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課程來源:TED
     

 

Margaret Wertheim 談珊瑚的數學之美

The beautiful math of coral

 

Photo of
three lions hunting on the Serengeti.

講者:Margaret Wertheim

20092月演講,20094月在TED上線

 

翻譯:洪曉慧

編輯:劉契良

簡繁轉換:陳盈

後製:洪曉慧

字幕影片後制:謝旻均

 

影片請按此下載

閱讀中文字幕純文字版本

 

關於這場演講

Margaret Wertheim領導一個計畫,用一位數學家發明的鉤針技術重建珊瑚礁生物,以頌揚珊瑚礁的神奇,並深入探討以雙曲幾何為基礎的珊瑚創作。

 

關於Margaret Wertheim

Margaret Wertheim藉由策劃一個只使用鉤針形塑珊瑚礁的計畫,希望將一些體現在我們宇宙中最複雜的數學模型,帶進大眾的腦海(和雙手)之中。

 

為什麼要聽她演講:

雪花、碎形、葉片上的圖案-這些美景存在於自然和物理美學和數學的交叉路口。科學作家Margaret Wertheim(連同她的孿生姐妹,Christine)成立了塑形研究所,推動科學概念的審美觀,從雪花和碎形這些自然物體,到人類建構物,如伊斯蘭馬賽克、繩索圖形和編織。

 

塑形研究所的最新計劃可能是難以理解的奇怪-完全由鉤針編織建造出珊瑚礁。這個計畫利用了由珊瑚礁生物完美形塑的數學現象模型,以及如鉤針編織這樣的重複任務間美好的一致性-但得到的結果卻也完全符合雙曲空間模型。我們很容易沉迷於這個萬花筒般,毛線形塑的珊瑚礁之美,但這個珊瑚礁計畫有雙重目的:引起大眾對世界各地珊瑚礁危機的注意,它們因為海洋鹽水水面變化、過度捕撈、和無數的威脅而死亡;另外也展示出一個之前幾乎不可能圖示出的數學性質。藉由形塑這些物理空間中的複雜方程式,這種技術可以幫助數學家看到圖形模式和有所突破。

 

Wertheim目前正著手寫一本關於特立獨行科學家James Carter的著作。

 

Margaret Wertheim或許在技術上被歸類於科學傳播者的平庸頭銜上。但這位充滿熱情的澳大利亞原住民的作為,早已遠遠超出了只談論科學這個標準定義的範圍。」

Kristin Abkemeier,《Inkling》雜誌

 

Margaret Wertheim的英語網上資料

首頁: The Institute for Figuring

 

[TED科技娛樂設計]
已有中譯字幕的TED影片目錄(繁體)(簡體)。請注意繁簡目錄是不一樣的。

 

Margaret Wertheim 談珊瑚的數學之美

 

我今天來到這裡,如我在6月時所說的,要談論我的雙胞胎妹妹和我在過去三年半時間內,一直做的一個計畫:我們用鉤針編織珊瑚礁。事實上,世界各地已有數百人加入,和我們一起進行這個計畫。有數千人確實在許多不同方面參與了這個計畫。這個計畫目前已擴及三大洲,它紮根於數學、海洋生物學、女性手工業和生態環境活動主義領域,萬分不假。這也是一個以一種非常美麗的方式來發展的計畫。事實上,它與地球上生命的演化並行。有件特別棒的事,就是20092月,如同我們之前演講者所說的,正是達爾文誕生200周年。

 

我希望能在接下來的18分鐘內說明這一切工作。不過,讓我先向你展示一些這些東西的照片,只是要給你一些關於它規模的概念。這個裝置大約有六英尺寬,最高的模型約有二或三英尺高。這是更多的圖片,右邊的這個約五英尺高,這項作品包含數百個不同的鉤針模式。事實上,有數千個模式;世界各地的人都做了一部份貢獻。這個計畫共包含了成千上萬小時的人力,99 是由女性所做的。在右邊那一小塊是裝置的一部分,約 12英尺長。

 

