Margaret Wertheim: The beautiful math of coral (and crochet)

譯者: K. C. Peng 審譯者: Joan Liu

I'm here today, as June said,

我今天來到這裡

to talk about a project

是要談一個計畫

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

我和我的雙胞胎姊妹已經執行了三年半

We're crocheting a coral reef.

我們用鉤針織出珊瑚礁

And it's a project that we've actually

而這個計畫到目前為止

been now joined by hundreds of people around the world,

已經有從世界各地數以百計的人

who are doing it with us. Indeed thousands of people

和我們一起執行，而有數千人

have actually been involved in this project,

有實際參與計畫

in many of its different aspects.

從各種不同的面向

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

現在更推行到三大洲去

and its roots go into the fields of mathematics,

根基於數學

marine biology, feminine handicraft

海洋生物學、婦女手工藝

and environmental activism.

以及環境運動

It's true.

沒錯

It's also a project

這也是一個

that in a very beautiful way,

用一種很美麗的方式完成的計畫

the development of this

它的發展

has actually paralleled the evolution of life on earth,

就和地球生物演化平行發生

which is a particularly lovely thing to be saying

這件事情講起來很有趣

right here in February 2009 --

在這裡，2009年二月

which, as one of our previous speakers told us,

前一個講者已經告訴我們

is the 200th anniversary

這是達爾文的

of the birth of Charles Darwin.

200歲誕辰

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

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

But let me first begin by showing you

但首先我想先讓大家看

some pictures of what this thing looks like.

一些照片，了解這些東西長什麼樣子

Just to give you an idea of scale,

為了讓大家對大小有個概念

that installation there is about six feet across,

這個裝置大概有六呎寬

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

最高一個大概有兩到三呎高

This is some more images of it.

這裡有更多照片

That one on the right is about five feet high.

最右邊那個大約有五呎高

The work involves hundreds of different crochet models.

一共需要上百種不同的鉤針織模型

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

而現在更有大半是由人們

have contributed all over the world as part of this.

從世界各地提供的數千種模型組成的

The totality of this project

這個計畫總共

involves tens of thousands of hours

花費數萬小時

of human labor --

人力

99 percent of it done by women.

而99％都是女性完成的

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

在右邊，是這個裝置的一部分

that is about 12 feet long.

約有12呎長

My sister and I started this project in 2005

我的姊妹和我在2005年開始這項計畫

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

因為在這一年，至少是在科學出版裡

there was a lot of talk about global warming,

有很多對全球暖化

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

以及其對珊瑚礁影響的討論

Corals are very delicate organisms,

珊瑚是很脆弱的生物

and they are devastated by any rise in sea temperatures.

海溫的些微上升就會造成很大傷害

It causes these vast bleaching events

也就是所謂的白化現象

that are the first signs of corals of being sick.

這是珊瑚生病的第一項警訊

And if the bleaching doesn't go away --

如果白化一直持續

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

溫度沒有下降，珊瑚礁就會開始死亡

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

這樣的故事在很多地方都有發生，像大堡礁

particularly in coral reefs all over the world.

還有世界各地的珊瑚礁

This is our invocation in crochet of a bleached reef.

這是我們用鉤針織出的白化珊瑚，為珊瑚祈禱

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

我們成立了一個「圖示學院」

which is a little organization we started

宗旨是

to promote, to do projects about the

推廣與承接計畫

aesthetic and poetic dimensions of science and mathematics.

展示科學與數學上的美學與詩意

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

當我公佈了聲明於網頁上

asking for people to join us in this enterprise.

歡迎加入這創舉

To our surprise, one of the first people who called

相當意外的是一開始打來詢問的

was the Andy Warhol Museum.

是安地沃荷美術館

And they said they were having an exhibition

說將有一展出

about artists' response to global warming,

是藝術家對全球暖化的反應

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

他們希望我們的珊瑚礁也能參與

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

我笑著回答「我們才剛剛開始

you can have a little bit of it."

所以只能提供一些些」

So in 2007 we had an exhibition,

2007年我們展出

a small exhibition of this crochet reef.

只是小小的一片珊瑚礁

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

其中有些從芝加哥來的人說

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

「2007年底, 芝加哥人文藝術的主題是

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

全球暖化，而我們有3000平方英呎的展場

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

希望能全面佈置你們的珊瑚礁」

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

我天真的就回說「好的!沒問題」

Now I say "naively" because actually

我說自己「天真」

my profession is as a science writer.

