Chapter 7Numeric systems

So far we have focused on geometric structures rather than numeric systems.The only exception was in the windscreen, where the nonlinear scaling was created by counting a specific sequence of diagonal straw rows.But there are many other instances where the African approach to fractal geometry makes use of numbers.

Nonlinear additive series in Africa

The counting numbers (1,2,3...) can be thought of as a kind of iteration, but only in the most trivial way.^[1] It is true that we could produce the counting numbers from a recursive loop; that is, a function in which the output at one stage becomes the input for the next: . But this is a strictly linear series, increasing by the same amount each time -- the numeric equivalent of what we saw in the linear concentric circle and linear spiral.Addition can, however, produce nonlinear series^[2] , and there are at least two examples of nonlinear additive series in African cultures. The triangular numbers (1,3,6,10,15...) are used in a game called “tarumbeta” in east Africa (Zaslavsky 1973 pp. 111). Figure 7.1 shows how these numbers are derived from the shape of triangles of increasing size, and how the numeric series can be created by a recursive loop. As in the case of certain formal age-grade initiation practices (see chapters 5 and 8), the simple versions are used by smaller children, and the higher iterations picked up with increasing age. While there is no indication of a formal relationship in this instance, there is still an underlying parallel between the iterative concept of aging common to many Africa cultures -- each individual passing through multiple turns of the "life-cycle" -- and the iterative nature of the triangular number series.

Another nonlinear additive series in was found in archaeological evidence from north Africa. Badawy (1965) noted what appears to be use of the Fibonacci series in the layout of the temples of ancient Egypt. Using a slightly different approach, I found a visually distinct example of this series in the successive chambers of the temple of Karnak, as shown in figure 7.2a. Figure 7.2b shows how these numbers can be generated using a recursive loop. This formal scaling plan may have been derived from the non-numeric versions of scaling architecture we see throughout Africa.An ancient set of balance weights, apparently used in Egypt, Syria and Palestine circa 1200 B.C.E., also appear to employ the Fibonacci sequence (Petruso 1985).This is a particularly interesting use, since one of the striking mathematical properties of the sequence is that one can create any positive integer through addition of selected members -- a property that makes it ideal for application to balance measurements (Hoggatt 1969 pp 76). There is no evidence that ancient Greek mathematicians knew of the Fibonacci sequence. There was use of the Fibonacci sequence in Minoan design, but Preziosi (1968) cites evidence indicating that this could have been brought from Egypt by Minoan architectural workers employed at Kahun.

Doubling series in Africa

In some accounts authors have stated that Africans use a “primitive” number system in which they count by multiples of two.It is true that many cases of African arithmetic are based on multiples of two, but as we will see, base two systems are not crude artifacts from a forgotten past. They have surprising mathematical significance, not onlyin relation to African fractals, but to the western history of mathematics and computing as well.

The presence of doubling as a cultural theme occurs in many different African societies, and in many different social domains, connecting the sacredness of twins, spirit doubles, and double vision with material objects, like the blacksmith's twin bellows and the double iron hoe given in bridewealth (figure 7.3). Figure 7.4a shows the Ishango bone, which is dated around 8,000 years old and appears to show a doubling sequence. Doubling is fundamental to many of the counting systems of Africa in modern times as well. It is common, for example, to have the word for an even number 2N mean "N plus N" (e.g. the number 8 in the Shambaa language of Tanzania is “ne na ne,” literally “four and four.”)A similar doubling takes place for the precisely articulated system of number hand gestures (figure 7.4b); for example “four” represented by two groups of two fingers, and “eight” by two groups of four. Petitto (1982) found that doubling was used in multiplication and division techniques in west Africa (figure 7.4c).Gillings (1972) details the persistent use of powers of two in ancient Egyptian mathematics as well, andZaslavsky (1973) shows archaeological evidence suggesting that ancient Egypt’s use of base-two calculations derived from the use of base-two in sub-Saharan Africa.

