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Repetition Periods for Elementary Cellular Automata

A collection of rules and their repetition periods as a function of size

Originator: Stephen Wolfram

A plot of these repetition periods as originally provided on page 260 of A New Kind of Science (Wolfram, 2002) and a visualization was provided on page 259

(2 columns, 5 rows)

Examples

Basic Examples

Retrieve the ResourceObject:

In[1]:=
ResourceObject["Repetition Periods for Elementary Cellular Automata"]
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View the data:

In[2]:=
ResourceData["Repetition Periods for Elementary Cellular Automata"]
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Visualization

Find the systems provided:

In[3]:=
Normal@ResourceData[ "Repetition Periods for Elementary Cellular Automata"][[All, "Rule"]]
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Find the systems sizes for rule 90 with repetition period between 5 and 50:

In[4]:=
Select[ Select[Normal@ ResourceData[ "Repetition Periods for Elementary Cellular Automata"], #Rule == 90 &][[1, "RepetitionPeriods"]], Between@{5, 50} ]
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Plot these:

In[5]:=
Module[{rule = 90, repetitionPeriods}, repetitionPeriods = Select[Normal@ ResourceData[ "Repetition Periods for Elementary Cellular Automata"], #Rule == rule &][[1, "RepetitionPeriods"]]; Table[ ArrayPlot@ CellularAutomaton[rule, ReplacePart[ ConstantArray[0, n], Ceiling@(n/2) -> 1 ], 100 ], {n, Keys@ Reverse@Sort@ Select[repetitionPeriods, Between@{5, 50}] // Take[#, UpTo[5]] & } ] ]
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Highlight the repeating blocks:

In[6]:=
Module[{rule = 90, repetitionPeriods, repBlock, firstRep, evolution}, repetitionPeriods = Select[Normal@ ResourceData[ "Repetition Periods for Elementary Cellular Automata"], #Rule == rule &][[1, "RepetitionPeriods"]]; Table[ evolution = CellularAutomaton[rule, ReplacePart[ ConstantArray[0, n], Ceiling@(n/2) -> 1 ], 100 ]; firstRep = FirstCase[Range[100], _?( # <= (100 - repetitionPeriods[n]) && evolution[[#]] == evolution[[repetitionPeriods[n] + #]] &) ]; ArrayPlot[ ReplacePart[evolution, firstRep | _?(# >= firstRep && Mod[# - firstRep + 1, repetitionPeriods[n]] == 0 &) -> Table[2, n] ], ColorRules -> { 1 -> GrayLevel[0.1], 2 -> Hue[.6, 1, 1] } ], {n, Keys@ Reverse@Sort@ Select[repetitionPeriods, Between@{5, 50}] // Take[#, UpTo@5] & } ] ]
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Wolfram Research, "Repetition Periods for Elementary Cellular Automata" from the Wolfram Data Repository (2017)   https://doi.org/10.24097/wolfram.75508.data

License Information

CC BY-SA 4.0

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Publisher Information

  • Publisher of Record: Wolfram Research