我妹妹和我於2005年開始這個計畫。因為這一年,至少在科學刊物中,談論到很多關於全球暖化的議題;全球暖化確實會對珊瑚礁造成影響。珊瑚是很脆弱的有機體,一點點的海洋溫度上升都會破壞它,會導致這種大量的褪色的情形,這是珊瑚生病的第一個跡象。如果這種褪色現象不消失,如果氣溫不下降,珊瑚礁就會開始死亡。這情形已在大堡礁大規模發生,世界各地的珊瑚礁也是。這是我們用鉤針編織的褪色珊瑚礁。

 

我們有個叫做塑形研究所的新組織;這是個我們已開始推動的小組織,做一些關於科學和數學方面,富美學及詩意的作品。我在我們的網站上貼了些公告,徵求人們加入這份事業。出乎意料的是,最先回應者之一是安迪.沃荷博物館。他們說,他們有一個關於藝術家對全球暖化問題迴響的展覽,他們希望我們的珊瑚礁加入。我笑笑說:「嗯,我們現在才剛剛開始,目前只有一點點作品」。因此,在2007年,我們有了這個鉤針珊瑚礁的小展覽。有些芝加哥的人來看,並說:「2007年底的芝加哥人文藝術節主題是全球暖化,我們有個3000平方英尺的陳列室,我們希望你用你的珊瑚礁裝滿它」。當時我很天真的說「哦,是的,當然可以」。我會說「天真」,是因為其實我的專業是個科學作家,我所做的是寫一些關於自然科學的文化歷史書籍,也寫一些關於空間、自然科學及宗教歷史的書。我為紐約時報、洛杉磯時報寫些文章,所以我不知道擺滿3000平方英尺的陳列室會是什麼情形,所以我答應了這個提議。當我回家,告訴我妹妹Christine時,她簡直快暈了;因為Christine 是個教授,任教於洛杉磯一個主要藝術學院,即加州藝術學院。她清楚地知道擺滿3000平方英尺的陳列室意味著什麼,她認為我頭殼壞去了。但她加倍努力的進行鉤針工作。長話短說,8個月後,我們真的擺滿了芝加哥文化中心3000平方英尺的陳列室。

 

到這階段,這個計畫已經呈現了一種病毒式的擴散規模,完全超出了我們所預期。在芝加哥的人們決定,像展示我們的珊瑚礁一樣,他們也想要讓當地居民製作珊瑚礁。於是我們到那兒教鉤針技巧,開了講習班和講座。芝加哥的人民做出了自己的珊瑚礁,跟我們的一起展出;有數百人參與,我們被邀請進行這整個計畫。在紐約、倫敦和洛杉磯這些城市中,當地居民數百人做出了珊瑚礁。越來越多的人參與,大多數人我們從來沒見過。所以,整個計畫已經演變成這種有機的、不斷演化的生物,這事實上早已超出了Christine和我所預想的。

 

在座各位可能會想,「這些人頭腦到底裝些什麼啊?鉤珊瑚礁到底要幹嘛?毛織品和濕度是兩個完全不同的概念,為什麼不用大理石雕鑿珊瑚礁並用青銅澆鑄?」但我們有一個很好的理由來用鉤針編織:因為很多在珊瑚礁中的生物有機體,如裸鰓類動物、珊瑚礁、海藻及海綿中,有非常特別的一種結構,就是你所看到的裙襯褶邊形式。在裸鰓類動物中,是一種稱為雙曲幾何的幾何形式,數學家所知道唯一形塑這種結構的方法,就是用鉤針編織。確實如此,用任何其他方式幾乎都不可能形塑這種結構,在電腦上也幾乎不可能做到這一點。所以,這種珊瑚和海蛞蝓所體現的雙曲幾何到底是什麼?

