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IN 1988, RON EGLASH was studying aerial photographs of a traditional Tanzanian village when a strangely familiar pattern caught his eye. The thatched-roof huts were organized in a geometric pattern of circular clusters within circular clusters, an arrangement Eglash recognized from his former days as a Silicon Valley computer engineer. Stunned, Eglash digitized the images and fed the information into a computer. The computer's calculations agreed with his intuition: He was seeing fractals. Since then, Eglash has documented the use of fractal geometry-the geometry of similar shapes repeated on ever-shrinking scales-in everything from hairstyles and architecture to artwork and religious practices in African culture. The complicated designs and surprisingly complex mathematical processes involved in their creation may force researchers and historians to rethink their assumptions about traditional African mathematics. The discovery may also provide a new tool for teaching African-Americans about their mathematical heritage.
In contrast to the relatively ordered world of Euclidean geometry taught in most classrooms, fractal geometry yields less obvious patterns. These patterns appear everywhere in nature, yet mathematicians began deciphering them only about 30 years ago. Fractal shapes have the property of self-similarity, in which a small part of an object resembles the whole object. "If I look at a mountain from afar, it looks jagged and irregular, and if I start hiking up it, it still looks jagged and irregular," said Harold Hastings, a professor of mathematics at Hofstra University. "So it's a fractal object-its appearance is maintained across some scales." Nearly 20 years ago, Hastings documented fractal growth patterns among cypress trees in Georgia's Okefenokee Swamp. Others have observed fractal patterns in the irregular features of rocky coastlines, the ever-diminishing scaling of ferns, and even the human respiratory and circulatory systems with their myriad divisions into smaller and smaller branches. What all of these patterns share is a close-up versus a panoramic symmetry instead of the common right versus left symmetry seen in mirror images.
The principles of fractal geometry are offering scientists powerful new tools for biomedical, geological and graphic applications. A few years ago, Hastings and a team of medical researchers found that the clustering of pancreatic cells in the human body follows the same fractal rules that meteorologists have used to describe cloud formation and the shapes of snowflakes. But Eglash envisioned a different potential for the beautiful fractal patterns he saw in the photos from Tanzania: a window into the world of native cultures.
Eglash had been leafing through an edited collection of research articles on women and Third World development when he came across an article about a group of Tanzanian women and their loss of autonomy in village organization. The author blamed the women's plight on a shift from traditional architectural designs to a more rigid modernization program. In the past, the women had decided where their houses would go. But the modernization plan ordered the village structures like a grid-based Roman army camp, similar to tract housing. Eglash was just beginning a doctoral program in the history of consciousness at the University of California at Santa Cruz. Searching for a topic that would connect cultural issues like race, class and gender with technology, Eglash was intrigued by what he read and asked the researcher to send him pictures of the village. After detecting the surprising fractal patterns, Eglash began going to museums and libraries to study aerial photographs from other cultures around the world.
"My assumption was that all indigenous architecture would be more fractal," he said. "My reasoning was that all indigenous architecture tends to be organized from the bottom up." This bottom-up, or self-organized, plan contrasts with a top-down, or hierarchical, plan in which only a few people decide where all the houses will go. "As it turns out, though, my reasoning was wrong," he said. "For example, if you look at Native American architecture, you do not see fractals. In fact, they're quite rare." Instead, Native American architecture is based on a combination of circular and square symmetry, he said. Pueblo Bonito, an ancient ruin in northwestern New Mexico built by the Anasazi people, consists of a big circular shape made of connected squares. This architectural design theme is repeated in Native American pottery, weaving and even folklore, said Eglash. When Eglash looked elsewhere in the world, he saw different geometric design themes being used by native cultures. But he found widespread use of fractal geometry only in Africa and southern India, leading him to conclude that fractals weren't a universal design theme.
