを通る線で反転させたのと同じです
11:02
Now it matters what order you do the operations in.
この場合は 順番が問題になるのです
11:05
And this enables us to distinguish
この2つの図形は両方とも
11:08
why the symmetries of these objects --
6つの対称性を持っていましたが
11:10
they both have six symmetries. So why shouldn't we say
同じ対称性を持っていると 言えるのでしょうか?
11:12
they have the same symmetries?
同じ対称性を持っていると 言えるのでしょうか?
11:14
But the way the symmetries interact
対称性の相互関係を知ることで
11:16
enable us -- we've now got a language
実は根本的に違う対称性だということが
11:18
to distinguish why these symmetries are fundamentally different.
言えるようになったのです
11:20
And you can try this when you go down to the pub, later on.
ビールのコースターを使って簡単に自分でも試せます
11:23
Take a beer mat and rotate it by a quarter of a turn,
コースターを90°回転させて反転させます
11:26
then flip it. And then do it in the other order,
それから今度は逆の順序で同じことをすると
11:29
and the picture will be facing in the opposite direction.
絵柄が最初と上下逆になります
11:31
Now, Galois produced some laws for how these tables -- how symmetries interact.
ガロアはこの表のような対称性の相互関係について
法則を生み出しました
11:35
It's almost like little Sudoku tables.
それはまるで数独の枡目のように
11:39
You don't see any symmetry twice
同じ対称操作は縦・横の各列に
11:41
in any row or column.
ひとつしか現れません
11:43
And, using those rules, he was able to say
そして この法則を使うことで
11:45
that there are in fact only two objects
実は6つ対称性を持つ図形は
11:49
with six symmetries.
2種類だけだと結論付けたのです
11:51
And they'll be the same as the symmetries of the triangle,
例の三角形と同等の対称性を持つものか
11:53
or the symmetries of the six-pointed starfish.
あのヒトデと同等の対称性を持つものです
11:56
I think this is an amazing development.
実に素晴しい成果です
11:58
It's almost like the concept of number being developed for symmetry.
対称性を「数」の様に考える
概念の発明だと言えます
12:00
In the front here, I've got one, two, three people
この会場の前の列に 1、2、3人の人が
12:04
sitting on one, two, three chairs.
1、2、3つの椅子に座っています
12:06
The people and the chairs are very different,
椅子と人間とは全く違いますが
12:08
but the number, the abstract idea of the number, is the same.
その数 つまり数という抽象的な概念では同一です
12:11
And we can see this now: we go back to the walls in the Alhambra.
アルハンブラの壁のタイル模様でも同じ事がわかります
12:14
Here are two very different walls,
ここにあるのは二つの壁の
12:17
very different geometric pictures.
まったく違う模様です
12:19
But, using the language of Galois,
しかしガロアの言語を使うと
12:21
we can understand that the underlying abstract symmetries of these things
これらの根底にある抽象的な対称性は
12:23
are actually the same.
同じだと分かります
12:26
For example, let's take this beautiful wall
たとえば この少しねじれた三角形の
12:28
with the triangles with a little twist on them.
模様をご覧ください
12:30
You can rotate them by a sixth of a turn
色を無視することにすると
12:33
if you ignore the colors. We're not matching up the colors.
1/6回転させることができます
12:35
But the shapes match up if I rotate by a sixth of a turn
1/6回転させると 色は揃いませんが
12:37
around the point where all the triangles meet.
この中央の点を軸に すべての三角形が重なります
12:40
What about the center of a triangle? I can rotate
三角形の中心を軸にしたらどうでしょう
12:43
by a third of a turn around the center of the triangle,
1/3回転させれば
12:45
and everything matches up.
元の図形に重なります
12:47
And then there is an interesting place halfway along an edge,
それから 辺の中間にも興味深い場所があります
12:49
where I can rotate by 180 degrees.
180°回転させると
12:51
And all the tiles match up again.
タイルが重なります
12:53
So rotate along halfway along the edge, and they all match up.
つまり辺の中点で回転させても元の図形に重なるのです
12:56
Now, let's move to the very different-looking wall in the Alhambra.
今度は まったく違う模様の壁を見てみましょう
12:59
And we find the same symmetries here, and the same interaction.
同じ対称性と 同じ相関関係を見付けられます
13:03
So, there was a sixth of a turn. A third of a turn where the Z pieces meet.
1/6回転でも 1/3でもZ字型の部分が重なります
13:06
And the half a turn is halfway between the six pointed stars.
六芒星の中央で半回転させることもできます
13:11
And although these walls look very different,
これらの壁の模様はまったく違って見えますが
13:15
Galois has produced a language to say
ガロアの発明した言語を使えば
13:17
that in fact the symmetries underlying these are exactly the same.
