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Substitution Systems
More Explorations into Simple Programs
Christian JacobMarch 2003
Substitution Systems versus Cellular AutomataCellular automata work on a fixed array of cells, whether in one, two, three, or N dimensions.
This means, while the colours or states of the cells can be updated following a wide range of different possible rules, the underlying number and organization of cells always stays the same.
In substitution systems, on the other hand, the number of elements can change.
Substitution SystemsLet us have a look at some variants of substitution systems and their evolutionary dynamics.
Typical Simple Substitution System
In the typical case, we have a sequence of elements—each coloured say black or white—and at each step each one of these elements is replaced by a new block of elements.
‡ Implementing a Simple Substitution System
So, let's start to implement the following rules:
ruleSet = 81 Ø 81, 0<, 0 Ø 80, 1<<;
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We apply these rules to a "1" as the initial element, and apply the rule set several times:
1 ê. ruleSet
81, 0<% ê. ruleSet
881, 0<, 80, 1<<% ê. ruleSet
8881, 0<, 80, 1<<, 880, 1<, 81, 0<<<If we want to get rid of the inner brackets, we can flatten the expression:
Flatten@%D81, 0, 0, 1, 0, 1, 1, 0<
‡ Automization of the Step-by-Step Rule Application
We can automize this step-by-step application of rules by using the NestList function. What does NestList do?
NestList@f, a, 3D8a, f@aD, f@f@aDD, f@f@f@aDDD<
Now we could define f to perform the rule applications defined in ruleSet:
f@x_D := x ê. ruleSet
NestList@f, 81<, 3D881<, 881, 0<<, 8881, 0<, 80, 1<<<, 88881, 0<, 80, 1<<, 880, 1<, 81, 0<<<<<
2 SubstSystems-00-Print.nb
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% êê ColumnForm
81<881, 0<<8881, 0<, 80, 1<<<88881, 0<, 80, 1<<, 880, 1<, 81, 0<<<<‡ Flattening the Lists
With an additional Flatten command, we can get rid of all the inner brackets. The following definition for f overwrites the previous one:
f@x_D := Flatten@x ê. ruleSetDNestList@f, 81<, 3D881<, 81, 0<, 81, 0, 0, 1<, 81, 0, 0, 1, 0, 1, 1, 0<<% êê ColumnForm
81<81, 0<81, 0, 0, 1<81, 0, 0, 1, 0, 1, 1, 0<‡ Generalization to Automatic Substitution
This function f is, of course, tied to the definition of ruleSet, as we have assigned it above:
ruleSet
81 Ø 81, 0<, 0 Ø 80, 1<<In order to write a more general evolution function for a substitution system, we can proceed as follows. We implement a function that takes three arguments:
(1) the rule set (list of substitutions)
(2) an initial expression
(3) the number of iterations (how many times to apply the rules)
SubstSystems-00-Print.nb 3
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Here is such a function, which we call SSEvolveList:
SSEvolveList@rule_, init_List, t_IntegerD := NestList@Flatten@# ê. ruleD &, init, tD‡ Example 1: Checking out the New Substitution System Function
Let us check, whether it works:
SSEvolveList@81 Ø 81, 0<, 0 Ø 80, 1<<, 81<, 8D881<, 81, 0<, 81, 0, 0, 1<, 81, 0, 0, 1, 0, 1, 1, 0<,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1<, 81, 0, 0, 1, 0, 1, 1, 0,0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0<,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0,0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0,1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1<,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1,0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0,1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0,1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1,0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0<,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1,0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0,1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0,1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1,0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0,1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1,0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1,0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0,0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0,1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1<<
% êê ColumnForm
81<81, 0<81, 0, 0, 1<81, 0, 0, 1, 0, 1, 1, 0<81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1<81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0,81, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0,
The visualization of these evolving structures will need some work. Here is one way to visualize it.
