Ron Eglash: The fractals at the heart of African designs

翻译人员: Jiayi Li 校对人员: Weihua ZHANG

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

我的故事发生在1877年，

with a mathematician named Georg Cantor.

当时有位德国数学家叫乔治·康托（Georg Cantor）。

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

有一天，他做了这样一件事：把一条线段分成三份，擦掉中间一份，

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

然后对剩下的两条线段进行同样的操作，周而复始。

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

于是他从一条线段得到两条，

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

然后是四条，然后十六条，不断增加。

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

如果他这样重复操作无限次 （在数学中你可以做到），

he ends up with an infinite number of lines,

最终他就会得到无数条线，

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

而每条线又由无数个点组成。

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

于是他意识到，他拥有一个集合——这个集合的元素个数比无穷还要多。

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

这简直让他发疯了。我没有夸张，他为此进了疗养院。

And when he came out of the sanitarium,

当他从疗养院出来以后，

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

他坚信自己是被上帝派来寻找超限集合论的，

because the largest set of infinity would be God Himself.

因为最大的无限集便是上帝本身。

He was a very religious man.

他是一个虔诚的教徒，

He was a mathematician on a mission.

并把成为一名数学家当做自己的使命。

And other mathematicians did the same sort of thing.

其他数学家也做过类似的事。

A Swedish mathematician, von Koch,

例如，一位名为von Koch的瑞典数学家

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

有一天决定把线段相加，而不是想减。

And so he came up with this beautiful curve.

最终，他得到了这样一段美丽的曲线。

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

其实我们选择这个图形作为起始形状没有什么特殊原因；

we can use any seed shape we like.

我们可以选择任何图形作为起始。

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

让我把这把这个图形变一下，把这个放在--这下面，好--

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

现在经过反复的操作，这个形状就被延展成了一种看似不同的形状。

So these all have the property of self-similarity:

但这些图形都有自我相似的特点：

the part looks like the whole.

每一小部分都跟整体相似。

It's the same pattern at many different scales.

也可以说是同样的形状，只是大小不同。

Now, mathematicians thought this was very strange

数学家们觉得这个非常奇怪，

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

因为（勾勒图形的边缘）长度越来越长，而你的尺子看似越来越短。

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

这些图形经过无数次重复的变化，

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

它们的长度趋向于无穷大，而相比之下，原先用于衡量他们边缘长度的尺子则趋向于无穷小了。

This made no sense at all,

这一点道理也没有，

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

于是数学家们把这些曲线塞到数学书的背后，

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

然后说这些是不正常的曲线，我们不用讨论它们。

(Laughter)

（笑声）

And that worked for a hundred years.

就这样，一百年过去了，

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

直到1977年，一位名为Benoit Mandelbrot的法国数学家

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

意识到如果人们通过计算机来生成这些他叫做“分形”的图形，

you get the shapes of nature.

便可以得到大自然的形状。

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

人们可以得到肺，洋槐树，蕨类植物……

you get these beautiful natural forms.

各种美丽自然的形状。

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

如果你们看一看你们的拇指与与食指之间的部分--

go ahead and do that now --

现在就可以看一下--

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

把手放松，你们可以看到一段皱纹，

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

然后这皱纹扩展成更多的皱纹，然后更多，是吧？

Your body is covered with fractals.

你们全身都被“分形”包围着。

The mathematicians who were saying these were pathologically useless shapes?

那些认为“分形”不正常的数学家们，

They were breathing those words with fractal lungs.

他们用分形的肺部呼吸，却说着那样的话，

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

多讽刺！现在我给大家演示一段自然的循环过程。

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

跟之前一样，我们用几条线，然后重复用整体代替它们。

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

这是第二次循环，第三次，第四次……不断重复。

So nature has this self-similar structure.

可以看到，大自然也有这种自我相似性。

Nature uses self-organizing systems.

大自然是一个自组织系统。

Now in the 1980s, I happened to notice

到了20世纪80年代，我碰巧发现

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

在航拍的非洲部落照片中，存在着分形。

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

我惊叹道：“这简直太不可思议了！究竟是为什么呢？！”

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

于是我就去了非洲，去请教当地人这个问题。

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

我拿到了Fulbright奖学金，去非洲旅行一年，

asking people why they were building fractals,

询问那儿的人为什么按照分形来盖房子。

which is a great job if you can get it.

