Cantor Square Fractal


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]}.

The size of the unit element after the nth iteration is


and the number of elements is given by the recurrence relation


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


The capacity dimension is therefore


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

Explore with Wolfram|Alpha


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.

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