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

非洲的分形结构---Ron Eglash

Filmed:
1,740,687 views

“我是一名数学家,并且我希望站在你们的屋顶上。”当Ron Eglash遇到许多非洲家庭时,他总是这样介绍。当时,他正在研究非洲的村落中随处可见的分形图案。
- 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开始 my story故事 in Germany德国, in 1877,
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我的故事发生在1877年,
00:16
with a mathematician数学家 named命名 Georg乔治· Cantor领唱者.
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当时有位德国数学家叫乔治·康托(Georg Cantor)。
00:18
And Cantor领唱者 decided决定 he was going to take a line线 and erase抹去 the middle中间 third第三 of the line线,
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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,
00:23
and then take those two resulting造成 lines线 and bring带来 them back into the same相同 process处理, a recursive递归 process处理.
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然后对剩下的两条线段进行同样的操作,周而复始。
00:28
So he starts启动 out with one line线, and then two,
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于是他从一条线段得到两条,
00:30
and then four, and then 16, and so on.
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然后是四条,然后十六条,不断增加。
00:33
And if he does this an infinite无穷 number of times, which哪一个 you can do in mathematics数学,
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如果他这样重复操作无限次 (在数学中你可以做到),
00:36
he ends结束 up with an infinite无穷 number of lines线,
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最终他就会得到无数条线,
00:38
each of which哪一个 has an infinite无穷 number of points in it.
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而每条线又由无数个点组成。
00:41
So he realized实现 he had a set whose谁的 number of elements分子 was larger than infinity无穷.
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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。
00:45
And this blew自爆 his mind心神. Literally按照字面. He checked检查 into a sanitarium疗养院. (Laughter笑声)
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这简直让他发疯了。我没有夸张,他为此进了疗养院。
00:48
And when he came来了 out of the sanitarium疗养院,
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当他从疗养院出来以后,
00:50
he was convinced相信 that he had been put on earth地球 to found发现 transfinite超限 set theory理论
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他坚信自己是被上帝派来寻找超限集合论的,
00:56
because the largest最大 set of infinity无穷 would be God Himself他自己.
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因为最大的无限集便是上帝本身。
00:59
He was a very religious宗教 man.
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他是一个虔诚的教徒,
01:00
He was a mathematician数学家 on a mission任务.
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并把成为一名数学家当做自己的使命。
01:02
And other mathematicians数学家 did the same相同 sort分类 of thing.
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其他数学家也做过类似的事。
01:04
A Swedish瑞典 mathematician数学家, von Koch科赫,
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例如,一位名为von Koch的瑞典数学家
01:06
decided决定 that instead代替 of subtracting减法 lines线, he would add them.
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有一天决定把线段相加,而不是想减。
01:10
And so he came来了 up with this beautiful美丽 curve曲线.
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最终,他得到了这样一段美丽的曲线。
01:12
And there's no particular特定 reason原因 why we have to start开始 with this seed种子 shape形状;
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其实我们选择这个图形作为起始形状没有什么特殊原因;
01:15
we can use any seed种子 shape形状 we like.
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我们可以选择任何图形作为起始。
01:19
And I'll rearrange改编 this and I'll stick this somewhere某处 -- down there, OK --
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让我把这把这个图形变一下,把这个放在--这下面,好--
01:23
and now upon iteration迭代, that seed种子 shape形状 sort分类 of unfolds展开 into a very different不同 looking structure结构体.
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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。
01:30
So these all have the property属性 of self-similarity自相似性:
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但这些图形都有自我相似的特点:
01:32
the part部分 looks容貌 like the whole整个.
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每一小部分都跟整体相似。
01:34
It's the same相同 pattern模式 at many许多 different不同 scales.
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也可以说是同样的形状,只是大小不同。
01:37
Now, mathematicians数学家 thought this was very strange奇怪
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数学家们觉得这个非常奇怪,
01:39
because as you shrink收缩 a ruler统治者 down, you measure测量 a longer and longer length长度.
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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。
01:44
And since以来 they went through通过 the iterations迭代 an infinite无穷 number of times,
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这些图形经过无数次重复的变化,
01:46
as the ruler统治者 shrinks收缩 down to infinity无穷, the length长度 goes to infinity无穷.
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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。
01:52
This made制作 no sense at all,
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这一点道理也没有,
01:53
so they consigned委托 these curves曲线 to the back of the math数学 books图书.
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于是数学家们把这些曲线塞到数学书的背后,
01:56
They said these are pathological病态的 curves曲线, and we don't have to discuss讨论 them.
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然后说这些是不正常的曲线,我们不用讨论它们。
02:00
(Laughter笑声)
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(笑声)
02:01
And that worked工作 for a hundred years年份.
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就这样,一百年过去了,
02:04
And then in 1977, Benoit伯努瓦 Mandelbrot曼德尔布罗, a French法国 mathematician数学家,
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直到1977年,一位名为Benoit Mandelbrot的法国数学家
02:09
realized实现 that if you do computer电脑 graphics图像 and used these shapes形状 he called fractals分形,
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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,
02:14
you get the shapes形状 of nature性质.