Focusing on Africa, he sought to answer what property of fractals made them so widespread in the culture. "If they used circular houses, they would use circles within circles," he said. "If they used rectangles you would see rectangles within rectangles. I would see these huge plazas. Those would narrow down to broad avenues, those would narrow down to smaller streets, and those would keep branching down to tiny footpaths. From a European point of view, that may look like chaos, but from a mathematical view it's the chaos of chaos theory-it's fractal geometry." Eglash expanded on his work in Africa after he won a Fulbright Grant in 1993. He toured central and western Africa, going as far north as the Sahel, the area just south of the Sahara Desert, and as far south as the equator. He visited seven countries in all. "Basically I just toured around looking for fractals, and when I found something that had a scaling geometry, I would ask the folks what was going on-why they had made it that way," he said.
In some cases Eglash found that fractal designs were based purely on aesthetics-they simply looked good to the people who used them. In many cases, however, Eglash found that step-by-step mathematical procedures were producing these designs, many of them surprisingly sophisticated. While visiting the Mangbetu society in central Africa, he studied the tradition of using multiples of 45-degree angles in the native artwork. The concept is similar to the shapes that American geometry students produce using only a compass and a straight edge, he said. In the Mangbetu society, the uniform rules allowed the artisans to compete for the best design. Eglash found a more complex example of fractal geometry in the windscreens widely used in the Sahel region. Strong Sahara winds regularly sweep the dry, dusty land. For protection from the biting wind and swirling sand, local residents have fashioned screens woven with millet, a common crop in the area. The windscreens consist of about 10 diagonal rows of millet stalk bundles, each row shorter than the one below it. "The geometry of the screen is quite extraordinary," said Eglash. "I had never seen anything like it." In Mali, Eglash interviewed an artisan who had constructed one of the screens, asking him why he had settled on the fractal design.
The man told Eglash the long, loosely bound rows forming the bottom of the screen are very cheap to construct but do little to keep out wind and dust. The smaller, tighter rows at the top require more time and straw to make but also offer much more protection. The artisans had learned from experience that the wind blows more strongly higher off the ground, so they had made only what was needed. "What they had done is what an engineer would call a cost-benefit analysis," said Eglash. He measured the length of each row of the non-linear windscreen and plotted the data on a graph. "I could figure out what the lengths should be based on wind engineering values and compared those values to the actual lengths and discovered that they were quite close," he said. "Not only are they using a formal geometrical system to produce these scaling shapes, but they also have a nice practical value." Eglash realized that many of the fractal designs he was seeing were consciously created. "I began to understand that this is a knowledge system, perhaps not as formal as western fractal geometry but just as much a conscious use of those same geometric concepts," he said. "As we say in California, it blew my mind." In Senegal, Eglash learned about a fortune-telling system that relies on a mathematical operation reminiscent of error checks on contemporary computer systems.
In traditional Bamana fortune-telling, a divination priest begins by rapidly drawing four dashed lines in the sand. The priest then connects the dashes into pairs. For lines containing an odd number of dashes and a single leftover, he draws one stroke in the sand. For lines with even-paired dashes, he draws two strokes. Then he repeats the entire process. The mathematical operation is called addition modulo 2, which simply gives the remainder after division by two. But in this case, the two "words" produced by the priest, each consisting of four odd or even strokes, become the input for a new round of addition modulo 2. In other words, it's a pseudo random-number generator, the same thing computers do when they produce random numbers. It's also a numerical feedback loop, just as fractals are generated by a geometric feedback loop. "Here is this absolutely astonishing numerical feedback loop, which is indigenous," said Eglash. "So you can see the concepts of fractal geometry resonate throughout many facets of African culture." Lawrence Shirley, chairman of the mathematics department at Towson (Md.) University, lived in Nigeria for 15 years and taught at Ahmadu Bello University in Zaria, Nigeria. He said he's impressed with Eglash's observations of fractal geometry in Africa.