是因為我的職業是科學作家

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

是寫作有關物理科學的文化歷史

I've written books about the history of space,

我曾寫過太空歷史

the history of physics and religion,

物理與宗教的歷史

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

也為紐約時報與洛杉磯時報撰寫文章

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

所以我根本搞不清楚填滿3000平方英呎的大小

So I said yes to this proposition.

所以我只管答應這邀請

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

回家告訴我的姊妹Christine

And she nearly had a fit

她嚇到了

because Christine is a professor at one of

因為Christine任教於

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

CalArts是洛杉磯的重要藝術學院

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

她清楚明白什麼是3000平方英呎的展出

She thought I'd gone off my head.

她說我瘋了

But she went into crochet overdrive.

但她還是加速鉤針趕進度

And to cut a long story short, eight months later

長話短說，8個月後

we did fill the Chicago Cultural Center's

我們還是填滿了芝加哥文化中心

3,000 square foot gallery.

3000平方英呎的展出

By this stage the project had taken on

到這一步整個計畫

a viral dimension of its own,

自然地進入到一重要國度

which got completely beyond us.

且不是我們能操控

The people in Chicago decided

芝加哥人決定

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

除了展出我們的珊瑚

was have the local people there make a reef.

也希望當地百姓也能參與製作

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

所以我們前往指導技巧、接著工作坊與課程

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

芝加哥民眾也做出他們自己的珊瑚礁

And it was exhibited alongside ours.

同時在我們的作品旁展出

There were hundreds of people involved in that.

數以百計的民眾參與

We got invited to do the whole thing

我們又被邀請作同樣展出與傳授的過程

in New York, and in London,

於紐約 倫敦

and in Los Angeles.

和洛杉磯

In each of these cities, the local citizens,

在每個地點 當地的市民

hundreds and hundreds of them, have made a reef.

幾百人 一起做珊瑚

And more and more people get involved in this,

也吸引了更多人參與

most of whom we've never met.

都是些我們從未見過的人

So the whole thing has sort of morphed

所以整件事已自然的轉型

into this organic, ever-evolving creature,

更有生機 更多人參與

that's actually gone way beyond Christine and I.

遠超過Christine和我的貢獻

Now some of you are sitting here thinking,

現在你們可能坐著想

"What planet are these people on?

「這些人是從哪個星球來的?

Why on earth are you crocheting a reef?

為什麼要鉤織珊瑚

Woolenness and wetness aren't exactly

棉線與含水

two concepts that go together.

是無法相容的

Why not chisel a coral reef out of marble?

為什麼不用大理石雕刻珊瑚呢?

Cast it in bronze."

或是銅鑄?」

But it turns out there is a very good reason

實際上 是有非常充分的理由

why we are crocheting it

用編織來表現珊瑚

because many organisms in coral reefs

因為每種的珊瑚

have a very particular kind of structure.

多有著特別的結構

The frilly crenulated forms that you see

這種奏摺重疊的形式

in corals, and kelps, and sponges and nudibranchs,

出現在珊瑚 海帶 海綿 以及 海蛞蝓

is a form of geometry known as hyperbolic geometry.

是一種幾何上稱為雙曲線的形式

And the only way that mathematicians know

也是數學家認為唯一

how to model this structure

能展現此幾何的方式

is with crochet. It happens to be a fact.

就是針織 這是個事實

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

好像沒有其他方式能建構這樣幾何

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

也好像不可能在電腦上呈現

So what is this hyperbolic geometry

所以到底什麼是 雙曲線幾何

that corals and sea slugs embody?

在珊瑚與海蛞蝓身上?

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

接下來的幾分鐘 我們都能進化到

to the level of a sea slug.

海蛞蝓的等級

(Laughter)

(笑聲)

This sort of geometry revolutionized mathematics

在19世紀時 這種幾何的

when it was first discovered in the 19th century.

出現 在數學上是革命性的

But not until 1997 did mathematicians actually understand

一直是到1997年 數學家才真正明白

how they could model it.

要如何具體模擬它

In 1997 a mathematician

1997年 一個康乃爾數學家

at Cornell, Daina Taimina,

Daina Taimina

made the discovery that this structure

才發現這樣的結構

could actually be done in knitting and crochet.

能由針織與鉤編展現

The first one she did was knitting.

她先用針織

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

但太多針了 所以立刻明白

crochet was the better thing.

