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4. Ron Eglash- Recursive Numeric Sequences in Africa

1 ) Nonlinear additive series in African cultures. The counting numbers (1,2,3...) can be thought of as a kind of iteration, but only in the most trivial way. 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: X n + 1 - X n + 1 . But this is a strictly linear series, increasing by the same amount each time -- the numeric equivalent of a staircase. Addition can, however, produce nonlinear series, 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. 11 1 ). Figure 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, 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.
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  4. Ron Eglash - Recursive Numeric Sequences in Africa
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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 2a. Figure 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 (cf. Eglash 1995, Eglash et al 1994).

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 7f). 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.

2) Discrete self organization in Owari

Figure 3a 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 3b). 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 4a) 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 4b). 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 5), 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 6 shows how the owari marching group system can be used as a one-dim~nsional cellular automaton to demonstrate many of the dynamic phenomena produced on two-dimensional systems. The Akan and other Ghanaian societies had a remarkable pre-colonial use of logarithmic spirals in iconic representations for self organizing systems (figure 7a). The Ghanaian spirals and the four-armed computer graphic in figure 5b are quite distant in terms of the machine technologies that produced produced them, but there may well be mathematical 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 7b) 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.


Agudoawu, Kofi. Rules for Playing Oware. Kumasi: KofiTall 1991.
Badaway, A. Ancient Egyptian architectural design: a study of the harmonic system. Berkeley: University of California Press, 1965.
Eglash, R., Diatta, C., Badiane, N. "Fractal structure in jola material culture." Ekistics pp. 367 371, vol 61 no. 368/3fi9, sept-dec 1994. Eglash, R. "Fractal geometry in African material culture." Symmetry: Culture and Science. Vol 6-1, pp 174-177, 1995.
Fagg, W. "The Study of African Art." Bulletin of the Allen Memorial Art Museum, Winter 1955 56, 12, 44-61.
Gies, F., Gies, J. Leonard of Pisa and the New Mathematics of the Middle Ages. NY: Thomas Orowell 1969.
Heims, S.J John von Neumann and Norbert Wiener The MIT Press, Cambridge, 1980.
Herskovits, Melville. "Wari in the new world." paper read at the Americanist Congress, Hamburg 1930.
Hoggatt, V.E. Fibonacci and Lucas Numbers. NY: Houghton Mifflin 1969.
Markus, Mario. "Autonomous organization of a chaotic medium into spirals." pp. 165-186 In Istvan Hargittai and Clifford Pickover (eds) Spiral Syrmmetry, London: World Scientific 1991.
Petruso, K.M. "Additive Progression in Prehistoric Mathematics: A Conjecture." Historia Mathematica 12, 101-106, 1985.
Preziosi, D. Minoan Architectural Design. Mouton 1968.
Rucker, Rudy. CALAB. San Jose: Autodesk 1989.
Von Neumann, John. Collected works. General editor, A. H. Taub. New York, Pergamon Press, 1951.
Zaslavsky, Claudia. Africa Counts. Boston: Prindle, Weber & Schmidt inc. 1973.

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