with a mathematician named Georg Cantor.

我的故事要從1877年在德國

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

一位名叫Georg Cantor的數學家說起。

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

Cantor決定要把一個線段的中間三分之一擦掉，

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

將頭尾兩端接起來再重複，如此週而復始。

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

所以他一開始有一條線段，然後有兩條，

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

接著有四條，然後有十六條，這樣繼續下去。

he ends up with an infinite number of lines,

如果他做這個無限多次，在數學上是可以做到的，

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

他就會有無限多條線段，

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

其中每一條線段都有無限多點。

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

所以他發現他會有一個比無限多還大的集合。

And when he came out of the sanitarium,

他為之瘋狂。真的。他進了療養院。（笑聲）

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

當他離開療養院時，

because the largest set of infinity would be God Himself.

他認為他來到地球是為了理解超限理論，

He was a very religious man.

因為最大的無限就是神。

He was a mathematician on a mission.

他是個非常虔誠的人。

And other mathematicians did the same sort of thing.

他是個有使命的數學家。

A Swedish mathematician, von Koch,

而且其他數學家也做了類似的事情。

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

van Koch是個瑞典的數學家，

And so he came up with this beautiful curve.

他做了類似的事情，但是不用減法而改用加法。

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

所以他得到了這漂亮的弧線。

we can use any seed shape we like.

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，

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

我們可以用任何的圖形作起始。

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

我來重新整理一下，把這個放在某個地方--放到這裡，好--

So these all have the property of self-similarity:

經過無數重複後，種子圖形展開成一個非常不同的結構。

the part looks like the whole.

所以這些都有自體相似的特質：

It's the same pattern at many different scales.

各個部份跟整體相似。

Now, mathematicians thought this was very strange

在不同尺度上都是同一個圖形。

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

好，數學家覺得這很奇怪。

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

因為如果你把一把尺縮小，你量到的數據會越來越長。

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

然後因為重複了無限多次，

This made no sense at all,

量尺變成無限小，長度變成無限長。

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

這不合理，

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

所以他們把這個放在數學書籍最後面。

(Laughter)

他們說這是有問題的曲線，所以我們不討論。

And that worked for a hundred years.

（笑聲）

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

且成功的這麼做了一百年。

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

然後在1977年，一個法國數學家Benoit Mandelbrot

you get the shapes of nature.

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

你可以得到自然界的圖形。

you get these beautiful natural forms.

你可以得到人類的肺圖形、刺槐、蕨類，

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

你可以得到這些美麗的大自然形狀。

go ahead and do that now --

如果你看你大拇指和食指交界的地方--

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

拿起來看看--

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

手放鬆，你會看到波紋，

Your body is covered with fractals.

波紋中有皺紋，皺紋中有波紋。對吧？

The mathematicians who were saying these were pathologically useless shapes?

你們全身都被碎形包覆著。

They were breathing those words with fractal lungs.

而這些數學家竟然說這些是有問題且無意義的圖形？

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

他們正在用碎形組成的肺說這些話。

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

這是非常諷刺的。我可以給你們看一些自然的循環。

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

再次的，我們將這些線段作重複。

So nature has this self-similar structure.

這是第二次重複、第三次、第四次...

Nature uses self-organizing systems.

大自然有這樣的自體相似結構。

Now in the 1980s, I happened to notice

大自然利用自體組織系統。

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

在1980年代我發現

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

如果看這個非洲村落的空照圖，你會看到碎形。

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

我就想：「這太好了！我想要知道為什麼？」

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

所以當然我要去非洲問那些人為什麼。

asking people why they were building fractals,

所以我拿了Fulbright獎學金去非洲旅行一年

which is a great job if you can get it.

問當地人為什麼要建造碎形。

(Laughter)

其實是個很不錯的工作如果你可以拿到這個工作。

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

（笑聲）

just to see how it would sort of unfold --

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，

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

讓我可以更瞭解如何展開的--

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

我到了那裡，找到酋長的宮殿，

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

我的法文不大好，我說了像是這樣的話：「

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

我是個數學家，我想要站到你的屋頂上。」

and we talked about fractals.

但他覺得沒問題，然後帶我上去，

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

然後我們聊了一下碎形。

we know all about that."