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便可以得到大自然的形状。
02:16
You get the human人的 lungs, you get acacia刺槐 trees树木, you get ferns蕨类植物,
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人们可以得到肺,洋槐树,蕨类植物……
02:20
you get these beautiful美丽 natural自然 forms形式.
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各种美丽自然的形状。
02:22
If you take your thumb拇指 and your index指数 finger手指 and look right where they meet遇到 --
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如果你们看一看你们的拇指与与食指之间的部分--
02:26
go ahead and do that now --
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现在就可以看一下--
02:28
-- and relax放松 your hand, you'll你会 see a crinkle,
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把手放松,你们可以看到一段皱纹,
02:31
and then a wrinkle皱纹 within the crinkle, and a crinkle within the wrinkle皱纹. Right?
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然后这皱纹扩展成更多的皱纹,然后更多,是吧?
02:34
Your body身体 is covered覆盖 with fractals分形.
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你们全身都被“分形”包围着。
02:36
The mathematicians数学家 who were saying these were pathologically病理 useless无用 shapes形状?
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那些认为“分形”不正常的数学家们,
02:39
They were breathing呼吸 those words with fractal分形 lungs.
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他们用分形的肺部呼吸,却说着那样的话,
02:41
It's very ironic具有讽刺意味. And I'll show显示 you a little natural自然 recursion递归 here.
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多讽刺!现在我给大家演示一段自然的循环过程。
02:45
Again, we just take these lines线 and recursively递归 replace更换 them with the whole整个 shape形状.
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跟之前一样,我们用几条线,然后重复用整体代替它们。
02:50
So here's这里的 the second第二 iteration迭代, and the third第三, fourth第四 and so on.
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这是第二次循环,第三次,第四次……不断重复。
02:55
So nature性质 has this self-similar自相似 structure结构体.
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可以看到,大自然也有这种自我相似性。
02:57
Nature性质 uses使用 self-organizing自组织 systems系统.
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大自然是一个自组织系统。
02:59
Now in the 1980s, I happened发生 to notice注意
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到了20世纪80年代,我碰巧发现
03:02
that if you look at an aerial天线 photograph照片 of an African非洲人 village, you see fractals分形.
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在航拍的非洲部落照片中,存在着分形。
03:06
And I thought, "This is fabulous极好! I wonder奇迹 why?"
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我惊叹道:“这简直太不可思议了!究竟是为什么呢?!”
03:10
And of course课程 I had to go to Africa非洲 and ask folks乡亲 why.
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于是我就去了非洲,去请教当地人这个问题。
03:12
So I got a Fulbright富布赖特 scholarship奖学金 to just travel旅行 around Africa非洲 for a year
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我拿到了Fulbright奖学金,去非洲旅行一年,
03:18
asking people why they were building建造 fractals分形,
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询问那儿的人为什么按照分形来盖房子。
03:20
which哪一个 is a great job工作 if you can get it.
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这工作真的很棒,如果你能得到的话。
03:22
(Laughter笑声)
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(笑声)
03:23
And so I finally最后 got to this city, and I'd doneDONE a little fractal分形 model模型 for the city
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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,
03:30
just to see how it would sort分类 of unfold展开 --
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想看看它与实际情况的符合情况--
03:33
but when I got there, I got to the palace of the chief首席,
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当我到了那儿,我去了酋长的宫殿,
03:36
and my French法国 is not very good; I said something like,
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我的法语说得不太好,当时大概对他说:
03:39
"I am a mathematician数学家 and I would like to stand on your roof屋顶."
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“我是一名数学家,我想到你的屋顶上看看。”
03:42
But he was really cool about it, and he took me up there,
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对此他一点问题都没有,带我上到了屋顶,
03:45
and we talked about fractals分形.
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与我讨论起有关分形的问题。
03:46
And he said, "Oh yeah, yeah! We knew知道 about a rectangle长方形 within a rectangle长方形,
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他说:“对,对!我们知道一个方形可以嵌套一个方形,
03:49
we know all about that."
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我们知道有关的一切。”
03:51
And it turns out the royal王室的 insignia徽章 has a rectangle长方形 within a rectangle长方形 within a rectangle长方形,
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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,
03:55
and the path路径 through通过 that palace is actually其实 this spiral螺旋 here.
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而宫殿的走道也是类似的螺旋形状。
03:59
And as you go through通过 the path路径, you have to get more and more polite有礼貌.
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当你沿着宫殿的走道往里走,你必须表现得越来越礼貌。
04:03
So they're mapping制图 the social社会 scaling缩放 onto the geometric几何 scaling缩放;
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他们将社会的层级结构跟房屋的几何结构联系起来;
04:06
it's a conscious意识 pattern模式. It is not unconscious无意识 like a termite白蚁 mound fractal分形.
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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。
04:11
This is a village in southern南部的 Zambia赞比亚.
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这是赞比亚南部的一个村落,
04:13
The Ba-ilaBA-ILA built内置 this village about 400 meters in diameter直径.
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Ba-Ila人建造了这个直径约400米村子。
04:17
You have a huge巨大 ring.