鉤編是更容易的

But what she was doing was actually making a model

但她實際所為 就是完成

of a mathematical structure, that many mathematicians

許多數學家都難以完成的

had thought it was actually impossible to model.

實體模型建構

And indeed they thought that anything like this structure

多數都以為是無法

was impossible per se.

達成的

Some of the best mathematicians spent hundreds of years

過去數百年 頂尖的數學家

trying to prove that this structure was impossible.

也試著證明不可能

So what is this impossible hyperbolic structure?

所以到底什麼是雙曲線結構?

Before hyperbolic geometry, mathematicians knew

在雙曲線幾何之前 數學家慣用

about two kinds of space:

兩種空間

Euclidean space, and spherical space.

歐幾里得式空間與球面空間

And they have different properties.

各有著不同的性質

Mathematicians like to characterize things by being formalist.

數學家喜歡用形式主義來分類

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

你們都熟悉平整的空間 就是歐幾里得空間

But mathematicians formalize this in a particular way.

但數學家以不同的方式標記

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

他們的作法是利用

of parallel lines.

平行線條的概念

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

所以 假設一條直線 與直線外的一個點

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

歐幾里得就問:「如何定義平行線?」

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

我問一下 我能畫出幾條平行線

the point but never meet the original line?"

能經過那點 又不與原來的直線相交

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

你們都知道這個答案 有人願意喊出來嗎?

One. Great. Okay.

一個 對! OK

That's our definition of a parallel line.

那就是我們定義的平行線

It's a definition really of Euclidean space.

那就是歐幾里得空間

But there is another possibility that you all know of:

但也有另一種可能

spherical space.

球面空間

Think of the surface of a sphere --

想想一個球面的表面

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

就像是海灘球 就像是地球表面

I have a straight line on my spherical surface.

我有一個在球表面上的直線

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

和一個線外的點 那有多少直線

can I draw through the point

通過那點 又不會

but never meet the original line?

與原始直線相交?

What do we mean to talk about

到底什麼叫作

a straight line on a curved surface?

曲面上的直線呢?

Now mathematicians have answered that question.

數學家已經定義

They've understood there is a generalized concept

共通概念的曲面上之

of straightness, it's called a geodesic.

直線性 就叫作 測地線

And on the surface of a sphere,

若是在球面上

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

直線就是最大能畫出的圓

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

所以 就像是赤道 或是南北方向的緯線

So we ask the question again,

所以 再問一次問題

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

「我能畫出多少直線 經過那點

but never meet the original line?"

又不與原直線相交?」

Does someone want to guess?

有人要猜嗎?

Zero. Very good.

零 非常好

Now mathematicians thought that was the only alternative.

數學家以為只有這另一個答案

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

有些可疑不是嗎? 能有兩個答案:

Zero and one.

零或一

Two answers? There may possibly be a third alternative.

兩個解答 也有可能有第三個答案

To a mathematician if there are two answers,

對於數學家來說 若有兩個答案

and the first two are zero and one,

首先的回答 就是 零 與 一

there is another number that immediately suggests itself

同時 也自然而然 會以為

as the third alternative.

有第三種可能

Does anyone want to guess what it is?

有人要猜嗎?

Infinity. You all got it right. Exactly.

無限多 的確 你們都答對

There is, there's a third alternative.

有第三個解答

This is what it looks like.

這就是圖形表示

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

有一條直線 以及無線多條直線

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

通過那一點 又不會與原始線相交會

This is the drawing.

是這樣畫的

This nearly drove mathematicians bonkers

這幾乎逼數學家發瘋

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

因為 像你們一般 他們覺得被搞糊塗了

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

想一想 怎麼可能? 你是在作弊 這些直線是彎曲的

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

只因為我將這些直線 投射在

flat surface.

平坦表面

Mathematicians for several hundred years

數學家歷經幾百年

had to really struggle with this.

的掙扎困惑

How could they see this?

怎麼能明白呢?

What did it mean to actually have a physical model

怎樣能有一實際的具體模型

that looked like this?

能展現這樣的理論呢?

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

像這樣 想像我們只理解與經歷 歐式幾何空間

Then our mathematicians come along

然後 我們的數學家過來說

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

"有一種球面空間

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

線條伸展南北極後 會重合

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

但你不明白球面的長相

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

另一個人走來說 「看! 這就是個球」

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

你就會「啊! 我懂了 我能感受了

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

我能觸摸 也能翻弄」

And that's exactly what happened

這就是1997年

when Daina Taimina

當 Daina Taimina

in 1997, showed that you could crochet models

以鉤織品展示了

in hyperbolic space.

