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Date: 28-9-2021
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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
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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.
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The sequence is then 1, 5, 25, 125, 625, 3125, 15625, ... (OEIS A000351). The capacity dimension is therefore
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(OEIS A113209).
REFERENCES:
Sloane, N. J. A. Sequences A000351/M3937 and A113209 in "The On-Line Encyclopedia of Integer Sequences."
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