这工作真的很棒，如果你能得到的话。

(Laughter)

（笑声）

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

后来我终于来到这座城市，那时我对城市分形建筑已构建了一些模型，

just to see how it would sort of unfold --

想看看它与实际情况的符合情况--

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

当我到了那儿，我去了酋长的宫殿，

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

我的法语说得不太好，当时大概对他说：

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

“我是一名数学家，我想到你的屋顶上看看。”

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

对此他一点问题都没有，带我上到了屋顶，

and we talked about fractals.

与我讨论起有关分形的问题。

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

他说：“对，对！我们知道一个方形可以嵌套一个方形，

we know all about that."

我们知道有关的一切。”

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

后来我才知道，他们的皇家徽章图形就是由嵌套的方形构成的，

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

而宫殿的走道也是类似的螺旋形状。

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

当你沿着宫殿的走道往里走，你必须表现得越来越礼貌。

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

他们将社会的层级结构跟房屋的几何结构联系起来；

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

这些房屋的分形源自主动的构造，不像白蚁窝那样毫无意义。

This is a village in southern Zambia.

这是赞比亚南部的一个村落，

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

Ba-Ila人建造了这个直径约400米村子。

You have a huge ring.

首先我们有一个很大的环形。

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

代表家族大小的环形，越往后走越大。

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

最终属于首领（家族）的环形就在大环形的尾端，

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

而首领的直系亲属在那个环形里。

So here's a little fractal model for it.

这就是这个村落的分形模型。

Here's one house with the sacred altar,

这是一幢拥有圣坛的房子，

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

这是房子集合而成的“房子”，家族意义上的，

with the humans here where the sacred altar would be,

原先圣坛所在的地方被人所占据，

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

而这就是由先前层层叠叠房屋最终形成的村庄---

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

一个由环形组成的环形组成的环形，首领的旁系亲属住这儿，直系亲属住这儿，

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

在这儿，有一个只有丁点儿大的村庄。

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

你也许会问，人怎么可能住进这么小的村子？

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

原因呢，在于住在这儿的居民是一些灵魂。他们是村民们的祖先。

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

当然，这些灵魂居住的村子里也有一个更小的村子，对吧？

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

所以就像康托说的，这样的递推将不断循环下去。

This is in the Mandara mountains, near the Nigerian border in Cameroon, Mokoulek.

村庄Mokoulek坐落于曼达拉（Mandara）山脉中，接近尼日利亚与喀麦隆的交界处。

I saw this diagram drawn by a French architect,

我看到这幅出自一位法国建筑师之手的图时，

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

不禁惊叹：“哇！多么漂亮的分形！”

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

于是我就试着画出这幅图的初始图形，一个经过不断重复变换能够转变成现在图案的初始图形。

I came up with this structure here.

结果我画出了这个结构。

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

让我们来看一下：（这是）第一次循环，第二次，第三次，第四次……

Now, after I did the simulation,

在我完成了这个模拟之后，

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

我意识到这整个村庄就像螺旋一般盘旋环绕，就像这样，

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

而这就是那条不断复制的曲线--一条不断自我复制并最终延展成分形的螺旋。

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

我也注意到在那条曲线所在的附近，有着全村唯一的方形建筑。

So, when I got to the village,

于是当我到达那个村子后，

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

我就问：“你可以把我带到那个方形建筑所在的地方去吗？”

I think something's going on there."

“那儿一定有特别的故事。”

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

他们回答：“我们可以带你到建筑的外围，但你不能进去，”

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

“因为那里面是圣坛，每年我们都举行祭祀，

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

以祈祷每年土地的耕种、丰收遵守它固有的规律。”

And I started to realize that the cycles of fertility

我开始意识到，土地耕种、收获的循环过程

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

就像建立这个村落所运用的几何算法的循环过程一般。

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

在一些村落中，这样的循环会始终持续直到很小的尺度上。

So here's a Nankani village in Mali.

这是一个位于马里的村庄，名叫Nankani。

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

你可以看到，这些家族的层次结构，

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

以及这些壁炉中按照一定次序叠放的瓦罐。

Here's calabashes that Issa was just showing us,

这些是Issa展示给我们的葫芦，

and they're stacked recursively.

它们也被“循环”地叠放着。

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

在这最小的葫芦中，保存着一个女人的灵魂。

And when she dies, they have a ceremony

当她死去时，人们会给她举行一个仪式，

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

仪式中人们打破这个叫做zalanga的葫芦堆，使她的灵魂走向永恒。

Once again, infinity is important.

这再次说明，无限（永恒）是非常重要的。

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

现在，有三个问题需待解决。

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

第一，这些图案在原生态的建筑中是普遍存在的吗？

And that was actually my original hypothesis.

