Menger Sponge

$\begin{figure}\BoxedEPSF{menger_sponge3.epsf scaled 400}\end{figure}$

A Fractal which is the 3-D analog of the Sierpinski Carpet. Let be the number of filled boxes, the length of a side of a hole, and the fractional Volume after the th iteration.

$\displaystyle N_n$	$\textstyle =$	$\displaystyle 20^n$	(1)
$\displaystyle L_n$	$\textstyle =$	$\displaystyle ({\textstyle{1\over 3}})^n=3^{-n}$	(2)
$\displaystyle V_n$	$\textstyle =$	$\displaystyle {L_n}^3N_n = ({\textstyle{20\over 27}})^n.$	(3)

The Capacity Dimension is therefore

$\displaystyle d_{\rm cap}$	$\textstyle =$	$\displaystyle -\lim_{n\to \infty}{\ln N_n\over\ln L_n} = -\lim_{n\to\infty}{\ln(20^n)\over\ln(3^{-n})}$
	$\textstyle =$	$\displaystyle {\ln 20\over\ln 3} = {\ln(2^2\cdot 5)\over\ln 3} = {2\ln 2+\ln 5\over\ln 3}$
	$\textstyle =$	$\displaystyle 2.726833028\ldots.$	(4)

J. Mosely is leading an effort to construct a large Menger sponge out of old business cards.

See also Sierpinski Carpet, Tetrix

References

Dickau, R. M. ``Menger (Sierpinski) Sponge.'' http://forum.swarthmore.edu/advanced/robertd/sponge.html.

Mosely, J. ``Menger's Sponge (Depth 3).'' http://world.std.com/~j9/sponge/.