ABOUT THE SPEAKER
Ron Eglash - Mathematician
Ron Eglash is an ethno-mathematician: he studies the way math and cultures intersect. He has shown that many aspects of African design -- in architecture, art, even hair braiding -- are based on perfect fractal patterns.

Why you should listen

"Ethno-mathematician" Ron Eglash is the author of African Fractals, a book that examines the fractal patterns underpinning architecture, art and design in many parts of Africa. By looking at aerial-view photos -- and then following up with detailed research on the ground -- Eglash discovered that many African villages are purposely laid out to form perfect fractals, with self-similar shapes repeated in the rooms of the house, and the house itself, and the clusters of houses in the village, in mathematically predictable patterns.

As he puts it: "When Europeans first came to Africa, they considered the architecture very disorganized and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."

His other areas of study are equally fascinating, including research into African and Native American cybernetics, teaching kids math through culturally specific design tools (such as the Virtual Breakdancer applet, which explores rotation and sine functions), and race and ethnicity issues in science and technology. Eglash teaches in the Department of Science and Technology Studies at Rensselaer Polytechnic Institute in New York, and he recently co-edited the book Appropriating Technology, about how we reinvent consumer tech for our own uses.

 

More profile about the speaker
Ron Eglash | Speaker | TED.com
TEDGlobal 2007

Ron Eglash: The fractals at the heart of African designs

Ron Eglash談非洲碎形

Filmed:
1,740,687 views

「我是個數學家,我想要站在你屋頂上。」這是Ron Eglash在研究非洲村落間碎形圖樣時對當地居名打招呼的方式。
- Mathematician
Ron Eglash is an ethno-mathematician: he studies the way math and cultures intersect. He has shown that many aspects of African design -- in architecture, art, even hair braiding -- are based on perfect fractal patterns. Full bio

Double-click the English transcript below to play the video.

