African Origins of Mathematics: Dr. Ron Eglash Part 1

I want to start my story in Germany in 1877, the mathematician named George Cantor. And Cantor decided he was going to take a line and erase the middle third of the line, and then take those two resulting lines and bring them back into the same process, a recursive process. So he starts out with one line and then two and then four and then six teams on. And if he does this in infinite number of times, which you can do in mathematics, he ends up with an infinite number of lines, each of which has an infinite number of points in it. So he realized he had a set whose number of elements was larger than infinity. And this blew his mind, literally, he checked in the sanitary. And when he came out of the sanitary him, he was convinced that he had been put on earth to found transfinite set theory because the largest set of infinity would be God himself. He was a very religious man. So mathematician on a mission. And other mathematicians did the same sort of thing, a Swedish mathematician, one cook, decided that instead of subtracting lines, he would add them. And so he came up with this beautiful curve. And there's no particular reason why we have to start with this C-shaped. We can use any C-shaped we like. And I'll rearrange this and I'll stick this somewhere down there. OK. And now upon iteration, that C-shaped sort of unfolds into a very different looking structure. So these all have the property of self-similarity. The part looks like the holes, the same pattern at many different scales. Now, mathematicians thought this was very strange because, as you shrink a ruler down, you measure a longer and longer length. And since they went through the iterations in infinite number of times, as the ruler shrinks down to infinity, the length goes to infinity. This made no sense at all. So they consign these curves to the back of the math books. They said, these are pathological curves, and we don't have to discuss them. And that worked for 100 years. And then in 1977, Benoit Manobrot, a French mathematician, realized that if you do computer graphics and use these shapes, you call fractals, you get the shapes of nature. You get the human lungs. You get a k-shitchery. You get ferns. You get these beautiful natural forms. If you take your thumb and your index finger and look right where they need, go ahead and do that now. And relax your hand. You'll see a crinkle, and then a wrinkle within the crinkle, and a crinkle within the wrinkle within the right, your body is covered with fractals. The mathematicians who were saying these are pathological useless shapes, they were breathing those words with fractal lungs. That's very ironic. And I'll show you a little natural recursion here. Again, we just take these lines and recursively replace them with the whole shape. So here's the second iteration, third, fourth, and so on. So nature has this self-similar structure. Nature uses self-organizing systems. Now in the 1980s, I happened to notice that if you look at an aerial photograph of an African village, you see fractals. And I thought this is fabulous. I'm wondering why. And of course, I had to go to Africa and ask folks. Why? So I got a full bright scholarship to just travel around Africa for your asking people why they were building fractals. Which is a great job if you can get it. And so I finally got to this city. And I done a little fractal model for the city, just to see how it unfold. But when I got there, I got to this palace of the chief. And my French is not very good. I said something like, I'm a mathematician and I would like to stand in your roof. But he was really cool about it. He took me up there and we talked about fractals. And he said, oh yeah, yeah. We know it. We're rectangle with an rectangle with a rectangle. And we know all about that. And it turns out the royal insignia has a rectangle with an rectangle, and the path through that palace is actually this spiral here. And as you go through the path, you have to get more and more polite. So they're mapping the social scaling onto the geometric scan. It's a conscious pattern. It is not unconscious like a termite mount fractal. This is a village in southern Zambia, the Baida. Build this village is about 400 meters in diameter. You have a huge ring that represents the family enclosures. Get larger and larger. You go towards the back. And then you have the chiefs ring here and towards the back. And then the chiefs immediate family in that ring. So here's a little fractal model for it. Here's one house with a sacred altar. Here's the house of houses. The family enclosure with the humans here, where the sacred altar would be. And then here's the village as a whole, a ring of rings, with the chiefs extended family here, the chiefs immediate family here. And then here, there's a tiny village, only this big. Now you might wonder, how can people fit in a tiny village only this big? That's because there's spirit people. It's the ancestors. And of course, the spirit people have a little miniature village in their village. So it's just like George Canterse, the recursion continues forever. This is in the Mandar Mountains near that