4 SubstSystems-00-Print.nb
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showRectangleEvolution@%%D;
‡ Example 2
For now, let's have a look at another simple substitution system, this time one that does not grow as fast:
SSEvolveList@81 Ø 0, 0 Ø 80, 1<<, 81<, 8D881<, 80<, 80, 1<, 80, 1, 0<, 80, 1, 0, 0, 1<,80, 1, 0, 0, 1, 0, 1, 0<, 80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1<,80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0<, 80, 1, 0, 0, 1, 0,1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1<<
SubstSystems-00-Print.nb 5
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% êê ColumnForm
81<80<80, 1<80, 1, 0<80, 1, 0, 0, 1<80, 1, 0, 0, 1, 0, 1, 0<80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1<80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0<80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0,
showRectangleEvolution@%%D;
In these simple cases, we are dealing with contextfree rules. That is, each element of a particular colour is replaced by a fixed block of new elements, independent of the colours of the neighbouring elements.
6 SubstSystems-00-Print.nb
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‡ Example 3
example@3D = SSEvolveList@81 Ø 81, 0, 1<, 0 Ø 80, 0, 0<<, 81<, 5D881<, 81, 0, 1<, 81, 0, 1, 0, 0, 0, 1, 0, 1<,81, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1<,81, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1<<
% êê ColumnForm
81<81, 0, 1<81, 0, 1, 0, 0, 0, 1, 0, 1<81, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1<81, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 4
NKS, p. 84 (a)
SubstSystems-00-Print.nb 7
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example@4D = SSEvolveList@81 Ø 80, 0<, 0 Ø 80, 1<<, 81<, 7D881<, 80, 0<, 80, 1, 0, 1<, 80, 1, 0, 0, 0, 1, 0, 0<,80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1<, 80, 1, 0, 0, 0, 1, 0, 1,0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0<,80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1,0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0,0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1<,80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0,1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0,0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0,1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0,1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0<<
% êê ColumnForm
81<80, 0<80, 1, 0, 1<80, 1, 0, 0, 0, 1, 0, 0<80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1<80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0,80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0,80, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 5
NKS, p. 84 (b)
8 SubstSystems-00-Print.nb
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example@5D = SSEvolveList@81 Ø 0, 0 Ø 81, 0, 1<<, 81<, 7D881<, 80<, 81, 0, 1<, 80, 1, 0, 1, 0<, 81, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1<,80, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0<,81, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1<,80, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0<<
% êê ColumnForm
81<80<81, 0, 1<80, 1, 0, 1, 0<81, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1<80, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0<81, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1,80, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 6
NKS, p. 84 (c)
SubstSystems-00-Print.nb 9
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example@6D = SSEvolveList@81 Ø 0, 0 Ø 80, 1, 1<<, 81<, 7D881<, 80<, 80, 1, 1<, 80, 1, 1, 0, 0<, 80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1<,80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0<,80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0,0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1<,80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0,1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1,0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0,0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0<<
% êê ColumnForm
81<80<80, 1, 1<80, 1, 1, 0, 0<80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1<80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0<80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1,80, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 7
NKS, p. 84 (d)