在我的最初假设中答案是肯定的。

When I first saw those African fractals,

当我第一次看到那些非洲的分形建筑时，

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

我想：“哇，那些没有形成正规国家社会与等级制度的土著族群，

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

一定都有那种‘自下而上’的建筑形式咯！”

But that turns out not to be true.

然而事实并非如此。

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

在我收集的美洲土著、南太平洋建筑的航拍照片中，

only the African ones were fractal.

只有非洲建筑具有分形结构。

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

如果你仔细回想，会发现所有这些社会都具有不同的几何设计作为它们的主题。

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

就如美洲土著用的是一种圆形对称和四方对称的组合图案，

You can see on the pottery and the baskets.

你可以在陶器和篮子上看到它们。

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

这是部分Anasazi废墟（Anasazi ruins）的航拍照片，

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

你可以发现，粗略看时它呈圆形，而细看时它是方形的，对吧？

It is not the same pattern at two different scales.

对于这种图形，在不同的尺度上，它有着不同的结构形态。

Second, you might ask,

第二点，你也许会奇怪，

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

“Eglash博士（演讲者），你是不是忽略了非洲文化的多样性呢？”

And three times, the answer is no.

我坚决地告诉你：不。

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

首先，我同意Mudimbe《非洲的发明》一书的说法，

that Africa is an artificial invention of first colonialism,

即非洲是第一次殖民主义及殖民抗争的

and then oppositional movements.

非自然的产物。

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

但分形建筑在非洲的普遍性却与此无太大关联。建筑形态的普遍性不代表文化的一致性---

and it definitely is not "in the DNA."

DNA绝没有决定人们的文化须是一致的。

And finally, the fractals have self-similarity --

最后一点，分形是具有自我相似性的---

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

可是它们只需自我相似，互相之间却未必是相似的---

you see very different uses for fractals.

对于分形的不同应用有很多种，

It's a shared technology in Africa.

在非洲这是一种众人皆知的技术。

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

再回想一下，恩，难道这不是某种直觉产生的技术吗？

It's not really mathematical knowledge.

它恐怕没有运用到什么真正意义上的数学知识。

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

非洲人不可能真的在运用“分形几何学”，对吧？

It wasn't invented until the 1970s.

因为分形几何学直到20世纪70年代才被发明出来。

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

的确，就我理解，一些非洲的分形不过来源于单纯的直觉罢了。

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

对于这些东西，如果我在达喀尔（Dakar)的街上闲逛

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

并且问当地人“有什么算法吗？构造这些的规则是什么？”，

and they'd say,

他们会回答说：

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

“嘿，我们这样做因为它们好看，傻瓜。”（笑声）

But sometimes, that's not the case.

但有些时候，情况则不尽相同。

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

对于一些图形的绘制，算法是必要的，而且是非常复杂的算法。

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

在Manghetu雕塑中，你可以看到这样有重复结构的几何图形。

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

在Ethiopian十字中，有这样美妙的延展而成的图形。

In Angola, the Chokwe people draw lines in the sand,

在安哥拉，Chokwe人在沙中绘制图线，

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

而这就是德国数学家欧拉（Euler）称作“图”（graph）的东西。

we now call it an Eulerian path --

现在，我们称之为欧拉路径（Eulerian path）---

you can never lift your stylus from the surface

你的笔尖始终不能离开纸平面，

and you can never go over the same line twice.

并且不能穿过同一条线两次。

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

Chokwe人反复学习绘图，并根据年龄区分他们所学的内容：

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

因而幼龄的孩子学习这个，稍年长的学习这个，

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

再下一个年龄层的，学习这个。

And with each iteration of that algorithm,

随着算法的迭代，

you learn the iterations of the myth.

你将瞥见奇妙事物的发生发展，

You learn the next level of knowledge.

并习得更深层次的知识。

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

再说一点，在整个非洲，你都可以看到这种棋牌游戏。

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

在我研究它的地方，加纳（Ghana）， 它被称作Owari.

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

在东海岸它被称为Mancala，在肯尼亚叫Bao,在其他地方则是Sogo.

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

在这个游戏中，你会发现自组织图案很自然的产生 。

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

加纳人知道并了解它们，

and would use them strategically.

并有策略地应用它们。

So this is very conscious knowledge.

对他们来说，这是一种有意义（而非不明不白获取）的知识。

Here's a wonderful fractal.