00:13
I want to start開始 my story故事 in Germany德國, in 1877,
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我的故事要從1877年在德國
00:16
with a mathematician數學家 named命名 Georg喬治· Cantor領唱者.
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一位名叫Georg Cantor的數學家說起。
00:18
And Cantor領唱者 decided決定 he was going to take a line and erase抹去 the middle中間 third第三 of the line,
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Cantor決定要把一個線段的中間三分之一擦掉,
00:23
and then take those two resulting造成 lines and bring帶來 them back into the same相同 process處理, a recursive遞歸 process處理.
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將頭尾兩端接起來再重複,如此週而復始。
00:28
So he starts啟動 out with one line, and then two,
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所以他一開始有一條線段,然後有兩條,
00:30
and then four, and then 16, and so on.
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接著有四條,然後有十六條,這樣繼續下去。
00:33
And if he does this an infinite無窮 number of times, which哪一個 you can do in mathematics數學,
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如果他做這個無限多次,在數學上是可以做到的,
00:36
he ends結束 up with an infinite無窮 number of lines,
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他就會有無限多條線段,
00:38
each of which哪一個 has an infinite無窮 number of points in it.
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其中每一條線段都有無限多點。
00:41
So he realized實現 he had a set whose誰的 number of elements分子 was larger than infinity無窮.
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所以他發現他會有一個比無限多還大的集合。
00:45
And this blew自爆 his mind心神. Literally按照字面. He checked檢查 into a sanitarium療養院. (Laughter笑聲)
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他為之瘋狂。真的。他進了療養院。(笑聲)
00:48
And when he came來了 out of the sanitarium療養院,
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當他離開療養院時,
00:50
he was convinced相信 that he had been put on earth地球 to found發現 transfinite超限 set theory理論
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他認為他來到地球是為了理解超限理論,
00:56
because the largest最大 set of infinity無窮 would be God Himself他自己.
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因為最大的無限就是神。
00:59
He was a very religious宗教 man.
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他是個非常虔誠的人。
01:00
He was a mathematician數學家 on a mission任務.
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他是個有使命的數學家。
01:02
And other mathematicians數學家 did the same相同 sort分類 of thing.
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而且其他數學家也做了類似的事情。
01:04
A Swedish瑞典 mathematician數學家, von Koch科赫,
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van Koch是個瑞典的數學家,
01:06
decided決定 that instead代替 of subtracting減法 lines, he would add them.
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他做了類似的事情,但是不用減法而改用加法。
01:10
And so he came來了 up with this beautiful美麗 curve曲線.
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所以他得到了這漂亮的弧線。
01:12
And there's no particular特定 reason原因 why we have to start開始 with this seed種子 shape形狀;
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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,
01:15
we can use any seed種子 shape形狀 we like.
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我們可以用任何的圖形作起始。
01:19
And I'll rearrange改編 this and I'll stick this somewhere某處 -- down there, OK --
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我來重新整理一下,把這個放在某個地方--放到這裡,好--
01:23
and now upon iteration迭代, that seed種子 shape形狀 sort分類 of unfolds展開 into a very different不同 looking structure結構體.
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經過無數重複後,種子圖形展開成一個非常不同的結構。
01:30
So these all have the property屬性 of self-similarity自相似性:
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所以這些都有自體相似的特質:
01:32
the part部分 looks容貌 like the whole整個.
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各個部份跟整體相似。
01:34
It's the same相同 pattern模式 at many許多 different不同 scales.
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在不同尺度上都是同一個圖形。
01:37
Now, mathematicians數學家 thought this was very strange奇怪
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好,數學家覺得這很奇怪。
01:39
because as you shrink收縮 a ruler統治者 down, you measure測量 a longer and longer length長度.
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因為如果你把一把尺縮小,你量到的數據會越來越長。
01:44
And since以來 they went through通過 the iterations迭代 an infinite無窮 number of times,
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然後因為重複了無限多次,
01:46
as the ruler統治者 shrinks收縮 down to infinity無窮, the length長度 goes to infinity無窮.
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量尺變成無限小,長度變成無限長。
01:52
This made製作 no sense at all,
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這不合理,
01:53
so they consigned委託 these curves曲線 to the back of the math數學 books圖書.
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所以他們把這個放在數學書籍最後面。
01:56
They said these are pathological病態的 curves曲線, and we don't have to discuss討論 them.
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他們說這是有問題的曲線,所以我們不討論。
02:00
(Laughter笑聲)
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(笑聲)
02:01
And that worked工作 for a hundred years年份.
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且成功的這麼做了一百年。
02:04
And then in 1977, Benoit伯努瓦 Mandelbrot曼德爾布羅, a French法國 mathematician數學家,
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然後在1977年,一個法國數學家Benoit Mandelbrot
02:09
realized實現 that if you do computer電腦 graphics圖像 and used these shapes形狀 he called fractals分形,
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發現如果利用電腦繪圖繪出這些他叫做碎形的圖樣,
02:14
you get the shapes形狀 of nature性質.
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你可以得到自然界的圖形。
02:16
You get the human人的 lungs, you get acacia刺槐 trees樹木, you get ferns蕨類植物,
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你可以得到人類的肺圖形、刺槐、蕨類,
02:20
you get these beautiful美麗 natural自然 forms形式.
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你可以得到這些美麗的大自然形狀。
02:22
If you take your thumb拇指 and your index指數 finger手指 and look right where they meet遇到 --
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如果你看你大拇指和食指交界的地方--
02:26
go ahead and do that now --
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拿起來看看--
02:28
-- and relax放鬆 your hand, you'll你會 see a crinkle,
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手放鬆,你會看到波紋,
02:31
and then a wrinkle皺紋 within the crinkle, and a crinkle within the wrinkle皺紋. Right?
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波紋中有皺紋,皺紋中有波紋。對吧?
02:34
Your body身體 is covered覆蓋 with fractals分形.
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你們全身都被碎形包覆著。
02:36
The mathematicians數學家 who were saying these were pathologically病理 useless無用 shapes形狀?