 

接下來幾分鐘,我們都將提升到海蛞蝓層級(笑聲)。這種數學上的幾何革命於19世紀首次發展出來,但直到1997年,數學家才真正瞭解它們如何形塑出這一切。1997年,一位康奈爾大學的數學家Daina Taimina發現了這個結構,事實上可以用棒針和鉤針編織出來。她首先所做的是棒針編織,但會有太多的針縫。於是,她很快意識到鉤針是更好的方式;但她事實上做出了一個許多數學家認為,這實際上是不可能形塑出來的數學結構模型。事實上,他們認為任何像這樣的結構,本質上是不可能存在的。一些最好的數學家花了數百年,試圖證明不可能有這種結構。

 

因此,這個不可能的雙曲結構到底什麼?在有雙曲幾何前,數學家知道有兩種空間-歐幾里德空間和球面空間;它們具有不同的性質。數學家喜歡將事物賦予形式主義的特徵;大家都多少知道平面空間是什麼,就是歐幾里德空間。但數學家用一種特殊方式將這個形式化,他們所做的就是藉由平行線的概念。這裡有一條線和線外的點,歐幾里德說:「我如何定義平行線?我問個問題,我可以繪製多少條通過這個點的線,但這些線永遠不會和原來的線相交?」大家都知道答案。是否有人要說出來?一條,沒錯。Okay,這是我們對平行線的定義,這是一個真正歐幾里德空間的定義。

 

但有另一種大家都知道的可能性,就是球形空間。想想一個球體的表面,就像一個沙灘球、地球的表面。球形表面有條直線,有個點在線外;我可以畫多少條通過點的直線,但它們永遠不會與原來的線相交?關於曲面上的直線,我們指的是什麼?數學家已經回答了這個問題。他們已經知道有一個廣義概念的直線,它被稱為測地線。在球體表面上的一條直線,就是你可能畫出的最大圓,就像是赤道或經度線。因此,我們再問個問題:「我可以畫多少條通過這個點的直線,但這些線永遠不會和原來的線相交?」是否有人要猜看看?零。好極了!

 

現在,數學家認為這是唯一的選擇,這有點令人懷疑,是不是?到目前為止,問題有兩個答案:零和一兩個答案。也可能有第三個選擇。對一位數學家來說,如果有兩個答案,前兩個是零和一,另外一個數字立刻就會自己出現,作為第三個選擇。有沒有人要猜是什麼?無限。你們都對了。沒錯,還有第三個選擇,這就是它的圖形。有一條直線,上有無數穿過點的線,且永遠不和原始的線相交;這是它的繪圖。這幾乎使數學家陷入困境,就像你們一樣,他們坐在那裡感到迷惑,心想,怎麼可能呢?你騙人。那些線是彎曲的。但這只是因為我將它投影到一個平面上。幾百年來,數學家必須真正與它抗爭。他們怎麼理解這個?這代表著什麼呢?若確實有一個實體模型看起來像這樣?

 

這有點像這樣:想像我們只知道歐幾里德空間,然後我們的數學家出現,說:「這是叫做球體的東西,這些線在南北極相交」但你不知道球體會像什麼模樣,有人來了,說:「看,這裡有個球」你說,「啊!我可以看到它,我能感覺到它,我可以觸摸它,我可以玩它」。這正是當Daina Taimina1997年,證明可以用鉤針編織形塑出雙曲空間時所發生的情形。這是鉤針編織塑形。我將歐幾里德平行公設編織到表面上,這些線條看起來是彎曲的,但你看,我可以向你證明它們是直線;因為我可以拿起任何一條線,我可以沿線折疊,這是一條直線。所以這個用羊毛編織的東西,藉由家庭女性藝術證明,最著名的數學假設是錯誤的。(掌聲)

 

在這表面上,你可以編織所有種類的數學定理;這雙曲空間的發現開創了數學領域,就是所謂的非歐幾里德幾何;這實際上屬於數學領域中廣義相對論的基礎。事實上,它最終將向我們顯示關於宇宙的形狀。因此,這就是銜接於女性工藝品、歐幾里德和廣義相對論間的直接關聯。

 