他說：「喔對對，我們知道這個長方形裡面的長方形，

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

我們知道那個。」

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

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

且皇宮中的走廊也是這樣迴旋著的。

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

而且順著這些走廊走下去，你必須越來越有禮貌。

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

所以他們是用這樣幾何縮放的方式來建立社會地位，

This is a village in southern Zambia.

是故意這麼做的，並不是像飛蟻丘那樣無意識的。

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

這是在南尚比亞的一個村莊。

You have a huge ring.

Ba-Ila人建造了一個直徑約400公尺的村莊。

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

首先你有一個很大的圈圈。

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

這些代表家族的圈圈越往後面越大，

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

在後面這邊有酋長的圈圈，

So here's a little fractal model for it.

圈圈旁邊是酋長的家人圈。

Here's one house with the sacred altar,

所以這是個小型的碎形模型。

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

這裡是一棟擁有神檀的屋子。

with the humans here where the sacred altar would be,

這裡是房子的房子，家庭圈圈，

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

這邊神壇的位置有人在，

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

這是整個村莊--

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

一圈一圈地在這裡，這是酋長的遠親，這裡是酋長的近親--

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

而這裡是一個非常小只有這麼大的村莊。

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

你們可能會問，這麼小的村莊怎麼住得下人？

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

那是因為這些是神魂人物，是祖先們。

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

而且當然的這迷你的村落裡有另一個更小的村落，對吧？

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

所以就像Georg Cantor說的，一再地重複著。

I saw this diagram drawn by a French architect,

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

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

我看到這幅法國建築家畫的圖，

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

然後我想：「哇！真是漂亮的碎形阿！」

I came up with this structure here.

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。

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

我想到這樣的一個結構。

Now, after I did the simulation,

讓我們看看，第一次重複、第二次、第三次、第四次。

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

經過模擬後，

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

我發現整個村莊像螺旋般環繞著，就像這樣，

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

且這邊是重複線：一條融入到碎形裡的自我複製線。

So, when I got to the village,

我發現這也是整個村莊唯一一棟正方形建築物所在地。

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

所以我到了這個村莊，

I think something's going on there."

我問：「你可以帶我到這棟正方形建築那裡嗎？

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

我覺得那裡有些什麼東西。」

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

他們說：「恩，我們可以帶你去那裡，但你不能進去，

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

因為那是聖壇也就是我們每年為了

And I started to realize that the cycles of fertility

保持土地肥沃做祭祀的地方。」

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

我開始瞭解到這肥沃土壤的循環

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

就跟建造這個的幾何算式循環一樣。

So here's a Nankani village in Mali.

且這樣的循環一直延續到非常小的尺度。

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

這裡是Mali的一個Nankani村莊。

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

你們可以看到，人們可以進到家庭圈圈中--

Here's calabashes that Issa was just showing us,

你可以進去然後這裡是壁爐中的鍋子，也是循環堆疊的。

and they're stacked recursively.

這是Issa剛剛給我們看得葫蘆，

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

它們也是循環堆疊的。

And when she dies, they have a ceremony

這裡，最小的葫蘆裡面保存著女人的靈魂。

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

她離開人世時，他們有一個儀式

Once again, infinity is important.

會打壞這個叫做zalanga的的堆疊讓她的靈魂可以達到永恆。

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

再次的，無限是非常重要的。

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

到此，你們可能會問自己三個問題。

And that was actually my original hypothesis.

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？

When I first saw those African fractals,

這事實上是我一開始的假設。

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

當我第一次看到非洲碎形時，

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

我想：「哇，所以任何一個沒有制式的階層結構的的原始族群

But that turns out not to be true.

都應該有類似的自下而上的建築形態。」

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

但後來發現這是不正確的。

only the African ones were fractal.

我開始蒐集美國原住民和南太平洋建築的空照圖，

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

只有非洲的有碎形。

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

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。

You can see on the pottery and the baskets.

美國原住民用了圓形對稱和四方對稱的組合。

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

你可以在陶器和籃子上看出來。

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

這是Anasazi殘骸的空照圖。

It is not the same pattern at two different scales.

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？

Second, you might ask,

在這兩個尺度上不是一樣的圖形。

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

第二，你可能會問：

And three times, the answer is no.

