Ron Eglash: The fractals at the heart of African designs

譯者: Joan Liu 審譯者: Nova Upinel Altesse

I want to start my story in Germany, in 1877,

我的故事要從1877年在德國

with a mathematician named Georg Cantor.

一位名叫Georg Cantor的數學家說起。

And Cantor decided he was going to take a line and erase the middle third of the line,

Cantor決定要把一個線段的中間三分之一擦掉，

and then take those two resulting lines and bring them back into the same process, a recursive process.

將頭尾兩端接起來再重複，如此週而復始。

So he starts out with one line, and then two,

所以他一開始有一條線段，然後有兩條，

and then four, and then 16, and so on.

接著有四條，然後有十六條，這樣繼續下去。

And if he does this an infinite number of times, which you can do in mathematics,

如果他做這個無限多次，在數學上是可以做到的，

he ends up with an infinite number of lines,

他就會有無限多條線段，

each of which has an infinite number of points in it.

其中每一條線段都有無限多點。

So he realized he had a set whose number of elements was larger than infinity.

所以他發現他會有一個比無限多還大的集合。

And this blew his mind. Literally. He checked into a sanitarium. (Laughter)

他為之瘋狂。真的。他進了療養院。（笑聲）

And when he came out of the sanitarium,

當他離開療養院時，

he was convinced that he had been put on earth to found transfinite set theory

他認為他來到地球是為了理解超限理論，

because the largest set of infinity would be God Himself.

因為最大的無限就是神。

He was a very religious man.

他是個非常虔誠的人。

He was a mathematician on a mission.

他是個有使命的數學家。

And other mathematicians did the same sort of thing.

而且其他數學家也做了類似的事情。

A Swedish mathematician, von Koch,

van Koch是個瑞典的數學家，

decided that instead of subtracting lines, he would add them.

他做了類似的事情，但是不用減法而改用加法。

And so he came up with this beautiful curve.

所以他得到了這漂亮的弧線。

And there's no particular reason why we have to start with this seed shape;

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，

we can use any seed shape we like.

我們可以用任何的圖形作起始。

And I'll rearrange this and I'll stick this somewhere -- down there, OK --

我來重新整理一下，把這個放在某個地方--放到這裡，好--

and now upon iteration, that seed shape sort of unfolds into a very different looking structure.

經過無數重複後，種子圖形展開成一個非常不同的結構。

So these all have the property of self-similarity:

所以這些都有自體相似的特質：

the part looks like the whole.

各個部份跟整體相似。

It's the same pattern at many different scales.

在不同尺度上都是同一個圖形。

Now, mathematicians thought this was very strange

好，數學家覺得這很奇怪。

because as you shrink a ruler down, you measure a longer and longer length.

因為如果你把一把尺縮小，你量到的數據會越來越長。

And since they went through the iterations an infinite number of times,

然後因為重複了無限多次，

as the ruler shrinks down to infinity, the length goes to infinity.

量尺變成無限小，長度變成無限長。

This made no sense at all,

這不合理，

so they consigned these curves to the back of the math books.

所以他們把這個放在數學書籍最後面。

They said these are pathological curves, and we don't have to discuss them.

他們說這是有問題的曲線，所以我們不討論。

(Laughter)

（笑聲）

And that worked for a hundred years.

且成功的這麼做了一百年。

And then in 1977, Benoit Mandelbrot, a French mathematician,

然後在1977年，一個法國數學家Benoit Mandelbrot

realized that if you do computer graphics and used these shapes he called fractals,

發現如果利用電腦繪圖繪出這些他叫做碎形的圖樣，

you get the shapes of nature.

你可以得到自然界的圖形。

You get the human lungs, you get acacia trees, you get ferns,

你可以得到人類的肺圖形、刺槐、蕨類，

you get these beautiful natural forms.

你可以得到這些美麗的大自然形狀。

If you take your thumb and your index finger and look right where they meet --

如果你看你大拇指和食指交界的地方--

go ahead and do that now --

拿起來看看--

-- and relax your hand, you'll see a crinkle,

手放鬆，你會看到波紋，

and then a wrinkle within the crinkle, and a crinkle within the wrinkle. Right?