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而這些數學家竟然說這些是有問題且無意義的圖形?
02:39
They were breathing呼吸 those words with fractal分形 lungs.
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他們正在用碎形組成的肺說這些話。
02:41
It's very ironic具有諷刺意味. And I'll show顯示 you a little natural自然 recursion遞歸 here.
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這是非常諷刺的。我可以給你們看一些自然的循環。
02:45
Again, we just take these lines and recursively遞歸 replace更換 them with the whole整個 shape形狀.
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再次的,我們將這些線段作重複。
02:50
So here's這裡的 the second第二 iteration迭代, and the third第三, fourth第四 and so on.
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這是第二次重複、第三次、第四次...
02:55
So nature性質 has this self-similar自相似 structure結構體.
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大自然有這樣的自體相似結構。
02:57
Nature性質 uses使用 self-organizing自組織 systems系統.
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大自然利用自體組織系統。
02:59
Now in the 1980s, I happened發生 to notice注意
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在1980年代我發現
03:02
that if you look at an aerial天線 photograph照片 of an African非洲人 village, you see fractals分形.
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如果看這個非洲村落的空照圖,你會看到碎形。
03:06
And I thought, "This is fabulous極好! I wonder奇蹟 why?"
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我就想:「這太好了!我想要知道為什麼?」
03:10
And of course課程 I had to go to Africa非洲 and ask folks鄉親 why.
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所以當然我要去非洲問那些人為什麼。
03:12
So I got a Fulbright富布賴特 scholarship獎學金 to just travel旅行 around Africa非洲 for a year
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所以我拿了Fulbright獎學金去非洲旅行一年
03:18
asking people why they were building建造 fractals分形,
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問當地人為什麼要建造碎形。
03:20
which哪一個 is a great job工作 if you can get it.
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其實是個很不錯的工作如果你可以拿到這個工作。
03:22
(Laughter笑聲)
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(笑聲)
03:23
And so I finally最後 got to this city, and I'd doneDONE a little fractal分形 model模型 for the city
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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,
03:30
just to see how it would sort分類 of unfold展開 --
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讓我可以更瞭解如何展開的--
03:33
but when I got there, I got to the palace of the chief首席,
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我到了那裡,找到酋長的宮殿,
03:36
and my French法國 is not very good; I said something like,
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我的法文不大好,我說了像是這樣的話:「
03:39
"I am a mathematician數學家 and I would like to stand on your roof屋頂."
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我是個數學家,我想要站到你的屋頂上。」
03:42
But he was really cool about it, and he took me up there,
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但他覺得沒問題,然後帶我上去,
03:45
and we talked about fractals分形.
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然後我們聊了一下碎形。
03:46
And he said, "Oh yeah, yeah! We knew知道 about a rectangle長方形 within a rectangle長方形,
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他說:「喔對對,我們知道這個長方形裡面的長方形,
03:49
we know all about that."
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我們知道那個。」
03:51
And it turns out the royal王室的 insignia徽章 has a rectangle長方形 within a rectangle長方形 within a rectangle長方形,
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而且事實上皇家徽章就是長方形裡面有長方形有長方形,
03:55
and the path路徑 through通過 that palace is actually其實 this spiral螺旋 here.
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且皇宮中的走廊也是這樣迴旋著的。
03:59
And as you go through通過 the path路徑, you have to get more and more polite有禮貌.
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而且順著這些走廊走下去,你必須越來越有禮貌。
04:03
So they're mapping製圖 the social社會 scaling縮放 onto the geometric幾何 scaling縮放;
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所以他們是用這樣幾何縮放的方式來建立社會地位,
04:06
it's a conscious意識 pattern模式. It is not unconscious無意識 like a termite白蟻 mound fractal分形.
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是故意這麼做的,並不是像飛蟻丘那樣無意識的。
04:11
This is a village in southern南部的 Zambia贊比亞.
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這是在南尚比亞的一個村莊。
04:13
The Ba-ilaBA-ILA built內置 this village about 400 meters in diameter直徑.
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Ba-Ila人建造了一個直徑約400公尺的村莊。
04:17
You have a huge巨大 ring.
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首先你有一個很大的圈圈。
04:19
The rings戒指 that represent代表 the family家庭 enclosures機箱 get larger and larger as you go towards the back,
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這些代表家族的圈圈越往後面越大,
04:26
and then you have the chief's酋長的 ring here towards the back
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在後面這邊有酋長的圈圈,
04:30
and then the chief's酋長的 immediate即時 family家庭 in that ring.
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圈圈旁邊是酋長的家人圈。
04:33
So here's這裡的 a little fractal分形 model模型 for it.
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所以這是個小型的碎形模型。
04:34
Here's這裡的 one house with the sacred神聖 altar,
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這裡是一棟擁有神檀的屋子。
04:37
here's這裡的 the house of houses房屋, the family家庭 enclosure附件,
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這裡是房子的房子,家庭圈圈,
04:40
with the humans人類 here where the sacred神聖 altar would be,
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這邊神壇的位置有人在,
04:43
and then here's這裡的 the village as a whole整個 --
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這是整個村莊--
04:45
a ring of ring of rings戒指 with the chief's酋長的 extended擴展 family家庭 here, the chief's酋長的 immediate即時 family家庭 here,
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一圈一圈地在這裡,這是酋長的遠親,這裡是酋長的近親--
04:50
and here there's a tiny village only this big.
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而這裡是一個非常小只有這麼大的村莊。
04:53
Now you might威力 wonder奇蹟, how can people fit適合 in a tiny village only this big?
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你們可能會問,這麼小的村莊怎麼住得下人?
04:57
That's because they're spirit精神 people. It's the ancestors祖先.
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那是因為這些是神魂人物,是祖先們。
05:00
And of course課程 the spirit精神 people have a little miniature微型 village in their village, right?
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而且當然的這迷你的村落裡有另一個更小的村落,對吧?
05:05
So it's just like Georg喬治· Cantor領唱者 said, the recursion遞歸 continues繼續 forever永遠.
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所以就像Georg Cantor說的,一再地重複著。
05:08