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首先我们有一个很大的环形。
04:19
The rings戒指 that represent代表 the family家庭 enclosures机箱 get larger and larger as you go towards the back,
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代表家族大小的环形,越往后走越大。
04:26
and then you have the chief's酋长的 ring here towards the back
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最终属于首领(家族)的环形就在大环形的尾端,
04:30
and then the chief's酋长的 immediate即时 family家庭 in that ring.
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而首领的直系亲属在那个环形里。
04:33
So here's这里的 a little fractal分形 model模型 for it.
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这就是这个村落的分形模型。
04:34
Here's这里的 one house with the sacred神圣 altar,
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这是一幢拥有圣坛的房子,
04:37
here's这里的 the house of houses房屋, the family家庭 enclosure附件,
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这是房子集合而成的“房子”,家族意义上的,
04:40
with the humans人类 here where the sacred神圣 altar would be,
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原先圣坛所在的地方被人所占据,
04:43
and then here's这里的 the village as a whole整个 --
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而这就是由先前层层叠叠房屋最终形成的村庄---
04:45
a ring of ring of rings戒指 with the chief's酋长的 extended扩展 family家庭 here, the chief's酋长的 immediate即时 family家庭 here,
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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,
04:50
and here there's a tiny village only this big.
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在这儿,有一个只有丁点儿大的村庄。
04:53
Now you might威力 wonder奇迹, how can people fit适合 in a tiny village only this big?
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你也许会问,人怎么可能住进这么小的村子?
04:57
That's because they're spirit精神 people. It's the ancestors祖先.
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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。
05:00
And of course课程 the spirit精神 people have a little miniature微型 village in their village, right?
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当然,这些灵魂居住的村子里也有一个更小的村子,对吧?
05:05
So it's just like Georg乔治· Cantor领唱者 said, the recursion递归 continues继续 forever永远.
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所以就像康托说的,这样的递推将不断循环下去。
05:08
This is in the Mandara曼达拉 mountains, near the Nigerian尼日利亚 border边境 in Cameroon喀麦隆, MokoulekMokoulek.
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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。
05:12
I saw this diagram drawn by a French法国 architect建筑师,
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我看到这幅出自一位法国建筑师之手的图时,
05:15
and I thought, "Wow! What a beautiful美丽 fractal分形!"
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不禁惊叹:“哇!多么漂亮的分形!”
05:17
So I tried试着 to come up with a seed种子 shape形状, which哪一个, upon iteration迭代, would unfold展开 into this thing.
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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。
05:23
I came来了 up with this structure结构体 here.
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结果我画出了这个结构。
05:25
Let's see, first iteration迭代, second第二, third第三, fourth第四.
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让我们来看一下:(这是)第一次循环,第二次,第三次,第四次……
05:29
Now, after I did the simulation模拟,
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在我完成了这个模拟之后,
05:31
I realized实现 the whole整个 village kind of spirals螺旋 around, just like this,
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我意识到这整个村庄就像螺旋一般盘旋环绕,就像这样,
05:34
and here's这里的 that replicating复制 line线 -- a self-replicating自我复制 line线 that unfolds展开 into the fractal分形.
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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。
05:40
Well, I noticed注意到 that line线 is about where the only square广场 building建造 in the village is at.
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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。
05:45
So, when I got to the village,
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于是当我到达那个村子后,
05:47
I said, "Can you take me to the square广场 building建造?
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我就问:“你可以把我带到那个方形建筑所在的地方去吗?”
05:49
I think something's什么是 going on there."
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“那儿一定有特别的故事。”
05:51
And they said, "Well, we can take you there, but you can't go inside
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他们回答:“我们可以带你到建筑的外围,但你不能进去,”
05:54
because that's the sacred神圣 altar, where we do sacrifices牺牲 every一切 year
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“因为那里面是圣坛,每年我们都举行祭祀,
05:57
to keep up those annual全年 cycles周期 of fertility生育能力 for the fields领域."
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以祈祷每年土地的耕种、丰收遵守它固有的规律。”
06:00
And I started开始 to realize实现 that the cycles周期 of fertility生育能力
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我开始意识到,土地耕种、收获的循环过程
06:02
were just like the recursive递归 cycles周期 in the geometric几何 algorithm算法 that builds建立 this.
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就像建立这个村落所运用的几何算法的循环过程一般。
06:06
And the recursion递归 in some of these villages村庄 continues继续 down into very tiny scales.
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在一些村落中,这样的循环会始终持续直到很小的尺度上。
06:10
So here's这里的 a Nankani南卡尼 village in Mali马里.
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这是一个位于马里的村庄,名叫Nankani。
06:12
And you can see, you go inside the family家庭 enclosure附件 --
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你可以看到,这些家族的层次结构,
06:15
you go inside and here's这里的 pots in the fireplace壁炉, stacked堆叠 recursively递归.
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以及这些壁炉中按照一定次序叠放的瓦罐。
06:19
Here's这里的 calabashes葫芦 that Issa伊萨 was just showing展示 us,
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这些是Issa展示给我们的葫芦,
06:23
and they're stacked堆叠 recursively递归.
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它们也被“循环”地叠放着。
06:25
Now, the tiniest最小的 calabash in here keeps保持 the woman's女人的 soul灵魂.
