and 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 sixteen and so on
so he realized he had a set whose number of elements was larger than infinity and this blew his mind literally
and when he came out of the sanitarium 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 he was a mathematician on a mission
this seed shape we can use any seed shape we like and i'll rearrange this and stick this somewhere down there ok and
now upon iteration that seed shape sort of unfolds into a very different looking structure so these all have the property of self similarity the part looks like the whole it's 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
this made no sense at all so they consigned these curves to the back of the math books they said these are pathological curves and we don't have to discuss
realized that if you do computer graphics and used these shapes he called fractals you get the shapes of nature
you get the human lungs you get acacia trees you get ferns you get these beautiful natural forms if you take your thumb and your index finger and look right where they meet 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 right
your body is covered with fractals the mathematicians who were saying these were pathologically useless shapes they were breathing those words with fractal lungs
and i'll show you a little natural recursion here again we just take these lines and recursively replace them with the whole shape
in the nineteen eighties 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 wonder why and of course i had to go to africa and ask folks why so i got a fulbright
to just travel around africa for a year asking people why they were building fractals which is a great job if you can get
and so i finally got to this city and i'd done a little fractal model for the city just to see how it would sort of unfold but when i got there
i got to the palace of the chief and my french is not very good i said something like i am a mathematician and i would like to stand on your roof
and it turns out the royal insignia has a rectangle within a rectangle within a 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 scaling it's a conscious
pattern it is not unconscious like a termite mound fractal this is a village in southern zambia the ba ila built this village about four hundred
you have a huge ring the rings that represent the family enclosures get larger and larger as you go towards the back and then you have the chief 's ring here towards the back and the chief 's immediate family
in that ring so here 's a little fractal model for it here 's one house with the 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 ring of rings with the chief 's extended family here the chief 's immediate family here and here there's a tiny village only this
now you might wonder how can people fit in a tiny village only this big that's because they're spirit people it's the ancestors and of course the spirit people
have a little miniature village in their village right so it's just like georg cantor said the recursion continues forever this is in the mandara mountains near the nigerian border in cameroon mokoulek 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
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 a self replicating line that unfolds into the fractal well i noticed that line is about where the only square building in the
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 to keep up those annual cycles of fertility
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 into very tiny scales so here 's a nankani village in mali
and you can see you go inside the family enclosure you go inside and here 's pots in the fireplace stacked recursively here 's calabashes that
eternity once again infinity is important now you might ask yourself three questions at this point
architecture but that turns out not to be true i started collecting aerial photographs of native american and south pacificarchitecture only the african ones were fractal and if you think about it all these different societies have different geometric design themes they use
so native americans use a combination of circular symmetry and fourfold symmetry you can see on the pottery and the baskets
here 's an aerial photograph of one of the anasazi ruins you can see it's circular at the largest scale but it's rectangular at the smaller scale right it is not the same pattern at two different scales
second you might ask well doctor eglash aren't you ignoring the diversity of african cultures and three times the answer is no
because a widely shared design practice doesn't necessarily give you a unity of culture and it definitely is not in the dna
africa
and finally well isn't this just intuition it's not really mathematical knowledge africans can't possibly really be using fractal geometry right it wasn't invented until the nineteen seventies
we just make it that way because it looks pretty stupid but sometimes
in angola the chokwe people draw lines in the sand and it's what the german mathematician euler called a graph we now call it an eulerian path you can never lift your stylus from 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 then the next age grade initiation
and with each iteration of that algorithm you learn the iterations of the myth you learn the next level of knowledge
see self organizing patterns that spontaneously occur in this board game and the folks in ghana knew about these self organizing patterns and would use them strategically so this is very conscious knowledge here 's a wonderful fractal
fences so i tracked down one of the folks who makes these things a guy in mali just outside of bamako and i asked him how come you're making fractal fences because nobody else is and his answer was very
he said but wind and dust goes through pretty easily now the tight rows up at the very top they really hold out the wind and dust but it takes a lot of time and it takes a lot of straw because they're really tight now he said we know from experience
that the farther up from the ground you go the stronger the wind blows right it's just like a cost benefit analysis and i measured out the lengths of straw put it on a log log plot got the scaling exponent
you can find it on the east coast as well as the west coast and often the symbols are very well preserved so
each of these symbols has four bits it's a four bit binary word you draw these lines in the sand randomly
and then you count off and if it's an odd number you put down one stroke and if it's an even number you put down two strokes and they did this very rapidly
and i couldn't understand where they were getting they only did the randomness four times i couldn't understand where they were getting the other twelve symbols
sideways so even plus odd gives you odd odd plus even gives you odd even plus even gives you even odd plus odd gives you even it's addition modulo two just like in the parity bit check on your computer
and then you take this symbol and you put it back in so it's a self generating diversity of symbols they're truly using
a kind of deterministic chaos in doing this now because it's a binary code you can actuallyimplement this in hardware what a fantastic teaching tool that should be
in african engineering schools and the most interesting thing i found out about it was historical in the twelfth century hugo of santalla brought it from islamic mystics into spain
the german mathematician talked about geomancy in his dissertation called de combinatoria and he said well instead of using one stroke and two strokes let 's use a one and a zero and we can count by powers of two
right ones and zeros the binary code george boole took leibniz 's binary code and created boolean algebra and john von neumann took boolean algebra and created the digital computer so all these little pdas and laptops every digital circuit in the world
started in africa and i know brian eno says there's not enough africa in computers you know i don't think there's enough african history in
just a few words about applications that we've found for this and you can go to our website the applets are all free they just run in the browser
in the world can use them the national science foundation 's broadening participation in computing program recently awarded us a grant to make
a programmable version of these design tools so hopefully in three years anybody 'll be able to go on the web and create their own simulations and their own artifacts we've focused in the u s on african american students as well as native american
really very successful teaching children they have a heritage that's about mathematics that it's not just about singing and dancing
a pilot program in ghana we got a small seed grant just to see if folks would be willing to work with us on this we're very excited about the
i just wanted to point out that this idea of self organization as we heard earlier it's in the brain it's in the
it's in google 's search engine actually the reason that google was such a success is because they were the first ones to take advantage of the self organizing properties of the web it's in ecological sustainability it's in the developmental power of entrepreneurship the ethical power of democracy
it's also in some bad things self organization is why the aids virus is spreading so fast and if you don't think that capitalism which is self organizing can have destructive effects you haven't opened your eyes enough so we need to think about
as was spoken earlier the traditional african methods for doing self organization these are robust algorithms these are ways of doing self organization of doing entrepreneurship that are gentle that are egalitarian
so if we want to find a better way of doing that kind of work we need look only no farther than africa to find these robust self organizing algorithms thank you
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