"It's amazing how he was able to pull things out of the culture and fit them into mathematics developed in the West," Shirley said. "He really did see a lot of interesting new mathematics that others had missed." Eglash said the fractal design themes reveal that traditional African mathematics may be much more complicated than previously thought. Now an assistant professor of science and technology studies at Rensselaer Polytechnic Institute in Troy, Eglash has written about the revelation in a new book, "African Fractals: Modern Computing and Indigenous Design." "We used to think of mathematics as a kind of ladder that you climb," Eglash said. "And we would think of counting systems-one plus one equals two-as the first step and simple shapes as the second step." Recent mathematical developments like fractal geometry represented the top of the ladder in most western thinking, he said. "But it's much more useful to think about the development of mathematics as a kind of branching structure and that what blossomed very late on European branches might have bloomed much earlier on the limbs of others.
"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." Eglash said educators also need to rethink the way in which disciplines like African studies have tended to skip over mathematics and related areas.
To remedy that oversight, Eglash said he's been working with African-American math teachers in the United States on ways to get minorities more interested in the subject. Eglash has consulted with Gloria Gilmer, a well-respected African-American mathematics educator who now runs her own company, Math-Tech, Inc., based in Milwaukee. Gilmer suggested that Eglash focus on the geometry of black hairstyles. Eglash had included some fractal models of corn-row hair styles in his book and agreed they presented a good way to connect with contemporary African-American culture. [Patterns in African American Hairstyles by Gloria Gilmer]
Jim Barta, an assistant professor of education at Utah State University in Logan, remembers a recent conference in which Eglash gave a talk on integrating hair braiding techniques into math education. The talk drew so many people the conference organizers worried about fire code regulations.
"What Ron is helping us understand is how mathematics pervades all that we do," said Barta. "Mathematics in and of itself just is, but as different cultures of human beings use it, they impart their cultural identities on it-they make it theirs." Joanna Masingila, president of the North American chapter of the International Study Group on Ethnomathematics, said Eglash's research has shed light on a type of mathematical thinking and creativity that has often been ignored by western concepts of mathematics. "It's challenging stereotypes on what people think of as advanced versus primitive approaches to solving problems," she said. "Sometimes we're limited by our own ideas of what counts as mathematics." Eglash has now written a program for his Web site that allows students to interactively explore scaling models for a photograph of a corn-row hair style.
Eventually, he'd like to create a CD ROM-based math lab thatcombines his African fractal materials with African-American hair styles and other design elements such as quilts.
One of the benefits of including familiar cultural icons in mathematics education is that it helps combat the notion of biological determinism, Eglash said.
Biological determinism is the theory that our thinking is limited by our racial genetics. This theory gets reinforced every time a parent dismisses a child's poor math scores as nothing more than a continuation of bad math skills in the family, said Eglash. "So for Americans, this myth of biological determinism is a very prevalent myth," he said. "We repeat it even when we don't realize it." Eglash said using the African fractals research to combat the biological determinism myth benefits all students. "On the other hand, there is a lot of interest in how this might fit in with African-American cultural identity," he said."Traditionally, black kids have been told, 'Your heritage is from the land of song and dance.' It might make a difference for them to see that their heritage is also from the land of mathematics."
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This article is very brief, but mentions several tantalizing examples of fractals and recursive similarity in Africa. He gives an example of fractals in the layout of the settlement of Mokoulek in Cameroon. There are apparently also hints of fractal architecture in ancient Egypt. The author tells us that recursive scaling (infinite self-similar structures) is also seen in Ethiopian crosses, Egyptian cosmological icons, and Cameroon bronzeware. The author also tells us that "specific scaling techniques are particularly evident in Ghana, where the use of log spirals to represent self-organizing systems (biological morphogenesis and fluid turbulence is common", and that "binary recursion is used in Bambara sand divination" [in Mali].
Fractals in African Art
Ancient Egyptian cosmogony often used the lotus blossom as an image for the development of the universe. The petals within petals within petals of the lotus represented the cosmos on smaller and smaller scales. On the left are the capitals of an Egyptian temple column. On the right note the stylized lotus petals and the similarity of this representation to the first few stages of a Cantor set.
Mastabas come to mind in early Pyramid construction as a Possible Fractal
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Obenga, I have already cited Ron Eglash's findings in this thread here, as well as another book about African mathetimatics by a mathematician.
