Substitution system

by dividing a square into 9 subsquares
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Wolfgang Hitzl
31.12.2018
In[]:=

Definition how substitution is done

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these definitions can easily be modified: simply try other matrices with values 1,2 or 3.

​
f:=ReplaceAll#,1->
2
2
2
2
2
2
2
2
2
,2
2
1
2
1
2
1
2
1
2
,3
1
3
1
3
1
3
1
3
1
&
​
​

the dimensions of the matrices can also be reduced, e.g.

(*f:=ReplaceAll#,1->
2
2
2
2
2
1
1
2
2
1
1
2
2
2
2
2
,2
2
2
2
2
3
3
2
2
3
2
3
2
2
3
3
2
,3
2
3
3
2
3
1
1
3
3
1
1
3
2
3
3
2
&*)
In[]:=
Flatten2D[list_]:=Apply[Join,Map[MapThread[Join,#]&,list]];​​system[n_,initial_]:=Nest[Flatten2D,Nest[f[#]&,initial,n],n-1];​​Illustration[n_,initial_]:=ArrayPlot[system[n,initial],ImageSizeMedium,ColorRules{1Lighter[Gray,0.5],2Darker[ColorData["Crayola"]["Mauvelous"],0.9],3Lighter[ColorData["Crayola"]["Maroon"],0.1]}];
In[]:=
Manipulate[Illustration[n,initial],{{initial,1,Style["Initial state",10,Bold]},{1,2,3},ControlTypeSetterBar,AppearanceTiny},{{n,3,Style["Iteration no.",10,Bold]},{1,2,3,4,5,6},ControlTypeSetterBar,AppearanceTiny}]
Out[]=
​
Initial state
1
2
3
Iteration no.
1
2
3
4
5
6
Illustration[5,1]