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
![{0->[0 0 0; 0 0 0; 0 0 0],1->[1 0 1; 0 1 0; 1 0 1]}.](/images/equations/BoxFractal/NumberedEquation1.svg) |
(1)
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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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(2)
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(3)
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(4)
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(5)
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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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(6)
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(7)
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(8)
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(9)
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(OEIS A113209).
