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Haferman Carpet

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HafermanCarpet

The Haferman carpet is a fractal constructed using string rewriting beginning with a black cell [1] and iterating the rules

{0->[1 1 1; 1 1 1; 1 1 1],1->[0 1 0; 1 0 1; 0 1 0]}
(1)

(Allouche and Shallit 2003, p. 407).

Haferman carpet

Taking five iterations gives the beautiful pattern illustrated above.

This fractal also appears on the cover of Allouche and Shallit (2003).

For a nonnegative integer n, let N_n be the number of black cells and L_n the side length of a cell after the nth iteration. Then

N_n=1/(14)[(-1)^n5^(n+1)+9^(n+1)]
(2)
L_n=3^(-n).
(3)

The numbers of black cells after n=0, 1, 2, ... iterations are therefore 1, 4, 61, 424, 4441, 36844, ... (OEIS A118005). The capacity dimension is therefore

d_(cap)=-lim_(n->infty)(lnN_n)/(lnL_n)
(4)
=2.
(5)

See also

Box Fractal, Cantor Dust, Cantor Square Fractal, Sierpiński Carpet

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References

Allouche, J.-P. and Shallit, J. Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, 2003.Sloane, N. J. A. Sequence A118005 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Haferman Carpet

Cite this as:

Weisstein, Eric W. "Haferman Carpet." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HafermanCarpet.html

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