雙曲面空間

Here is this diagram in crochetness.

這是以鉤織品來展現

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

我已將歐式的平行線設在這個表面

And the lines look curved.

線條看起來是彎曲的

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

我能證明這是一條線

because I can take any one of these lines,

因為我能以任一條線

and I can fold along it.

沿著它折

And it's a straight line.

是一條直線

So here, in wool,

所以呢 經由一

through a domestic feminine art,

家庭婦女的藝術棉織品

is the proof that the most famous postulate

證明數學界最有名的假設

in mathematics is wrong.

(無法建出雙曲面模型) 是錯的

(Applause)

（掌聲）

And you can stitch all sorts of mathematical

你能鉤織各式的數學定理

theorems onto these surfaces.

在這些表面上顯現

The discovery of hyperbolic space ushered in the field of mathematics

而雙曲面引領了其他數學

that is called non-Euclidean geometry.

稱為 非歐式幾何

And this is actually the field of mathematics

這類數學也是

that underlies general relativity

廣義相對論的基礎

and is actually ultimately going to show us

終極地為我們

about the shape of the universe.

引導出宇宙的形狀

So there is this direct line

所以有一直接關聯線

between feminine handicraft,

連結女性手工藝

Euclid and general relativity.

歐基里得 與 廣義相對論

Now, I said that mathematicians thought that this was impossible.

我剛說數學家原本認為是不可能

Here's two creatures who've never heard of Euclid's parallel postulate --

這裡有兩種生物從來沒有聽過 歐基里得 的平行假設

didn't know it was impossible to violate,

也就不知道不能違反

and they're simply getting on with it.

它們卻與 非歐幾何 相處融洽

They've been doing it for hundreds of millions of years.

他們已存在 數億年之久

I once asked the mathematicians why it was

我曾問過數學家怎麼會這樣

that mathematicians thought this structure was impossible

數學專家沒能具體建構的模型

when sea slugs have been doing it since the Silurian age.

而海蛞蝓 卻已經從志留纪就擁有著

Their answer was interesting.

他們的回答是有趣的

They said, "Well I guess there aren't that many mathematicians

他們說「可能沒有足夠的數學家

sitting around looking at sea slugs."

四處坐著看到海蛞蝓」

And that's true. But it also goes deeper than that.

或許是 但這件事也能更深入

It also says a whole lot of things

也說明 整體數學家

about what mathematicians thought mathematics was,

以為的數學是什麼

what they thought it could and couldn't do,

以為數學能做到與做不到

what they thought it could and couldn't represent.

以為數學能呈現到與不能呈現

Even mathematicians, who in some sense

就連數學家 在某些角度

are the freest of all thinkers,

是最自由的思考者

literally couldn't see

沒能看到

not only the sea slugs around them,

身旁的海蛞蝓

but the lettuce on their plate --

也沒留意到 餐盤中的 萵苣

because lettuces, and all those curly vegetables,

因為 像萵苣這些彎曲的蔬菜

they also are embodiments of hyperbolic geometry.

都是雙曲面幾何的體現

And so in some sense they literally,

某種程度數學家

they had such a symbolic view of mathematics,

他們有著對數學的符號式的觀點

they couldn't actually see what was going on

卻不能察覺

on the lettuce in front of them.

在眼前的萵苣

It turns out that the natural world is full of hyperbolic wonders.

事實上 自然界中 充滿著太多符號式 驚奇

And so, too, we've discovered

基於此 我們也發現

that there is an infinite taxonomy

有無限多分類

of crochet hyperbolic creatures.

來鉤織雙曲面的生物

We started out, Chrissy and I and our contributors,

我們姊妹加上其他參與者 開始

doing the simple mathematically perfect models.

作出簡單數學上的完美模型

But we found that when we deviated from the specific

我們發現當我們偏離特定

setness of the mathematical code

數學符號設定

that underlies it -- the simple algorithm

就是原本簡單的規律:

crochet three, increase one --

鉤織三針 加一針

when we deviated from that and made embellishments to the code,

當我們偏離 做了些規律上的額外裝飾變化

the models immediately started to look more natural.

模型立即呈現更佳的自然

And all of our contributors, who are an amazing

所有來自世界各地的參與者

collection of people around the world,

無不覺得驚奇

do their own embellishments.

也開始了他們的裝飾變化

As it were, we have this ever-evolving,

就這樣 我們開始了

crochet taxonomic tree of life.