「恩，Eglash博士，你是不是忽略了非洲文化的多樣性？」

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

第三次的，答案是否定的。

that Africa is an artificial invention of first colonialism,

首先，我完全贊同Mudimbe在他很棒的書《非洲創立》中寫到

and then oppositional movements.

非洲是個是個人類殖明主義的開始，

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

接著是對抗性運動。

and it definitely is not "in the DNA."

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 finally, the fractals have self-similarity --

亦不代表是包含在DNA中的。

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

而且這些碎形是自體相似的，

you see very different uses for fractals.

也就是說他們跟自己像而跟其它的碎形不像，

It's a shared technology in Africa.

你可以看到非常不同的碎形使用方式。

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

這是在非洲的一個共同的科技。

It's not really mathematical knowledge.

最後，恩，會不會這只是直覺？

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

事實上跟數學知識一點關係都沒有？

It wasn't invented until the 1970s.

非洲人不可能真的使用碎形幾何對吧？

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

碎形幾何一直到1970年代才發明的。

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

是的，我認為非洲碎形有很大一部份是直覺。

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

有時候我會在Dakar的街上遊蕩

and they'd say,

問當地人：「這背後的算式是什麼？規則是什麼？」

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

他們會說：「

But sometimes, that's not the case.

我們把它建造成這樣所以好看阿！你這個笨蛋。」（笑聲）

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

但有些時候不是這樣的。

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

有些時候，背後真的有算式，且是非常複雜的算式。

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

你可以在Manghetu的雕像上看到重複的幾何圖形。

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

在Ehiopian的十字架上也可以看到這些無限展開的形狀。

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

在Angola，Chokwe人會在沙上畫線，

we now call it an Eulerian path --

也就是德國數學家Euler叫做圖像的東西。

you can never lift your stylus from the surface

我們把它叫做Eulerian道路--

and you can never go over the same line twice.

你永遠不可以將你的筆從表面上提起，

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

也不可以重複同一條線段。

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

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，

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

所以小朋友會學這個，大一點的學這個，

And with each iteration of that algorithm,

在大一點的學這個。

you learn the iterations of the myth.

而且在每一次重複這些算式時

You learn the next level of knowledge.

他們會學這些重複背後的意義。

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

他們會學到下一層的知識。

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

最後，在整個非洲你都可以看到這樣的棋盤遊戲。

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

這遊戲在我研究的加那叫作Owari，

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

在東岸叫做Mancaia，在肯亞叫做Bao，在其他地方叫做Sogo。

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

你可以在這些棋盤遊戲中看到自體重複的圖形。

and would use them strategically.

在加那的人知道這些圖形，

So this is very conscious knowledge.

且會有策略地運用它們。

Here's a wonderful fractal.

所以是個有意識的知識。

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

這是個很棒的碎形。

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

在Sahel的各個地方，你都可以看到這樣的擋風玻璃。

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

當然的世界上任何籬笆都是笛卡爾式的，都是直線的。

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

但在非洲，你也可以看到這些不是直線的籬笆。

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

所以我找到設計這些籬笆的人，

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

他是一個住在Bamako外面的Mali的人，我問他：

And his answer was very interesting.

「為什麼你用碎形法製造籬笆？因為沒有其他人這麼做。」

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

他的答覆非常有趣。

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

他說：「恩，當我走在叢林中時，我只會用長條的稻草，

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

因為使用它們既快又便宜。

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

不需要花太多時間且不需要太多稻草。」

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

他說：「但風和塵土很容易穿過。

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

最上層很緊的那排可以擋住風和塵土。

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

但這需要花很多時間、很多稻草，因為他們需要非常緊。」

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

「現在」他說：「我們從經驗中得知，

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

越高的地方風越強。」

And I measured out the lengths of straw,

對吧？就有點像是成本效益分析。

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

我量了稻草的長度，

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

把它放進對數圖形，得到尺度指數，

in the wind engineering handbook.

發現他幾乎完全和風速工程書上的

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

風速與高度的指數相同。

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

這些人在利用尺度科技上正中目標。

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

我找到最復雜的算式碎形

and this was Bamana sand divination.

並不是幾何圖形，而是符號象徵，

And the same divination system is found all over Africa.

而這是在Bamana的沙占卜。

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

在整個非洲都有同樣的占卜系統。

and often the symbols are very well preserved,

你可以在東岸西岸都找得到這個占卜，

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