Cantor Square Fractal

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

(1)

The size of the unit element after the th iteration is

(2)

and the number of elements is given by the recurrence relation

(3)

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

(4)

The capacity dimension is therefore

			(5)
			(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.

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."