A Fractal which can be constructed using String Rewriting by creating a matrix three times the size of the 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

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.

**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.

© 1996-9

1999-05-26