I want to start my story in Germany, in 1877, with a mathematician named Georg Cantor. And Cantor decided he was going to take a line and erase the middle third of the line, and then take those two resulting lines and bring them back into the same process, a recursive process. So he starts out with one line, and then two, and then four, and then 16, and so on. And if he does this an infinite number of times, which you can do in mathematics, he ends up with an infinite number of lines, each of which has an infinite number of points in it. So he realized he had a set whose number of elements was larger than infinity. And this blew his mind. Literally. He checked into a sanitarium. (Laughter) And when he came out of the sanitarium, he was convinced that he had been put on earth to found transfinite set theory because the largest set of infinity would be God Himself. He was a very religious man. He was a mathematician on a mission.
我的故事要從1877年在德國 一位名叫Georg Cantor的數學家說起。 Cantor決定要把一個線段的中間三分之一擦掉, 將頭尾兩端接起來再重複,如此週而復始。 所以他一開始有一條線段,然後有兩條, 接著有四條,然後有十六條,這樣繼續下去。 如果他做這個無限多次,在數學上是可以做到的, 他就會有無限多條線段, 其中每一條線段都有無限多點。 所以他發現他會有一個比無限多還大的集合。 他為之瘋狂。真的。他進了療養院。(笑聲) 當他離開療養院時, 他認為他來到地球是為了理解超限理論, 因為最大的無限就是神。 他是個非常虔誠的人。 他是個有使命的數學家。
And other mathematicians did the same sort of thing. A Swedish mathematician, von Koch, decided that instead of subtracting lines, he would add them. And so he came up with this beautiful curve. And there's no particular reason why we have to start with this seed shape; we can use any seed shape we like. And I'll rearrange this and I'll stick this somewhere -- down there, OK -- and now upon iteration, that seed shape sort of unfolds into a very different looking structure. So these all have the property of self-similarity: the part looks like the whole. It's the same pattern at many different scales.
而且其他數學家也做了類似的事情。 van Koch是個瑞典的數學家, 他做了類似的事情,但是不用減法而改用加法。 所以他得到了這漂亮的弧線。 並沒有什麼特定的原因讓我們必須從這樣的種子圖形開始, 我們可以用任何的圖形作起始。 我來重新整理一下,把這個放在某個地方--放到這裡,好-- 經過無數重複後,種子圖形展開成一個非常不同的結構。 所以這些都有自體相似的特質: 各個部份跟整體相似。 在不同尺度上都是同一個圖形。
Now, mathematicians thought this was very strange because as you shrink a ruler down, you measure a longer and longer length. And since they went through the iterations an infinite number of times, as the ruler shrinks down to infinity, the length goes to infinity. This made no sense at all, so they consigned these curves to the back of the math books. They said these are pathological curves, and we don't have to discuss them. (Laughter) And that worked for a hundred years.
好,數學家覺得這很奇怪。 因為如果你把一把尺縮小,你量到的數據會越來越長。 然後因為重複了無限多次, 量尺變成無限小,長度變成無限長。 這不合理, 所以他們把這個放在數學書籍最後面。 他們說這是有問題的曲線,所以我們不討論。 (笑聲) 且成功的這麼做了一百年。
And then in 1977, Benoit Mandelbrot, a French mathematician, realized that if you do computer graphics and used these shapes he called fractals, you get the shapes of nature. You get the human lungs, you get acacia trees, you get ferns, you get these beautiful natural forms. If you take your thumb and your index finger and look right where they meet -- go ahead and do that now -- -- and relax your hand, you'll see a crinkle, and then a wrinkle within the crinkle, and a crinkle within the wrinkle. Right? Your body is covered with fractals. The mathematicians who were saying these were pathologically useless shapes? They were breathing those words with fractal lungs. It's very ironic. And I'll show you a little natural recursion here. Again, we just take these lines and recursively replace them with the whole shape. So here's the second iteration, and the third, fourth and so on.
