Box Fractal

The box fractal is a fractal also called the anticross-stitch curve which can be constructed using string rewriting beginning with a cell [1] and iterating the rules

(1)

An outline of the box fractal can encoded as a Lindenmayer system with initial string "F-F-F-F", string rewriting rule "F" -> "F-F+F+F-F", and angle  (J. Updike, pers. comm., Oct. 26, 2004).

Let  be the number of black boxes,  the length of a side of a white box, and  the fractional area of black boxes after the th iteration.

			(2)
			(3)
			(4)
			(5)

The sequence  is then 1, 5, 25, 125, 625, 3125, 15625, ... (OEIS A000351). The capacity dimension is therefore

			(6)
			(7)
			(8)
			(9)

(OEIS A113209).

REFERENCES:

Sloane, N. J. A. Sequences A000351/M3937 and A113209 in "The On-Line Encyclopedia of Integer Sequences."