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© Copyright 1993 Mario Hilgemeier

Element development as equation system - an exemplary L-system

Now let's determine where the abundances come from. First, we 
have to see that Conway's audioactive decay can be understood as a 
system of equations. The solution of this system of equations (by 
iteration) gives the abundances.

      Let me clarify this by a simpler string-rewriting system (like 
the GZR). The GZR operates on ciphers, this example uses characters. 
There are five somewhat unusual characters to be iterated: "F", "[", 
"]",  "+" "-". The iteration rule is: "F" is expanded to 
"F[+F][-F]". All other characters remain as they are.

      Consider the pseudocode of the following system of equations. 
It describes the growth of the number (abundance)  for each of the 
five characters from one development stage to the next. One "F" 
generates three new "F"s. The number of left brackets "[" is the sum 
of the number of extisting (old) left brackets plus those (new) 
generated by the expansion of the "F" characters.

Fnew = 3 * Fold
[new = [old + 2*Fold
]new = ]old + 2*Fold
+new =+old + Fold
-new = -old + Fold

Figure (7.)7 This system is visualized in figure 7 (in the style of figure 4). If you iterate this system, what will the abundances of the five characters be? If you don't see it at once, you could do the computation on a computer and see that the abundances of the characters eventually settle to 1/4 for F and each of the brackets, and 1/8 for the + and - characters. Of course, the growth factor is 3 because only F is responsible for the development of the system. This string rewriting system is one of the simpler L-systems [ 2, 3, 11]. When you experiment with pseudocode 1 (see appendix) you will generate fractal images of trees and bushes. Can you find similar interpretations for the strings of the GZR? Equation systems (almost always) have characteristic limit vectors and growth factors. This is exemplified by the above system and also by the GZR. The abundances of the GZR can be computed by pseudocode 2. Now you can see the relation of the exemplary L-system to the audioactive development. For instance, interpret pairs of ciphers of the GZR as drawing instructions. There are eight different pairs: 11, 12, 13, 21, 22, 23, 31 and 32 (33 never occurs, proof is given in [13]). Now interpret these eight pairs as letters of a graphical alphabet analogous to pseudocode 1, possibly with some extensions or redundancies. I've tried some simple interpretations, but they didn't show the (fractal) self-similarity which should be there because the iteration rule of the GZR is pure self-description. Can you find a beautiful interpretation of the GZR? L-systems are used in - Biology: modelling cell division patterns and morphogenesis research - Robotics: structural pattern recognition, speech recognition - Computer Science: semantics of programming languages [2] Beautiful pictures of plants can be found in the book by Lindenmayer and Prusinkiewicz[3]. The complex L-systems described there model the development patterns of plants, their growth and flowering patterns.

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