波紋中有皺紋，皺紋中有波紋。對吧？

Your body is covered with fractals.

你們全身都被碎形包覆著。

The mathematicians who were saying these were pathologically useless shapes?

而這些數學家竟然說這些是有問題且無意義的圖形？

They were breathing those words with fractal lungs.

他們正在用碎形組成的肺說這些話。

It's very ironic. And I'll show you a little natural recursion here.

這是非常諷刺的。我可以給你們看一些自然的循環。

Again, we just take these lines and recursively replace them with the whole shape.

再次的，我們將這些線段作重複。

So here's the second iteration, and the third, fourth and so on.

這是第二次重複、第三次、第四次...

So nature has this self-similar structure.

大自然有這樣的自體相似結構。

Nature uses self-organizing systems.

大自然利用自體組織系統。

Now in the 1980s, I happened to notice

在1980年代我發現

that if you look at an aerial photograph of an African village, you see fractals.

如果看這個非洲村落的空照圖，你會看到碎形。

And I thought, "This is fabulous! I wonder why?"

我就想：「這太好了！我想要知道為什麼？」

And of course I had to go to Africa and ask folks why.

所以當然我要去非洲問那些人為什麼。

So I got a Fulbright scholarship to just travel around Africa for a year

所以我拿了Fulbright獎學金去非洲旅行一年

asking people why they were building fractals,

問當地人為什麼要建造碎形。

which is a great job if you can get it.

其實是個很不錯的工作如果你可以拿到這個工作。

(Laughter)

（笑聲）

And so I finally got to this city, and I'd done a little fractal model for the city

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，

just to see how it would sort of unfold --

讓我可以更瞭解如何展開的--

but when I got there, I got to the palace of the chief,

我到了那裡，找到酋長的宮殿，

and my French is not very good; I said something like,

我的法文不大好，我說了像是這樣的話：「

"I am a mathematician and I would like to stand on your roof."

我是個數學家，我想要站到你的屋頂上。」

But he was really cool about it, and he took me up there,

但他覺得沒問題，然後帶我上去，

and we talked about fractals.

然後我們聊了一下碎形。

And he said, "Oh yeah, yeah! We knew about a rectangle within a rectangle,

他說：「喔對對，我們知道這個長方形裡面的長方形，

we know all about that."

我們知道那個。」

And it turns out the royal insignia has a rectangle within a rectangle within a rectangle,

而且事實上皇家徽章就是長方形裡面有長方形有長方形，

and the path through that palace is actually this spiral here.

且皇宮中的走廊也是這樣迴旋著的。

And as you go through the path, you have to get more and more polite.

而且順著這些走廊走下去，你必須越來越有禮貌。

So they're mapping the social scaling onto the geometric scaling;

所以他們是用這樣幾何縮放的方式來建立社會地位，

it's a conscious pattern. It is not unconscious like a termite mound fractal.

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.

這是在南尚比亞的一個村莊。

The Ba-ila built this village about 400 meters in diameter.

Ba-Ila人建造了一個直徑約400公尺的村莊。

You have a huge ring.

首先你有一個很大的圈圈。

The rings that represent the family enclosures get larger and larger as you go towards the back,

這些代表家族的圈圈越往後面越大，

and then you have the chief's ring here towards the back

在後面這邊有酋長的圈圈，

and then the chief's immediate family in that ring.

圈圈旁邊是酋長的家人圈。

So here's a little fractal model for it.

所以這是個小型的碎形模型。

Here's one house with the sacred altar,

這裡是一棟擁有神檀的屋子。

here's the house of houses, the family enclosure,

這裡是房子的房子，家庭圈圈，

with the humans here where the sacred altar would be,

這邊神壇的位置有人在，

and then here's the village as a whole --

這是整個村莊--

a ring of ring of rings with the chief's extended family here, the chief's immediate family here,

一圈一圈地在這裡，這是酋長的遠親，這裡是酋長的近親--

and here there's a tiny village only this big.

