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).

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