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在这最小的葫芦中,保存着一个女人的灵魂。
06:27
And when she dies, they have a ceremony仪式
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当她死去时,人们会给她举行一个仪式,
06:29
where they break打破 this stack called the zalangazalanga and her soul灵魂 goes off to eternity永恒.
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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。
06:34
Once一旦 again, infinity无穷 is important重要.
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这再次说明,无限(永恒)是非常重要的。
06:38
Now, you might威力 ask yourself你自己 three questions问题 at this point.
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现在,有三个问题需待解决。
06:42
Aren't是不是 these scaling缩放 patterns模式 just universal普遍 to all indigenous土著 architecture建筑?
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第一,这些图案在原生态的建筑中是普遍存在的吗?
06:46
And that was actually其实 my original原版的 hypothesis假设.
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在我的最初假设中答案是肯定的。
06:48
When I first saw those African非洲人 fractals分形,
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当我第一次看到那些非洲的分形建筑时,
06:50
I thought, "Wow, so any indigenous土著 group that doesn't have a state society社会,
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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,
06:54
that sort分类 of hierarchy等级制度, must必须 have a kind of bottom-up自下而上 architecture建筑."
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一定都有那种‘自下而上’的建筑形式咯!”
06:57
But that turns out not to be true真正.
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然而事实并非如此。
06:59
I started开始 collecting搜集 aerial天线 photographs照片 of Native本地人 American美国 and South Pacific和平的 architecture建筑;
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在我收集的美洲土著、南太平洋建筑的航拍照片中,
07:03
only the African非洲人 ones那些 were fractal分形.
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只有非洲建筑具有分形结构。
07:05
And if you think about it, all these different不同 societies社会 have different不同 geometric几何 design设计 themes主题 that they use.
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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。
07:11
So Native本地人 Americans美国人 use a combination组合 of circular symmetry对称 and fourfold四倍 symmetry对称.
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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,
07:17
You can see on the pottery陶器 and the baskets.
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你可以在陶器和篮子上看到它们。
07:19
Here's这里的 an aerial天线 photograph照片 of one of the Anasazi阿纳萨齐 ruins废墟;
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这是部分Anasazi废墟(Anasazi ruins)的航拍照片,
07:22
you can see it's circular at the largest最大 scale规模, but it's rectangular长方形 at the smaller scale规模, right?
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你可以发现,粗略看时它呈圆形,而细看时它是方形的,对吧?
07:27
It is not the same相同 pattern模式 at two different不同 scales.
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对于这种图形,在不同的尺度上,它有着不同的结构形态。
07:31
Second第二, you might威力 ask,
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第二点,你也许会奇怪,
07:32
"Well, Dr博士. EglashEglash, aren't you ignoring无视 the diversity多样 of African非洲人 cultures文化?"
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“Eglash博士(演讲者),你是不是忽略了非洲文化的多样性呢?”
07:36
And three times, the answer回答 is no.
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我坚决地告诉你:不。
07:38
First of all, I agree同意 with Mudimbe'sMudimbe的 wonderful精彩 book, "The Invention发明 of Africa非洲,"
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首先,我同意Mudimbe《非洲的发明》一书的说法,
07:42
that Africa非洲 is an artificial人造 invention发明 of first colonialism殖民主义,
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即非洲是第一次殖民主义及殖民抗争的
07:45
and then oppositional对立 movements运动.
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非自然的产物。
07:47
No, because a widely广泛 shared共享 design设计 practice实践 doesn't necessarily一定 give you a unity统一 of culture文化 --
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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---
07:52
and it definitely无疑 is not "in the DNA脱氧核糖核酸."
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DNA绝没有决定人们的文化须是一致的。
07:55
And finally最后, the fractals分形 have self-similarity自相似性 --
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最后一点,分形是具有自我相似性的---
07:57
so they're similar类似 to themselves他们自己, but they're not necessarily一定 similar类似 to each other --
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可是它们只需自我相似,互相之间却未必是相似的---
08:01
you see very different不同 uses使用 for fractals分形.
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对于分形的不同应用有很多种,
08:03
It's a shared共享 technology技术 in Africa非洲.
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在非洲这是一种众人皆知的技术。
08:06
And finally最后, well, isn't this just intuition直觉?
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再回想一下,恩,难道这不是某种直觉产生的技术吗?
08:09
It's not really mathematical数学的 knowledge知识.
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它恐怕没有运用到什么真正意义上的数学知识。
08:11
Africans非洲人 can't possibly或者 really be using运用 fractal分形 geometry几何, right?
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非洲人不可能真的在运用“分形几何学”,对吧?
08:14
It wasn't invented发明 until直到 the 1970s.
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因为分形几何学直到20世纪70年代才被发明出来。
08:17
Well, it's true真正 that some African非洲人 fractals分形 are, as far as I'm concerned关心, just pure intuition直觉.
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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。
08:22
So some of these things, I'd wander漫步 around the streets街道 of Dakar达喀尔
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对于这些东西,如果我在达喀尔(Dakar)的街上闲逛
08:25
asking people, "What's the algorithm算法? What's the rule规则 for making制造 this?"