而這裡是一個非常小只有這麼大的村莊。

Now you might wonder, how can people fit in a tiny village only this big?

你們可能會問，這麼小的村莊怎麼住得下人？

That's because they're spirit people. It's the ancestors.

那是因為這些是神魂人物，是祖先們。

And of course the spirit people have a little miniature village in their village, right?

而且當然的這迷你的村落裡有另一個更小的村落，對吧？

So it's just like Georg Cantor said, the recursion continues forever.

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.

這是在奈吉利亞邊界Mokoulek地區Cameroon的Mandara山中的景象。

I saw this diagram drawn by a French architect,

我看到這幅法國建築家畫的圖，

and I thought, "Wow! What a beautiful fractal!"

然後我想：「哇！真是漂亮的碎形阿！」

So I tried to come up with a seed shape, which, upon iteration, would unfold into this thing.

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。

I came up with this structure here.

我想到這樣的一個結構。

Let's see, first iteration, second, third, fourth.

讓我們看看，第一次重複、第二次、第三次、第四次。

Now, after I did the simulation,

經過模擬後，

I realized the whole village kind of spirals around, just like this,

我發現整個村莊像螺旋般環繞著，就像這樣，

and here's that replicating line -- a self-replicating line that unfolds into the fractal.

且這邊是重複線：一條融入到碎形裡的自我複製線。

Well, I noticed that line is about where the only square building in the village is at.

我發現這也是整個村莊唯一一棟正方形建築物所在地。

So, when I got to the village,

所以我到了這個村莊，

I said, "Can you take me to the square building?

我問：「你可以帶我到這棟正方形建築那裡嗎？

I think something's going on there."

我覺得那裡有些什麼東西。」

And they said, "Well, we can take you there, but you can't go inside

他們說：「恩，我們可以帶你去那裡，但你不能進去，

because that's the sacred altar, where we do sacrifices every year

因為那是聖壇也就是我們每年為了

to keep up those annual cycles of fertility for the fields."

保持土地肥沃做祭祀的地方。」

And I started to realize that the cycles of fertility

我開始瞭解到這肥沃土壤的循環

were just like the recursive cycles in the geometric algorithm that builds this.

就跟建造這個的幾何算式循環一樣。

And the recursion in some of these villages continues down into very tiny scales.

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.

這裡是Mali的一個Nankani村莊。

And you can see, you go inside the family enclosure --

你們可以看到，人們可以進到家庭圈圈中--

you go inside and here's pots in the fireplace, stacked recursively.

你可以進去然後這裡是壁爐中的鍋子，也是循環堆疊的。

Here's calabashes that Issa was just showing us,

這是Issa剛剛給我們看得葫蘆，

and they're stacked recursively.

它們也是循環堆疊的。

Now, the tiniest calabash in here keeps the woman's soul.

這裡，最小的葫蘆裡面保存著女人的靈魂。

And when she dies, they have a ceremony

她離開人世時，他們有一個儀式

where they break this stack called the zalanga and her soul goes off to eternity.

會打壞這個叫做zalanga的的堆疊讓她的靈魂可以達到永恆。

Once again, infinity is important.

再次的，無限是非常重要的。

Now, you might ask yourself three questions at this point.

到此，你們可能會問自己三個問題。

Aren't these scaling patterns just universal to all indigenous architecture?

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？

And that was actually my original hypothesis.

這事實上是我一開始的假設。

When I first saw those African fractals,

當我第一次看到非洲碎形時，

I thought, "Wow, so any indigenous group that doesn't have a state society,

我想：「哇，所以任何一個沒有制式的階層結構的的原始族群

that sort of hierarchy, must have a kind of bottom-up architecture."

都應該有類似的自下而上的建築形態。」

But that turns out not to be true.

但後來發現這是不正確的。

I started collecting aerial photographs of Native American and South Pacific architecture;

我開始蒐集美國原住民和南太平洋建築的空照圖，

only the African ones were fractal.

只有非洲的有碎形。

And if you think about it, all these different societies have different geometric design themes that they use.