數學家認為這是不可能的。下面是兩種從來沒有聽說過歐幾里德平行公設的生物,不知道這個公設是不可能違反的;他們就只是長成這樣,他們已經這樣子億萬年了。我曾問過數學家,為什麼他們認為這種結構是不可能的?因為海蛞蝓從志留紀時代就是這樣。他們的回答很有趣。他們說「嗯,我猜有沒有那麼多數學家,閒閒沒事來看海蛞蝓」。這是真的。但它也有更深層意義,這還蘊含著很多意義。關於數學家眼中的數學,他們認為可能或不可能的事,他們認為可能或不可能設想的事。即使數學家在某種意義上,是所有思想家中最自由的一群,但他們確實不僅看不到身邊的海蛞蝓,也看不到他們盤中的生菜。因為生菜,以及所有那些捲曲的蔬菜,也都體現了雙曲幾何。從某種意義上說,他們確實有如此象徵性的數學觀點,他們無法真正看到發生的事情,即使是就在他們眼前的生菜。自然世界確實充滿雙曲奇蹟。

 

我們也已經發現,有一個無限分類的雙曲生物鉤針形式。Chrissy和我及我們的贊助者,開始著手做簡單的完美數學模型。但我們發現,當我們偏離了特定的數學編碼設定:這是基於簡單的運算規則,每鉤3針,增加1針。當我們偏離了這些,將編碼做了潤飾,這個模型立刻變得更自然。我們所有的貢獻者,來自全世界了不起人們的集合,他們自行做出潤飾;可以說,我們有這個不斷演化生命的鉤針分類樹,就像形態學和地球上生命的複雜性一樣,是永無止境的。在DNA編碼中放入一些小潤飾和複雜性,就衍生了新事物,像長頸鹿或蘭花。所以鉤針編碼中的小潤飾,在鉤針生命的演化樹中,衍生出新的和令人驚奇的生物。所以,這個計畫確實呈現了有機生命本身的內在,及所有參與者整體,加上他們的個人觀點和他們對這個數學編碼的參與工作。

 

我們擁有這些技術,我們使用它們。但,為什麼?攸關什麼利益?有什麼重要性?對Chrissy和我來說,重要的是這些東西表現出體現知識的重要與價值。我們生活在一個社會中,完全傾向於穩定價值的象徵形式表現、代數表現方程、編碼,我們生活的社會沉溺於用這種方式提供資訊,用這種方式教導資訊。但是藉由這種形態,就是鉤針及其他塑形的做法,人們可以從事最抽象的高功率的理論概念。這類的概念,通常你必須去大學念相關學系才能學到;如高等數學,這正是我第一次學習雙曲空間之處。但你可以藉由與實體物質交流而做到這一點。其中一個我們考慮的方式就是我們正試圖在塑形研究做的:我們試圖以幼稚園的學習方式來教育成人。

 

幼稚園事實上是非常正式的教育系體之一,是由一位名為福祿.貝爾的人所建立。他是一位19世紀的晶體學家,他認為晶體是所有東西的代表模型。他發展了一個全新的替代系統,給予最年幼的孩子最抽象的概念,藉由遊戲的物理形式達成。值得用整個演講來說明他的成就。教育的價值是福祿所宣導的,這藉由遊戲模式的形塑達成。

 

我們生活在一個社會,擁有大量的智庫,有很棒的心智來思考整個世界;他們寫了傑出的象徵性專著,就是所謂的書籍、論文和專欄特稿。Chrissy和我想做個建議,藉由塑形研究所提供另一種行事方式選擇,那就是遊戲庫。遊戲庫跟智庫一樣,是一個人們可以從事傑出想法的地方,但我們希望進行的是最高層級的抽象物質,如數學、計算、邏輯等等,所有這一切都可以進行。不只是藉由純粹大腦運算的象徵性方法,而是實際上將這些想法身體力行。感謝聆聽。(掌聲)

 

以下為系統擷取之英文原文

About this talk

Margaret Wertheim leads a project to re-create the creatures of the coral reefs using a crochet technique invented by a mathematician -- celebrating the amazements of the reef, and deep-diving into the hyperbolic geometry underlying coral creation.

