Sierpinski Carpet

$\begin{figure}\begin{center}\BoxedEPSF{Sierpinski_carpet.epsf scaled 730}\end{center}\end{figure}$

A Fractal which is constructed analogously to the Sierpinski Sieve, but using squares instead of triangles. 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. Then

$\displaystyle N_n$	$\textstyle =$	$\displaystyle 8^n$	(1)
$\displaystyle L_n$	$\textstyle =$	$\displaystyle ({\textstyle{1\over 3}})^n=3^{-n}$	(2)
$\displaystyle A_n$	$\textstyle =$	$\displaystyle {L_n}^2N_n = ({\textstyle{8\over 9}})^n.$	(3)

The Capacity Dimension is therefore

$\displaystyle d_{\rm cap}$	$\textstyle =$	$\displaystyle -\lim_{n\to\infty}{\ln N_n\over\ln L_n}=-\lim_{n\to\infty}{\ln(8^n)\over\ln(3^{-n})}={\ln 8\over\ln 3}$
	$\textstyle =$	$\displaystyle {3\ln 2\over\ln 3} = 1.892789260\ldots.$	(4)

See also Menger Sponge, Sierpinski Sieve

References

Dickau, R. M. ``The Sierpinski Carpet.'' http://forum.swarthmore.edu/advanced/robertd/carpet.html.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 112-121, 1992.

Weisstein, E. W. ``Fractals.'' Mathematica notebook Fractal.m.