然後在1977年,一個法國數學家Benoit Mandelbrot 發現如果利用電腦繪圖繪出這些他叫做碎形的圖樣, 你可以得到自然界的圖形。 你可以得到人類的肺圖形、刺槐、蕨類, 你可以得到這些美麗的大自然形狀。 如果你看你大拇指和食指交界的地方-- 拿起來看看-- 手放鬆,你會看到波紋, 波紋中有皺紋,皺紋中有波紋。對吧? 你們全身都被碎形包覆著。 而這些數學家竟然說這些是有問題且無意義的圖形? 他們正在用碎形組成的肺說這些話。 這是非常諷刺的。我可以給你們看一些自然的循環。 再次的,我們將這些線段作重複。 這是第二次重複、第三次、第四次...
So nature has this self-similar structure. Nature uses self-organizing systems. Now in the 1980s, I happened to notice that if you look at an aerial photograph of an African village, you see fractals. And I thought, "This is fabulous! I wonder why?" And of course I had to go to Africa and ask folks why. So I got a Fulbright scholarship to just travel around Africa for a year asking people why they were building fractals, which is a great job if you can get it. (Laughter)
大自然有這樣的自體相似結構。 大自然利用自體組織系統。 在1980年代我發現 如果看這個非洲村落的空照圖,你會看到碎形。 我就想:「這太好了!我想要知道為什麼?」 所以當然我要去非洲問那些人為什麼。 所以我拿了Fulbright獎學金去非洲旅行一年 問當地人為什麼要建造碎形。 其實是個很不錯的工作如果你可以拿到這個工作。 (笑聲)
And so I finally got to this city, and I'd done a little fractal model for the city just to see how it would sort of unfold -- but when I got there, I got to the palace of the chief, and my French is not very good; I said something like, "I am a mathematician and I would like to stand on your roof." But he was really cool about it, and he took me up there, and we talked about fractals. And he said, "Oh yeah, yeah! We knew about a rectangle within a rectangle, we know all about that." And it turns out the royal insignia has a rectangle within a rectangle within a rectangle, and the path through that palace is actually this spiral here. And as you go through the path, you have to get more and more polite. So they're mapping the social scaling onto the geometric scaling; it's a conscious pattern. It is not unconscious like a termite mound fractal.
所以我終於到達了這個城市。我做了一個這個城市的小型碎形模型, 讓我可以更瞭解如何展開的-- 我到了那裡,找到酋長的宮殿, 我的法文不大好,我說了像是這樣的話:「 我是個數學家,我想要站到你的屋頂上。」 但他覺得沒問題,然後帶我上去, 然後我們聊了一下碎形。 他說:「喔對對,我們知道這個長方形裡面的長方形, 我們知道那個。」 而且事實上皇家徽章就是長方形裡面有長方形有長方形, 且皇宮中的走廊也是這樣迴旋著的。 而且順著這些走廊走下去,你必須越來越有禮貌。 所以他們是用這樣幾何縮放的方式來建立社會地位, 是故意這麼做的,並不是像飛蟻丘那樣無意識的。
This is a village in southern Zambia. The Ba-ila built this village about 400 meters in diameter. You have a huge ring. The rings that represent the family enclosures get larger and larger as you go towards the back, and then you have the chief's ring here towards the back and then the chief's immediate family in that ring. So here's a little fractal model for it. Here's one house with the sacred altar, here's the house of houses, the family enclosure, with the humans here where the sacred altar would be, and then here's the village as a whole -- a ring of ring of rings with the chief's extended family here, the chief's immediate family here, and here there's a tiny village only this big. Now you might wonder, how can people fit in a tiny village only this big? That's because they're spirit people. It's the ancestors. And of course the spirit people have a little miniature village in their village, right? So it's just like Georg Cantor said, the recursion continues forever.
