to talk about a project

私が今日

that my twin sister and I have been doing for the past three and half years.

お話しするのは

We're crocheting a coral reef.

私が双子の妹と３年半続けている企画に関してです

And it's a project that we've actually

私たちは珊瑚礁をかぎ針編みで作っています

been now joined by hundreds of people around the world,

この企画は現在 世界中の

who are doing it with us. Indeed thousands of people

何百人という人たちが参加し

have actually been involved in this project,

何千人もの人たちが

in many of its different aspects.

この企画にいろいろな面で

It's a project that now reaches across three continents,

関わっています

and its roots go into the fields of mathematics,

この企画は三大陸にまたがり進められています

marine biology, feminine handicraft

ルーツは 数学や海洋生物

and environmental activism.

女性による手芸や環境活動まで

It's true.

様々な分野が含まれます

It's also a project

本当です

that in a very beautiful way,

また この企画は

the development of this

地球上の

has actually paralleled the evolution of life on earth,

命の進化に対して

which is a particularly lovely thing to be saying

理解を深めることができます

right here in February 2009 --

特に 2009年2月現在

which, as one of our previous speakers told us,

これは最高のテーマだと思います

is the 200th anniversary

先ほど ある方も言っていたように

of the birth of Charles Darwin.

2009年はチャールズ ダーウィンの

All of this I'm going to get to in the next 18 minutes, I hope.

生誕200年にあたるからです

But let me first begin by showing you

以上のことを18分間でお伝えしますが

some pictures of what this thing looks like.

あらましをつかむのに

Just to give you an idea of scale,

最初に写真をお見せします

that installation there is about six feet across,

ここに見える作品は

and the tallest models are about two or three feet high.

幅1.8mの大きさです

This is some more images of it.

高さは一番丈のあるもので約60～90cmです

That one on the right is about five feet high.

こんなのも あります

The work involves hundreds of different crochet models.

右端の作品は高さが約1.5m

And indeed there are now thousands and thousands of models that people

かぎ針編みの珊瑚は何百種類もあって

have contributed all over the world as part of this.

今ではこの一部として世界中から

The totality of this project

かぎ針編みの珊瑚が何千と寄付されました

involves tens of thousands of hours

この企画に関わる人たちの

of human labor --

労働時間は合計で

99 percent of it done by women.

何万時間にもなり

On the right hand side, that bit there is part of an installation

99％が女性によるものです

that is about 12 feet long.

右側にあるのは 幅が約3.6mの

My sister and I started this project in 2005

作品の一部です

because in that year, at least in the science press,

私たち姉妹がこの企画を始めたのは2005年

there was a lot of talk about global warming,

その年 少なくとも科学系の出版物には

and the effect that global warming was having on coral reefs.

地球温暖化と珊瑚礁への影響について

Corals are very delicate organisms,

多くの記事が書かれていました

and they are devastated by any rise in sea temperatures.

珊瑚は非常に繊細な生物で

It causes these vast bleaching events

海水温度が少しでも上昇すると死滅してしまいます

that are the first signs of corals of being sick.

珊瑚が退色するのは

And if the bleaching doesn't go away --

病気になっている最初のサインです

if the temperatures don't go down -- reefs start to die.

白化したままであったり

A great deal of this has been happening in the Great Barrier Reef,

水温が下がらなければ 珊瑚礁は死滅に向かいます

particularly in coral reefs all over the world.

グレートバリアリーフの状況は酷く

This is our invocation in crochet of a bleached reef.

世界中の珊瑚礁が衰退しています

We have a new organization together called The Institute for Figuring,

これは白化した珊瑚礁を作品にしたものです

which is a little organization we started

私たちは IFFという組織を運営しており

to promote, to do projects about the

科学と数学の

aesthetic and poetic dimensions of science and mathematics.

美しくてロマンチックな様相を

And I went and put a little announcement up on our site,

広めることを趣旨に活動しています

asking for people to join us in this enterprise.

この企画への参加をネット上で

To our surprise, one of the first people who called

公募したところ

was the Andy Warhol Museum.