Nigerian border in Cameroon, Moukulek. I saw this diagram drawn by a French architect. And I thought, wow, what a beautiful fractal. So I tried to come up with a seed shape, which upon iteration would unfold into this thing. I came up with this structure here. I'd say it's first iteration, second, third, fourth. Now after I did the simulation, I realized the whole village kind of spirals around, just like this. And here's that replicating line, self-replicating line, that unfolds into the fractal. I noticed that line is about where they only square building and the village is at. So when I got to the village, I said, can you take me to the square building? I think something's going on there. And they said, well, we can take you there, but you can't go inside. Because that's the sacred altar where we do sacrifices every year. We keep up those annual cycles of fertility from the fields. And I started to realize that the cycles of fertility were just like the recursive cycles in the geometric algorithm that builds this. And the recursion in some of these villages continues down to very tiny scales. So here's a non-cony village in Molly. And you can see you go inside the family enclosure. You go inside. And here's pots in the fireplace. Stack recursively. Here's the calabashes that ESA was just showing us. And they're stacked recursively. Now the tiniest calabash in here keeps the woman's soul. And when she dies, they have her ceremony where they break this stack called this Alonga. And her soul goes off to eternity. What's again? Once again, infinity is important. Now you might ask yourself three questions at this point. Aren't these scaling counters just universal to all indigenous architecture? And that was actually my original hypothesis. When I first saw those African fractals, I thought, wow. So any indigenous group that doesn't have a state society that's in a hierarchy must have a kind of bottom-up architecture. But that turns out not to be true. I started collecting aerial photographs of Native America and South Pacific architecture. Only the African ones were fractal. And if you think about it, all these different societies have different geometric design themes that they use. So Native Americans use a combination of circular symmetry and forefold symmetry. And you can see them in the pottery and the baskets. Here's an aerial photograph of one of the Anasasi ruins. You can see it's circular at the largest scale, but it's rectangular at the smaller scale. It is not the same pattern at two different scales. Second, you might ask, well, Dr. Egglash, aren't you ignoring the diversity of African cultures? And three times, the answer is no. First of all, I agree with Moudinbe's wonderful book, the Invention of Africa, that Africa is an artificial invention of first colonialism and then oppositional movements. No, because a widely shared design practice doesn't necessarily give you a unity of culture. And definitely is not in the DNA. And finally, the fractals have self-similarity. So they're similar to themselves, but they're not necessarily similar to each other. You see very different uses for fractals. It's a shared technology in Africa. And finally, well, isn't this just intuition? It's not really mathematical knowledge. Africans can't possibly really be using fractal geometry. It wasn't invented until the 1970s. Well, it's true that some African fractals are as far as I'm concerned, just pure intuition. So some of these things, you know, I water around the streets of Dakar asking people, well, what's the algorithm? What's the rule for making this? And they'd say, well, you know, we just make it that way because it looks pretty stupid. But sometimes, sometimes that's not the case. In some cases, they would actually be algorithms and very sophisticated algorithms. So in May, what do you sculpture? You see this recursive geometry. In Ethiopian crosses, you see this wonderful unfolding of the shape. In Angola, the chocolate people draw lines in the sand. And it's what German mathematician Euler called a graph. We now call an Olerian path. You can never lift your stylus in the surface. And you can never go over the same line twice. But they do it recursively. And they do it with an age grade system. So the little kids learn this one. And then the older kids learn this one. And then the next age grade initiation you're learning this one. And with each iteration of that algorithm, you learn the iterations of the myth. You learn the next level of knowledge. And finally, all of Africa, you see this board game. It's called a worry in Ghana, where I studied it. It's called Moncala here on the East Coast, Bau in Kenya, Sogo, elsewhere. Well, you see self-organizing patterns that spontaneously occur. And it's board game. And the folks in Ghana knew about these self-organizing patterns, and we'd use them strategically. So this is very conscious knowledge. Here's a wonderful fractal. Anywhere you go in this hall, you'll see this windscreen. And of course, fences around the world are all Cartesian, all strictly linear. But here in Africa, you've got these nonlinear scaling fences. So I tracked down one of the folks who makes these things.