這是在南尚比亞的一個村莊。 Ba-Ila人建造了一個直徑約400公尺的村莊。 首先你有一個很大的圈圈。 這些代表家族的圈圈越往後面越大, 在後面這邊有酋長的圈圈, 圈圈旁邊是酋長的家人圈。 所以這是個小型的碎形模型。 這裡是一棟擁有神檀的屋子。 這裡是房子的房子,家庭圈圈, 這邊神壇的位置有人在, 這是整個村莊-- 一圈一圈地在這裡,這是酋長的遠親,這裡是酋長的近親-- 而這裡是一個非常小只有這麼大的村莊。 你們可能會問,這麼小的村莊怎麼住得下人? 那是因為這些是神魂人物,是祖先們。 而且當然的這迷你的村落裡有另一個更小的村落,對吧? 所以就像Georg Cantor說的,一再地重複著。
This is in the Mandara mountains, near the Nigerian border in Cameroon, Mokoulek. I saw this diagram drawn by a French architect, and I thought, "Wow! What a beautiful fractal!" So I tried to come up with a seed shape, which, upon iteration, would unfold into this thing. I came up with this structure here. Let's see, first iteration, second, third, fourth. Now, after I did the simulation, I realized the whole village kind of spirals around, just like this, and here's that replicating line -- a self-replicating line that unfolds into the fractal. Well, I noticed that line is about where the only square building in the village is at. So, when I got to the village, I said, "Can you take me to the square building? I think something's going on there." And they said, "Well, we can take you there, but you can't go inside because that's the sacred altar, where we do sacrifices every year to keep up those annual cycles of fertility for the fields." And I started to realize that the cycles of fertility were just like the recursive cycles in the geometric algorithm that builds this. And the recursion in some of these villages continues down into very tiny scales.
這是在奈吉利亞邊界Mokoulek地區Cameroon的Mandara山中的景象。 我看到這幅法國建築家畫的圖, 然後我想:「哇!真是漂亮的碎形阿!」 所以我試著找出一個種子圖形在經過重複後可以展開成這樣的東西。 我想到這樣的一個結構。 讓我們看看,第一次重複、第二次、第三次、第四次。 經過模擬後, 我發現整個村莊像螺旋般環繞著,就像這樣, 且這邊是重複線:一條融入到碎形裡的自我複製線。 我發現這也是整個村莊唯一一棟正方形建築物所在地。 所以我到了這個村莊, 我問:「你可以帶我到這棟正方形建築那裡嗎? 我覺得那裡有些什麼東西。」 他們說:「恩,我們可以帶你去那裡,但你不能進去, 因為那是聖壇也就是我們每年為了 保持土地肥沃做祭祀的地方。」 我開始瞭解到這肥沃土壤的循環 就跟建造這個的幾何算式循環一樣。 且這樣的循環一直延續到非常小的尺度。
So here's a Nankani village in Mali. And you can see, you go inside the family enclosure -- you go inside and here's pots in the fireplace, stacked recursively. Here's calabashes that Issa was just showing us, and they're stacked recursively. Now, the tiniest calabash in here keeps the woman's soul. And when she dies, they have a ceremony where they break this stack called the zalanga and her soul goes off to eternity. Once again, infinity is important.
這裡是Mali的一個Nankani村莊。 你們可以看到,人們可以進到家庭圈圈中-- 你可以進去然後這裡是壁爐中的鍋子,也是循環堆疊的。 這是Issa剛剛給我們看得葫蘆, 它們也是循環堆疊的。 這裡,最小的葫蘆裡面保存著女人的靈魂。 她離開人世時,他們有一個儀式 會打壞這個叫做zalanga的的堆疊讓她的靈魂可以達到永恆。 再次的,無限是非常重要的。
Now, you might ask yourself three questions at this point. Aren't these scaling patterns just universal to all indigenous architecture? And that was actually my original hypothesis. When I first saw those African fractals, I thought, "Wow, so any indigenous group that doesn't have a state society, that sort of hierarchy, must have a kind of bottom-up architecture." But that turns out not to be true.
到此,你們可能會問自己三個問題。 這樣的不同尺度間呼應的圖形不是在每個原始建築中都存在嗎? 這事實上是我一開始的假設。 當我第一次看到非洲碎形時, 我想:「哇,所以任何一個沒有制式的階層結構的的原始族群 都應該有類似的自下而上的建築形態。」 但後來發現這是不正確的。
I started collecting aerial photographs of Native American and South Pacific architecture; only the African ones were fractal. And if you think about it, all these different societies have different geometric design themes that they use. So Native Americans use a combination of circular symmetry and fourfold symmetry. You can see on the pottery and the baskets. Here's an aerial photograph of one of the Anasazi ruins; you can see it's circular at the largest scale, but it's rectangular at the smaller scale, right? It is not the same pattern at two different scales.