This is in the Mandara曼達拉 mountains, near the Nigerian尼日利亞 border邊境 in Cameroon喀麥隆, MokoulekMokoulek.
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這是在奈吉利亞邊界Mokoulek地區Cameroon的Mandara山中的景象。
05:12
I saw this diagram drawn by a French法國 architect建築師,
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我看到這幅法國建築家畫的圖,
05:15
and I thought, "Wow! What a beautiful美麗 fractal分形!"
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然後我想:「哇!真是漂亮的碎形阿!」
05:17
So I tried試著 to come up with a seed種子 shape形狀, which哪一個, upon iteration迭代, would unfold展開 into this thing.
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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。
05:23
I came來了 up with this structure結構體 here.
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我想到這樣的一個結構。
05:25
Let's see, first iteration迭代, second第二, third第三, fourth第四.
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讓我們看看,第一次重複、第二次、第三次、第四次。
05:29
Now, after I did the simulation模擬,
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經過模擬後,
05:31
I realized實現 the whole整個 village kind of spirals螺旋 around, just like this,
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我發現整個村莊像螺旋般環繞著,就像這樣,
05:34
and here's這裡的 that replicating複製 line -- a self-replicating自我複製 line that unfolds展開 into the fractal分形.
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且這邊是重複線:一條融入到碎形裡的自我複製線。
05:40
Well, I noticed注意到 that line is about where the only square廣場 building建造 in the village is at.
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我發現這也是整個村莊唯一一棟正方形建築物所在地。
05:45
So, when I got to the village,
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所以我到了這個村莊,
05:47
I said, "Can you take me to the square廣場 building建造?
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我問:「你可以帶我到這棟正方形建築那裡嗎?
05:49
I think something's什麼是 going on there."
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我覺得那裡有些什麼東西。」
05:51
And they said, "Well, we can take you there, but you can't go inside
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他們說:「恩,我們可以帶你去那裡,但你不能進去,
05:54
because that's the sacred神聖 altar, where we do sacrifices犧牲 every一切 year
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因為那是聖壇也就是我們每年為了
05:57
to keep up those annual全年 cycles週期 of fertility生育能力 for the fields領域."
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保持土地肥沃做祭祀的地方。」
06:00
And I started開始 to realize實現 that the cycles週期 of fertility生育能力
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我開始瞭解到這肥沃土壤的循環
06:02
were just like the recursive遞歸 cycles週期 in the geometric幾何 algorithm算法 that builds建立 this.
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就跟建造這個的幾何算式循環一樣。
06:06
And the recursion遞歸 in some of these villages村莊 continues繼續 down into very tiny scales.
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且這樣的循環一直延續到非常小的尺度。
06:10
So here's這裡的 a Nankani南卡尼 village in Mali馬里.
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這裡是Mali的一個Nankani村莊。
06:12
And you can see, you go inside the family家庭 enclosure附件 --
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你們可以看到,人們可以進到家庭圈圈中--
06:15
you go inside and here's這裡的 pots in the fireplace壁爐, stacked堆疊 recursively遞歸.
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你可以進去然後這裡是壁爐中的鍋子,也是循環堆疊的。
06:19
Here's這裡的 calabashes葫蘆 that Issa伊薩 was just showing展示 us,
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這是Issa剛剛給我們看得葫蘆,
06:23
and they're stacked堆疊 recursively遞歸.
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它們也是循環堆疊的。
06:25
Now, the tiniest最小的 calabash in here keeps保持 the woman's女人的 soul靈魂.
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這裡,最小的葫蘆裡面保存著女人的靈魂。
06:27
And when she dies, they have a ceremony儀式
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她離開人世時,他們有一個儀式
06:29
where they break打破 this stack called the zalangazalanga and her soul靈魂 goes off to eternity永恆.
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會打壞這個叫做zalanga的的堆疊讓她的靈魂可以達到永恆。
06:34
Once一旦 again, infinity無窮 is important重要.
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再次的,無限是非常重要的。
06:38
Now, you might威力 ask yourself你自己 three questions問題 at this point.
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到此,你們可能會問自己三個問題。
06:42
Aren't是不是 these scaling縮放 patterns模式 just universal普遍 to all indigenous土著 architecture建築?
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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?
06:46
And that was actually其實 my original原版的 hypothesis假設.
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這事實上是我一開始的假設。
06:48
When I first saw those African非洲人 fractals分形,
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當我第一次看到非洲碎形時,
06:50
I thought, "Wow, so any indigenous土著 group that doesn't have a state society社會,
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我想:「哇,所以任何一個沒有制式的階層結構的的原始族群
06:54
that sort分類 of hierarchy等級制度, must必須 have a kind of bottom-up自下而上 architecture建築."
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都應該有類似的自下而上的建築形態。」
06:57
But that turns out not to be true真正.
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但後來發現這是不正確的。
06:59
I started開始 collecting蒐集 aerial天線 photographs照片 of Native本地人 American美國 and South Pacific和平的 architecture建築;
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我開始蒐集美國原住民和南太平洋建築的空照圖,
07:03
only the African非洲人 ones那些 were fractal分形.
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只有非洲的有碎形。
07:05
And if you think about it, all these different不同 societies社會 have different不同 geometric幾何 design設計 themes主題 that they use.
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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。
07:11
So Native本地人 Americans美國人 use a combination組合 of circular symmetry對稱 and fourfold四倍 symmetry對稱.
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美國原住民用了圓形對稱和四方對稱的組合。
07:17
You can see on the pottery陶器 and the baskets.
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你可以在陶器和籃子上看出來。
07:19
Here's這裡的 an aerial天線 photograph照片 of one of the Anasazi阿納薩齊 ruins廢墟;
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這是Anasazi殘骸的空照圖。
07:22
you can see it's circular at the largest最大 scale規模, but it's rectangular長方形 at the smaller scale規模, right?
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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?
07:27
It is not the same相同 pattern模式 at two different不同 scales.
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在這兩個尺度上不是一樣的圖形。
07:31
Second第二, you might威力 ask,
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第二,你可能會問:
07:32