About Margaret Wertheim

By masterminding a project to model a coral reef armed only with crochet hooks, Margaret Wertheim hopes to bring some of the most complicated mathematical models embodied in our universe into… Full bio and more links

Transcript

I'm here today, as June said, to talk about a project that my twin sister and I have been doing for the past three and half years. We're crocheting a coral reef. And it's a project that we've actually been now joined by hundreds of people around the world who are doing it with us. Indeed thousands of people have actually been involved in this project, in many of its different aspects. It's a project that now reaches across three continents. Its roots go into the fields of mathematics, marine biology, feminine handicraft and environmental activism. It's true. It's also a project that in a very beautiful way, the development of this has actually paralleled the evolution of life on earth, which is a particularly lovely thing to be saying right here in February 2009 -- which, as one of our previous speakers told us, is the 200th anniversary of the birth of Charles Darwin.

All of this I'm going to get to in the next 18 minutes, I hope. But let me first begin by showing you some pictures of what this thing looks like. Just to give you an idea of scale, that installation there is about six feet across. And the tallest models are about two or three feet high. This is some more images of it. That one on the right is about five feet high. The work involves hundreds of different crochet models. And indeed there are now thousands and thousands of models that people have contributed all over the world as part of this. The totality of this project involves tens of thousands of hours of human labor -- 99 percent of it done by women. On the right hand side, that bit there is part of an installation that is about 12 feet long.

My sister and I started this project in 2005 because in that year, at least in the science press, there was a lot of talk about global warming, and the effect that global warming was having on coral reefs. Corals are very delicate organisms. And they are devastated by any rise in sea temperatures. It causes these vast bleaching events that are the first signs of corals of being sick. And if the bleaching doesn't go away, if the temperatures don't go down, reefs start to die. A great deal of this has been happening in the Great Barrier Reef, particularly in coral reefs all over the world. This is our invocation in crochet of a bleached reef.

We have a new organization together called The Institute For Figuring, which is a little organization we started to promote, to do projects about aesthetic and poetic dimensions of science and mathematics. And I went and put a little announcement up on our site, asking for people to join us in this enterprise. To our surprise, one of the first people who called was the Andy Warhol Museum. And they said they were having an exhibition about artists' response to global warming, and they'd like our coral reef to be part of it. I laughed and said, "Well we've only just started it, you can have a little bit of it." So in 2007 we had an exhibition, a small exhibition of this crochet reef. And then some people in Chicago came along and they said, "In late 2007, the theme of the Chicago Humanities Festival is global warming. And we've got this 3,000 square foot gallery and we want you to fill it with your reef." And I, naively by this stage, said, "Oh, yes. Sure." Now I say "naively" because actually my profession is as a science writer. What I do is I write books about the cultural history of physics. I've written books about the history of space, the history of physics and religion, and I write articles for people like the New York Times, and the L.A. Times. So I had no idea what it meant to fill a 3,000 square foot gallery. So I said yes to this proposition. And I went home, and I told my sister Christine. And she nearly had a fit because Christine is a professor at one of L.A.'s major art colleges, CalArts, and she knew exactly what it meant to fill a 3,000 square foot gallery. She thought I'd gone off my head. But she went into crochet overdrive. And to cut a long story short, eight months later we did fill the Chicago Cultural Center's 3,000 square foot gallery.

By this stage the project had taken on a viral dimension of its own, which got completely beyond us. The people in Chicago decided that as well as exhibiting our reefs, what they wanted to do was have the local people there make a reef. So we went and taught the techniques. We did workshops and lectures. And the people in Chicago made a reef of their own. And it was exhibited alongside ours. There were hundreds of people involved in that. We got invited to do the whole thing in New York, and in London, and in Los Angeles. In each of these cities, the local citizens, hundreds and hundreds of them, have made reefs. And more and more people get involved in this, most of whom we've never met. So the whole thing has sort of morphed into this organic, ever evolving creature, that's actually gone way beyond Christine and I.