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。

So Native Americans use a combination of circular symmetry and fourfold symmetry.

美國原住民用了圓形對稱和四方對稱的組合。

You can see on the pottery and the baskets.

你可以在陶器和籃子上看出來。

Here's an aerial photograph of one of the Anasazi ruins;

這是Anasazi殘骸的空照圖。

you can see it's circular at the largest scale, but it's rectangular at the smaller scale, right?

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？

It is not the same pattern at two different scales.

在這兩個尺度上不是一樣的圖形。

Second, you might ask,

第二，你可能會問：

"Well, Dr. Eglash, aren't you ignoring the diversity of African cultures?"

「恩，Eglash博士，你是不是忽略了非洲文化的多樣性？」

And three times, the answer is no.

第三次的，答案是否定的。

First of all, I agree with Mudimbe's wonderful book, "The Invention of Africa,"

首先，我完全贊同Mudimbe在他很棒的書《非洲創立》中寫到

that Africa is an artificial invention of first colonialism,

非洲是個是個人類殖明主義的開始，

and then oppositional movements.

接著是對抗性運動。

No, because a widely shared design practice doesn't necessarily give you a unity of culture --

不，因為一個廣泛被使用的設計並不代表文化上是統一的，

and it definitely is not "in the DNA."

亦不代表是包含在DNA中的。

And finally, the fractals have self-similarity --

而且這些碎形是自體相似的，

so they're similar to themselves, but they're not necessarily similar to each other --

也就是說他們跟自己像而跟其它的碎形不像，

you see very different uses for fractals.

你可以看到非常不同的碎形使用方式。

It's a shared technology in Africa.

這是在非洲的一個共同的科技。

And finally, well, isn't this just intuition?

最後，恩，會不會這只是直覺？

It's not really mathematical knowledge.

事實上跟數學知識一點關係都沒有？

Africans can't possibly really be using fractal geometry, right?

非洲人不可能真的使用碎形幾何對吧？

It wasn't invented until the 1970s.

碎形幾何一直到1970年代才發明的。

Well, it's true that some African fractals are, as far as I'm concerned, just pure intuition.

是的，我認為非洲碎形有很大一部份是直覺。

So some of these things, I'd wander around the streets of Dakar

有時候我會在Dakar的街上遊蕩

asking people, "What's the algorithm? What's the rule for making this?"

問當地人：「這背後的算式是什麼？規則是什麼？」

and they'd say,

他們會說：「

"Well, we just make it that way because it looks pretty, stupid." (Laughter)

我們把它建造成這樣所以好看阿！你這個笨蛋。」（笑聲）

But sometimes, that's not the case.

但有些時候不是這樣的。

In some cases, there would actually be algorithms, and very sophisticated algorithms.

有些時候，背後真的有算式，且是非常複雜的算式。

So in Manghetu sculpture, you'd see this recursive geometry.

你可以在Manghetu的雕像上看到重複的幾何圖形。

In Ethiopian crosses, you see this wonderful unfolding of the shape.

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,

在Angola，Chokwe人會在沙上畫線，

and it's what the German mathematician Euler called a graph;

也就是德國數學家Euler叫做圖像的東西。

we now call it an Eulerian path --

我們把它叫做Eulerian道路--

you can never lift your stylus from the surface

你永遠不可以將你的筆從表面上提起，

and you can never go over the same line twice.

也不可以重複同一條線段。

But they do it recursively, and they do it with an age-grade system,

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，

so the little kids learn this one, and then the older kids learn this one,

所以小朋友會學這個，大一點的學這個，

then the next age-grade initiation, you learn this one.

在大一點的學這個。

And with each iteration of that algorithm,

而且在每一次重複這些算式時

you learn the iterations of the myth.

他們會學這些重複背後的意義。

You learn the next level of knowledge.

他們會學到下一層的知識。

And finally, all over Africa, you see this board game.

最後，在整個非洲你都可以看到這樣的棋盤遊戲。

It's called Owari in Ghana, where I studied it;

這遊戲在我研究的加那叫作Owari，

it's called Mancala here on the East Coast, Bao in Kenya, Sogo elsewhere.

