Posted by sociolingo on March 8th, 2007
With a Malian colleague I visited a sand diviner in his village along the Guinea road some years ago. It was an interesting experience. The man was introduced to me as a ‘marabout’ or muslim holy man. My colleague was visiting him to find some information out about his future as he was worried about government reorganisation. I was invited to video the reading and afterwards the marabout gave me some information about his methodology. He insisted that it was not a religious thing, but that it was a technical skill using mathematics.
I was exploring to find more information about sand divination used by the Bambara in Mali when I found the following excerpt from a book. In this chapter the author describes a system very similar to the one I witnessed.
Excerpt from Chapter 7
Eglash, R. African Fractals: Modem Computing and Indigenous Design (Rutgers University Press, 1999.)
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.
African Fractals: modern computing and indigenous design
Order by phone from Rutgers University Press: 800-446-9323
Fractal geometry has emerged as one of the most exciting frontiers in the fusion between mathematics and information technology. Fractals can be seen in many of the swirling patterns produced by computer graphics, and have become an important new tool for modeling in biology, geology, and other natural sciences. While fractal geometry can take us into the far reaches of high tech science, its patterns are surprisingly common in traditional African designs, and some of its basic concepts are fundamental to African knowledge systems. African Fractals introduces readers to fractal geometry and explores the ways it is expressed in African cultures. Drawing on interviews with African designers, artists, and scientists, Ron Eglash investigates fractals in African architecture, traditional hairstyling, textiles, sculpture, painting, carving, metalwork, religion, games, quantitative techniques, and symbolic systems. He also examines the political and social implications of the existence of African fractal geometry. Both clear and complex, this book makes a unique contribution to the study of mathematics, African culture, anthropology, and aesthetic design. 
