 TOPICS  # 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  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).

Cantor Dust, Cantor Square Fractal, Cross-Stitch Curve, Haferman Carpet, Sierpiński Carpet, Sierpiński Sieve

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## References

Sloane, N. J. A. Sequences A000351/M3937 and A113209 in "The On-Line Encyclopedia of Integer Sequences."

Box Fractal

## Cite this as:

Weisstein, Eric W. "Box Fractal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BoxFractal.html