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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 $N_n$ 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 $n$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


Dickau, R. M. ``The Sierpinski Carpet.''

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

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

© 1996-9 Eric W. Weisstein