在東岸叫做Mancaia，在肯亞叫做Bao，在其他地方叫做Sogo。

Well, you see self-organizing patterns that spontaneously occur in this board game.

你可以在這些棋盤遊戲中看到自體重複的圖形。

And the folks in Ghana knew about these self-organizing patterns

在加那的人知道這些圖形，

and would use them strategically.

且會有策略地運用它們。

So this is very conscious knowledge.

所以是個有意識的知識。

Here's a wonderful fractal.

這是個很棒的碎形。

Anywhere you go in the Sahel, you'll see this windscreen.

在Sahel的各個地方，你都可以看到這樣的擋風玻璃。

And of course fences around the world are all Cartesian, all strictly linear.

當然的世界上任何籬笆都是笛卡爾式的，都是直線的。

But here in Africa, you've got these nonlinear scaling fences.

但在非洲，你也可以看到這些不是直線的籬笆。

So I tracked down one of the folks who makes these things,

所以我找到設計這些籬笆的人，

this guy in Mali just outside of Bamako, and I asked him,

他是一個住在Bamako外面的Mali的人，我問他：

"How come you're making fractal fences? Because nobody else is."

「為什麼你用碎形法製造籬笆？因為沒有其他人這麼做。」

And his answer was very interesting.

他的答覆非常有趣。

He said, "Well, if I lived in the jungle, I would only use the long rows of straw

他說：「恩，當我走在叢林中時，我只會用長條的稻草，

because they're very quick and they're very cheap.

因為使用它們既快又便宜。

It doesn't take much time, doesn't take much straw."

不需要花太多時間且不需要太多稻草。」

He said, "but wind and dust goes through pretty easily.

他說：「但風和塵土很容易穿過。

Now, the tight rows up at the very top, they really hold out the wind and dust.

最上層很緊的那排可以擋住風和塵土。

But it takes a lot of time, and it takes a lot of straw because they're really tight."

但這需要花很多時間、很多稻草，因為他們需要非常緊。」

"Now," he said, "we know from experience

「現在」他說：「我們從經驗中得知，

that the farther up from the ground you go, the stronger the wind blows."

越高的地方風越強。」

Right? It's just like a cost-benefit analysis.

對吧？就有點像是成本效益分析。

And I measured out the lengths of straw,

我量了稻草的長度，

put it on a log-log plot, got the scaling exponent,

把它放進對數圖形，得到尺度指數，

and it almost exactly matches the scaling exponent for the relationship between wind speed and height

發現他幾乎完全和風速工程書上的

in the wind engineering handbook.

風速與高度的指數相同。

So these guys are right on target for a practical use of scaling technology.

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

我找到最復雜的算式碎形

was actually not in geometry, it was in a symbolic code,

並不是幾何圖形，而是符號象徵，

and this was Bamana sand divination.

而這是在Bamana的沙占卜。

And the same divination system is found all over Africa.

在整個非洲都有同樣的占卜系統。

You can find it on the East Coast as well as the West Coast,

你可以在東岸西岸都找得到這個占卜，

and often the symbols are very well preserved,

而且大部份的時候這些符號是保存得很好的。

so each of these symbols has four bits -- it's a four-bit binary word --

每一個符號有四個小部份：是四個二進法組成的字。

you draw these lines in the sand randomly, and then you count off,

你隨意畫這些線，然後數一下，

and if it's an odd number, you put down one stroke,

如果是奇數，就畫一條線；

and if it's an even number, you put down two strokes.

如果是偶數，就畫兩條線。

And they did this very rapidly,

且他們很迅速地這麼做，

and I couldn't understand where they were getting --

我無法瞭解他們怎麼做到的，

they only did the randomness four times --

他們在隨意的部份只做了四次，

I couldn't understand where they were getting the other 12 symbols.

我不懂他們另外十二個符號怎麼來的。

And they wouldn't tell me.

他們也不告訴我。

They said, "No, no, I can't tell you about this."