这儿有一个美丽的分形。

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

在萨赫勒（Sahel）地区，你到哪儿都可看到这样的篱笆。

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

人们通常认为篱笆在全世界都是"笛卡尔"式的，严格的直线型排列。

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

但在非洲，你会发现这些不笔直排列的篱笆。

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

我找到了一个做这种篱笆的人，

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

他住在Bamako外的Mali（马里）.我问他：

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

“为什么你做分形的篱笆，而别人都没有？”

And his answer was very interesting.

他的回答相当有趣。

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

他说：“如果我住在丛林里，我会只用那些长麦秆来做篱笆，

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

因为它们易完成，并且很廉价。

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

不需要花费太多时间，也不需要太多麦秆”

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

他继续道：“但是风沙和尘土很容易穿过那些篱笆。

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

而如果篱笆顶部（的麦秆）排列比较紧密，防风尘的效 果会非常好。

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

但制作它们花费很多时间，也需要很多麦秆，因为它们排列真的很紧密。

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

从经验中我们也知道，

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

从地面往上越靠近篱笆顶部，风力越强劲。”

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

他说的很正确，是吧？这就像是成本效益分析。

And I measured out the lengths of straw,

于是我测量了篱笆麦秆的长度，

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

把数据放到重对数坐标中，得到了一个标度指数，

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

这个标度指数几乎跟风力工程手册中

in the wind engineering handbook.

风速与高度的标度指数完全匹配。

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

所以，这些当地人把分形很好地应用在了实际中。

The most complex example of an algorithmic approach to fractals that I found

在众多形成分形的算法中，我所发现的最为复杂的

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

并不是几何图形的算法，而是这个符号代码的，

and this was Bamana sand divination.

用于Bamana沙地占卜。

And the same divination system is found all over Africa.

类似的占卜系统在整个非洲都可见到，

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

东、西海岸都有。

and often the symbols are very well preserved,

这些符号通常都被良好的保存下来，

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

每个符号分为四部分，可看做四个二进制位组成的单元---

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

你在沙地里随意画下这样的线段，然后数一下，

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

（一行中）如果有奇数条线段，划下一条线，

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

而如果有偶数条，划两条线。

And they did this very rapidly,

他们非常快速的完成这工作，

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

可我不明白他们究竟做了些什么---

they only did the randomness four times --

他们仅仅随意画四行线段---

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

我不知道剩下的十二个（占卜）符号他们是怎样得来的，

And they wouldn't tell me.

而他们也不愿意告诉我。

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

他们说：“不，不，我们不能告诉你这些。”

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

我回答说：“这样吧，你们可以做我的老师，我付你们工钱，

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

我每天都上你们这儿来，并每日付薪水。”

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

他们说：“这不是钱的问题。这涉及到宗教与信仰。”

And finally, out of desperation, I said,

最终，我绝望地说道：

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

“好吧，那最后请让我向你们介绍一下康托。（Georg Cantor）”

And I started explaining why I was there in Africa,

于是我开始向他们解释我来非洲的原因。

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

当他们听说康托集时，显得异常兴奋。

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

他们中的一个说道：“来吧，我想我能解决你的问题。”

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

于是他带我完成了Bamana教的入会仪式。

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

当然，我只对其中的数学问题感兴趣。

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

整个过程中，他始终摇头晃脑，说着

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

“你知道吗，我原来可不知道这其中的奥秘。”

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

而我得和埋在床边沙子中的可乐树果子（kola nut）睡一块儿，

and give seven coins to seven lepers and so on.

将七枚硬币给予七个麻风病人，等等。

And finally, he revealed the truth of the matter.

最终，他向我揭示了那些符号的奥秘。

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

事实是，那些符号产生自确定性混沌---一个伪随机过程。

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

你将一个已有的4位（four-bit）的符号与另一个放在一起。

So even plus odd gives you odd.

于是偶数加奇数得奇数；

Odd plus even gives you odd.

奇数加偶数得奇；

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

偶数加偶数得偶；奇数加奇数得偶。

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

这是一位加和的二进制数，就像计算机奇偶校验中的一位加和编码一样。

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

然后你用新得到的符号替换原有的，

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

于是你就“自我繁衍”出一系列的符号。

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

他们真真确确在运用确定性混沌的理论。

Now, because it's a binary code,

由于这些是二值码，

you can actually implement this in hardware --

事实上你可以将他们运用到硬件中---

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

多么有趣的案例，真该运用到非洲的工程学校的教学中。

And the most interesting thing I found out about it was historical.

对于这些符号，我发现的最有趣的事还是关于它们的历史。

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

在十二世纪，桑塔拉的休（Hugo of Santalla）将来源于伊斯兰神话的它们带到西班牙。

And there it entered into the alchemy community as geomancy:

在那儿，它进入炼金术士的团体，用于看风水：

divination through the earth.