But I do appreciate showing us examples of African fractals at work in Egyptian architecture. Another knotch to add to the many on the belt of Egypt's African identity and relation.
Posts: 26238 | From: Atlanta, Georgia, USA | Registered: Feb 2005
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Thanx for the link Djehuti...search didnt come up with anything for me.
For those who read the book did they discuss any other types of relationships to Kmt....Mastabas perhaps. The rectangle within a rectangle seems to be a Fractal to me.
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I haven't read it all, but have had glances at teh book. Great book. Infact, I own it and have been planning to start a thread myself with facts and pictures from the book.
But it will probably be later, and in this thread. ( I'll just use the good 'ole search )
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Wow ... its been a long time Obenga ... I thought you were gone.
Posts: 951 | From: where rules end and freedom begins | Registered: Jun 2004
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Dont post much but I try to pop in to read a few times a week as there is often something useful posted I can add to what I teach my pupils.
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I was browsing a similar topic for information and ran across a couple of links with loads of citations and summaries over viewing the history of mathematics on the African continent [I think Obenga posted a summary and citation from one of them].. Of interest to me, though I've read about it before and they mention it above, is the binary recursion as part of the sand divination of the Bambara, similar to that of the IFA tradition among the Yoruba. Amazingly, Africans have already been using the language of computers well before their actual invention.
I take IT so this is of particular interest to me.. Here's an excerpt from a slightly more recent book by Eglash:
quote: Chapter 7 - My introduction to Cedena, or sand divination, took place in Dakar, Senegal, where the local Islamic culture credits the Bamana (also known as “Bambara”) with a potent pagan mysticism. Almost all diviners had some kind of physical deformity — “the price paid for their power.”One diviner seemed quite willing to teach me about the system, suggesting that it “would be just like school.”The first few sessions went smoothly, with the diviner showing me a symbolic code in which each symbol, represented by a set of four vertical dashed lines drawn in the sand, stood for some archetypical concept (travel, desire, health, etc.) with which he assembled narratives about the future.But when I finally asked how he derived the symbols — in particular the meaning of some patterns drawn prior to the symbol writing — they all laughed at me and shook their heads.“That’s the secret!”My offers of increasingly high payments were met with disinterest.Finally, I tried to explain the social significance of cross-cultural mathematics.I happened to have a copy of Linda Garcia’s Fractal Explorer with me, and began by showing a graph of the Cantor set, explaining its recursive construction.The head diviner, with an expression of excitement, suddenly stopped me, snapped the book shut and said “show him what he wants!”
As it turns out, the recursive construction of the Cantor set was just the right thing to show, because the Bamana divination is also based on recursion. The divination begins with four horizontal dashed lines, drawn rapidly, so that there is some random variation in the number of dashes in each.The dashes are then connected in pairs, such that each of the four lines are left with either one single dash (in the case of an odd number) or no dashes (all pairs, the case of an even number).The narrative symbol is then constructed as a column of four vertical marks, with double vertical lines representing an even number of dashes and single lines representing an odd number of dashes.At this point the system is similar to the famous Ifa divination: there are two possible marks in four positions, so 16 possible symbols.Unlike Ifa, however, the random symbol production is repeated four times rather than two.The difference is quite significant. Each of the Ifa symbol pairs are interpreted as one of256 possible Odu, or verses.The Ifa diviner must memorize the Odu; hence four symbols would be too cumbersome (65,536 possible verses).But the Bamana divination does not require any verse memorization; as we will see, its use of recursion allows for verse self-assembly.
As in the additive sequences we examined, the divination code is generated by an iterative loop in which the output of the operation is used as the input for the next stage.In this case the operation is addition modulo 2 (”mod 2″ for short), which simply gives the remainde after division by two. This is the same even/odd distinction used in the parity bit operation which checks for errors on contemporary computer systems. There is nothing particularly complex about mod 2; in fact I was quite disappointed at first because its reapplication destroyed the potential for a binary placeholder representation in the Bamana divination.Rather than interpret each position in the column as having some meaning (as would our binary number 1011, which means one 1, one 2, zero 4s, and one 8), the diviners reapplied mod 2 to each row of the first two symbols, and each row of the last two symbols.The results were then assembled into two new symbols, and mod 2 was applied again to generate a third symbol.Another four symbols were created by reading the rows of the original four as columns, and mod 2 was again recursively applied to generate another three symbols.