Doubling practices were also used by African descendants in the Americas. Benjamin Banneker, for example, made unusual use of doubling in his calculations, which may have derived from the teachings of his African father and grandfather (Eglash 1997c). Gates (1988) examined the cultural significance of doubling in west African religions such as vodun, and its transfer to “voodoo” in the Americas. In the religion of Shango, for example, the vodun god of thunder and lightning is represented by a double-bladed axe (figure 7.5a), used by Shango devotees in the new world as well (Thompson 1983). Figure 7.5b shows the use of a doubling sequence in the structure of a Shango temple, and in religious ceremonies (ritual choreography aligning two priests, four children, eight legs). A curator at the Musée Ethnographique in Porto Novo, Benin who specialized in Shango explained to me that these doubling structures were used because the god of lightning required a portrait of the forked structure of a lightning bolt. The model is particularly interesting in that the lengths of each iteration are shortened, so that one could have infinite doublings in a finite space-- a true fractal. The self-similar structure of lightning has been a favorite example for fractal geometry texts (cf. Mandelbrot 1977). The doubling sequence used to model the fractal structure of lightning in Shango would not give an accurate value for the empirical fractal dimension -- real lightning tends to branch much more than doubling allows for -- but it’s enough to know that the vodun representation offers a testable quantitative model.

The most mathematically significant aspect of doubling in African religion occurs in the divination (“fortune-telling”) techniques of vodun and its religious relatives (Eglash 1997b). The famous Ifa divination system (figure 7.6) is based on tossing pairs of flat shells or seeds split in two. Each lands open-side or closed-side (like "heads or tails" in a coin toss). They are connected by a doubled chain to make four pairs.Each group of four pairs gives one of the 16 divination symbols, which tell the future of the diviner’s client. The Ifa system is what a mathematician would call “stochastic,” that is, it operates by pure chance.But a closely related divination system, Cedena, has a non-stochastic element -- it is closer to what mathematicians call “deterministic chaos.”

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." ^[3] 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 (figure 7.7). 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.

A similar system for self-generated variety was developed as a model for the "chaos" of nonlinear dynamics by Marston Morse (1892-1977). Previous to the 1970s, mathematicians had assumed that other than a few esoteric exceptions (the algorithms for producing irrational numbers such as  ), the output of an equation would eventually start repeating. That assumption was partly based on European cultural ideas about free will: complex behavior could not be the result of pre-determined systems (cf. Porter 1986). It was not until the 1960s-70s that mathematicians realized that even simple, common equations describing things like population growth or fluid flow, could result in what they called “deterministic chaos” -- an output that never repeats; giving the appearance of random numbers from a non-random (deterministic) equation.Morse developed the minimal case for such behavior.

The construction of the Morse sequence begins by counting from zero in binary notation: 000, 001, 010, 011....It then takes the sum of the digits in each number -- 0 + 0 + 0 = 0, 0 + 0 + 1 = 1, etc. -- and finally mod 2 of each sum.The result is a sequence with many recursive properties^[4] , but also endless variety.Morse did the same "misreading" of the binary number as did the Bamana -- although he did not have an anthropologist scowling at him for ignoring place-value -- and he did it for the same reason; because combined with the mod 2 operation it maximizes variety.

In my reading of divination literature I eventually came across the duplicate of the Bamana technique 5,000 miles to the east in Malagasy sikidy (Sussman and Sussman 1977), which inspired a study of the history of its diffusion.The strong similarity of both symbolic technique and semantic categories to what Europeans termed geomancy was first noted by Flacourt (1661), but it was not until Trautmann (1939) that a serious claim was made for a common source for this Arabic, European, West African, and East African divination technique.The commonality was confirmed in a detailed formal analysis by Jaulin (1966). But where did it originate?

Skinner (1980) provides a well-documented history of the diffusion evidence, from the first specific written record, a ninth century Jewish commentary by Aran ben Joseph, to its modern use in Aleister Crowley's Liber 777.The oldest Arabic documents (those of az-Zanti in the thirteenth century) claim the origin of geomancy (ilm al-raml, "the science of sand") through the Egyptian god Idris (Hermes Trismegistus), and while we need not take that as anything more than a claim to antiquity, a Nilotic influence is not unreasonable.Budge (1961)attempts to connect the use of sand in ancient Egyptian rituals to African geomancy, but it is hard to see this as unique.Mathematically, however, geomancy is strikingly out of place in non-African systems.

Like other linguistic codes, number bases tend to have an extremely long historical persistence.Even under Platonic rationalism, the ancient Greeks held 10 to be the most sacred of all numbers; the Kabbalah's Ayin Sof emanates by 10 Sefirot; and the Christian west counts on its "Hindu-Arabic" decimal notation.In Africa, on the other hand, base two calculation was ubiquitous, even for multiplication and division. And it is here that we find the cultural connotations of doubling that ground the divination practice in its religious significance.