驚いたことに アンディーウォーホル美術館から

And they said they were having an exhibition

連絡が来ました

about artists' response to global warming,

地球温暖化を題材に

and they'd like our coral reef to be part of it.

展覧会をするので

I laughed and said, "Well we've only just started it,

出展してほしい というものでした

you can have a little bit of it."

私は笑って “始めたばかりだから

So in 2007 we had an exhibition,

少しだけならば” と言いました

a small exhibition of this crochet reef.

それで2007年に展覧会をしました

And then some people in Chicago came along and they said,

かぎ針編み珊瑚の小さな展覧会です

"In late 2007, the theme of the Chicago Humanities Festival is

その後 別の依頼がありました

global warming. And we've got this 3,000 square-foot gallery

“シカゴで開催する芸術文化祭典の2007年のテーマは地球温暖化です

and we want you to fill it with your reef."

84坪のギャラリーがあるので

And I, naively by this stage, said, "Oh, yes, sure."

貴女が作る珊瑚礁で埋め尽くしてほしいの”

Now I say "naively" because actually

この時の私の考えは甘く “もちろんよ” と答えました

my profession is as a science writer.

考えが甘いと言ったのは

What I do is I write books about the cultural history of physics.

私の本職は科学作家なのです

I've written books about the history of space,

物理の文化的歴史に関して執筆していて

the history of physics and religion,

宇宙の歴史に関する本や

and I write articles for people like the New York Times and the L.A. Times.

物理や宗教の歴史の本を書いてきました

So I had no idea what it meant to fill a 3,000 square-foot gallery.

ニューヨークタイムズやロサンゼルスタイムズにも記事を書いています

So I said yes to this proposition.

ですから84坪を埋め尽くす大変さが分からず

And I went home, and I told my sister Christine.

二つ返事をしたのです

And she nearly had a fit

帰宅して 妹に伝えると

because Christine is a professor at one of

彼女はカンカンに怒りました

L.A.'s major art colleges, CalArts,

なぜなら妹は

and she knew exactly what it meant to fill a 3,000 square-foot gallery.

カリフォルニア芸術大学で教授をしていて

She thought I'd gone off my head.

84坪を埋め尽くす大変さを知っていたからです

But she went into crochet overdrive.

妹は怒りながらも 猛烈に

And to cut a long story short, eight months later

かぎ針編みを始めました

we did fill the Chicago Cultural Center's

結論を言ってしまうと ８ヶ月後

3,000 square foot gallery.

シカゴ文化センターの84坪のギャラリーを

By this stage the project had taken on

埋め尽くしました

a viral dimension of its own,

この時点で この企画は

which got completely beyond us.

想像以上に

The people in Chicago decided

どんどん広がっていきました

that as well as exhibiting our reefs, what they wanted to do

シカゴの人たちは

was have the local people there make a reef.

私たちの珊瑚礁の展示に加えて 彼ら自身の

So we went and taught the techniques. We did workshops and lectures.

珊瑚礁づくりをしたいと言いました

And the people in Chicago made a reef of their own.

私たちは講習会を開いて作り方を教え

And it was exhibited alongside ours.

シカゴの人たちは自ら珊瑚礁を作ったのです

There were hundreds of people involved in that.

私たちの作品と並べて展示されました

We got invited to do the whole thing

何百人もの人が関わりました

in New York, and in London,

私たちは招待されて同じことを

and in Los Angeles.

ニューヨークとロンドン

In each of these cities, the local citizens,

ロサンゼルスでも行いました

hundreds and hundreds of them, have made a reef.

その各都市で何百人という

And more and more people get involved in this,

地元の市民が珊瑚礁を作りました

most of whom we've never met.

参加者は どんどん増えています

So the whole thing has sort of morphed

大半が初対面の人たちです

into this organic, ever-evolving creature,

ですから この企画は

that's actually gone way beyond Christine and I.

妹と私の域を超えて

Now some of you are sitting here thinking,

成長し続ける生命体になっています

"What planet are these people on?

さて こう思っている方はいませんか？

Why on earth are you crocheting a reef?

“なぜ この人たちは

Woolenness and wetness aren't exactly

珊瑚をかぎ針編みしてるのか？

two concepts that go together.

