Box Fractal

The box fractal is a fractal also called the anticross-stitch curve which can be constructed using cell substitution beginning with a single black unit square, represented by [1], and iterating the rules below, with each replacement block scaled to occupy its parent cell

(1)

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

Let be the number of black boxes, the side length of an elementary black box, and the fractional area covered by black boxes after iteration , where is a nonnegative integer and the initial square corresponds to .

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

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

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

(OEIS A113209).