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Cantor Square Fractal


CantorsSquare

A fractal which can be constructed using string rewriting beginning with a cell [1] and iterating the rules

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

The size of the unit element after the nth iteration is

 L_n=(1/3)^n
(2)

and the number of elements is given by the recurrence relation

 N_n=4N_(n-1)+5(9^n)
(3)

where N_1=5, and the first few numbers of elements are 5, 65, 665, 6305, ... (OEIS A118004). Expanding out gives

 N_n=5sum_(k=0)^n4^(n-k)9^(k-1)=9^n-4^n.
(4)

The capacity dimension is therefore

D=-lim_(n->infty)(lnN_n)/(lnL_n)
(5)
=2.
(6)

Since the dimension of the filled part is 2 (i.e., the square is completely filled), Cantor's square fractal is not a true fractal.


See also

Box Fractal, Cantor Dust, Haferman Carpet, Sierpiński Carpet

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References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 82-83, 1991.Sloane, N. J. A. Sequence A118004 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Cantor Square Fractal

Cite this as:

Weisstein, Eric W. "Cantor Square Fractal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CantorSquareFractal.html

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