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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 $N_n$ be the number of filled boxes, $L_n$ the length of a side of a hole, and $V_n$ the fractional Volume after the $n$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


Dickau, R. M. ``Menger (Sierpinski) Sponge.''

Mosely, J. ``Menger's Sponge (Depth 3).''

© 1996-9 Eric W. Weisstein