我開始蒐集美國原住民和南太平洋建築的空照圖, 只有非洲的有碎形。 而且如果你仔細想,這些不同的文民都有不同的幾何設計主題。 美國原住民用了圓形對稱和四方對稱的組合。 你可以在陶器和籃子上看出來。 這是Anasazi殘骸的空照圖。 你們可以看到在大尺度上是圓環的,但在較小的尺度上是長方形的,對吧? 在這兩個尺度上不是一樣的圖形。
Second, you might ask, "Well, Dr. Eglash, aren't you ignoring the diversity of African cultures?" And three times, the answer is no. First of all, I agree with Mudimbe's wonderful book, "The Invention of Africa," that Africa is an artificial invention of first colonialism, and then oppositional movements. No, because a widely shared design practice doesn't necessarily give you a unity of culture -- and it definitely is not "in the DNA." And finally, the fractals have self-similarity -- so they're similar to themselves, but they're not necessarily similar to each other -- you see very different uses for fractals. It's a shared technology in Africa.
第二,你可能會問: 「恩,Eglash博士,你是不是忽略了非洲文化的多樣性?」 第三次的,答案是否定的。 首先,我完全贊同Mudimbe在他很棒的書《非洲創立》中寫到 非洲是個是個人類殖明主義的開始, 接著是對抗性運動。 不,因為一個廣泛被使用的設計並不代表文化上是統一的, 亦不代表是包含在DNA中的。 而且這些碎形是自體相似的, 也就是說他們跟自己像而跟其它的碎形不像, 你可以看到非常不同的碎形使用方式。 這是在非洲的一個共同的科技。
And finally, well, isn't this just intuition? It's not really mathematical knowledge. Africans can't possibly really be using fractal geometry, right? It wasn't invented until the 1970s. Well, it's true that some African fractals are, as far as I'm concerned, just pure intuition. So some of these things, I'd wander around the streets of Dakar asking people, "What's the algorithm? What's the rule for making this?" and they'd say, "Well, we just make it that way because it looks pretty, stupid." (Laughter) But sometimes, that's not the case. In some cases, there would actually be algorithms, and very sophisticated algorithms. So in Manghetu sculpture, you'd see this recursive geometry. In Ethiopian crosses, you see this wonderful unfolding of the shape.
最後,恩,會不會這只是直覺? 事實上跟數學知識一點關係都沒有? 非洲人不可能真的使用碎形幾何對吧? 碎形幾何一直到1970年代才發明的。 是的,我認為非洲碎形有很大一部份是直覺。 有時候我會在Dakar的街上遊蕩 問當地人:「這背後的算式是什麼?規則是什麼?」 他們會說:「 我們把它建造成這樣所以好看阿!你這個笨蛋。」(笑聲) 但有些時候不是這樣的。 有些時候,背後真的有算式,且是非常複雜的算式。 你可以在Manghetu的雕像上看到重複的幾何圖形。 在Ehiopian的十字架上也可以看到這些無限展開的形狀。
In Angola, the Chokwe people draw lines in the sand, and it's what the German mathematician Euler called a graph; we now call it an Eulerian path -- you can never lift your stylus from the surface and you can never go over the same line twice. But they do it recursively, and they do it with an age-grade system, so the little kids learn this one, and then the older kids learn this one, then the next age-grade initiation, you learn this one. And with each iteration of that algorithm, you learn the iterations of the myth. You learn the next level of knowledge.
在Angola,Chokwe人會在沙上畫線, 也就是德國數學家Euler叫做圖像的東西。 我們把它叫做Eulerian道路-- 你永遠不可以將你的筆從表面上提起, 也不可以重複同一條線段。 但他們可以重複地這個做,且以一個年紀劃分的方式這麼做, 所以小朋友會學這個,大一點的學這個, 在大一點的學這個。 而且在每一次重複這些算式時 他們會學這些重複背後的意義。 他們會學到下一層的知識。
And finally, all over Africa, you see this board game. It's called Owari in Ghana, where I studied it; it's called Mancala here on the East Coast, Bao in Kenya, Sogo elsewhere. Well, you see self-organizing patterns that spontaneously occur in this board game. And the folks in Ghana knew about these self-organizing patterns and would use them strategically. So this is very conscious knowledge.