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example@7D = SSEvolveList@81 Ø 0, 0 Ø 80, 1, 0<<, 81<, 7D881<, 80<, 80, 1, 0<, 80, 1, 0, 0, 0, 1, 0<,80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0<,80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0,0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0<,80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0,1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1,0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0,0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0<,80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1,0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1,0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0,1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0,1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0,1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0,0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1,0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0,1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0<<
% êê ColumnForm
81<80<80, 1, 0<80, 1, 0, 0, 0, 1, 0<80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0<80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0,80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0,80, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0,
showRectangleEvolution@%%D;showRectangleEvolutionStrech@%%%D;
‡ Example 8
NKS, p. 84 (e)
SubstSystems-00-Print.nb 11
Page 12
example@8D = SSEvolveList@81 Ø 80, 0, 0<, 0 Ø 80, 1<<, 81<, 5D881<, 80, 0, 0<, 80, 1, 0, 1, 0, 1<,80, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0<, 80, 1, 0, 0, 0, 0, 1, 0, 1,0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1<,80, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0,0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0,0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0<<
% êê ColumnForm
81<80, 0, 0<80, 1, 0, 1, 0, 1<80, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0<80, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1,80, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 9
NKS, p. 84 (f)
12 SubstSystems-00-Print.nb
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example@9D = SSEvolveList@81 Ø 80, 1, 0<, 0 Ø 80, 1<<, 81<, 5D881<, 80, 1, 0<, 80, 1, 0, 1, 0, 0, 1<,80, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0<,80, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0,1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1<,80, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0,1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1,0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0,0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0<<
% êê ColumnForm
81<80, 1, 0<80, 1, 0, 1, 0, 0, 1<80, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0<80, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0,80, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 10
NKS, p. 84 (g)
SubstSystems-00-Print.nb 13
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example@10D = SSEvolveList@81 Ø 81, 0, 0<, 0 Ø 80, 1<<, 81<, 5D881<, 81, 0, 0<, 81, 0, 0, 0, 1, 0, 1<,81, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0<,81, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1,0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1<,81, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0,0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0,1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0,1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0<<
% êê ColumnForm
81<81, 0, 0<81, 0, 0, 0, 1, 0, 1<81, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0<81, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1,81, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 11
NKS, p. 84 (h)
14 SubstSystems-00-Print.nb
Page 15
example@11D = SSEvolveList@81 Ø 81, 0, 1<, 0 Ø 80, 0<<, 81<, 5D881<, 81, 0, 1<, 81, 0, 1, 0, 0, 1, 0, 1<,81, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1<,81, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0,0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1<,81, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1,0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1<<
% êê ColumnForm
81<81, 0, 1<81, 0, 1, 0, 0, 1, 0, 1<81, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1<81, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,81, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
‡ Example 12
NKS, p. 84 (i)
SubstSystems-00-Print.nb 15
Page 16
example@12D = SSEvolveList@81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<<, 81<, 5D881<, 81, 0, 0<, 81, 0, 0, 1, 0, 1, 1, 0, 1<,81, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0<,81, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1,0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1,0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1<,81, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1,1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0<<
% êê ColumnForm