毛糸と水なんて

Why not chisel a coral reef out of marble?

ミスマッチな感じがする

Cast it in bronze."

なぜ大理石を彫ったり

But it turns out there is a very good reason

ブロンズで作らないのか？”

why we are crocheting it

それには かぎ針編みをする

because many organisms in coral reefs

大きな理由があるのです

have a very particular kind of structure.

珊瑚礁には独特な形をした―

The frilly crenulated forms that you see

生き物がたくさんいます

in corals, and kelps, and sponges and nudibranchs,

珊瑚 海藻 スポンジ

is a form of geometry known as hyperbolic geometry.

ウミウシに見られる ひだがついた形は

And the only way that mathematicians know

双曲幾何学と呼ばれる幾何学の一種です

how to model this structure

数学者にとって この構造を再現する―

is with crochet. It happens to be a fact.

唯一の方法は かぎ針編みです

It's almost impossible to model this structure any other way,

これは事実なんです

and it's almost impossible to do it on computers.

この構造は コンピュータを含め

So what is this hyperbolic geometry

別の方法で作るのは ほぼ不可能です

that corals and sea slugs embody?

では 珊瑚やウミウシが表している―

The next few minutes is, we're all going to get raised up

双曲幾何学とは何かを探るため

to the level of a sea slug.

次の数分 皆でレベルを上げましょう

(Laughter)

ウミウシのレベルまで

This sort of geometry revolutionized mathematics

（笑）

when it was first discovered in the 19th century.

双曲幾何学が19世紀に発見されたとき

But not until 1997 did mathematicians actually understand

数学に大変革が起こりましたが

how they could model it.

数学者は 1997年まで

In 1997 a mathematician

形に表すことができませんでした

at Cornell, Daina Taimina,

1997年にコーネル大学の

made the discovery that this structure

数学者 デーナ タイミナが

could actually be done in knitting and crochet.

この構造を編み物で再現できると

The first one she did was knitting.

発見しました

But you get too many stitches on the needle. So she quickly realized

彼女は最初 棒針を使いましたが

crochet was the better thing.

目があまりにも多すぎるため

But what she was doing was actually making a model

かぎ針が適していると気づきました

of a mathematical structure, that many mathematicians

彼女は再現は不可能だと言う

had thought it was actually impossible to model.

数学者の概念を覆す―

And indeed they thought that anything like this structure

数学的構造を作り上げました

was impossible per se.

本来 この種の構造は

Some of the best mathematicians spent hundreds of years

無理だと考えられていました

trying to prove that this structure was impossible.

数学の権威と言われる人たちも

So what is this impossible hyperbolic structure?

不可能性を証明するのに何百時間と費やしました

Before hyperbolic geometry, mathematicians knew

双曲構造の何がそうさせるのでしょう？

about two kinds of space:

双曲幾何学が出てくる前

Euclidean space, and spherical space.

数学者が認識していた二つの空間は

And they have different properties.

ユークリッド空間と球状空間で

Mathematicians like to characterize things by being formalist.

どちらも異なる特質をもっています

You all have a sense of what a flat space is, Euclidean space is.

数学者は形式主義者なので 特徴づけが好きです

But mathematicians formalize this in a particular way.

皆さんは平面 いわゆるユークリッド空間の意味はわかりますね？

And what they do is, they do it through the concept

でも 数学者は平行線の概念を使って

of parallel lines.

ユークリッド空間を

So here we have a line and a point outside the line.

定義します

And Euclid said, "How can I define parallel lines?

ここに直線があり その上に点があります

I ask the question, how many lines can I draw through

ユークリッドの定義で質問します

the point but never meet the original line?"

この点を通過しつつ 直線に

And you all know the answer. Does someone want to shout it out?

交わらない線は 何本？

One. Great. Okay.

どなたか答えてくれますか？

That's our definition of a parallel line.

そう　１です

It's a definition really of Euclidean space.

それが平行線の定義で

But there is another possibility that you all know of:

ユークリッド空間の定義ですが

spherical space.

