to talk about a project

我今天來到這裡

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

是要談一個計畫

We're crocheting a coral reef.

我和我的雙胞胎姊妹已經執行了三年半

And it's a project that we've actually

我們用鉤針織出珊瑚礁

been now joined by hundreds of people around the world,

而這個計畫到目前為止

who are doing it with us. Indeed thousands of people

已經有從世界各地數以百計的人

have actually been involved in this project,

和我們一起執行，而有數千人

in many of its different aspects.

有實際參與計畫

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

從各種不同的面向

and its roots go into the fields of mathematics,

現在更推行到三大洲去

marine biology, feminine handicraft

根基於數學

and environmental activism.

海洋生物學、婦女手工藝

It's true.

以及環境運動

It's also a project

沒錯

that in a very beautiful way,

這也是一個

the development of this

用一種很美麗的方式完成的計畫

has actually paralleled the evolution of life on earth,

它的發展

which is a particularly lovely thing to be saying

就和地球生物演化平行發生

right here in February 2009 --

這件事情講起來很有趣

which, as one of our previous speakers told us,

在這裡，2009年二月

is the 200th anniversary

前一個講者已經告訴我們

of the birth of Charles Darwin.

這是達爾文的

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

200歲誕辰

But let me first begin by showing you

而在這接下來的18分鐘裡面，我希望可以把這些都帶過一遍

some pictures of what this thing looks like.

但首先我想先讓大家看

Just to give you an idea of scale,

一些照片，了解這些東西長什麼樣子

that installation there is about six feet across,

為了讓大家對大小有個概念

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

這個裝置大概有六呎寬

This is some more images of it.

最高一個大概有兩到三呎高

That one on the right is about five feet high.

這裡有更多照片

The work involves hundreds of different crochet models.

最右邊那個大約有五呎高

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

一共需要上百種不同的鉤針織模型

have contributed all over the world as part of this.

而現在更有大半是由人們

The totality of this project

從世界各地提供的數千種模型組成的

involves tens of thousands of hours

這個計畫總共

of human labor --

花費數萬小時

99 percent of it done by women.

人力

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

而99％都是女性完成的

that is about 12 feet long.

在右邊，是這個裝置的一部分

My sister and I started this project in 2005

約有12呎長

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

我的姊妹和我在2005年開始這項計畫

there was a lot of talk about global warming,

因為在這一年，至少是在科學出版裡

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

有很多對全球暖化

Corals are very delicate organisms,

以及其對珊瑚礁影響的討論

and they are devastated by any rise in sea temperatures.

珊瑚是很脆弱的生物

It causes these vast bleaching events

海溫的些微上升就會造成很大傷害

that are the first signs of corals of being sick.

也就是所謂的白化現象

And if the bleaching doesn't go away --

這是珊瑚生病的第一項警訊

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

如果白化一直持續

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

溫度沒有下降，珊瑚礁就會開始死亡

particularly in coral reefs all over the world.

這樣的故事在很多地方都有發生，像大堡礁

This is our invocation in crochet of a bleached reef.

還有世界各地的珊瑚礁

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

這是我們用鉤針織出的白化珊瑚，為珊瑚祈禱

which is a little organization we started

我們成立了一個「圖示學院」

to promote, to do projects about the

宗旨是

aesthetic and poetic dimensions of science and mathematics.

推廣與承接計畫

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

展示科學與數學上的美學與詩意

asking for people to join us in this enterprise.

當我公佈了聲明於網頁上

To our surprise, one of the first people who called

歡迎加入這創舉

was the Andy Warhol Museum.

相當意外的是一開始打來詢問的

And they said they were having an exhibition

是安地沃荷美術館

about artists' response to global warming,

說將有一展出

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

是藝術家對全球暖化的反應

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

他們希望我們的珊瑚礁也能參與

you can have a little bit of it."

我笑著回答「我們才剛剛開始

So in 2007 we had an exhibition,

所以只能提供一些些」

a small exhibition of this crochet reef.

2007年我們展出

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

只是小小的一片珊瑚礁

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

其中有些從芝加哥來的人說

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

「2007年底, 芝加哥人文藝術的主題是

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

全球暖化，而我們有3000平方英呎的展場

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

希望能全面佈置你們的珊瑚礁」

Now I say "naively" because actually

我天真的就回說「好的!沒問題」

my profession is as a science writer.

我說自己「天真」

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

是因為我的職業是科學作家

I've written books about the history of space,

是寫作有關物理科學的文化歷史

the history of physics and religion,

我曾寫過太空歷史

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

物理與宗教的歷史

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

也為紐約時報與洛杉磯時報撰寫文章

So I said yes to this proposition.