"Well, Dr博士. EglashEglash, aren't you ignoring無視 the diversity多樣 of African非洲人 cultures文化?"
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「恩,Eglash博士,你是不是忽略了非洲文化的多樣性?」
07:36
And three times, the answer回答 is no.
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第三次的,答案是否定的。
07:38
First of all, I agree同意 with Mudimbe'sMudimbe的 wonderful精彩 book, "The Invention發明 of Africa非洲,"
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首先,我完全贊同Mudimbe在他很棒的書《非洲創立》中寫到
07:42
that Africa非洲 is an artificial人造 invention發明 of first colonialism殖民主義,
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非洲是個是個人類殖明主義的開始,
07:45
and then oppositional對立 movements運動.
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接著是對抗性運動。
07:47
No, because a widely廣泛 shared共享 design設計 practice實踐 doesn't necessarily一定 give you a unity統一 of culture文化 --
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不,因為一個廣泛被使用的設計並不代表文化上是統一的,
07:52
and it definitely無疑 is not "in the DNA脫氧核糖核酸."
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亦不代表是包含在DNA中的。
07:55
And finally最後, the fractals分形 have self-similarity自相似性 --
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而且這些碎形是自體相似的,
07:57
so they're similar類似 to themselves他們自己, but they're not necessarily一定 similar類似 to each other --
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也就是說他們跟自己像而跟其它的碎形不像,
08:01
you see very different不同 uses使用 for fractals分形.
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你可以看到非常不同的碎形使用方式。
08:03
It's a shared共享 technology技術 in Africa非洲.
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這是在非洲的一個共同的科技。
08:06
And finally最後, well, isn't this just intuition直覺?
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最後,恩,會不會這只是直覺?
08:09
It's not really mathematical數學的 knowledge知識.
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事實上跟數學知識一點關係都沒有?
08:11
Africans非洲人 can't possibly或者 really be using運用 fractal分形 geometry幾何, right?
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非洲人不可能真的使用碎形幾何對吧?
08:14
It wasn't invented發明 until直到 the 1970s.
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碎形幾何一直到1970年代才發明的。
08:17
Well, it's true真正 that some African非洲人 fractals分形 are, as far as I'm concerned關心, just pure intuition直覺.
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是的,我認為非洲碎形有很大一部份是直覺。
08:22
So some of these things, I'd wander漫步 around the streets街道 of Dakar達喀爾
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有時候我會在Dakar的街上遊蕩
08:25
asking people, "What's the algorithm算法? What's the rule規則 for making製造 this?"
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問當地人:「這背後的算式是什麼?規則是什麼?」
08:28
and they'd他們會 say,
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他們會說:「
08:29
"Well, we just make it that way because it looks容貌 pretty漂亮, stupid." (Laughter笑聲)
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我們把它建造成這樣所以好看阿!你這個笨蛋。」(笑聲)
08:32
But sometimes有時, that's not the case案件.
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但有些時候不是這樣的。
08:35
In some cases, there would actually其實 be algorithms算法, and very sophisticated複雜的 algorithms算法.
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有些時候,背後真的有算式,且是非常複雜的算式。
08:40
So in ManghetuManghetu sculpture雕塑, you'd see this recursive遞歸 geometry幾何.
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你可以在Manghetu的雕像上看到重複的幾何圖形。
08:43
In Ethiopian埃塞俄比亞 crosses十字架, you see this wonderful精彩 unfolding展開 of the shape形狀.
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在Ehiopian的十字架上也可以看到這些無限展開的形狀。
08:48
In Angola安哥拉, the Chokwe紹奎 people draw lines in the sand,
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在Angola,Chokwe人會在沙上畫線,
08:52
and it's what the German德語 mathematician數學家 Euler歐拉 called a graph圖形;
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也就是德國數學家Euler叫做圖像的東西。
08:55
we now call it an Eulerian歐拉 path路徑 --
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我們把它叫做Eulerian道路--
08:57
you can never lift電梯 your stylus唱針 from the surface表面
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你永遠不可以將你的筆從表面上提起,
08:59
and you can never go over the same相同 line twice兩次.
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也不可以重複同一條線段。
09:02
But they do it recursively遞歸, and they do it with an age-grade年齡段 system系統,
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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,
09:05
so the little kids孩子 learn學習 this one, and then the older舊的 kids孩子 learn學習 this one,
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所以小朋友會學這個,大一點的學這個,
09:08
then the next下一個 age-grade年齡段 initiation引發, you learn學習 this one.
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在大一點的學這個。
09:11
And with each iteration迭代 of that algorithm算法,
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而且在每一次重複這些算式時
09:14
you learn學習 the iterations迭代 of the myth神話.
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他們會學這些重複背後的意義。
09:16
You learn學習 the next下一個 level水平 of knowledge知識.
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他們會學到下一層的知識。
09:19
And finally最後, all over Africa非洲, you see this board game遊戲.
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最後,在整個非洲你都可以看到這樣的棋盤遊戲。
09:21
It's called Owari尾張 in Ghana加納, where I studied研究 it;
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這遊戲在我研究的加那叫作Owari,
09:24
it's called Mancala寶石棋 here on the East Coast, Bao in Kenya肯尼亞, Sogo去啊 elsewhere別處.
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在東岸叫做Mancaia,在肯亞叫做Bao,在其他地方叫做Sogo。
09:29
Well, you see self-organizing自組織 patterns模式 that spontaneously自發 occur發生 in this board game遊戲.
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你可以在這些棋盤遊戲中看到自體重複的圖形。
09:34
And the folks鄉親 in Ghana加納 knew知道 about these self-organizing自組織 patterns模式
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在加那的人知道這些圖形,
09:37
and would use them strategically戰略性.
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且會有策略地運用它們。
09:39
So this is very conscious意識 knowledge知識.
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所以是個有意識的知識。
09:41
Here's這裡的 a wonderful精彩 fractal分形.
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這是個很棒的碎形。
09:43
Anywhere隨地 you go in the Sahel薩赫勒, you'll你會 see this windscreen風檔.
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在Sahel的各個地方,你都可以看到這樣的擋風玻璃。
09:47
And of course課程 fences圍欄 around the world世界 are all Cartesian笛卡爾, all strictly嚴格 linear線性.
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當然的世界上任何籬笆都是笛卡爾式的,都是直線的。
09:51
But here in Africa非洲, you've got these nonlinear非線性 scaling縮放 fences圍欄.
