Sierpiński Carpet

Author

Eric W. Weisstein
January 20, 2017

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Fractals/SierpinskiCarpet.nb.

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

©2017 Wolfram Research, Inc. except for portions noted otherwise

Graphic

V5

fractal=SierpinskiCarpet;

Show[GraphicsArray[Block[{$DisplayFunction=Identity},ArrayPlot/@fractal[{0,4}]]],ImageSize->500]

ArrayPlot[fractal[5],PixelConstrained->True]

V6

GraphicsRow[ArrayPlot[SierpinskiCarpet[#]]&/@Range[0,4]]

"SierpinskiCarpet"->,

V11

rules={
1->{{1,1,1},{1,0,1},{1,1,1}},
0->ConstantArray[0,{3,3}]
};

RulePlot[SubstitutionSystem[rules]]

its=SubstitutionSystem[rules,{{1}},5];

GraphicsGrid[Partition[ArrayPlot/@its,3]]

V11.1

GraphicsRow[Table[MengerMesh[n],{n,0,3}]]

Dimension

Count[#,1,2]&/@its

FindSequenceFunction[%,n+1]

Table[(8^n),{n,0,7}]

<<Utilities`Simplify`

Limit[-((Log[(8^n)])/(Log[(3^(-n))])),n->∞]//LogExpand

N[%]