81<81, 0, 0<81, 0, 0, 1, 0, 1, 1, 0, 1<81, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0<81, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0,81, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0,
showRectangleEvolution@%%D;showRectangleEvolutionStrech@%%%D;
‡ Example 13
NKS, p. 84 (i)
16 SubstSystems-00-Print.nb
Page 17
example@13D = SSEvolveList@81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<, 81<, 5D881<, 80, 0, 0<, 81, 0, 1, 1, 0, 1, 1, 0, 1<,80, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0<,81, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1<,80, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1,0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1,0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0,1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1,0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0<<
% êê ColumnForm
81<80, 0, 0<81, 0, 1, 1, 0, 1, 1, 0, 1<80, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0<81, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0,80, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0,
showRectangleEvolution@%%D;showRectangleEvolutionStrech@%%%D;
SubstSystems-00-Print.nb 17
Page 18
Summary of the Simple Substitution System Examples
‡ All the discussed rule systems
ruleSystems = 881 Ø 80, 0<, 0 Ø 80, 1<<, H* Example 4 *L81 Ø 0, 0 Ø 81, 0, 1<<, H* Example 5 *L81 Ø 0, 0 Ø 80, 1, 1<<, H* Example 6 *L81 Ø 0, 0 Ø 80, 1, 0<<, H* Example 7 *L81 Ø 80, 0, 0<, 0 Ø 80, 1<<, H* Example 8 *L81 Ø 80, 1, 0<, 0 Ø 80, 1<<, H* Example 9 *L81 Ø 81, 0, 0<, 0 Ø 80, 1<<, H* Example 10 *L81 Ø 81, 0, 1<, 0 Ø 80, 0<<, H* Example 11 *L81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<<, H* Example 12 *L81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<< H* Example 13 *L<;‡ Visualize for 3 Iteration Steps
Map@showRectangleEvolutionStrechWithLabel@SSEvolveList@#, 81<, 3D, #D &, ruleSystemsD;81 Ø 80, 0<, 0 Ø 80, 1<<
18 SubstSystems-00-Print.nb
Page 19
81 Ø 0, 0 Ø 81, 0, 1<<SubstSystems-00-Print.nb 19
Page 20
81 Ø 0, 0 Ø 80, 1, 1<<
81 Ø 0, 0 Ø 80, 1, 0<<
20 SubstSystems-00-Print.nb
Page 21
81 Ø 80, 0, 0<, 0 Ø 80, 1<<
81 Ø 80, 1, 0<, 0 Ø 80, 1<<
81 Ø 81, 0, 0<, 0 Ø 80, 1<<
81 Ø 81, 0, 1<, 0 Ø 80, 0<<
81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<<81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<
SubstSystems-00-Print.nb 21
Page 22
Show@GraphicsArray@Partition@%, 2DDD;
81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<< 81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<
81 Ø 81, 0, 0<, 0 Ø 80, 1<< 81 Ø 81, 0, 1<, 0 Ø 80, 0<<
81 Ø 80, 0, 0<, 0 Ø 80, 1<< 81 Ø 80, 1, 0<, 0 Ø 80, 1<<
81 Ø 0, 0 Ø 80, 1, 1<< 81 Ø 0, 0 Ø 80, 1, 0<<
81 Ø 80, 0<, 0 Ø 80, 1<< 81 Ø 0, 0 Ø 81, 0, 1<<
22 SubstSystems-00-Print.nb
Page 23
‡ Visualize for 4 Iteration Steps
Map@showRectangleEvolutionStrechWithLabel@SSEvolveList@#, 81<, 4D, #D &, ruleSystemsD;81 Ø 80, 0<, 0 Ø 80, 1<<
81 Ø 0, 0 Ø 81, 0, 1<<
SubstSystems-00-Print.nb 23
Page 24
81 Ø 0, 0 Ø 80, 1, 1<<
81 Ø 0, 0 Ø 80, 1, 0<<
81 Ø 80, 0, 0<, 0 Ø 80, 1<<81 Ø 80, 1, 0<, 0 Ø 80, 1<<81 Ø 81, 0, 0<, 0 Ø 80, 1<<81 Ø 81, 0, 1<, 0 Ø 80, 0<<
81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<<
24 SubstSystems-00-Print.nb
Page 25
81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<Show@GraphicsArray@Partition@%, 2DDD;
81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<< 81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<81 Ø 81, 0, 0<, 0 Ø 80, 1<< 81 Ø 81, 0, 1<, 0 Ø 80, 0<<81 Ø 80, 0, 0<, 0 Ø 80, 1<< 81 Ø 80, 1, 0<, 0 Ø 80, 1<<
81 Ø 0, 0 Ø 80, 1, 1<< 81 Ø 0, 0 Ø 80, 1, 0<<81 Ø 80, 0<, 0 Ø 80, 1<< 81 Ø 0, 0 Ø 81, 0, 1<<
‡ Visualize for 5 Iteration Steps
Map@showRectangleEvolutionStrechWithLabel@SSEvolveList@#, 81<, 5D, #D &, ruleSystemsD;81 Ø 80, 0<, 0 Ø 80, 1<<
SubstSystems-00-Print.nb 25
Page 26
81 Ø 0, 0 Ø 81, 0, 1<<
81 Ø 0, 0 Ø 80, 1, 1<<
81 Ø 0, 0 Ø 80, 1, 0<<81 Ø 80, 0, 0<, 0 Ø 80, 1<<81 Ø 80, 1, 0<, 0 Ø 80, 1<<81 Ø 81, 0, 0<, 0 Ø 80, 1<<81 Ø 81, 0, 1<, 0 Ø 80, 0<<81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<<81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<
26 SubstSystems-00-Print.nb
Page 27
Show@GraphicsArray@Partition@%, 2DDD;
81 Ø 81, 0, 0<, 0 Ø 81, 0, 1<< 81 Ø 80, 0, 0<, 0 Ø 81, 0, 1<<81 Ø 81, 0, 0<, 0 Ø 80, 1<< 81 Ø 81, 0, 1<, 0 Ø 80, 0<<81 Ø 80, 0, 0<, 0 Ø 80, 1<< 81 Ø 80, 1, 0<, 0 Ø 80, 1<<
81 Ø 0, 0 Ø 80, 1, 1<< 81 Ø 0, 0 Ø 80, 1, 0<<81 Ø 80, 0<, 0 Ø 80, 1<< 81 Ø 0, 0 Ø 81, 0, 1<<
More Complicated Substitution SystemsIn order to get more complicated behaviour than simple nesting, we must consider substitution systems whose rules depend not only on the colour of a single element.
