Cantor Square Fractal
A Fractal which can be constructed using String Rewriting by creating a matrix three times the size ofthe current matrix using the rules
The first three steps are illustrated above.
The size of the unit element after the 
th iteration is
and the number of elements is given by the Recurrence Relation
where
, and the first few numbers of elements are 5, 65, 665, 6305, .... Expanding out gives
The Capacity Dimension is therefore
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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
References
Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 82-83, 1991.
Weisstein, E. W. ``Fractals.'' Mathematica notebook Fractal.m.
- Polyhex
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