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但在非洲,你也可以看到這些不是直線的籬笆。
09:55
So I tracked追踪 down one of the folks鄉親 who makes品牌 these things,
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所以我找到設計這些籬笆的人,
09:57
this guy in Mali馬里 just outside of Bamako巴馬科, and I asked him,
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他是一個住在Bamako外面的Mali的人,我問他:
10:01
"How come you're making製造 fractal分形 fences圍欄? Because nobody沒有人 else其他 is."
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「為什麼你用碎形法製造籬笆?因為沒有其他人這麼做。」
10:03
And his answer回答 was very interesting有趣.
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他的答覆非常有趣。
10:05
He said, "Well, if I lived生活 in the jungle叢林, I would only use the long rows of straw稻草
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他說:「恩,當我走在叢林中時,我只會用長條的稻草,
10:10
because they're very quick and they're very cheap低廉.
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因為使用它們既快又便宜。
10:12
It doesn't take much time, doesn't take much straw稻草."
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不需要花太多時間且不需要太多稻草。」
10:15
He said, "but wind and dust灰塵 goes through通過 pretty漂亮 easily容易.
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他說:「但風和塵土很容易穿過。
10:17
Now, the tight rows up at the very top最佳, they really hold保持 out the wind and dust灰塵.
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最上層很緊的那排可以擋住風和塵土。
10:21
But it takes a lot of time, and it takes a lot of straw稻草 because they're really tight."
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但這需要花很多時間、很多稻草,因為他們需要非常緊。」
10:26
"Now," he said, "we know from experience經驗
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「現在」他說:「我們從經驗中得知,
10:28
that the farther更遠 up from the ground地面 you go, the stronger the wind blows打擊."
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越高的地方風越強。」
10:33
Right? It's just like a cost-benefit成本效益 analysis分析.
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對吧?就有點像是成本效益分析。
10:36
And I measured測量 out the lengths長度 of straw稻草,
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我量了稻草的長度,
10:38
put it on a log-log登錄日誌 plot情節, got the scaling縮放 exponent指數,
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把它放進對數圖形,得到尺度指數,
10:40
and it almost幾乎 exactly究竟 matches火柴 the scaling縮放 exponent指數 for the relationship關係 between之間 wind speed速度 and height高度
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發現他幾乎完全和風速工程書上的
10:45
in the wind engineering工程 handbook手冊.
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風速與高度的指數相同。
10:46
So these guys are right on target目標 for a practical實際的 use of scaling縮放 technology技術.
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這些人在利用尺度科技上正中目標。
10:51
The most complex複雜 example of an algorithmic算法 approach途徑 to fractals分形 that I found發現
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我找到最復雜的算式碎形
10:56
was actually其實 not in geometry幾何, it was in a symbolic象徵 code,
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並不是幾何圖形,而是符號象徵,
10:58
and this was BamanaBamana sand divination卜筮.
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而這是在Bamana的沙占卜。
11:01
And the same相同 divination卜筮 system系統 is found發現 all over Africa非洲.
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在整個非洲都有同樣的占卜系統。
11:04
You can find it on the East Coast as well as the West西 Coast,
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你可以在東岸西岸都找得到這個占卜,
11:09
and often經常 the symbols符號 are very well preserved罐頭,
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而且大部份的時候這些符號是保存得很好的。
11:11
so each of these symbols符號 has four bits -- it's a four-bit四位 binary二進制 word --
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每一個符號有四個小部份:是四個二進法組成的字。
11:17
you draw these lines in the sand randomly隨機, and then you count計數 off,
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你隨意畫這些線,然後數一下,
11:22
and if it's an odd number, you put down one stroke行程,
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如果是奇數,就畫一條線;
11:24
and if it's an even number, you put down two strokes.
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如果是偶數,就畫兩條線。
11:26
And they did this very rapidly急速,
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且他們很迅速地這麼做,
11:29
and I couldn't不能 understand理解 where they were getting得到 --
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我無法瞭解他們怎麼做到的,
11:31
they only did the randomness隨機性 four times --
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他們在隨意的部份只做了四次,
11:33
I couldn't不能 understand理解 where they were getting得到 the other 12 symbols符號.
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我不懂他們另外十二個符號怎麼來的。
11:35
And they wouldn't不會 tell me.
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他們也不告訴我。
11:37
They said, "No, no, I can't tell you about this."
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他們說:「不不,我不能告訴你這個。」
11:39
And I said, "Well look, I'll pay工資 you, you can be my teacher老師,
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然後我說:「恩,我可以付你錢,你可以當我的老師,
11:41
and I'll come each day and pay工資 you."
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然後我可以每天來付你學費。」
11:43
They said, "It's not a matter of money. This is a religious宗教 matter."
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他們說:「這不是錢的問題。這是宗教問題。」
11:46
And finally最後, out of desperation絕望, I said,
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最後在絕望中我說:「
11:47
"Well, let me explain說明 Georg喬治· Cantor領唱者 in 1877."
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恩,讓我來解釋一下1877年的Georg Cantor。」
11:50
And I started開始 explaining說明 why I was there in Africa非洲,
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所以我開始解釋我為什麼會在非洲,
11:54
and they got very excited興奮 when they saw the Cantor領唱者 set.
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他們看了Cantor組合後非常興奮。
11:56
And one of them said, "Come here. I think I can help you out here."
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他們之中其中一個說:「過來,我想我可以幫你一些。」
12:00
And so he took me through通過 the initiation引發 ritual儀式 for a BamanaBamana priest牧師.
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所以他代Bamana牧師帶我走過了一連串的起始儀式。
12:05
And of course課程, I was only interested有興趣 in the math數學,
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當然的,我只對數學的部份有興趣,
12:07
so the whole整個 time, he kept不停 shaking發抖 his head going,
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所以整個過程,他一直搖頭說:
12:09
"You know, I didn't learn學習 it this way."
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「我不是這樣學的。」
12:10
But I had to sleep睡覺 with a kola科拉 nut堅果 next下一個 to my bed, buried隱藏 in sand,
