Haferman Carpet

Author

Eric W. Weisstein
August 31, 2016

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

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

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

Initialization

fractal=HafermanCarpet;

Graphic

V5

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

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

V6

GraphicsRow[ArrayPlot/@HafermanCarpet[{0,4}]]

V11

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

RulePlot[SubstitutionSystem[rules]]

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

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

Dimension

0's

Count[#,0,2]&/@its

FindSequenceFunction[%,n+1]//FullSimplify

1's

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

FindSequenceFunction[%,n+1]//FullSimplify

Table[(1/14) ((((-1))^n) (5^(1+n))+(9^(1+n))),{n,0,7}]

Dimension

<<Utilities`Simplify`

Limit[-((Log[(1/14) ((((-1))^n) (5^(1+n))+(9^(1+n)))])/(Log[(3^(-n))])),n->∞]//LogExpand