Rule 110

Author

Eric W. Weisstein
August 31, 2016

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CellularAutomata/Rule110.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Rule110.html.

©2016 Wolfram Research, Inc. except for portions noted otherwise

Initialization

rule=110;

Plot

V5

ElementaryCADiagram[rule]

V6

ElementaryCADiagram[rule]

V11

RulePlot[CellularAutomaton[110]]

RulePlot[CellularAutomaton[110],{{1},0},15,Mesh->True,PlotLegends->"Icon"]

Variants

ElementaryCAVariants[rule]

Evolution

ArrayPlot[ElementaryCellularAutomaton[110,250],Frame->False,PixelConstrained->True,ImageSize->1000]

Triangle of Values

ColumnForm[t=ElementaryCATriangle[rule,5],Center]

Represented in Decimal

d=FromDigits[#,2]&/@With[{n=10},MapIndexed[Take[#1,n+#2[[1]]{-1,1}+{2,0}]&,ElementaryCellularAutomaton[rule,n]]]

FindSequenceFunction[{1,6,28,104,496,1568,7360,27520,130304,396800,1848320},n]

IntegerDigits[#,2]&/@%

d/2^Range[0,10]

FromDigits[#,2]&/@With[{n=40},MapIndexed[Take[#1,n+#2[[1]]{-1,0}+{2,1}]&,ElementaryCellularAutomaton[rule,n]]]

FindSequenceFunction[{1,3,7,13,31,49,115,215,509,775,1805,3359,7985,12659,29655,54909,130759,197581,460383,855793,2038675,3227319,7562237,14149127,33304077,50625055,118279729,220060275,523730647,830325757,1942439431,3595423245,8571017759,12951092785,30166680179,56073062103,133588389885,211506632711,495595632653,927282980895,2182630996017},n]

IntegerDigits[#,2]&/@%

Evoluation

Show[RasterGraphics[ElementaryCellularAutomaton[110,500]]]//Timing

Show[RasterGraphics[
ElementaryCellularAutomaton[110,1000]]]//Timing