他の可能性もあります

Think of the surface of a sphere --

球状空間です

just like a beach ball, the surface of the Earth.

ビーチボールや地球のような

I have a straight line on my spherical surface.

球の表面を考えてください

And I have a point outside the line. How many straight lines

球の表面に直線が引いてあり

can I draw through the point

その上に点があります　この点を通過しつつ

but never meet the original line?

直線に交わらない線は

What do we mean to talk about

何本あるでしょう？

a straight line on a curved surface?

丸みを帯びた表面における

Now mathematicians have answered that question.

直線とはどういう意味か―

They've understood there is a generalized concept

数学者が答えを出しています

of straightness, it's called a geodesic.

直線を一般化した―

And on the surface of a sphere,

測地線と呼ばれるものがあり

a straight line is the biggest possible circle you can draw.

球の表面に描ける

So it's like the equator or the lines of longitude.

最長の円が直線です

So we ask the question again,

赤道や経線を想像してください

"How many straight lines can I draw through the point,

もう一度質問します

but never meet the original line?"

点を通りつつ 元の線に

Does someone want to guess?

交わらない線は何本？

Zero. Very good.

どなたか当ててくれますか？

Now mathematicians thought that was the only alternative.

そうですね　０です

It's a bit suspicious isn't it? There is two answers to the question so far,

数学者はそれが唯一の選択肢だと思っていました

Zero and one.

この二つの答えは少し疑わしいですね

Two answers? There may possibly be a third alternative.

０と１です

To a mathematician if there are two answers,

三つめの答えがあるかもしれません

and the first two are zero and one,

数学者が答えを導き

there is another number that immediately suggests itself

それが０と１ならば

as the third alternative.

即座に提案できる 三つめの

Does anyone want to guess what it is?

選択肢があります

Infinity. You all got it right. Exactly.

どなたか当ててくれますか？

There is, there's a third alternative.

無限大　皆さん正解です

This is what it looks like.

三つめの選択肢は

There's a straight line, and there is an infinite number of lines

このように見えます

that go through the point and never meet the original line.

直線があり 無限の線が点を通り

This is the drawing.

直線とは交ることはありません

This nearly drove mathematicians bonkers

これは その図形です

because, like you, they're sitting there feeling bamboozled.

さすがに 数学者たちは頭を抱えました

Thinking, how can that be? You're cheating. The lines are curved.

線が弧を描いていることから

But that's only because I'm projecting it onto a

だまされたように感じていたのです

flat surface.

でも それは単に平面に

Mathematicians for several hundred years

映し出しているからです

had to really struggle with this.

数学者は数百年間

How could they see this?

どうやってこれを

What did it mean to actually have a physical model

形に出来るか格闘しました

that looked like this?

図形しかなければ想像するのも大変ですが

It's a bit like this: imagine that we'd only ever encountered Euclidean space.

このように考えてください

Then our mathematicians come along

ユークリッド空間しか知らない人に

and said, "There's this thing called a sphere,

北極点と南極点で

and the lines come together at the north and south pole."

必ず直線が交わる―

But you don't know what a sphere looks like.

球の存在を話したとします

And someone that comes along and says, "Look here's a ball."

どんな形なのか教えるために

And you go, "Ah! I can see it. I can feel it.

ボールを見てごらん と言ってあげれば

I can touch it. I can play with it."

視覚や触覚から

And that's exactly what happened

球とは何かを理解できるのです

when Daina Taimina

それがまさに1997年に

in 1997, showed that you could crochet models

デーナ タイミナがやってみせたことです

in hyperbolic space.

彼女はかぎ針編みで双曲線空間を

Here is this diagram in crochetness.

形成できると証明しました

I've stitched Euclid's parallel postulate on to the surface.

これがかぎ針編みで表した図解です

And the lines look curved.

表面にユークリッド幾何学の平行線公準を縫いました

But look, I can prove to you that they're straight

線はカーブしているように見えますが

because I can take any one of these lines,

直線だということを証明できます

and I can fold along it.

線を一つ選んで

And it's a straight line.

それに沿って折ると

So here, in wool,

直線が現れます

through a domestic feminine art,

数学で最も名高い公準が