Periodicity in Cellular Automata

The new function FindTransientRepeat identifies the transient and repeated parts of a list. It can be used to investigate cellular automata.

Look at the rule 30 cellular automaton (CA). The Version 11 function RulePlot is a useful tool to find the associated transformation rules.

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RulePlot[CellularAutomaton[30]]

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Define the initial conditions of the CA by using the new function CenterArray.

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initc = CenterArray[7]

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Compute the rule 30 CA with the above initial conditions and plot it in a discrete array of squares.

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ca30 = CellularAutomaton[30, initc, 20]; ArrayPlot[ca30, Mesh -> True]

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FindTransientRepeat gives the repeated part of this CA.

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tr30 = FindTransientRepeat[ca30, 2]; ArrayPlot[Last[tr30], Mesh -> True]

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Repeat the same steps for the rule 90 cellular automaton.

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RulePlot[CellularAutomaton[90]]

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ca90 = CellularAutomaton[90, initc, 20]; ArrayPlot[ca90, Mesh -> True]

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tr90 = FindTransientRepeat[ca90, 2]; ArrayPlot[Last[tr90], Mesh -> True]

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