根底にある対称性は 完全に同一だと言えるのです
13:20
And it's a symmetry we call 6-3-2.
この例は6-3-2の対称性と呼ばれています
13:23
Here is another example in the Alhambra.
これは アルハンブラ宮殿のまた別の図形です
13:26
This is a wall, a ceiling, and a floor.
壁、天井、そして床の模様です
13:28
They all look very different. But this language allows us to say
まったく違うように見えますが ガロアの言語によれば
13:31
that they are representations of the same symmetrical abstract object,
対象性では同等な抽象的モチーフの
異なる表現型なのです
13:34
which we call 4-4-2. Nothing to do with football,
この型は 1/4回転できる位置が二カ所
13:38
but because of the fact that there are two places where you can rotate
そして半回転できる位置が一カ所あるので
13:40
by a quarter of a turn, and one by half a turn.
4-4-2 と呼ばれています
13:43
Now, this power of the language is even more,
このガロアの言語は 更に強力です
13:47
because Galois can say,
「ムーア人の芸術家は
13:49
"Did the Moorish artists discover all of the possible symmetries
あり得る全ての対称性を見付けだしたのか?」
13:51
on the walls in the Alhambra?"
という質問をしたり その質問に
13:54
And it turns out they almost did.
「ほぼ全て見付けた」と答えられるのです
13:56
You can prove, using Galois' language,
ガロアの言語を使うことで
13:58
there are actually only 17
アルハンブラ宮殿の壁では
14:00
different symmetries that you can do in the walls in the Alhambra.
全部で17の異なる対称性が存在可能であり
14:02
And they, if you try to produce a different wall with this 18th one,
もしも18番目の模様を考え出したとしても
14:06
it will have to have the same symmetries as one of these 17.
それは必ず 先程の17の模様のどれかと
14:09
But these are things that we can see.
同じ対称性になってしまう と分かるのです
14:14
And the power of Galois' mathematical language
さらにこのガロアの言語を使うと
14:16
is it also allows us to create
見たこともない世界の
14:18
symmetrical objects in the unseen world,
対称的な図形を作り出すこともできます
14:20
beyond the two-dimensional, three-dimensional,
二次元、三次元を越えて
14:23
all the way through to the four- or five- or infinite-dimensional space.
四次元、五次元、そして無限の次元空間までも
14:25
And that's where I work. I create
それが私の研究対象です
14:28
mathematical objects, symmetrical objects,
高次元空間に 数学的な物体 対称的な物体を
14:30
using Galois' language,
ガロアの言語を使って
14:32
in very high dimensional spaces.
作り出しているのです
14:34
So I think it's a great example of things unseen,
見えないものを作り出せる
14:36
which the power of mathematical language allows you to create.
それが数学の力のすばらしい例だと思います
14:38
So, like Galois, I stayed up all last night
そこで私も ガロアのように昨晩徹夜して
14:42
creating a new mathematical symmetrical object for you,
皆さんのために新しい数学的な
対称的物体を作ってみました
14:44
and I've got a picture of it here.
これがその図です
14:48
Well, unfortunately it isn't really a picture. If I could have my board
そう 図とは言えませんね
14:50
at the side here, great, excellent.
そのボードを持ってきてくれる?
14:53
Here we are. Unfortunately, I can't show you
残念ながら この対称的物体の図を
14:55
a picture of this symmetrical object.
お見せすることは不可能です
14:57
But here is the language which describes
しかし ここにある言語で
どんな対称的性質があるか
14:59
how the symmetries interact.
記述してあります
15:02
Now, this new symmetrical object
さて この新しい対称的物体には
15:04
does not have a name yet.
まだ名前がついていません
15:06
Now, people like getting their names on things,
月のクレータや 動物の新種に
15:08
on craters on the moon
自分の名前をつけるのが好きな
15:10
or new species of animals.
人々がいますよね
15:12
So I'm going to give you the chance to get your name on a new symmetrical object
ですから みなさんの名前を
新しい対称的物体につける
15:14
which hasn't been named before.
チャンスをさし上げましょう
15:18
And this thing -- species die away,
生物の種は絶滅しますし
15:20
and moons kind of get hit by meteors and explode --
クレータは別の隕石の衝突で消滅しますが
15:22
but this mathematical object will live forever.
この数学的な物体は永遠のものです
15:25
It will make you immortal.
あなたを永遠不滅にする力を持っています
15:27
In order to win this symmetrical object,
この対称的物体を勝ち取るために
15:29
what you have to do is to answer the question I asked you at the beginning.
みなさんには 私が冒頭でお聞きした
質問に答えていただきたい
15:32
How many symmetries does a Rubik's Cube have?
ルービックキューブには対称性がいくつあるでしょうか?
15:35
Okay, I'm going to sort you out.