所以我根本搞不清楚填滿3000平方英呎的大小

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

所以我只管答應這邀請

And she nearly had a fit

回家告訴我的姊妹Christine

because Christine is a professor at one of

她嚇到了

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

因為Christine任教於

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

CalArts是洛杉磯的重要藝術學院

She thought I'd gone off my head.

她清楚明白什麼是3000平方英呎的展出

But she went into crochet overdrive.

她說我瘋了

And to cut a long story short, eight months later

但她還是加速鉤針趕進度

we did fill the Chicago Cultural Center's

長話短說，8個月後

3,000 square foot gallery.

我們還是填滿了芝加哥文化中心

By this stage the project had taken on

3000平方英呎的展出

a viral dimension of its own,

到這一步整個計畫

which got completely beyond us.

自然地進入到一重要國度

The people in Chicago decided

且不是我們能操控

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

芝加哥人決定

was have the local people there make a reef.

除了展出我們的珊瑚

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

也希望當地百姓也能參與製作

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

所以我們前往指導技巧、接著工作坊與課程

And it was exhibited alongside ours.

芝加哥民眾也做出他們自己的珊瑚礁

There were hundreds of people involved in that.

同時在我們的作品旁展出

We got invited to do the whole thing

數以百計的民眾參與

in New York, and in London,

我們又被邀請作同樣展出與傳授的過程

and in Los Angeles.

於紐約 倫敦

In each of these cities, the local citizens,

和洛杉磯

hundreds and hundreds of them, have made a reef.

在每個地點 當地的市民

And more and more people get involved in this,

幾百人 一起做珊瑚

most of whom we've never met.

也吸引了更多人參與

So the whole thing has sort of morphed

都是些我們從未見過的人

into this organic, ever-evolving creature,

所以整件事已自然的轉型

that's actually gone way beyond Christine and I.

更有生機 更多人參與

Now some of you are sitting here thinking,

遠超過Christine和我的貢獻

"What planet are these people on?

現在你們可能坐著想

Why on earth are you crocheting a reef?

「這些人是從哪個星球來的?

Woolenness and wetness aren't exactly

為什麼要鉤織珊瑚

two concepts that go together.

棉線與含水

Why not chisel a coral reef out of marble?

是無法相容的

Cast it in bronze."

為什麼不用大理石雕刻珊瑚呢?

But it turns out there is a very good reason

或是銅鑄?」

why we are crocheting it

實際上 是有非常充分的理由

because many organisms in coral reefs

用編織來表現珊瑚

have a very particular kind of structure.

因為每種的珊瑚

The frilly crenulated forms that you see

多有著特別的結構

in corals, and kelps, and sponges and nudibranchs,

這種奏摺重疊的形式

is a form of geometry known as hyperbolic geometry.

出現在珊瑚 海帶 海綿 以及 海蛞蝓

And the only way that mathematicians know

是一種幾何上稱為雙曲線的形式

how to model this structure

也是數學家認為唯一

is with crochet. It happens to be a fact.

能展現此幾何的方式

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

就是針織 這是個事實

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

好像沒有其他方式能建構這樣幾何

So what is this hyperbolic geometry

也好像不可能在電腦上呈現

that corals and sea slugs embody?

所以到底什麼是 雙曲線幾何

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

在珊瑚與海蛞蝓身上?

to the level of a sea slug.

接下來的幾分鐘 我們都能進化到

(Laughter)

海蛞蝓的等級

This sort of geometry revolutionized mathematics

(笑聲)

when it was first discovered in the 19th century.

在19世紀時 這種幾何的

But not until 1997 did mathematicians actually understand

出現 在數學上是革命性的

how they could model it.

一直是到1997年 數學家才真正明白

In 1997 a mathematician

要如何具體模擬它

at Cornell, Daina Taimina,

1997年 一個康乃爾數學家

made the discovery that this structure

Daina Taimina

could actually be done in knitting and crochet.

才發現這樣的結構

The first one she did was knitting.

能由針織與鉤編展現

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

她先用針織

crochet was the better thing.

但太多針了 所以立刻明白

But what she was doing was actually making a model

鉤編是更容易的

of a mathematical structure, that many mathematicians

但她實際所為 就是完成

had thought it was actually impossible to model.

許多數學家都難以完成的

And indeed they thought that anything like this structure

實體模型建構

was impossible per se.

多數都以為是無法

Some of the best mathematicians spent hundreds of years

達成的

trying to prove that this structure was impossible.

過去數百年 頂尖的數學家

So what is this impossible hyperbolic structure?