The implications of this trajectory -- from sub-Saharan Africa, to North Africa, to Europe -- are quite significant for the history of mathematics.Following the introduction of geomancy to Europe by Hugo of Santalla in twelfth century Spain, it was taken up with great interest by the pre-science mystics of those times -- alchemists, hermeticists, and Rosicrucians (figure 7.9). But these European geomancers -- Raymond Lull, Robert Fludd, de Peruchio, Henry de Pisis and others -- persistently replaced the deterministic aspects of the system with chance.By mounting the sixteen figures on a wheel and spinning it, they maintained their society's exclusion of any connections between determinism and unpredictability.The Africans, on the other hand, seem to have emphasized such connections. In chapter 10 we will explore one source of this difference: the African concept of a “trickster” god, one who is both deterministic and unpredictable.

On a video recording I made of the Bamana divination, I later noticed that they had used a shortcut method in some demonstrations (this may have been a parting gift, as the video was shot on my last day).As first taught to me, when they count off the pairs of random dashes, they link them by drawing short curves.The shortcut method then links those curves with larger curves, and those below with even larger curves.This upside-down Cantor set shows that they are not simply applying mod 2 again and again in a mindless fashion.The self-similar physical structure of the shortcut method vividly illustrates a recursive process, and as a non-traditional invention (there is no record of its use elsewhere) it shows active mathematical practice.Other African divination practices can be linked to recursion as well; for example Devisch (1991) describes the Yaka diviners' "self-generative" initiation and uterine symbolism.

Before leaving divination, there is one more important connection to mathematical history. While Raymond Lull did not like the idea that the complexity of life was linked to deterministic generation, he did continue to experiment with geomancy, and used it to develop his “logic machine,” a categorizing system based on iterative binary distinctions. Around 1670 German mathematician Leibnitz took Lull’s category system and applied it to counting, creating the system we now call the binary code. In other words all those ones and zeros, running around in every digital circuit from alarm clocks to super-computers, originate in African divination.

In a 1995 interview in Wired magazine, techno-pop musician Brain Eno claimed that the problem with computers is that “they don’t have enough African in them.”Eno was, no doubt, trying to be complimentary, saying that there is some intuitive quality that is a valuable attribute of African culture.But in doing so he obscured the cultural origins of digital computing, and did an injustice to the very concept he was trying to convey.

Discrete self-organization in Owari

Figure 7.10a shows a board game that is played throughout Africa in many different versions variously termed "ayo," "bao," "giuthi," "lela," "mancala," "omweso," "owari," “tei,” and "songo" (among many other names).Boards cut into stones, some of extreme antiquity, have been found from Zimbabwe to Ethopia (see Zaslavsky 1973 figure 11-6).The game is played by scooping pebble or seed counters from one cup, and sequentially placing one each in the cups that follow.The goal is to have the last counter land in a cup with only one or two counters already in it, which allows the player to capture those counters.In the Ghanaian game of Owari, players are known for utilizing a series of moves they call a“marching group.”They note that if the number of counters in a series of cups each decrease by one (e.g. 4-3-2-1) the entire pattern can be replicated with a right-shift by scooping from the largest cup, and that if left uninterrupted it can propagate in this way as far as needed (figure 7.10b).As simple as it seems, this concept of a self-replicating pattern is at the heart of some sophisticated mathematical concepts.

John von Neumann, who played a pivotal role in the development of the modern digital computer, was also a founder of the mathematical theory of self-organizing systems. Initially von Neumann’s theory was to be based on self-reproducing physical robots. Why work on a theory of self-reproducing machines?I believe the answer can be found in von Neumann's social outlook. Heims' (1984) biography emphasizes how the disorder of von Neumann’s precarious youth as a Hungarian Jew was reflected in his adult efforts to impose a strict mathematical order in various aspects of the world. In von Neumann's application of game theory to social science, for example, Heims writes that his "Hobbesian" assumptions were "conditioned by the harsh political realities of his Hungarian existence."His enthusiasm for the use of nuclear weapons against the Soviet Union is also attributed to this experience.

During the Hixon Symposium (von Neumann 1951) he was asked if computing machines could be built such that they could repair themselves if "damaged in air raids," and replied that "there is no doubt that one can design machines which, under suitable circumstances, will repair themselves."His work on nuclear radiation tolerance for the AEC in 1954-5 included biological effects as well as machine operation. Putting these facts together, I cannot escape the creepy conclusion that von Neumann’s interest in self-reproducing automata originated in fantasies about having a more perfect mechanical progeny survive the nuclear purging of organic life on this planet.