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并且问当地人“有什么算法吗?构造这些的规则是什么?”,
08:28
and they'd他们会 say,
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他们会回答说:
08:29
"Well, we just make it that way because it looks容貌 pretty漂亮, stupid." (Laughter笑声)
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“嘿,我们这样做因为它们好看,傻瓜。”(笑声)
08:32
But sometimes有时, that's not the case案件.
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但有些时候,情况则不尽相同。
08:35
In some cases, there would actually其实 be algorithms算法, and very sophisticated复杂的 algorithms算法.
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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。
08:40
So in ManghetuManghetu sculpture雕塑, you'd see this recursive递归 geometry几何.
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在Manghetu雕塑中,你可以看到这样有重复结构的几何图形。
08:43
In Ethiopian埃塞俄比亚 crosses十字架, you see this wonderful精彩 unfolding展开 of the shape形状.
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在Ethiopian十字中,有这样美妙的延展而成的图形。
08:48
In Angola安哥拉, the Chokwe绍奎 people draw lines线 in the sand,
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在安哥拉,Chokwe人在沙中绘制图线,
08:52
and it's what the German德语 mathematician数学家 Euler欧拉 called a graph图形;
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而这就是德国数学家欧拉(Euler)称作“图”(graph)的东西。
08:55
we now call it an Eulerian欧拉 path路径 --
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现在,我们称之为欧拉路径(Eulerian path)---
08:57
you can never lift电梯 your stylus唱针 from the surface表面
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你的笔尖始终不能离开纸平面,
08:59
and you can never go over the same相同 line线 twice两次.
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并且不能穿过同一条线两次。
09:02
But they do it recursively递归, and they do it with an age-grade年龄段 system系统,
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Chokwe人反复学习绘图,并根据年龄区分他们所学的内容:
09:05
so the little kids孩子 learn学习 this one, and then the older旧的 kids孩子 learn学习 this one,
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因而幼龄的孩子学习这个,稍年长的学习这个,
09:08
then the next下一个 age-grade年龄段 initiation引发, you learn学习 this one.
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再下一个年龄层的,学习这个。
09:11
And with each iteration迭代 of that algorithm算法,
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随着算法的迭代,
09:14
you learn学习 the iterations迭代 of the myth神话.
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你将瞥见奇妙事物的发生发展,
09:16
You learn学习 the next下一个 level水平 of knowledge知识.
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并习得更深层次的知识。
09:19
And finally最后, all over Africa非洲, you see this board game游戏.
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再说一点,在整个非洲,你都可以看到这种棋牌游戏。
09:21
It's called Owari尾张 in Ghana加纳, where I studied研究 it;
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在我研究它的地方,加纳(Ghana), 它被称作Owari.
09:24
it's called Mancala宝石棋 here on the East Coast, Bao in Kenya肯尼亚, Sogo去啊 elsewhere别处.
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在东海岸它被称为Mancala,在肯尼亚叫Bao,在其他地方则是Sogo.
09:29
Well, you see self-organizing自组织 patterns模式 that spontaneously自发 occur发生 in this board game游戏.
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在这个游戏中,你会发现自组织图案很自然的产生 。
09:34
And the folks乡亲 in Ghana加纳 knew知道 about these self-organizing自组织 patterns模式
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加纳人知道并了解它们,
09:37
and would use them strategically战略性.
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并有策略地应用它们。
09:39
So this is very conscious意识 knowledge知识.
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对他们来说,这是一种有意义(而非不明不白获取)的知识。
09:41
Here's这里的 a wonderful精彩 fractal分形.
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这儿有一个美丽的分形。
09:43
Anywhere随地 you go in the Sahel萨赫勒, you'll你会 see this windscreen风档.
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在萨赫勒(Sahel)地区,你到哪儿都可看到这样的篱笆。
09:47
And of course课程 fences围栏 around the world世界 are all Cartesian笛卡尔, all strictly严格 linear线性.
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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。
09:51
But here in Africa非洲, you've got these nonlinear非线性 scaling缩放 fences围栏.
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但在非洲,你会发现这些不笔直排列的篱笆。
09:55
So I tracked追踪 down one of the folks乡亲 who makes品牌 these things,
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我找到了一个做这种篱笆的人,
09:57
this guy in Mali马里 just outside of Bamako巴马科, and I asked him,
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他住在Bamako外的Mali(马里).我问他:
10:01
"How come you're making制造 fractal分形 fences围栏? Because nobody没有人 else其他 is."
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“为什么你做分形的篱笆,而别人都没有?”
10:03
And his answer回答 was very interesting有趣.
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他的回答相当有趣。
10:05
He said, "Well, if I lived生活 in the jungle丛林, I would only use the long rows of straw稻草
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他说:“如果我住在丛林里,我会只用那些长麦秆来做篱笆,
10:10
because they're very quick and they're very cheap低廉.
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因为它们易完成,并且很廉价。
10:12
It doesn't take much time, doesn't take much straw稻草."
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不需要花费太多时间,也不需要太多麦秆”
10:15
He said, "but wind and dust灰尘 goes through通过 pretty漂亮 easily容易.