最後,在整個非洲你都可以看到這樣的棋盤遊戲。 這遊戲在我研究的加那叫作Owari, 在東岸叫做Mancaia,在肯亞叫做Bao,在其他地方叫做Sogo。 你可以在這些棋盤遊戲中看到自體重複的圖形。 在加那的人知道這些圖形, 且會有策略地運用它們。 所以是個有意識的知識。
Here's a wonderful fractal. Anywhere you go in the Sahel, you'll see this windscreen. And of course fences around the world are all Cartesian, all strictly linear. But here in Africa, you've got these nonlinear scaling fences. So I tracked down one of the folks who makes these things, this guy in Mali just outside of Bamako, and I asked him, "How come you're making fractal fences? Because nobody else is." And his answer was very interesting. He said, "Well, if I lived in the jungle, I would only use the long rows of straw because they're very quick and they're very cheap. It doesn't take much time, doesn't take much straw." He said, "but wind and dust goes through pretty easily. Now, the tight rows up at the very top, they really hold out the wind and dust. But it takes a lot of time, and it takes a lot of straw because they're really tight." "Now," he said, "we know from experience that the farther up from the ground you go, the stronger the wind blows." Right? It's just like a cost-benefit analysis. And I measured out the lengths of straw, put it on a log-log plot, got the scaling exponent, and it almost exactly matches the scaling exponent for the relationship between wind speed and height in the wind engineering handbook. So these guys are right on target for a practical use of scaling technology.
這是個很棒的碎形。 在Sahel的各個地方,你都可以看到這樣的擋風玻璃。 當然的世界上任何籬笆都是笛卡爾式的,都是直線的。 但在非洲,你也可以看到這些不是直線的籬笆。 所以我找到設計這些籬笆的人, 他是一個住在Bamako外面的Mali的人,我問他: 「為什麼你用碎形法製造籬笆?因為沒有其他人這麼做。」 他的答覆非常有趣。 他說:「恩,當我走在叢林中時,我只會用長條的稻草, 因為使用它們既快又便宜。 不需要花太多時間且不需要太多稻草。」 他說:「但風和塵土很容易穿過。 最上層很緊的那排可以擋住風和塵土。 但這需要花很多時間、很多稻草,因為他們需要非常緊。」 「現在」他說:「我們從經驗中得知, 越高的地方風越強。」 對吧?就有點像是成本效益分析。 我量了稻草的長度, 把它放進對數圖形,得到尺度指數, 發現他幾乎完全和風速工程書上的 風速與高度的指數相同。 這些人在利用尺度科技上正中目標。
The most complex example of an algorithmic approach to fractals that I found was actually not in geometry, it was in a symbolic code, and this was Bamana sand divination. And the same divination system is found all over Africa. You can find it on the East Coast as well as the West Coast, and often the symbols are very well preserved, so each of these symbols has four bits -- it's a four-bit binary word -- you draw these lines in the sand randomly, and then you count off, and if it's an odd number, you put down one stroke, and if it's an even number, you put down two strokes. And they did this very rapidly, and I couldn't understand where they were getting -- they only did the randomness four times -- I couldn't understand where they were getting the other 12 symbols. And they wouldn't tell me. They said, "No, no, I can't tell you about this." And I said, "Well look, I'll pay you, you can be my teacher, and I'll come each day and pay you." They said, "It's not a matter of money. This is a religious matter."
