Function Repository Resource:

SubstitutionSystemPlot

Visualize the evolution of a one-dimensional neighbor-independent substitution system

Contributed by: Stephen Wolfram and Jan Mangaldan

ResourceFunction["SubstitutionSystemPlot"][rule,init,t]

visualizes the evolution of the list substitution system with the specified rule from initial condition init for t steps.

ResourceFunction["SubstitutionSystemPlot"][rule,init,{t₁,t₂}]

visualizes the evolution from steps t₁ through t₂.

Details and Options

In ResourceFunction["SubstitutionSystemPlot"][rule,…], rule must be of the form {i₁→rhs₁,i₂→rhs₂,…}, where the rhs_i can be lists or strings of any length.

ResourceFunction["SubstitutionSystemPlot"] has the same options as Graphics, with the following additions:

Alignment

Center

alignment of the boxes used to visualize the steps

overall appearance

rules for determining colors from values

Mesh

Automatic

whether to draw a mesh

MeshStyle

Automatic

the style to use for a mesh

Offset

Automatic

offsets for the boxes used to visualize the steps

Possible settings for the Appearance option include "Array", "Boxes" and "Tree".

Examples

Basic Examples (6)

Show five steps of a string substitution system:

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Show as an array:

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Show only the second to fourth steps:

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Show the second to fourth steps as an array:

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Show only the first step:

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Show the first step as an array:

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Scope (5)

Use any symbol:

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$ResourceFunction[ "SubstitutionSystemPlot"][{"\[FilledSquare]" -> {"\[FilledCircle]", "\[FilledSquare]"}, "\[FilledCircle]" -> {"\[FilledSquare]", "\[FilledCircle]"}}, {"\[FilledSquare]"}, 5]$

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$ResourceFunction[ "SubstitutionSystemPlot"][{"\[FilledSquare]" -> {"\[FilledCircle]", "\[FilledSquare]"}, "\[FilledCircle]" -> {"\[FilledSquare]", "\[FilledCircle]"}}, {"\[FilledSquare]"}, 5, Appearance -> "Array"]$

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Lists do not have to be the same length:

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The initial condition can be of any length:

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Rules can involve patterns:

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$ResourceFunction["SubstitutionSystemPlot"][{n_ :> Range[n]}, {3}, 3]$

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$ResourceFunction["SubstitutionSystemPlot"][{n_ :> Range[n]}, {3}, 3, Appearance -> "Array"]$

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Use a string substitution system:

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$ResourceFunction[ "SubstitutionSystemPlot"][{"\[EmptyCircle]" -> "\[EmptyCircle]\[FilledSquare]", "\[FilledSquare]" -> "\[FilledSquare]\[EmptyCircle]"}, "\[FilledSquare]", 3]$

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$ResourceFunction[ "SubstitutionSystemPlot"][{"\[EmptyCircle]" -> "\[EmptyCircle]\[FilledSquare]", "\[FilledSquare]" -> "\[FilledSquare]\[EmptyCircle]"}, "\[FilledSquare]", 3, Appearance -> "Array"]$

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Options (9)

Alignment (1)

Different alignment formats for SubstitutionSystemPlot:

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Appearance (3)

Show as boxes:

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Show as an array:

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Show as a tree:

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ColorRules (2)

Specify color rules for explicit values or patterns:

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Implement a “default color” by adding a rule for _:

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$ResourceFunction[ "SubstitutionSystemPlot"][{0 | 1 -> {1, 2}, 2 -> {0, 1}}, {0}, 3, ColorRules -> {0 -> Blue, _ -> Orange}]$

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$ResourceFunction[ "SubstitutionSystemPlot"][{"A" -> "B", "B" -> "BA"}, "BAB", {2, 4}, Appearance -> "Array", ColorRules -> {"A" -> RGBColor[1., 0.843137, 0.6], _ -> RGBColor[ 0.711529, 0.181058, 0.]}]$

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$ResourceFunction[ "SubstitutionSystemPlot"][{"B" -> "BAB", "A" -> "AAA"}, {"B"}, 4, Appearance -> "Tree", ColorRules -> {"A" -> RGBColor[1., 0.843137, 0.6], _ -> RGBColor[ 0.711529, 0.181058, 0.]}]$