We now investigate context-sensitive substitution systems, whose rules depend on the colour of an element and on the colour of at least one of its neighbours.
Here are two simple examples.
Implementation
For 1-context-sensitive rules we have to slightly modify our evolution function.
SS2EvolveList@rule_, init_List, t_IntegerD :=NestList@Flatten@Partition@#, 2, 1D ê. ruleD &, init, tD
Example 1
NKS, p. 85 (a)
SubstSystems-00-Print.nb 27
Page 28
SS2EvolveList@881, 1< Ø 80, 1<,81, 0< Ø 81, 0<,80, 1< Ø 0,80, 0< Ø 80, 1<<,80, 1, 1, 0<, 8D880, 1, 1, 0<, 80, 0, 1, 1, 0<, 80, 1, 0, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 0, 1, 1, 0<, 80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1,0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<, 80, 1, 0, 0, 1, 0, 1, 0, 0, 1,0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1,0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1,0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0,1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0,1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<<
% êê ColumnForm
80, 1, 1, 0<80, 0, 1, 1, 0<80, 1, 0, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0,80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0,80, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0,
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
28 SubstSystems-00-Print.nb
Page 29
Example 2
NKS, p. 85 (b)
SS2EvolveList@881, 1< Ø 80, 0<,81, 0< Ø 81, 1<,80, 1< Ø 1,80, 0< Ø 0<,80, 1, 1, 0<, 12D880, 1, 1, 0<, 81, 0, 0, 1, 1<, 81, 1, 0, 1, 0, 0<, 80, 0, 1, 1, 1, 1, 1, 0<,80, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1<, 81, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0<,80, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0<,80, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1<,80, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0<,80, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0,0, 1, 1, 1, 0<, 81, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0,1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1<,80, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1,1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0<,80, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1,1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1,0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0,1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0<<
% êê ColumnForm
80, 1, 1, 0<81, 0, 0, 1, 1<81, 1, 0, 1, 0, 0<80, 0, 1, 1, 1, 1, 1, 0<80, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1<81, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0<80, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0<80, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1<80, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0<80, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0,81, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1,80, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1,80, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0,
SubstSystems-00-Print.nb 29
Page 30
showRectangleEvolution@%%D;
showRectangleEvolutionStrech@%%%D;
Example 3: Creation and Destruction
NKS, p. 86
The following rule system allows both for creation and destruction of elements.
SS2EvolveList@881, 1< Ø 81, 1<,81, 0< Ø 80<,80, 1< Ø 81, 0<,80, 0< Ø 8<<,80, 1, 1, 0<, 12D880, 1, 1, 0<, 81, 0, 1, 1, 0<, 80, 1, 0, 1, 1, 0<, 81, 0, 0, 1, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 1, 1, 0<, 81, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<, 81, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<,81, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<,81, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<,80, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<<
30 SubstSystems-00-Print.nb
Page 31
% êê ColumnForm
80, 1, 1, 0<81, 0, 1, 1, 0<80, 1, 0, 1, 1, 0<81, 0, 0, 1, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 1, 1, 0<81, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<81, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<81, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<81, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<80, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0<showRectangleEvolution@%%D;
SubstSystems-00-Print.nb 31
Page 32
showRectangleEvolutionStrech@%%%D;
Example 4: Creation and Destruction (3 and 4 possible colours)
NKS, p. 87
Here are some examples of substitution systems that have three or four possible colours for each element. The rules shown lead to slow growth in the total number of elements.
