Cantor Square Fractal

Author

Eric W. Weisstein
August 31, 2016

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

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

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

Plot

V5

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

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

V6

GraphicsRow[ArrayPlot/@CantorSquareFractal/@Range[0,4]]

V11

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

RulePlot[SubstitutionSystem[rules]]

its=SubstitutionSystem[rules,{{0}},5];

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

Dimension

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

FindSequenceFunction[%,n+1]//FullSimplify

Table[9^n-4^n,{n,0,7}]

<<Utilities`Simplify`

Limit[-((Log[9^n-4^n])/(Log[(3^(-n))])),n->∞]//LogExpand