鉤織品物種族譜的生命演化

Just as the morphology

就像是地球生物

and the complexity of life on earth is never ending,

生生不息的變化與複雜化

little embellishments and complexifications

基因些微的變化與複雜

in the DNA code

才演化出

lead to new things like giraffes, or orchids --

長頸鹿 或是 蘭花

so too, do little embellishments in the crochet code

同樣地 鉤織中小小裝飾變化

lead to new and wondrous creatures

產出了全新的品種

in the evolutionary tree of crochet life.

鉤織品物種族譜的生命演化

So this project really has

所以這個計畫

taken on this inner organic life of its own.

真的開始其內在的有機生命

There is the totality of all the people who have come to it.

統整了所有參與者的

And their individual visions,

各自願景

and their engagement with this mathematical mode.

加上各自以數學形式的參與

We have these technologies. We use them.

我們已有各式科技 能被使用

But why? What's at stake here? What does it matter?

那為什麼要用手工呢? 有什麼重要的?

For Chrissy and I, one of the things that's important here

對我們姊妹而言 最重要的一點是

is that these things suggest

這樣的實作顯示出

the importance and value of embodied knowledge.

將內隱知識的具體展現 之重要性與價值

We live in a society

我們生活在這樣的社會

that completely tends to valorize

總是傾向於使用

symbolic forms of representation --

象徵符號的表達

algebraic representations,

如代數

equations, codes.

函數式 程式 等

We live in a society that's obsessed

我們著魔於

with presenting information in this way,

將資訊如此表達

teaching information in this way.

也傳授資訊用這樣的方式

But through this sort of modality,

但是利用鉤織的形式

crochet, other plastic forms of play --

或是其他種遊戲

people can be engaged with the most abstract,

人們能更體會最抽象的

high-powered, theoretical ideas,

最高層的 理論的概念

the kinds of ideas that normally you have to go

而這些概念 通常都是要

to university departments to study in higher mathematics,

就學於高等教育才會聽到

which is where I first learned about hyperbolic space.

那也是我過去第一次 學到雙曲面空間 的地方

But you can do it through playing with material objects.

但是 你可以經由操弄實體物質了

One of the ways that we've come to think about this

在我們的數字研究中心

is that what we're trying to do with the Institute for Figuring

我們也想出一套邏輯去實踐

and projects like this, we're trying to have

就是設計出

kindergarten for grown-ups.

成人式的幼稚園

And kindergarten was actually a very formalized

幼稚園事實上是一個非常制式的

system of education,

教育系統

established by a man named Friedrich Froebel,

當初創始的是 Friedrich Froebel

who was a crystallographer in the 19th century.

而他原本是19世紀的結晶學家

He believed that the crystal was the model

他認為 結晶結構

for all kinds of representation.

是所有事務的規律表現

He developed a radical alternative system

他也就發展出嶄新不同既往

of engaging the smallest children

的幼兒教育系統

with the most abstract ideas

經由身體操作的遊戲

through physical forms of play.

試著傳遞抽象意念

And he is worthy of an entire talk on his own right.

他這個題材故事 本身就值得另闢一場演講

The value of education

Froebel 引領的

is something that Froebel championed,

教育價值的傳遞

through plastic modes of play.

是經由 物質模式的遊戲

We live in a society now

現今的社會

where we have lots of think tanks,

我們有一大堆的 智庫

where great minds go to think about the world.

有著一群聰明腦袋 為世界想像

They write these great symbolic treatises

撰述許多偉大的抽象論文

called books, and papers,

像 書籍 論文

and op-ed articles.

專欄 等等

We want to propose, Chrissy and I,

我們姊妹倆 想提議

through The Institute for Figuring, another alternative way of doing things,

經由 數字研究中心 的提倡 另一種不同的作法

which is the play tank.

就是 「玩庫」

And the play tank, like the think tank,

玩庫 就像是智庫一般

is a place where people can go

是個人們可聚集

and engage with great ideas.

激發出偉大想法

But what we want to propose,

但我們要強調的是

is that the highest levels of abstraction,

最抽象的學問

things like mathematics, computing, logic, etc. --

像 數學 電腦 邏輯 等等

all of this can be engaged with,

不只能

not just through purely cerebral algebraic

靠純粹的智力演算

symbolic methods,

抽象符號

but by literally, physically playing with ideas.

也能用玩的方式 產出想法

Thank you very much.

謝謝

(Applause)

(掌聲)