The box fractal is a fractal also called the anticross-stitch curve which can be constructed using string
rewriting beginning with a cell  and iterating the rules
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
is then 1, 5, 25, 125, 625, 3125, 15625, ... (OEIS A000351).
The capacity dimension is therefore
See alsoCantor Dust
, Cantor Square Fractal
, Cross-Stitch Curve
, Haferman Carpet
, Sierpiński Sieve
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ReferencesSloane, N. J. A. Sequences A000351/M3937 and A113209 in "The On-Line Encyclopedia
of Integer Sequences."
Referenced on Wolfram|AlphaBox Fractal
Cite this as:
Weisstein, Eric W. "Box Fractal." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoxFractal.html