Now some of you are sitting here thinking, "What planet are these people on? Why on earth are you crocheting a reef? Woolenness and wetness aren't exactly two concepts that go together. Why not chisel a coral reef out of marble? Cast it in bronze." But it turns out there is a very good reason why we are crocheting it because many organisms in coral reefs have a very particular kind of structure. The frilly crinolated forms that you see in corals, and kelps, and sponges, and nudibranchs, is a form of geometry known as hyperbolic geometry. And the only way that mathematicians know how to model this structure, is with crochet. It happens to be a fact. It's almost impossible to model this structure any other way. And it's almost impossible to do it on computers. So what is this hyperbolic geometry that corals and sea slugs embody?

The next few minutes is, we're all going to get raised up to the level of a sea slug. (Laughter) This sort of geometry revolutionized mathematics when it was first discovered in the 19th century. But not until 1997 did mathematicians actually understand how they could model it. In 1997 a mathematician at Cornell, Daina Taimina, made the discovery that this structure could actually be done in knitting and crochet. The first one she did was knitting. But you get too many stitches on the needle. So she quickly realized crochet was the better thing. But what she was doing was actually making a model of a mathematical structure, that many mathematicians had thought it was actually impossible to model. Indeed they thought that anything like this structure was impossible per se. Some of the best mathematicians spent hundreds of years trying to prove that this structure was impossible.

So what is this impossible hyperbolic structure? Before hyperbolic geometry, mathematicians knew about two kinds of space, Euclidean space and spherical space. And they have different properties. Mathematicians like to characterize things by being formalist. You all have a sense of what a flat space is, Euclidean space is. But mathematicians formalize this in a particular way. And what they do is, they do it through the concept of parallel lines. So here we have a line and a point outside the line. Euclid said, "How can I define parallel lines? I ask the question, how many lines can I draw through the point but never meet the original line?" And you all know the answer. Does someone want to shout it out? One. Right. Okay. That's our definition of a parallel line. It's a definition really of Euclidean space.

But there is another possibility that you all know of -- spherical space. Think of the surface of a sphere -- just like a beach ball, the surface of the Earth. I have a straight line on my spherical surface. And I have a point outside the line. How many straight lines can I draw through the point but never meet the original line? What do we mean to talk about a straight line on a curved surface? Now mathematicians have answered that question. They've understood there is a generalized concept of straightness. It's called a geodesic. And on the surface of a sphere, a straight line is the biggest possible circle you can draw. So it's like the equator or the lines of longitude. So we ask the question again, "How many straight lines can I draw through the point, but never meet the original line?" Does someone want to guess? Zero. Very good.

Now mathematicians thought that was the only alternative. It's a bit suspicious isn't it? There is two answers to the question so far, Zero and one. Two answers? There may possibly be a third alternative. To a mathematician if there are two answers, and the first two are zero and one, there is another number that immediately suggests itself, as the third alternative. Does anyone want to guess what it is? Infinity. You all got it right. Exactly. There is a third alternative. This is what it looks like. It has a straight line, and there is an infinite number of lines that go through the point and never meet the original line. This is the drawing. This nearly drove mathematicians bonkers because, like you, they're sitting there feeling bamboozled. Thinking, how can that be? You're cheating. The lines are curved. But that's only because I'm projecting it onto a flat surface. Mathematicians for several hundred years had to really struggle with this. How could they see this? What did it mean to actually have a physical model that looked like this?

It's a bit like this: imagine that we'd only ever encountered Euclidean space. Then our mathematicians come along and said, "There's this thing called a sphere, and the lines come together at the north and south pole." But you don't know what a sphere looks like. And someone that comes along and says, "Look here's a ball." You go, "Ah! I can see it. I can feel it. I can touch it. I can play with it." And that's exactly what happened when Daina Taimina in 1997, showed that you could crochet models in hyperbolic space. Here is this diagram in crochetness. I've stitched Euclid's parallel postulate on to the surface. And the lines look curved. But look, I can prove to you that they're straight because I can take any one of these lines, and I can fold along it. It's a straight line. So here, in wool, through a domestic feminine art, is the proof that the most famous postulate in mathematics is wrong. (Applause)

You can stitch all sorts of mathematical theorems onto these surfaces. The discovery of hyperbolic space ushered in the field of mathematics that is called non-Euclidean geometry. This is actually the field of mathematics that underlies general relativity and is actually ultimately going to show us about the shape of the universe. So there is this direct line between feminine handicraft, Euclid and general relativity.