他們說：「不不，我不能告訴你這個。」

And I said, "Well look, I'll pay you, you can be my teacher,

然後我說：「恩，我可以付你錢，你可以當我的老師，

and I'll come each day and pay you."

然後我可以每天來付你學費。」

They said, "It's not a matter of money. This is a religious matter."

他們說：「這不是錢的問題。這是宗教問題。」

And finally, out of desperation, I said,

最後在絕望中我說：「

"Well, let me explain Georg Cantor in 1877."

恩，讓我來解釋一下1877年的Georg Cantor。」

And I started explaining why I was there in Africa,

所以我開始解釋我為什麼會在非洲，

and they got very excited when they saw the Cantor set.

他們看了Cantor組合後非常興奮。

And one of them said, "Come here. I think I can help you out here."

他們之中其中一個說：「過來，我想我可以幫你一些。」

And so he took me through the initiation ritual for a Bamana priest.

所以他代Bamana牧師帶我走過了一連串的起始儀式。

And of course, I was only interested in the math,

當然的，我只對數學的部份有興趣，

so the whole time, he kept shaking his head going,

所以整個過程，他一直搖頭說：

"You know, I didn't learn it this way."

「我不是這樣學的。」

But I had to sleep with a kola nut next to my bed, buried in sand,

但我必須在床邊放一顆埋在沙裡的可樂果，

and give seven coins to seven lepers and so on.

然後給七個痲瘋病人七個銅板之類的事情。

And finally, he revealed the truth of the matter.

最後，他終於告訴我這後面的祕密。

And it turns out it's a pseudo-random number generator using deterministic chaos.

事實上這是一個偽渾沌的產生數字的過程。

When you have a four-bit symbol, you then put it together with another one sideways.

當你有一個四位符號，你把它們並排排起來。

So even plus odd gives you odd.

所以偶數加奇數會得到奇數。

Odd plus even gives you odd.

奇數加偶數會得到奇數。

Even plus even gives you even. Odd plus odd gives you even.

偶數加偶數會得到偶數。奇數加奇數得到偶數。

It's addition modulo 2, just like in the parity bit check on your computer.

這是加法定理，就像是電腦裡的配對法一樣。

And then you take this symbol, and you put it back in

然後你拿所得到的符號，再放回去，

so it's a self-generating diversity of symbols.

就得到一個自我生成的多樣性符號。

They're truly using a kind of deterministic chaos in doing this.

他們真的在使用決定性混度來產生這些符號。

Now, because it's a binary code,

好，因為是二進位符號，

you can actually implement this in hardware --

事實上你可以將這個置入到硬體裡面--

what a fantastic teaching tool that should be in African engineering schools.

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.

我發現最有趣的是它的歷史。

In the 12th century, Hugo of Santalla brought it from Islamic mystics into Spain.

在十二世紀，Santalla的Hugu將這個從西班牙的回教傳統中引進的。

And there it entered into the alchemy community as geomancy:

在那裡，碎形以看風水的身分

divination through the earth.

進入了煉金術的世界。

This is a geomantic chart drawn for King Richard II in 1390.

這是1390年理查國王二世所畫的幾何圖表。

Leibniz, the German mathematician,

德國數學家Leibniz在他的論文中

talked about geomancy in his dissertation called "De Combinatoria."

提到「De Combinatoria」的幾何性。

And he said, "Well, instead of using one stroke and two strokes,

他說：「恩，讓我們用零和一取代

let's use a one and a zero, and we can count by powers of two."

一條線和兩條線，這樣我們可以以二的指數數下去。」

Right? Ones and zeros, the binary code.

對吧？零和一，二進位法。

George Boole took Leibniz's binary code and created Boolean algebra,

George Boole拿了Leibniz的二進位法而創造了Boolean算式，

and John von Neumann took Boolean algebra and created the digital computer.

然後John von Neumann拿了Boolean算式而創造了數位電腦。

So all these little PDAs and laptops --

所以這些掌上型電腦和筆記型電腦--

every digital circuit in the world -- started in Africa.