我找到最復雜的算式碎形 並不是幾何圖形,而是符號象徵, 而這是在Bamana的沙占卜。 在整個非洲都有同樣的占卜系統。 你可以在東岸西岸都找得到這個占卜, 而且大部份的時候這些符號是保存得很好的。 每一個符號有四個小部份:是四個二進法組成的字。 你隨意畫這些線,然後數一下, 如果是奇數,就畫一條線; 如果是偶數,就畫兩條線。 且他們很迅速地這麼做, 我無法瞭解他們怎麼做到的, 他們在隨意的部份只做了四次, 我不懂他們另外十二個符號怎麼來的。 他們也不告訴我。 他們說:「不不,我不能告訴你這個。」 然後我說:「恩,我可以付你錢,你可以當我的老師, 然後我可以每天來付你學費。」 他們說:「這不是錢的問題。這是宗教問題。」
And finally, out of desperation, I said, "Well, let me explain Georg Cantor in 1877." And I started explaining why I was there in Africa, and they got very excited when they saw the Cantor set. And one of them said, "Come here. I think I can help you out here." And so he took me through the initiation ritual for a Bamana priest. And of course, I was only interested in the math, so the whole time, he kept shaking his head going, "You know, I didn't learn it this way." But I had to sleep with a kola nut next to my bed, buried in sand, and give seven coins to seven lepers and so on. And finally, he revealed the truth of the matter. And it turns out it's a pseudo-random number generator using deterministic chaos. When you have a four-bit symbol, you then put it together with another one sideways. So even plus odd gives you odd. Odd plus even gives you odd. Even plus even gives you even. Odd plus odd gives you even. It's addition modulo 2, just like in the parity bit check on your computer. And then you take this symbol, and you put it back in so it's a self-generating diversity of symbols. They're truly using a kind of deterministic chaos in doing this. Now, because it's a binary code, you can actually implement this in hardware -- what a fantastic teaching tool that should be in African engineering schools.
最後在絕望中我說:「 恩,讓我來解釋一下1877年的Georg Cantor。」 所以我開始解釋我為什麼會在非洲, 他們看了Cantor組合後非常興奮。 他們之中其中一個說:「過來,我想我可以幫你一些。」 所以他代Bamana牧師帶我走過了一連串的起始儀式。 當然的,我只對數學的部份有興趣, 所以整個過程,他一直搖頭說: 「我不是這樣學的。」 但我必須在床邊放一顆埋在沙裡的可樂果, 然後給七個痲瘋病人七個銅板之類的事情。 最後,他終於告訴我這後面的祕密。 事實上這是一個偽渾沌的產生數字的過程。 當你有一個四位符號,你把它們並排排起來。 所以偶數加奇數會得到奇數。 奇數加偶數會得到奇數。 偶數加偶數會得到偶數。奇數加奇數得到偶數。 這是加法定理,就像是電腦裡的配對法一樣。 然後你拿所得到的符號,再放回去, 就得到一個自我生成的多樣性符號。 他們真的在使用決定性混度來產生這些符號。 好,因為是二進位符號, 事實上你可以將這個置入到硬體裡面-- 多麼適合給非洲工程學校的教材阿!
And the most interesting thing I found out about it was historical. In the 12th century, Hugo of Santalla brought it from Islamic mystics into Spain. And there it entered into the alchemy community as geomancy: divination through the earth. This is a geomantic chart drawn for King Richard II in 1390. Leibniz, the German mathematician, talked about geomancy in his dissertation called "De Combinatoria." And he said, "Well, instead of using one stroke and two strokes, let's use a one and a zero, and we can count by powers of two." Right? Ones and zeros, the binary code. George Boole took Leibniz's binary code and created Boolean algebra, and John von Neumann took Boolean algebra and created the digital computer. So all these little PDAs and laptops -- every digital circuit in the world -- started in Africa. And I know Brian Eno says there's not enough Africa in computers, but you know, I don't think there's enough African history in Brian Eno. (Laughter) (Applause)
我發現最有趣的是它的歷史。 在十二世紀,Santalla的Hugu將這個從西班牙的回教傳統中引進的。 在那裡,碎形以看風水的身分 進入了煉金術的世界。 這是1390年理查國王二世所畫的幾何圖表。 德國數學家Leibniz在他的論文中 提到「De Combinatoria」的幾何性。 他說:「恩,讓我們用零和一取代 一條線和兩條線,這樣我們可以以二的指數數下去。」 對吧?零和一,二進位法。 George Boole拿了Leibniz的二進位法而創造了Boolean算式, 然後John von Neumann拿了Boolean算式而創造了數位電腦。 所以這些掌上型電腦和筆記型電腦-- 所有利用數位迴路的東西--都是從非洲開始的。 我知道Brian Eno說非洲的電腦不夠, 但你知道嗎?我覺得Brian Eno的非洲歷史知識不夠。 (掌聲)
So let me end with just a few words about applications that we've found for this. And you can go to our website, the applets are all free; they just run in the browser. Anybody in the world can use them. The National Science Foundation's Broadening Participation in Computing program recently awarded us a grant to make a programmable version of these design tools, so hopefully in three years, anybody'll be able to go on the Web and create their own simulations and their own artifacts. We've focused in the U.S. on African-American students as well as Native American and Latino. We've found statistically significant improvement with children using this software in a mathematics class in comparison with a control group that did not have the software. So it's really very successful teaching children that they have a heritage that's about mathematics, that it's not just about singing and dancing. We've started a pilot program in Ghana. We got a small seed grant, just to see if folks would be willing to work with us on this; we're very excited about the future possibilities for that.