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他继续道:“但是风沙和尘土很容易穿过那些篱笆。
10:17
Now, the tight rows up at the very top最佳, they really hold保持 out the wind and dust灰尘.
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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。
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。
10:26
"Now," he said, "we know from experience经验
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从经验中我们也知道,
10:28
that the farther更远 up from the ground地面 you go, the stronger the wind blows打击."
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从地面往上越靠近篱笆顶部,风力越强劲。”
10:33
Right? It's just like a cost-benefit成本效益 analysis分析.
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他说的很正确,是吧?这就像是成本效益分析。
10:36
And I measured测量 out the lengths长度 of straw稻草,
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于是我测量了篱笆麦秆的长度,
10:38
put it on a log-log登录日志 plot情节, got the scaling缩放 exponent指数,
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把数据放到重对数坐标中,得到了一个标度指数,
10:40
and it almost几乎 exactly究竟 matches火柴 the scaling缩放 exponent指数 for the relationship关系 between之间 wind speed速度 and height高度
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这个标度指数几乎跟风力工程手册中
10:45
in the wind engineering工程 handbook手册.
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风速与高度的标度指数完全匹配。
10:46
So these guys are right on target目标 for a practical实际的 use of scaling缩放 technology技术.
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所以,这些当地人把分形很好地应用在了实际中。
10:51
The most complex复杂 example of an algorithmic算法 approach途径 to fractals分形 that I found发现
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在众多形成分形的算法中,我所发现的最为复杂的
10:56
was actually其实 not in geometry几何, it was in a symbolic象征 code,
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并不是几何图形的算法,而是这个符号代码的,
10:58
and this was BamanaBamana sand divination卜筮.
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用于Bamana沙地占卜。
11:01
And the same相同 divination卜筮 system系统 is found发现 all over Africa非洲.
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类似的占卜系统在整个非洲都可见到,
11:04
You can find it on the East Coast as well as the West西 Coast,
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东、西海岸都有。
11:09
and often经常 the symbols符号 are very well preserved罐头,
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这些符号通常都被良好的保存下来,
11:11
so each of these symbols符号 has four bits -- it's a four-bit四位 binary二进制 word --
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每个符号分为四部分,可看做四个二进制位组成的单元---
11:17
you draw these lines线 in the sand randomly随机, and then you count计数 off,
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你在沙地里随意画下这样的线段,然后数一下,
11:22
and if it's an odd number, you put down one stroke行程,
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(一行中)如果有奇数条线段,划下一条线,
11:24
and if it's an even number, you put down two strokes.
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而如果有偶数条,划两条线。
11:26
And they did this very rapidly急速,
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他们非常快速的完成这工作,
11:29
and I couldn't不能 understand理解 where they were getting得到 --
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可我不明白他们究竟做了些什么---
11:31
they only did the randomness随机性 four times --
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他们仅仅随意画四行线段---
11:33
I couldn't不能 understand理解 where they were getting得到 the other 12 symbols符号.
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我不知道剩下的十二个(占卜)符号他们是怎样得来的,
11:35
And they wouldn't不会 tell me.
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而他们也不愿意告诉我。
11:37
They said, "No, no, I can't tell you about this."
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他们说:“不,不,我们不能告诉你这些。”
11:39
And I said, "Well look, I'll pay工资 you, you can be my teacher老师,
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我回答说:“这样吧,你们可以做我的老师,我付你们工钱,
11:41
and I'll come each day and pay工资 you."
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我每天都上你们这儿来,并每日付薪水。”
11:43
They said, "It's not a matter of money. This is a religious宗教 matter."
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他们说:“这不是钱的问题。这涉及到宗教与信仰。”
11:46
And finally最后, out of desperation绝望, I said,
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最终,我绝望地说道:
11:47
"Well, let me explain说明 Georg乔治· Cantor领唱者 in 1877."
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“好吧,那最后请让我向你们介绍一下康托。(Georg Cantor)”
11:50
And I started开始 explaining说明 why I was there in Africa非洲,
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于是我开始向他们解释我来非洲的原因。
11:54
and they got very excited兴奋 when they saw the Cantor领唱者 set.
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当他们听说康托集时,显得异常兴奋。
11:56
And one of them said, "Come here. I think I can help you out here."
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他们中的一个说道:“来吧,我想我能解决你的问题。”
12:00
And so he took me through通过 the initiation引发 ritual仪式 for a BamanaBamana priest牧师.
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于是他带我完成了Bamana教的入会仪式。
12:05
And of course课程, I was only interested有兴趣 in the math数学,
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当然,我只对其中的数学问题感兴趣。
12:07
so the whole整个 time, he kept不停 shaking发抖 his head going,
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整个过程中,他始终摇头晃脑,说着
12:09
"You know, I didn't learn学习 it this way."
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“你知道吗,我原来可不知道这其中的奥秘。”
12:10
But I had to sleep睡觉 with a kola科拉 nut坚果 next下一个 to my bed, buried隐藏 in sand,
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而我得和埋在床边沙子中的可乐树果子(kola nut)睡一块儿,
12:14
and give seven coins硬币 to seven lepers麻风病人 and so on.
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将七枚硬币给予七个麻风病人,等等。
12:17
And finally最后, he revealed透露 the truth真相 of the matter.