也試著證明不可能

Before hyperbolic geometry, mathematicians knew

所以到底什麼是雙曲線結構?

about two kinds of space:

在雙曲線幾何之前 數學家慣用

Euclidean space, and spherical space.

兩種空間

And they have different properties.

歐幾里得式空間與球面空間

Mathematicians like to characterize things by being formalist.

各有著不同的性質

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

數學家喜歡用形式主義來分類

But mathematicians formalize this in a particular way.

你們都熟悉平整的空間 就是歐幾里得空間

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

但數學家以不同的方式標記

of parallel lines.

他們的作法是利用

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

平行線條的概念

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

所以 假設一條直線 與直線外的一個點

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

歐幾里得就問:「如何定義平行線?」

the point but never meet the original line?"

我問一下 我能畫出幾條平行線

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

能經過那點 又不與原來的直線相交

One. Great. Okay.

你們都知道這個答案 有人願意喊出來嗎?

That's our definition of a parallel line.

一個 對! OK

It's a definition really of Euclidean space.

那就是我們定義的平行線

But there is another possibility that you all know of:

那就是歐幾里得空間

spherical space.

但也有另一種可能

Think of the surface of a sphere --

球面空間

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

想想一個球面的表面

I have a straight line on my spherical surface.

就像是海灘球 就像是地球表面

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

我有一個在球表面上的直線

can I draw through the point

和一個線外的點 那有多少直線

but never meet the original line?

通過那點 又不會

What do we mean to talk about

與原始直線相交?

a straight line on a curved surface?

到底什麼叫作

Now mathematicians have answered that question.

曲面上的直線呢?

They've understood there is a generalized concept

數學家已經定義

of straightness, it's called a geodesic.

共通概念的曲面上之

And on the surface of a sphere,

直線性 就叫作 測地線

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

若是在球面上

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

直線就是最大能畫出的圓

So we ask the question again,

所以 就像是赤道 或是南北方向的緯線

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

所以 再問一次問題

but never meet the original line?"

「我能畫出多少直線 經過那點

Does someone want to guess?

又不與原直線相交?」

Zero. Very good.

有人要猜嗎?

Now mathematicians thought that was the only alternative.

零 非常好

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

數學家以為只有這另一個答案

Zero and one.

有些可疑不是嗎? 能有兩個答案:

Two answers? There may possibly be a third alternative.

零或一

To a mathematician if there are two answers,

兩個解答 也有可能有第三個答案

and the first two are zero and one,

對於數學家來說 若有兩個答案

there is another number that immediately suggests itself

首先的回答 就是 零 與 一

as the third alternative.

同時 也自然而然 會以為

Does anyone want to guess what it is?

有第三種可能

Infinity. You all got it right. Exactly.

有人要猜嗎?

There is, there's a third alternative.

無限多 的確 你們都答對

This is what it looks like.

有第三個解答

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

這就是圖形表示

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

有一條直線 以及無線多條直線

This is the drawing.

通過那一點 又不會與原始線相交會

This nearly drove mathematicians bonkers

是這樣畫的

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

這幾乎逼數學家發瘋

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

因為 像你們一般 他們覺得被搞糊塗了

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

想一想 怎麼可能? 你是在作弊 這些直線是彎曲的

flat surface.

只因為我將這些直線 投射在

Mathematicians for several hundred years

平坦表面

had to really struggle with this.

數學家歷經幾百年

How could they see this?

的掙扎困惑

What did it mean to actually have a physical model

怎麼能明白呢?

that looked like this?

怎樣能有一實際的具體模型

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

能展現這樣的理論呢?

Then our mathematicians come along

像這樣 想像我們只理解與經歷 歐式幾何空間

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

然後 我們的數學家過來說

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

"有一種球面空間

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

線條伸展南北極後 會重合

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

但你不明白球面的長相

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

另一個人走來說 「看! 這就是個球」

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

你就會「啊! 我懂了 我能感受了

And that's exactly what happened

我能觸摸 也能翻弄」

when Daina Taimina

這就是1997年

in 1997, showed that you could crochet models

當 Daina Taimina

in hyperbolic space.

以鉤織品展示了

Here is this diagram in crochetness.

雙曲面空間

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

這是以鉤織品來展現

And the lines look curved.

我已將歐式的平行線設在這個表面

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

線條看起來是彎曲的

because I can take any one of these lines,

我能證明這是一條線

and I can fold along it.

因為我能以任一條線

And it's a straight line.

沿著它折

So here, in wool,

是一條直線

through a domestic feminine art,

所以呢 經由一