Appendix 2.4 [re rule 193] ...

...[The]... spectrum of CA behaviour ... rang[es] progressively through ordered, complex and chaotic dynamics, corresponding to Wolfram's ... classes

**Ron Eglash** (in his book "African Fractals" (Rutgers 1999)
and on his web site at www.csdt.rpi.edu) says: "...

... the owari marching-group system can be used as a one-diemnsional cellular automaton ...

... transients of many different lengths can be produced. ... the constant pattern is called a "point attractor", and the transients would be said to lie in the "basin of attraction".

The marching group rule can also produce periodic behavior (a "limit cycle" or "periodic attractor" ...). Here is a period-3 system using only four conters:

211 -> 22 -> 31 -> 211

... The numbers which lead to marching groups - 1 , 3 , 6 , 10 , 15 ... - ...[are]... the triangular numbers ...

[ the triangular numbers correspond to the dimension of the grade-2 bivectors in Clifford Algebras -

- for the case of the 2^8 = 256 Elementary CA Rules, there are 28 grade-2 CA Rules ]

... One-dimensional versions can ... be used as a kind of parallel computer. Consider, for example, a rule that in each iteration the number of counters in a cup is replaced by the sum of itself and its left neighbor. Starting with one:

0100000 -> 0110000 -> 0121000 -> 0133100 -> 0146410

This fourth iteration gives the us the binomial coefficients for expansion of (a+b)^4 ,

which equals to a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 .

[ Such a rule reproduces at each step succeeding rows of the Yang Hui triangle. ]

... If we think of the two-strokes as zero and single stroke as one, the Bamana divination system is almost identical to the process of pseudorandom number generation used by digital circuits called "shift registers". Here the circuit takes mod 2 of the last two bits in the register and places the result in the first position. The other bits are shifted to the right, with the last discarded. ...

1111 0111 0011 0001 1000 0100 0010 1001 1100 0110 1011 0101 1010 1101 1110

Skinner ... "Terrestrial Astrology: Divination by Geomancy". London: Routledge and Kegan Paul, 1980 ... provides a well-documented history of the diffusion evidence ... for ... Arabic, European, West African, and East African ... "geomancy" ...divination technique. ...

[ Such diffusion seems also to have extended

from Africa to India, China, and Japan. ]

... The implications of this trajectory - from sub-Saharan Africa to North Africa to Europe - are quite significant for the history of mathematics.

... a historical path for base-2 calculation ... begins with African divination, runs through the geomancy of European alchemists, and is finally transformed into binary calculation, where it is now applied in every digital circuit ...

... Following the introduction of geomancy to Europe by Hugo of Santalla in twelfth-century Spain ... European geomancers ... Ramon Lull ... and others ... persistently replaced the deterministic aspects of the system with chance. By mounting the 16 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 ...[with]]... a "trickster" god, one who is both deterministic and unpredictable. ...

The fractal settlement patterns

of Africa stand in sharp contrast to the Cartesian grids of Euro-American settlements. ... Euro-American cultures are organized by ... "top-down" organization. Precolonial African cultures included ... societies that are organized "bottom-up" rather than "top-down". ... African architecture tends to be fractal because that is a prominent design theme in African culture ... most of the indigenous African societies were neither utterly anarchic, nor frozen in static order; rather they utilized an adaptive flexibility ... African traditions of decentralized decision making could ... be combined with new information technologies, creating new forms that combine democratic rule with collective informaiton sharing ... what is needed is not ... "small is beautiful", but rather a self-organized approach to changes in the relations between scale and the socioenvironmental systems - not just appropriate technology, but appropriate scaling. ...

we are trapped between the periodic stasis of the preservationists' limit cycle,

and the white noise of the profiteering positive feedback loop.

... both are lacking in flexible interactions with memory;

the ... preservationists' ... limit cycle being too tied to it, and the ... profiteering ... white noise being too free from it. ...".