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但我必須在床邊放一顆埋在沙裡的可樂果,
12:14
and give seven coins硬幣 to seven lepers麻風病人 and so on.
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然後給七個痲瘋病人七個銅板之類的事情。
12:17
And finally最後, he revealed透露 the truth真相 of the matter.
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最後,他終於告訴我這後面的祕密。
12:22
And it turns out it's a pseudo-random偽隨機 number generator發電機 using運用 deterministic確定性 chaos混沌.
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事實上這是一個偽渾沌的產生數字的過程。
12:26
When you have a four-bit四位 symbol符號, you then put it together一起 with another另一個 one sideways側身.
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當你有一個四位符號,你把它們並排排起來。
12:32
So even plus odd gives you odd.
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所以偶數加奇數會得到奇數。
12:34
Odd plus even gives you odd.
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奇數加偶數會得到奇數。
12:36
Even plus even gives you even. Odd plus odd gives you even.
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偶數加偶數會得到偶數。奇數加奇數得到偶數。
12:39
It's addition加成 modulo 2, just like in the parity平價 bit check on your computer電腦.
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這是加法定理,就像是電腦裡的配對法一樣。
12:43
And then you take this symbol符號, and you put it back in
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然後你拿所得到的符號,再放回去,
12:47
so it's a self-generating自發電 diversity多樣 of symbols符號.
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就得到一個自我生成的多樣性符號。
12:49
They're truly using運用 a kind of deterministic確定性 chaos混沌 in doing this.
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他們真的在使用決定性混度來產生這些符號。
12:53
Now, because it's a binary二進制 code,
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好,因為是二進位符號,
12:55
you can actually其實 implement實行 this in hardware硬件 --
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事實上你可以將這個置入到硬體裡面--
12:57
what a fantastic奇妙 teaching教學 tool工具 that should be in African非洲人 engineering工程 schools學校.
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多麼適合給非洲工程學校的教材阿!
13:02
And the most interesting有趣 thing I found發現 out about it was historical歷史的.
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我發現最有趣的是它的歷史。
13:05
In the 12th century世紀, Hugo雨果 of SantallaSantalla brought it from Islamic清真 mystics神秘主義者 into Spain西班牙.
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在十二世紀,Santalla的Hugu將這個從西班牙的回教傳統中引進的。
13:11
And there it entered進入 into the alchemy煉金術 community社區 as geomancy風水:
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在那裡,碎形以看風水的身分
13:17
divination卜筮 through通過 the earth地球.
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進入了煉金術的世界。
13:19
This is a geomantic風水 chart圖表 drawn for King國王 Richard理查德 IIII in 1390.
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這是1390年理查國王二世所畫的幾何圖表。
13:24
Leibniz萊布尼茨, the German德語 mathematician數學家,
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德國數學家Leibniz在他的論文中
13:27
talked about geomancy風水 in his dissertation論文 called "De CombinatoriaCombinatoria."
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提到「De Combinatoria」的幾何性。
13:31
And he said, "Well, instead代替 of using運用 one stroke行程 and two strokes,
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他說:「恩,讓我們用零和一取代
13:35
let's use a one and a zero, and we can count計數 by powers權力 of two."
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一條線和兩條線,這樣我們可以以二的指數數下去。」
13:39
Right? Ones那些 and zeros, the binary二進制 code.
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對吧?零和一,二進位法。
13:41
George喬治 Boole布爾 took Leibniz's萊布尼茲 binary二進制 code and created創建 Boolean布爾 algebra代數,
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George Boole拿了Leibniz的二進位法而創造了Boolean算式,
13:44
and John約翰 von Neumann諾伊曼 took Boolean布爾 algebra代數 and created創建 the digital數字 computer電腦.
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然後John von Neumann拿了Boolean算式而創造了數位電腦。
13:47
So all these little PDAs掌上電腦 and laptops筆記本電腦 --
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所以這些掌上型電腦和筆記型電腦--
13:50
every一切 digital數字 circuit電路 in the world世界 -- started開始 in Africa非洲.
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所有利用數位迴路的東西--都是從非洲開始的。
13:53
And I know Brian布賴恩 Eno伊諾 says there's not enough足夠 Africa非洲 in computers電腦,
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我知道Brian Eno說非洲的電腦不夠,
13:58
but you know, I don't think there's enough足夠 African非洲人 history歷史 in Brian布賴恩 Eno伊諾.
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但你知道嗎?我覺得Brian Eno的非洲歷史知識不夠。
14:03
(Laughter笑聲) (Applause掌聲)
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(掌聲)
14:06
So let me end結束 with just a few少數 words about applications應用 that we've我們已經 found發現 for this.
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所以讓我在結束前談談我們做的一些程式。
14:10
And you can go to our website網站,
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你們可以到我們的網站,
14:12
the applets小程序 are all free自由; they just run in the browser瀏覽器.
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使用免費在瀏覽器中始用的程式。
14:14
Anybody任何人 in the world世界 can use them.
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世界上任何人都可以使用它。
14:16
The National國民 Science科學 Foundation's基金會 Broadening擴大 Participation參與 in Computing計算 program程序
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美國國家科學基金會的擴大計算機計畫
14:21
recently最近 awarded頒發 us a grant發放 to make a programmable可編程的 version of these design設計 tools工具,
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最近給我們一筆經費來設計一個可編輯的設計工具,
14:28
so hopefully希望 in three years年份, anybody'llanybody'll be able能夠 to go on the Web捲筒紙
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希望在三年內,任何人都可以上網
14:30
and create創建 their own擁有 simulations模擬 and their own擁有 artifacts文物.
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去作自己的模擬和設計自己的藝品。
14:33
We've我們已經 focused重點 in the U.S. on African-American非裔美國人 students學生們 as well as Native本地人 American美國 and Latino拉丁美洲人.
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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。
14:38
We've我們已經 found發現 statistically統計學 significant重大 improvement起色 with children孩子 using運用 this software軟件 in a mathematics數學 class
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相較於沒有使用這些程式的控制組,
14:44
in comparison對照 with a control控制 group that did not have the software軟件.
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我們發現有使用的孩子在數學課尚有顯著地進步。
14:47
So it's really very successful成功 teaching教學 children孩子 that they have a heritage遺產 that's about mathematics數學,
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所以教孩子們他們有數學的傳統是非常有效的,
14:53