通过泥土来占卜(抓沙散地,按其所成像以断吉凶)。

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

这是一幅在1390年为理查二世（King Richard II）绘制的占卜图。

Leibniz, the German mathematician,

德国数学家莱布尼兹（Leibniz）

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

在他名为"De Combinatoria"的论文中谈论到了泥土占卜。

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

在文章中他说：“我们不使用一条或两条的划线

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

而是使用数字0和1，于是我们可以把它们作二进制数来对待。”

Right? Ones and zeros, the binary code.

这不就是吗？由很多0和1组成了二进制码。

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

布尔（George Boole）运用莱布尼兹的二进制码创造了布尔代数（Boolean algebra），

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

约翰.冯.诺依曼（John von Neumann)则利用布尔代数创造了电脑.

So all these little PDAs and laptops --

因而所有这些小器件---PDA,便携式电脑---

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

所有世间的数字电路---都起源于非洲。

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

据我所知布莱恩·伊诺（Brian Eno）说非洲在数字化进程中没有多大贡献；

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

而我认为事实是Brian Eno脑中没有足够的非洲历史。

(Laughter) (Applause)

（掌声）

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

请让我简单地用这些分形的实际应用结束这场演讲。

And you can go to our website,

你也可以浏览我们的网站，

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

程序都是免费的，可以直接运行，

Anybody in the world can use them.

世界上的任何人都可以使用它们。

The National Science Foundation's Broadening Participation in Computing program

The National Science Foundation's Broadening Participation in Computing program（某基金会）

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

近日授予我们一笔资金，来将这些图形设计工具制作成可编辑版本，

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

顺利的话，在三年内，所有人都能在网上

and create their own simulations and their own artifacts.

创造出属于自己的分形与艺术品。

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

在美国，我们特别关注了非洲裔美国学生、美国土著居民和拉丁美洲人，

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

并通过统计发现在数学课中使用这款软件的孩子与一批作为对照组、

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

不使用该软件的孩子相比，学术表现有了极大提高。

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

因而教授学生，告知他们自己所具有的数学传统，是非常有意义的，

that it's not just about singing and dancing.

而不仅仅教他们唱歌、跳舞。

We've started a pilot program in Ghana.

我们在加纳启动了一个试验项目。

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

我们先提供一小笔种子资金，看人们是否愿意与我们合作；

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

对于未来（更大规模）的合作，我们都充满期待。

We've also been working in design.

我们也在设计方面不断努力。

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

我没把我这位同事的名字放上来---肯尼亚的Kerry,是他想出了这个绝妙的点子：

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

在具有分形结构的村落中应用具有分形结构的邮政网络，

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

因为一个方格状的邮递系统很难适应

it doesn't quite fit.

分形的村落结构。

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

哥伦比亚大学的Bernard Tschumi运用分形（及其衍生品）完成了对非洲艺术博物馆的设计。

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

俄亥俄州立大学的David Hughes完成了一本有关非洲中心架构（Afrocentric architecture）的入门读物，

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

在其中他运用到了一些分形结构。

And finally, I just wanted to point out that this idea of self-organization,

最后，我想指出这种自组织（self-organization）的思想---

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

我之前也提到过---是牢固存在大脑里的。

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

它也存在于谷歌（Google）的搜索引擎中。

Actually, the reason that Google was such a success

事实上，谷歌能够获得如此巨大的成功，

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

就在于它第一个利用了网络的这种自组织性质。

It's in ecological sustainability.

它体现于生态的可持续性，

It's in the developmental power of entrepreneurship,

体现于企业的发展力，

the ethical power of democracy.

也体现于民主思想的道德约束力。

It's also in some bad things.

它也体现在一些坏的事情当中。

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

自我组织是艾滋病毒传播如此迅速的原因。

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

此外，如果你不认为具有自组织性质的资本主义能产生毁灭性的影响，

you haven't opened your eyes enough.

那么你还没有真正看清这个世界。

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

因而我们需要思考，如我之前所说的，

the traditional African methods for doing self-organization.

非洲传统的自组织的方式。

These are robust algorithms.

这些才是强健的算法（方法）。

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

这些才是进行自组织的方式---发展企业的方式---

that are gentle, that are egalitarian.

它们温和、平缓。

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

因此如果我们想寻找一个更好的涉及此类工作的方式，

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

只需从非洲就能找寻到这些强健的自组织算法。

Thank you.

谢谢大家。