Models for physical robots turned out to be too complex, and at the suggestion of his colleague Stanislaw Ulam, von Neumann settled for a graphic abstraction; “cellular automata” as they came to be called.In this model (figure 7.11a) each square in a grid is said to be either alive or dead (that is, in one of two possible states). The iterative rules for changing the state of any one square are based on the eight nearest neighbors (e.g. if 3 or more nearest-neighbors are full, the cell becomes full in the next iteration). At first researchers carried out on these cellular automata experiments on checkered table cloths with poker chips and dozens of human helpers (Mayer-Kress, personal communication), but by 1970 it had been developed into a simple computer program (Conway's "game of life") which was described by Martin Gardner in his famous "Mathematical Games" column in Scientific American.The "game of life" column was an instant hit, and computer screens all over the world began to pulsate with a bizarre array of patterns (figure 7.11b). As these activities drew increasing professional attention, a wide range of mathematically-oriented scientists began to realize that the spontaneous emergence of self-sustaining patterns created in certain cellular automata were excellent models for the kinds of self-organizing patterns that had been so elusive in studies of fluid flow and biological growth.

Since scaling structures are one of the hallmarks of both fluid turbulence and biological growth, the occurrence of fractal patterns in cellular automata attracted a great deal of interest. But more simple scaling structure, the logarithmic spiral (figure 7.12), has garnered much of the attention. Even back in the 1950s mathematician Alan Turing, whose theory of computation provided von Neumann with the inspiration for the first digital computer, began his research on “biological morphogenesis” with an analysis of logarithmic spirals in growth patterns. Markus (1991) notes that the application areas for cellular automata models of spiral waves include nerve axons, the retina, the surface of fertilized eggs, the cerebral cortex, heart tissue, and aggregating slime molds. In the text for CALAB, the first comprehensive software for experimenting with cellular automata, mathematician Rudy Rucker (1989, pp. 168) refers to systems which produce paired log spirals as “Zhabotinsky CAs,” after the chemist who first observed such self-organizing patterns in artificial media:

When you look at Zhabotinsky CAs, you are seeing very striking three dimensional structures; things like paired vortex sheets in the surface of a river below a dam, the scroll pair stretching all the way down to the river bottom.... In three dimensions, a Zhabotinsky reaction would be like two paired nautilus shells, facing each other with their lips blending. The successive layers of such a growing pattern would build up very like a fetus!

Figure 7.13 shows how the owari marching group system can be used as a one-dimensional cellular automaton to demonstrate many of the dynamic phenomena produced on two-dimensional systems. ^[5] Earlier we noted that the Akan and other Ghanaian societies had a remarkable pre-colonial use of logarithmic spirals in iconic representations for living systems. The Ghanaian four-fold spiral (figure 6.4a) and the four-armed computer graphic in figure 7.12b are quite distant in terms of the technologies that produced produced them, but there may well be some subtle connections between the two.Since cellular automata model the emergence of such patterns in modern scientific studies of living systems, and certain Ghanaian log spiral icons were also intended as generalized models for organic growth, it is not unreasonable to consider the possibility that the self-organizing dynamics observable in owari were also linked to concepts of biological morphogenesis in traditional Ghanaian knowledge systems.

Rattray’s classic volume on the Asante culture of Ghana includes a chapter on owari, but unfortunately it only covers the rules and strategies of the game. Recently Kofi Agudoawu (1991) of Ghana has written a booklet on owari “dedicated to Africans who are engaged in the formidable task of reclaiming their heritage,”and he does note its association with reproduction: “wari” in the Ghanaian language Twi means “he/she marries.” Herskovits (1930), noting that the “awari” game played by the descendants of African slaves in the new world had retained some of the pre-colonial cultural associations from Africa, reports that awari had a distinct “sacred character” to it, particularly involving the carving of the board. Owari boards with carvings of logarithmic spirals (figure 7.14) can be commonly found in Ghana today, suggesting that western scientists may not be the only ones who developed an association between discrete self-organizing patterns and biological reproduction. It is a bit vindictive, but I can’t help enjoying the thought of von Neumann, apostle of a mechanistic New World Order that would wipe out the irrational cacophony of living systems, spinning in his grave every time we watch a cellular automaton -- whether in pixels or owari cups -- bring forth chaos in the games of life.

Conclusion

Both tarumbeta and owari’s marching group dynamics are governed by the triangular numbers. There is nothing special about the triangular number series -- similar nonlinear growth properties can be found in the numbers which form successively larger rectangles, pentagons, or other shapes. Nor is there anything special about the powers of two we found in divination -- similar aperiodic properties can be produced by applications of mod 3, mod 4, etc. What is special is the underlying concept of recursion -- the ways in which a kind of mathematical feedback loop can generate new structures in space and new dynamics in time. In the next chapter, we will see how this underlying process is found in both practical applications and abstract symbolics of African cultures.