that it's not just about singing唱歌 and dancing跳舞.
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讓他們知道他們的傳統不只是唱歌與跳舞而已。
14:57
We've我們已經 started開始 a pilot飛行員 program程序 in Ghana加納.
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我們也在加納開始了一個前驅計畫,
15:00
We got a small seed種子 grant發放, just to see if folks鄉親 would be willing願意 to work with us on this;
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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,
15:05
we're very excited興奮 about the future未來 possibilities可能性 for that.
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我們對於這個計畫的未來性感到興奮。
15:08
We've我們已經 also been working加工 in design設計.
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我們也在設計上下工夫。
15:10
I didn't put his name名稱 up here -- my colleague同事, Kerry黑色的小乳牛, in Kenya肯尼亞, has come up with this great idea理念
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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,
15:15
for using運用 fractal分形 structure結構體 for postal郵政 address地址 in villages村莊 that have fractal分形 structure結構體,
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就是用碎形結構在碎形村莊中作郵遞區號,
15:20
because if you try to impose強加 a grid structure結構體 postal郵政 system系統 on a fractal分形 village,
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因為如果你想要將格子式的郵遞區號放入碎形的村莊中
15:24
it doesn't quite相當 fit適合.
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是不大適合的。
15:26
Bernard伯納德 Tschumi屈米 at Columbia哥倫比亞 University大學 has finished using運用 this in a design設計 for a museum博物館 of African非洲人 art藝術.
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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。
15:31
David大衛 Hughes休斯 at Ohio俄亥俄州 State University大學 has written書面 a primer底漆 on Afrocentric非洲中心 architecture建築
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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,
15:39
in which哪一個 he's used some of these fractal分形 structures結構.
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裡面包括了一些碎形結構。
15:41
And finally最後, I just wanted to point out that this idea理念 of self-organization自組織,
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最後,我想要指出這個自體組織的想法,
15:46
as we heard聽說 earlier, it's in the brain.
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就像我們早些兒聽到的,是在腦裡面的。
15:48
It's in the -- it's in Google's谷歌的 search搜索 engine發動機.
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這是在,有在Google的搜尋引擎中。
15:53
Actually其實, the reason原因 that Google谷歌 was such這樣 a success成功
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事實上,Google之所以這麼成功就是
15:55
is because they were the first ones那些 to take advantage優點 of the self-organizing自組織 properties性能 of the web捲筒紙.
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因為他是前幾個使用自體組織的優點建構的。
15:59
It's in ecological生態 sustainability可持續性.
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這是存在於生態持續性的。
16:01
It's in the developmental發展的 power功率 of entrepreneurship創業,
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這也是創業精神中發展的動力,
16:03
the ethical合乎道德的 power功率 of democracy民主.
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民主的道德力量。
16:06
It's also in some bad things.
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它也存在於一些不大好的東西當中。
16:08
Self-organization自組織 is why the AIDS艾滋病 virus病毒 is spreading傳播 so fast快速.
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自體組織是為什麼愛滋病可以如此迅速的擴散。
16:11
And if you don't think that capitalism資本主義, which哪一個 is self-organizing自組織, can have destructive有害 effects效果,
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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,
16:15
you haven't沒有 opened打開 your eyes眼睛 enough足夠.
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你看得還不夠多。
16:17
So we need to think about, as was spoken earlier,
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所以我們需要想想,就像我們之前說的,
16:21
the traditional傳統 African非洲人 methods方法 for doing self-organization自組織.
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這個非洲的自體組織的方式。
16:23
These are robust強大的 algorithms算法.
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這是非常有力的計算方法。
16:26
These are ways方法 of doing self-organization自組織 -- of doing entrepreneurship創業 --
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自體組織有很多種方式--就像創業一樣--
16:29
that are gentle溫和, that are egalitarian平均主義.
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可以是溫柔的,可以是平均的。
16:31
So if we want to find a better way of doing that kind of work,
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所以如果我們想要找到一個更好的方式來做這件事情,
16:35
we need look only no farther更遠 than Africa非洲 to find these robust強大的 self-organizing自組織 algorithms算法.
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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。
16:40
Thank you.
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謝謝。
Translated by Joan Liu
Reviewed by Nova Upinel Altesse

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ABOUT THE SPEAKER
Ron Eglash - Mathematician
Ron Eglash is an ethno-mathematician: he studies the way math and cultures intersect. He has shown that many aspects of African design -- in architecture, art, even hair braiding -- are based on perfect fractal patterns.

Why you should listen

"Ethno-mathematician" Ron Eglash is the author of African Fractals, a book that examines the fractal patterns underpinning architecture, art and design in many parts of Africa. By looking at aerial-view photos -- and then following up with detailed research on the ground -- Eglash discovered that many African villages are purposely laid out to form perfect fractals, with self-similar shapes repeated in the rooms of the house, and the house itself, and the clusters of houses in the village, in mathematically predictable patterns.

As he puts it: "When Europeans first came to Africa, they considered the architecture very disorganized and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet."

His other areas of study are equally fascinating, including research into African and Native American cybernetics, teaching kids math through culturally specific design tools (such as the Virtual Breakdancer applet, which explores rotation and sine functions), and race and ethnicity issues in science and technology. Eglash teaches in the Department of Science and Technology Studies at Rensselaer Polytechnic Institute in New York, and he recently co-edited the book Appropriating Technology, about how we reinvent consumer tech for our own uses.

 

More profile about the speaker
Ron Eglash | Speaker | TED.com