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Linear and nonlinear systems A vast number of different applications of traditional mathematics are ultimately based on linear equations of the form u  m . v where u and v are vectors (lists) and m is a matrix (list of lists), all containing ordinary continuous numbers.
Note that in the approximation that the colors of all cells in the pattern are assumed completely independent and random there should be motion by 0.25 cells per step.
In all cases, s[n, k] is divisible by n .
Note that rules which yield maximal size networks are in a sense close to allowing all possible sequences.
All sorts of schemes have been invented for getting unbiased output from such systems, and acceptable randomness can often be obtained at kilohertz rates, but obvious correlations almost always appear at higher rates.
Branching in plants Almost all kinds of plants exhibit some form of branching, and particularly in smaller plants the branching is often extremely regular.
Beyond a few thousand cells, however, individual cells seem to be less relevant, and instead what appears to happen is that chemicals such as retinoic acid (a derivative of vitamin A) produced by particular cells diffuse to affect all cells in a region a tenth of a millimeter or so across.
All generalized additive rules ultimately yield nested patterns.
This principle has worked well in physics, where it has often proven to be the case, for example, that out of all possible terms in an equation the only ones that actually occur are the very simplest.
And in fact, intuition from traditional science and mathematics has always tended to suggest that unless one adds all sorts of complications, most systems will never be able to exhibit any very relevant behavior.
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