ためしてみましょう
15:39
Rather than you all shouting out, I want you to count how many digits there are
みなさんにそれぞれ答えてもらう代わりに
15:41
in that number. Okay?
思った数が何桁か数えてください
15:44
If you've got it as a factorial, you've got to expand the factorials.
もし答えを数の階乗で考えているなら
展開しておいてください
15:46
Okay, now if you want to play,
では このゲームに参加したい方は
15:49
I want you to stand up, okay?
桁の数の見当がついたら
15:51
If you think you've got an estimate for how many digits,
立ち上がっていただけますか?
15:53
right -- we've already got one competitor here.
一人目の参加者がこちらにいます
15:55
If you all stay down he wins it automatically.
他に誰もいなければ 彼が勝者になりますよ
15:58
Okay. Excellent. So we've got four here, five, six.
よし 4人目...5、6人参加です
16:00
Great. Excellent. That should get us going. All right.
すばらしい そろそろ始められますね
16:03
Anybody with five or less digits, you've got to sit down,
5桁よりも小さな数を考えた方は座ってください
16:08
because you've underestimated.
見積もりが少なすぎです
16:11
Five or less digits. So, if you're in the tens of thousands you've got to sit down.
5桁以下_つまり 10000以下の方は座ってください
16:13
60 digits or more, you've got to sit down.
60桁よりも大きい方も座ってください
16:17
You've overestimated.
大きすぎです
16:20
20 digits or less, sit down.
20桁より下のみなさんもハズレです
16:22
How many digits are there in your number?
あなたの考えた数字は何桁?
16:26
Two? So you should have sat down earlier.
2桁? では もっと前に座らないと
16:31
(Laughter)
(笑)
16:33
Let's have the other ones, who sat down during the 20, up again. Okay?
他のみなさんも確認しましょう
20桁と言った時に座ったみなさん立ってください
16:34
If I told you 20 or less, stand up.
いま座った方は立ってください
16:38
Because this one. I think there were a few here.
こっちにも何人かいましたよね?
16:40
The people who just last sat down.
いま座ったみなさんですよ
16:42
Okay, how many digits do you have in your number?
ではあなたが考えた数字は何桁ですか?
16:45
(Laughs)
(笑)
16:50
21. Okay good. How many do you have in yours?
21ですね いいでしょう
あなたは?
16:53
18. So it goes to this lady here.
18_そうすると こちらのご婦人の答え
16:55
21 is the closest.
21桁が一番近い数字です
16:58
It actually has -- the number of symmetries in the Rubik's cube
実際の答え ルービックキューブの対称性の数は
17:00
has 25 digits.
25桁です
17:02
So now I need to name this object.
それでは この物体に命名しましょう
17:04
So, what is your name?
あなたのお名前は?
17:06
I need your surname. Symmetrical objects generally --
苗字を教えてもらえますか?
通常このような命名では...
17:08
spell it for me.
綴りは?
17:11
G-H-E-Z
G-H-E-Z
17:13
No, SO2 has already been used, actually,
残念 SO2は既に別の数学的言語で使われてるので
17:20
in the mathematical language. So you can't have that one.
その名前を使うことはできません
17:22
So Ghez, there we go. That's your new symmetrical object.
さあGhezさん あなたの新しい対称的物体をどうぞ
17:24
You are now immortal.
これで あなたは永遠不滅になりました
17:26
(Applause)
(拍手)
17:28
And if you'd like your own symmetrical object,
もしみなさんも対称的物体を欲しければ
17:34
I have a project raising money for a charity in Guatemala,
グァテマラへの
教育援助プロジェクトをやっていますので
17:36
where I will stay up all night and devise an object for you,
グァテマラの子供たちへの募金をしていただければ
17:39
for a donation to this charity to help kids get into education in Guatemala.
私が徹夜して
新しい対称的物体を作ってさし上げましょう
17:42
And I think what drives me, as a mathematician,
数学者としての私を駆り立てるのは
17:46
are those things which are not seen, the things that we haven't discovered.
このような見たこともない まだ発見されてないものです
17:49
It's all the unanswered questions which make mathematics a living subject.
答えが見つかっていない疑問が数学を
生き生きとした学問にするのです
17:53
And I will always come back to this quote from the Japanese "Essays in Idleness":
そしていつも「徒然草」の一節を思い出します
17:57
"In everything, uniformity is undesirable.
「何でも全部が完全に整っているのはよくない
18:00
Leaving something incomplete makes it interesting,
やり残したことを そのままにしておくのが面白く
18:03
and gives one the feeling that there is room for growth." Thank you.
先に楽しみを残すことにもなる」
以上 ありがとうございました
18:06
(Applause)
(拍手)
18:09
Translator:Ryoichi KATO
Reviewer:Akiko Hicks

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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.

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.

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