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Box Fractal


BoxFractal

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]}.
(1)
BoxFractalLSystem

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 90 degrees (J. Updike, pers. comm., Oct. 26, 2004).

Let N_n be the number of black boxes, L_n the length of a side of a white box, and A_n the fractional area of black boxes after the nth iteration.

N_n=5^n
(2)
L_n=3^(-n)
(3)
A_n=L_n^2N_n
(4)
=(5/9)^n.
(5)

The sequence N_n is then 1, 5, 25, 125, 625, 3125, 15625, ... (OEIS A000351). The capacity dimension is therefore

d_(cap)=-lim_(n->infty)(lnN_n)/(lnL_n)
(6)
=log_35
(7)
=(ln5)/(ln3)
(8)
=1.464973521...
(9)

(OEIS A113209).


See also

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."

Referenced on Wolfram|Alpha

Box Fractal

Cite this as:

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

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