所有利用數位迴路的東西--都是從非洲開始的。

And I know Brian Eno says there's not enough Africa in computers,

我知道Brian Eno說非洲的電腦不夠，

but you know, I don't think there's enough African history in Brian Eno.

但你知道嗎？我覺得Brian Eno的非洲歷史知識不夠。

(Laughter) (Applause)

（掌聲）

So let me end with just a few words about applications that we've found for this.

所以讓我在結束前談談我們做的一些程式。

And you can go to our website,

你們可以到我們的網站，

the applets are all free; they just run in the browser.

使用免費在瀏覽器中始用的程式。

Anybody in the world can use them.

世界上任何人都可以使用它。

The National Science Foundation's Broadening Participation in Computing program

美國國家科學基金會的擴大計算機計畫

recently awarded us a grant to make a programmable version of these design tools,

最近給我們一筆經費來設計一個可編輯的設計工具，

so hopefully in three years, anybody'll be able to go on the Web

希望在三年內，任何人都可以上網

and create their own simulations and their own artifacts.

去作自己的模擬和設計自己的藝品。

We've focused in the U.S. on African-American students as well as Native American and Latino.

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。

We've found statistically significant improvement with children using this software in a mathematics class

相較於沒有使用這些程式的控制組，

in comparison with a control group that did not have the software.

我們發現有使用的孩子在數學課尚有顯著地進步。

So it's really very successful teaching children that they have a heritage that's about mathematics,

所以教孩子們他們有數學的傳統是非常有效的，

that it's not just about singing and dancing.

讓他們知道他們的傳統不只是唱歌與跳舞而已。

We've started a pilot program in Ghana.

我們也在加納開始了一個前驅計畫，

We got a small seed grant, just to see if folks would be willing to work with us on this;

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，

we're very excited about the future possibilities for that.

我們對於這個計畫的未來性感到興奮。

We've also been working in design.

我們也在設計上下工夫。

I didn't put his name up here -- my colleague, Kerry, in Kenya, has come up with this great idea

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，

for using fractal structure for postal address in villages that have fractal structure,

就是用碎形結構在碎形村莊中作郵遞區號，

because if you try to impose a grid structure postal system on a fractal village,

因為如果你想要將格子式的郵遞區號放入碎形的村莊中

it doesn't quite fit.

是不大適合的。

Bernard Tschumi at Columbia University has finished using this in a design for a museum of African art.

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。

David Hughes at Ohio State University has written a primer on Afrocentric architecture

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，

in which he's used some of these fractal structures.

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,

最後，我想要指出這個自體組織的想法，

as we heard earlier, it's in the brain.

就像我們早些兒聽到的，是在腦裡面的。

It's in the -- it's in Google's search engine.

這是在，有在Google的搜尋引擎中。

Actually, the reason that Google was such a success

事實上，Google之所以這麼成功就是

is because they were the first ones to take advantage of the self-organizing properties of the web.

因為他是前幾個使用自體組織的優點建構的。

It's in ecological sustainability.

這是存在於生態持續性的。

It's in the developmental power of entrepreneurship,

這也是創業精神中發展的動力，

the ethical power of democracy.

民主的道德力量。

It's also in some bad things.

它也存在於一些不大好的東西當中。

Self-organization is why the AIDS virus is spreading so fast.

自體組織是為什麼愛滋病可以如此迅速的擴散。

And if you don't think that capitalism, which is self-organizing, can have destructive effects,

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，

you haven't opened your eyes enough.

你看得還不夠多。

So we need to think about, as was spoken earlier,

所以我們需要想想，就像我們之前說的，

the traditional African methods for doing self-organization.

這個非洲的自體組織的方式。

These are robust algorithms.

這是非常有力的計算方法。

These are ways of doing self-organization -- of doing entrepreneurship --

自體組織有很多種方式--就像創業一樣--

that are gentle, that are egalitarian.

可以是溫柔的，可以是平均的。

So if we want to find a better way of doing that kind of work,

所以如果我們想要找到一個更好的方式來做這件事情，

we need look only no farther than Africa to find these robust self-organizing algorithms.

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。

Thank you.

謝謝。