32 SubstSystems-00-Print.nb
Page 33
‡ Growth (a)
ssEvolve = SS2EvolveList@881, 1< Ø 80<,81, .5< Ø 80<,81, 0< Ø 81<,8.5, 1< Ø 80, 0<,8.5, .5< Ø 80, .5<,8.5, 0< Ø 8.5, .5<,80, 1< Ø 81<,80, .5< Ø 81<,80, 0< Ø 80<<,80, .5, .5, 0<, 100D;showRectangleEvolution@ssEvolveD;
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showRectangleEvolutionStrech@ssEvolveD;
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‡ Growth (b)
ssEvolve = SS2EvolveList@881, 1< Ø 81<,81, .5< Ø 80, .5<,81, 0< Ø 80<,8.5, 1< Ø 8<,8.5, .5< Ø 8<,8.5, 0< Ø 81<,80, 1< Ø 80<,80, .5< Ø 80, .5<,80, 0< Ø 81<<,80, .5, 1, .5<, 100D;
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showRectangleEvolution@ssEvolveD;36 SubstSystems-00-Print.nb
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showRectangleEvolutionStrech@ssEvolveD;
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‡ Growth (c)
ssEvolve = SS2EvolveList@881, 1< Ø 8.5<,81, .5< Ø 8<,81, 0< Ø 80, .5<,8.5, 1< Ø 81, .5<,8.5, .5< Ø 80, 1<,8.5, 0< Ø 81, 1<,80, 1< Ø 8<,80, .5< Ø 8.5, 1<,80, 0< Ø 80<<,80, .5, .5, 0<, 100D;showRectangleEvolution@ssEvolveD;
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showRectangleEvolutionStrech@ssEvolveD;
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‡ Growth (d)
ssEvolve = SS2EvolveList@881, 1< Ø 81, 0<,81, .5< Ø 8.5, .5<,81, 0< Ø 81, 0<,8.5, 1< Ø 81<,8.5, .5< Ø 8.5<,8.5, 0< Ø 8<,80, 1< Ø 80<,80, .5< Ø 80<,80, 0< Ø 81, .5<<,80, .5, 1, 0<, 100D;showRectangleEvolution@ssEvolveD;
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showRectangleEvolutionStrech@ssEvolveD;
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‡ Growth (e)
4 colours
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ssEvolve = SS2EvolveList@881, 1< Ø 8.3, .6<,81, .6< Ø 8.6, 1<,81, .3< Ø 80<,81, 0< Ø 8.3, 0<,8.6, 1< Ø 8<,8.6, .6< Ø 8.6<,8.6, .3< Ø 8.3, 1<,8.6, 0< Ø 8<,8.3, 1< Ø 8.6, 0<,8.3, .6< Ø 8<,8.3, .3< Ø 8.3<,8.3, 0< Ø 81, 0<,80, 1< Ø 8<,80, .6< Ø 8.6, 0<,80, .3< Ø 81, 1<,80, 0< Ø 8.6, .6<<,80, .3, 0, 0<, 100D;showRectangleEvolution@ssEvolveD;
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showRectangleEvolutionStrech@ssEvolveD;
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‡ Growth (f)
4 colours
ssEvolve = SS2EvolveList@881, 1< Ø 8.6, 1<,81, .6< Ø 80, 1<,81, .3< Ø 8.3<,81, 0< Ø 8<,8.6, 1< Ø 80, .6<,8.6, .6< Ø 8<,8.6, .3< Ø 80, .3<,8.6, 0< Ø 81<,8.3, 1< Ø 80, 1<,8.3, .6< Ø 80<,8.3, .3< Ø 8.6, .3<,8.3, 0< Ø 8.6, .6<,80, 1< Ø 8<,80, .6< Ø 81<,80, .3< Ø 8.3, 0<,80, 0< Ø 8.3, 1<<,80, .3, 0, 0<, 100D;showRectangleEvolution@ssEvolveD;
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showRectangleEvolutionStrech@ssEvolveD;
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Code
ReferenceS. Wolfram, A New Kind of Science, Wolfram Media, 2002.
SubstSystems-00-Print.nb 53