The use of an iterative loop, passing outputs of an operation back as inputs for the next stage, was a shock to me; I was at least as taken aback by the sand symbols as the diviners had been by the Cantor set.It would be naive to claim that this was somehow a leap outside of our cultural barriers and power differences — in fact that’s just the sort of pretension that the last two decades of reflexive anthropology has been dedicated against — but it would also be ethnocentric to rule out those aspects that would be attributed to mathematical collaboration elsewhere in the world:the mutual delight of two recursion fanatics discovering each other.And the appearance of the symbols laid out in two groups of seven — the Rosicrucian’s mystic number — added some numerological icing on the cake.
The following day I found that the presentation had not been complete.There were an additional two symbols that were left out; these were also generated by mod 2 recursion using the two bottom symbols to create a 15th, and using that last symbol with the first symbol to create a 16th (bringing the total depth of recursion to five iterations).The 15th symbol is called “this world,” and the 16th is “the next world,” so there was good reason to separate them from the others.The final part of the system — creating a narrative from the symbols — was still unclear, but I was assured that it could be learned if I carefully followed their instructions.I was to give seven coins to seven lepers, place a kola nut on a pile of sand next to my bed at night, and in the morning bring a white cock, which would have to be sacrificed to compensate for the harmful energy released in the telling of the secret.I followed all the instructions, and the next morning bought a large white cock at the market. They held the chicken over the divination sand, and I was told to eat the bitter kola nut as they marked divination symbols on its feet with an ink pen.A little sand was thrown in its mouth, and then I was told to hold it down as prayers were chanted.There was no action on the part of the diviner; the chicken simply died in my hands.
While still a bit shaken by the chicken’s demise (as well as a respectable buzz from the kola nut), I was told the remaining mystery.Each symbol has a “house” in which it belongs — for example, the position of the 16th symbol is “the next world”– but in any given divination most symbols will not be located in their own house.Thus the 16th symbol generated might be “desire,” so we would have desire in the house of the next world, and so on.Obviously this still leaves room for creative narration on the part of the diviner, but the beauty of the system is that no verses need to be memorized or books consulted; the system creates its own complex variety.
The most elegant part of the method is that it only requires four random drawings; after that the entire symbolic array is quickly self-generated.Self-generated variety is important in modern computing, where it is called pseudorandom number generation (figure 7.8). These algorithms take little memory, but can generate very long lists of what appear to be random numbers, although the list will eventually start over again (this length is called the “period” of the algorithm). Although the Bamana only require an additional 12 symbols to be generated in this fashion, a maximum-length pseudorandom number generator using their initial four symbols will produce 65,535 symbols before it begins to repeat.
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Look at those columns, their mathematical perfection and beauty,imagine these were made,hundreds of years before greec or any other western civilisation,no wonder all other people during that period,credited the egyptians as the begginers of civilisation.
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^ But the mathematical ideas or methods are even much older, dating back millennia perhaps to prehistoric times as many other peoples in Africa have mathematica similar schemes that go back to times immemorial.
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I wonder if anything of signifigance was written down in Mali and Sankore University....it would be interesting to see if Africans of Timbuctou had developed new Mathmatical and Philosophical procedures independently and was lost after the decline of Songhai dynasty.
This blows a hole in the whole Western "IQ" test...if African are able to achieve a form of Mathmatics that Europeans and the West could'nt even understand until the development of computers...this shows that people develop their own understanding and approach to mathmatics, science...ect.
Yet, for some reason I doubt this will gather much attention.