Now, I said that mathematicians thought that this was impossible. Here's two creatures who've never heard of Euclid's parallel postulate -- didn't know it was impossible to violate, and they're simply getting on with it. They've been doing it for hundreds of millions of years. I once asked the mathematicians why it was that mathematicians thought this structure was impossible when sea slugs have been doing it since the Silurian age. Their answer was interesting. They said, "Well I guess there aren't that many mathematicians sitting around looking at sea slugs." And that's true. But it also goes deeper than that. It also says a whole lot of things about what mathematicians thought mathematics was. What they thought it could and couldn't do. What they thought it could and couldn't represent. Even mathematicians, who in some sense are the freest of all thinkers, literally couldn't see not only the sea slugs around them, but the lettuce on their plate because lettuces, and all those curly vegetables, they also are embodiments of hyperbolic geometry. In some sense they literally -- they had such a symbolic view of mathematics -- they couldn't actually see what was going on on the lettuce in front of them. It turns out that the natural world is full of hyperbolic wonders.

And so too, we've discovered that there is an infinite taxonomy of crochet hyperbolic creatures. We started out, Chrissy and I and our contributors, doing the simple mathematically perfect models. But we found that when we deviated from the specific setness of the mathematical code that underlies is the simple algorithm, crochet three, increase one. When we deviated from that and made embellishments to the code, the models immediately started to look more natural. And all of our contributors, who are an amazing collection of people around the world, do their own embellishments. As it were, we have this ever evolving, crochet taxonomic tree of life. Just as the morphology and the complexity of life on earth is never ending, little embellishments and complexifications in the DNA code, lead to new things like giraffes or orchids. So too, little embellishments in the crochet code lead to new and wondrous creatures in the evolutionary tree of crochet life. So this project really has taken on this inner organic life of its own. There is the totality of all the people who have come to it. And their individual visions, and their engagement with this mathematical mode.

We have these technologies. We use them. But why? What is at stake here? What does it matter? For Chrissy and I, one of the things that's important here is that these things suggest the importance and value of embodied knowledge. We live in a society that completely tends to valorize symbolic forms of representation -- algebraic representations, equations, codes. We live in a society that's obsessed with presenting information in this way, teaching information in this way. But through this sort of modality, crochet, other plastic forms of play, people can be engaged with the most abstract, high powered, theoretical ideas -- the kinds of ideas that normally you have to go to university departments to study in higher mathematics, which is where I first learned about hyperbolic space. But you can do it through playing with material objects. One of the ways that we've come to think about this is that what we're trying to do with the Institute for Figuring, and projects like this, we're trying to have kindergarten for grown-ups.

Kindergarten was actually a very formalized system of education, established by a man named Friedrich Froebel, who was a crystallographer in the 19th century. He believed that the crystal was the model for all kinds of representation. He developed a radical alternative system of engaging the smallest children with the most abstract ideas through physical forms of play. He is worthy of an entire talk on his own right. The value of education is something that Froebel championed, through plastic modes of play.

We live in a society now where we have lots of think tanks, where great minds go to think about the world. They write these great symbolic treatises called books, and papers, and op-ed articles. We want to propose, Chrissy and I, through The Institute For Figuring, another alternative way of doing things, which is the play tank. The play tank, like the think tank, is a place where people can go and engage with great ideas. But what we want to propose, is that the highest levels of abstraction, things like mathematics, computing, logic, et cetera -- all of this can be engaged with, not just through purely cerebral algebraic symbolic methods, but by literally, physically playing with ideas. Thank you very much. (Applause)


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