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最终,他向我揭示了那些符号的奥秘。
12:22
And it turns out it's a pseudo-random伪随机 number generator发电机 using运用 deterministic确定性 chaos混沌.
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事实是,那些符号产生自确定性混沌---一个伪随机过程。
12:26
When you have a four-bit四位 symbol符号, you then put it together一起 with another另一个 one sideways侧身.
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你将一个已有的4位(four-bit)的符号与另一个放在一起。
12:32
So even plus odd gives you odd.
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于是偶数加奇数得奇数;
12:34
Odd plus even gives you odd.
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奇数加偶数得奇;
12:36
Even plus even gives you even. Odd plus odd gives you even.
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偶数加偶数得偶;奇数加奇数得偶。
12:39
It's addition加成 modulo 2, just like in the parity平价 bit check on your computer电脑.
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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。
12:43
And then you take this symbol符号, and you put it back in
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然后你用新得到的符号替换原有的,
12:47
so it's a self-generating自发电 diversity多样 of symbols符号.
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于是你就“自我繁衍”出一系列的符号。
12:49
They're truly using运用 a kind of deterministic确定性 chaos混沌 in doing this.
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他们真真确确在运用确定性混沌的理论。
12:53
Now, because it's a binary二进制 code,
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由于这些是二值码,
12:55
you can actually其实 implement实行 this in hardware硬件 --
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事实上你可以将他们运用到硬件中---
12:57
what a fantastic奇妙 teaching教学 tool工具 that should be in African非洲人 engineering工程 schools学校.
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多么有趣的案例,真该运用到非洲的工程学校的教学中。
13:02
And the most interesting有趣 thing I found发现 out about it was historical历史的.
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对于这些符号,我发现的最有趣的事还是关于它们的历史。
13:05
In the 12th century世纪, Hugo雨果 of SantallaSantalla brought it from Islamic清真 mystics神秘主义者 into Spain西班牙.
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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。
13:11
And there it entered进入 into the alchemy炼金术 community社区 as geomancy风水:
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在那儿,它进入炼金术士的团体,用于看风水:
13:17
divination卜筮 through通过 the earth地球.
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通过泥土来占卜(抓沙散地,按其所成像以断吉凶)。
13:19
This is a geomantic风水 chart图表 drawn for King国王 Richard理查德 IIII in 1390.
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这是一幅在1390年为理查二世(King Richard II)绘制的占卜图。
13:24
Leibniz莱布尼茨, the German德语 mathematician数学家,
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德国数学家莱布尼兹(Leibniz)
13:27
talked about geomancy风水 in his dissertation论文 called "De CombinatoriaCombinatoria."
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在他名为"De Combinatoria"的论文中谈论到了泥土占卜。
13:31
And he said, "Well, instead代替 of using运用 one stroke行程 and two strokes,
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在文章中他说:“我们不使用一条或两条的划线
13:35
let's use a one and a zero, and we can count计数 by powers权力 of two."
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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。”
13:39
Right? Ones那些 and zeros, the binary二进制 code.
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这不就是吗?由很多0和1组成了二进制码。
13:41
George乔治 Boole布尔 took Leibniz's莱布尼兹 binary二进制 code and created创建 Boolean布尔 algebra代数,
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布尔(George Boole)运用莱布尼兹的二进制码创造了布尔代数(Boolean algebra),
13:44
and John约翰 von Neumann诺伊曼 took Boolean布尔 algebra代数 and created创建 the digital数字 computer电脑.
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约翰.冯.诺依曼(John von Neumann)则利用布尔代数创造了电脑.
13:47
So all these little PDAs掌上电脑 and laptops笔记本电脑 --
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因而所有这些小器件---PDA,便携式电脑---
13:50
every一切 digital数字 circuit电路 in the world世界 -- started开始 in Africa非洲.
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所有世间的数字电路---都起源于非洲。
13:53
And I know Brian布赖恩 Eno伊诺 says there's not enough足够 Africa非洲 in computers电脑,
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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;
13:58
but you know, I don't think there's enough足够 African非洲人 history历史 in Brian布赖恩 Eno伊诺.
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而我认为事实是Brian Eno脑中没有足够的非洲历史。
14:03
(Laughter笑声) (Applause掌声)
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(掌声)
14:06
So let me end结束 with just a few少数 words about applications应用 that we've我们已经 found发现 for this.
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请让我简单地用这些分形的实际应用结束这场演讲。
14:10
And you can go to our website网站,
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你也可以浏览我们的网站,
14:12
the applets小程序 are all free自由; they just run in the browser浏览器.
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程序都是免费的,可以直接运行,
14:14
Anybody任何人 in the world世界 can use them.
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世界上的任何人都可以使用它们。
14:16
The National国民 Science科学 Foundation's基金会 Broadening扩大 Participation参与 in Computing计算 program程序
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The National Science Foundation's Broadening Participation in Computing program(某基金会)
14:21
recently最近 awarded颁发 us a grant发放 to make a programmable可编程的 version of these design设计 tools工具,
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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,
14:28
so hopefully希望 in three years年份, anybody'llanybody'll be able能够 to go on the Web卷筒纸
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顺利的话,在三年内,所有人都能在网上
14:30
and create创建 their own拥有 simulations模拟 and their own拥有 artifacts文物.