Posts: 8804 | From: The fear of his majesty had entered their hearts, they were powerless | Registered: Nov 2007
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My mother always told me that Africans were in Africa using math science..ect. At first I did'nt believe her...I was young and was under the impression Africans were tribal until Euros came and the Egyptians were "Black" but did'nt count and besides they only erected the pyramids...RIGHT. It was'nt until I caught a show on History Channel that very briefly spoke about Africans using Medicine and Mathmatics...I began to research it and came upon some very interesting info....One site I wish I could have saved talked about the misconception the Western World(I.E Whites) have the idea that Western civilization is a European invention founded on European ONLY(WHITE) developments...while the world is ass backward...This site produced by a whiteman...told of developments from various cutures and descibed African doing Surgery and other Medical Practices along with conributions from India, Americas, Asia..ect.
The more I research the more I learn...and I just learned something new....
What I would love is to read some of the Manuscripts from Mali and Timbuctou...the should seriously make some books of reprinted Material for Western readers...
Posts: 8804 | From: The fear of his majesty had entered their hearts, they were powerless | Registered: Nov 2007
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^ Jari, it appears you have alot to learn. Africans being "tribal"?? Tribes are a social-political system not unique to Africans but other peoples around the world including some rural Europeans today. Also, virtually all of the African groups you hear about are NOT tribes but ethnicities.
Posts: 26238 | From: Atlanta, Georgia, USA | Registered: Feb 2005
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"The age of the sediment in which they were found suggests that the six ceramic fragments discovered between 2002 and 2005 are at least 11,400 years old. Most ancient ceramics from the Middle East and the central and eastern Sahara regions are 10,000 and between 9-10,000 years old, respectively."
"In fact, only in Africa do you find such a range of practices in the process of direct reduction [a method in which metal is obtained in a single operation without smelting],and metal workers who were so inventive that they could extract iron in furnaces made out of the trunks of banana trees," says Hamady Bocoum, one of the authors.
- The Origins of Iron Metallurgy in Africa, 2002; "Iron Roads in Africa" project c/o UNESCO]
I got the citations from Myra and from Super's compilation though (look through Mystery's started threads to find the 'African Chronology' thread). Those above threads are good reading for Jari nonetheless..
Posts: 5555 | From: Tha 5th Dimension. | Registered: Apr 2006
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quote:Originally posted by Djehuti: ^ Jari, it appears you have alot to learn. Africans being "tribal"?? Tribes are a social-political system not unique to Africans but other peoples around the world including some rural Europeans today. Also, virtually all of the African groups you hear about are NOT tribes but ethnicities.
I do have alot to learn but I USED to think like that...I now know Africa was full of civilization, higher learning, cultural organization..etc. however I I thought like that Imagine how MANY Americans blacks included view Black Africans like that.
Posts: 8804 | From: The fear of his majesty had entered their hearts, they were powerless | Registered: Nov 2007
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Ohh..I had just missed this..would have continued the discussion here..Oh well!!.. Posts: 6546 | From: japan | Registered: Feb 2009
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Great Topic!! Have a look at these vids. African Origins of Maths by Dr Ron Eglash at the TED Conference. (the same Dr Eglash that Djehuti mentioned).
OT.. I haven't posted for a long while but i lurk and take as much as i can from this fantastic forum So I want to say A BIG WUDUP to Djehuti, alTakruri, Rasol, Supercar, Brada, Mindovermatter, Whatbox, TheExplorer and all the vets who are holding down the fort n keepin 'em in check.
P.S. if Truthcentric is reading, I urge you not to let them win by driving you away.
Sorry for the off topic ramblings, i shall now return to my lurkings but check out that video, i found it very interesting.
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Hi RedShift! indeed it a great topic I also got a similar topic on the other side of E/S...while the biology of the ancient Kemetians and other Africans is important..it gets over-whelming coverage sometmes at the cost of their accomplishments..I hope level headed E/S members will shift the discussions from biology and more to technology,philosophy and historiography. Go here for a similar topic. http://www.egyptsearch.com/forums/ultimatebb.cgi?ubb=get_topic;f=15;t=002594African Fractals In Minoan Civilization.Posts: 6546 | From: japan | Registered: Feb 2009
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