所以讓我在結束前談談我們做的一些程式。 你們可以到我們的網站, 使用免費在瀏覽器中始用的程式。 世界上任何人都可以使用它。 美國國家科學基金會的擴大計算機計畫 最近給我們一筆經費來設計一個可編輯的設計工具, 希望在三年內,任何人都可以上網 去作自己的模擬和設計自己的藝品。 我們把重點放在美國和非裔美國學生和美國原住民和西班牙裔。 相較於沒有使用這些程式的控制組, 我們發現有使用的孩子在數學課尚有顯著地進步。 所以教孩子們他們有數學的傳統是非常有效的, 讓他們知道他們的傳統不只是唱歌與跳舞而已。 我們也在加納開始了一個前驅計畫, 我們拿到一小筆經費,只為了知道當地的人們有沒有興趣跟我們合作, 我們對於這個計畫的未來性感到興奮。
We've also been working in design. I didn't put his name up here -- my colleague, Kerry, in Kenya, has come up with this great idea for using fractal structure for postal address in villages that have fractal structure, because if you try to impose a grid structure postal system on a fractal village, it doesn't quite fit. Bernard Tschumi at Columbia University has finished using this in a design for a museum of African art. David Hughes at Ohio State University has written a primer on Afrocentric architecture in which he's used some of these fractal structures.
我們也在設計上下工夫。 我沒有把他的名字放上去--我的同事Kerry在肯亞想到一個很棒的點子, 就是用碎形結構在碎形村莊中作郵遞區號, 因為如果你想要將格子式的郵遞區號放入碎形的村莊中 是不大適合的。 哥倫比亞大學的Bernard Tschumi已經成功的利用碎形設計了一個非洲藝術博物館。 Ohio州立大學的David Hughes也寫了一本關於非洲中心建築的入門書籍, 裡面包括了一些碎形結構。
And finally, I just wanted to point out that this idea of self-organization, as we heard earlier, it's in the brain. It's in the -- it's in Google's search engine. Actually, the reason that Google was such a success is because they were the first ones to take advantage of the self-organizing properties of the web. It's in ecological sustainability. It's in the developmental power of entrepreneurship, the ethical power of democracy. It's also in some bad things. Self-organization is why the AIDS virus is spreading so fast. And if you don't think that capitalism, which is self-organizing, can have destructive effects, you haven't opened your eyes enough. So we need to think about, as was spoken earlier, the traditional African methods for doing self-organization. These are robust algorithms. These are ways of doing self-organization -- of doing entrepreneurship -- that are gentle, that are egalitarian. So if we want to find a better way of doing that kind of work, we need look only no farther than Africa to find these robust self-organizing algorithms. Thank you.
最後,我想要指出這個自體組織的想法, 就像我們早些兒聽到的,是在腦裡面的。 這是在,有在Google的搜尋引擎中。 事實上,Google之所以這麼成功就是 因為他是前幾個使用自體組織的優點建構的。 這是存在於生態持續性的。 這也是創業精神中發展的動力, 民主的道德力量。 它也存在於一些不大好的東西當中。 自體組織是為什麼愛滋病可以如此迅速的擴散。 而且如果你覺得資本主義,也是一種自體組織,不會有破壞性的影響的話, 你看得還不夠多。 所以我們需要想想,就像我們之前說的, 這個非洲的自體組織的方式。 這是非常有力的計算方法。 自體組織有很多種方式--就像創業一樣-- 可以是溫柔的,可以是平均的。 所以如果我們想要找到一個更好的方式來做這件事情, 我們只需要找到非洲這些強而有力的自體組織算式就夠了。 謝謝。