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创造出属于自己的分形与艺术品。
14:33
We've我们已经 focused重点 in the U.S. on African-American非裔美国人 students学生们 as well as Native本地人 American美国 and Latino拉丁美洲人.
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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,
14:38
We've我们已经 found发现 statistically统计学 significant重大 improvement起色 with children孩子 using运用 this software软件 in a mathematics数学 class
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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、
14:44
in comparison对照 with a control控制 group that did not have the software软件.
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不使用该软件的孩子相比,学术表现有了极大提高。
14:47
So it's really very successful成功 teaching教学 children孩子 that they have a heritage遗产 that's about mathematics数学,
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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,
14:53
that it's not just about singing唱歌 and dancing跳舞.
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而不仅仅教他们唱歌、跳舞。
14:57
We've我们已经 started开始 a pilot飞行员 program程序 in Ghana加纳.
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我们在加纳启动了一个试验项目。
15:00
We got a small seed种子 grant发放, just to see if folks乡亲 would be willing愿意 to work with us on this;
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我们先提供一小笔种子资金,看人们是否愿意与我们合作;
15:05
we're very excited兴奋 about the future未来 possibilities可能性 for that.
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对于未来(更大规模)的合作,我们都充满期待。
15:08
We've我们已经 also been working加工 in design设计.
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我们也在设计方面不断努力。
15:10
I didn't put his name名称 up here -- my colleague同事, Kerry黑色的小乳牛, in Kenya肯尼亚, has come up with this great idea理念
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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:
15:15
for using运用 fractal分形 structure结构体 for postal邮政 address地址 in villages村庄 that have fractal分形 structure结构体,
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在具有分形结构的村落中应用具有分形结构的邮政网络,
15:20
because if you try to impose强加 a grid structure结构体 postal邮政 system系统 on a fractal分形 village,
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因为一个方格状的邮递系统很难适应
15:24
it doesn't quite相当 fit适合.
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分形的村落结构。
15:26
Bernard伯纳德 Tschumi屈米 at Columbia哥伦比亚 University大学 has finished using运用 this in a design设计 for a museum博物馆 of African非洲人 art艺术.
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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。
15:31
David大卫 Hughes休斯 at Ohio俄亥俄州 State University大学 has written书面 a primer底漆 on Afrocentric非洲中心 architecture建筑
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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,
15:39
in which哪一个 he's used some of these fractal分形 structures结构.
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在其中他运用到了一些分形结构。
15:41
And finally最后, I just wanted to point out that this idea理念 of self-organization自组织,
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最后,我想指出这种自组织(self-organization)的思想---
15:46
as we heard听说 earlier, it's in the brain.
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我之前也提到过---是牢固存在大脑里的。
15:48
It's in the -- it's in Google's谷歌的 search搜索 engine发动机.
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它也存在于谷歌(Google)的搜索引擎中。
15:53
Actually其实, the reason原因 that Google谷歌 was such这样 a success成功
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事实上,谷歌能够获得如此巨大的成功,
15:55
is because they were the first ones那些 to take advantage优点 of the self-organizing自组织 properties性能 of the web卷筒纸.
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就在于它第一个利用了网络的这种自组织性质。
15:59
It's in ecological生态 sustainability可持续性.
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它体现于生态的可持续性,
16:01
It's in the developmental发展的 power功率 of entrepreneurship创业,
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体现于企业的发展力,
16:03
the ethical合乎道德的 power功率 of democracy民主.
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也体现于民主思想的道德约束力。
16:06
It's also in some bad things.
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它也体现在一些坏的事情当中。
16:08
Self-organization自组织 is why the AIDS艾滋病 virus病毒 is spreading传播 so fast快速.
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自我组织是艾滋病毒传播如此迅速的原因。
16:11
And if you don't think that capitalism资本主义, which哪一个 is self-organizing自组织, can have destructive有害 effects效果,
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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,
16:15
you haven't没有 opened打开 your eyes眼睛 enough足够.
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那么你还没有真正看清这个世界。
16:17
So we need to think about, as was spoken earlier,
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因而我们需要思考,如我之前所说的,
16:21
the traditional传统 African非洲人 methods方法 for doing self-organization自组织.
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非洲传统的自组织的方式。
16:23
These are robust强大的 algorithms算法.
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这些才是强健的算法(方法)。
16:26
These are ways方法 of doing self-organization自组织 -- of doing entrepreneurship创业 --
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这些才是进行自组织的方式---发展企业的方式---
16:29
that are gentle温和, that are egalitarian平均主义.
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它们温和、平缓。
16:31
So if we want to find a better way of doing that kind of work,
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因此如果我们想寻找一个更好的涉及此类工作的方式,
16:35
we need look only no farther更远 than Africa非洲 to find these robust强大的 self-organizing自组织 algorithms算法.
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只需从非洲就能找寻到这些强健的自组织算法。
16:40
Thank you.
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谢谢大家。
Translated by Jiayi Jessie Li
Reviewed by Weihua ZHANG

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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