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Cantor Dust

\begin{figure}\begin{center}\BoxedEPSF{CantorDustFractal.epsf scaled 700}\end{center}\end{figure}

A Fractal which can be constructed using String Rewriting by creating a matrix three times the size of the current matrix using the rules

\begin{figure}\begin{center}{\rm line\ 1}: {\tt ''*'' -> ''* *'','' '' -> ''\ \ \ ''}\end{center}\end{figure}

\begin{figure}\begin{center}{\rm line\ 2}: {\tt ''*'' -> ''\ \ \ '','' '' -> ''\ \ \ ''}\end{center}\end{figure}

\begin{figure}\begin{center}{\rm line\ 3}: {\tt ''*'' -> ''* *'','' '' -> ''\ \ \ ''}\end{center}\end{figure}


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.

$\displaystyle N_n$ $\textstyle =$ $\displaystyle 5^n$ (1)
$\displaystyle L_n$ $\textstyle =$ $\displaystyle ({\textstyle{1\over 3}})^n=3^{-n}$ (2)
$\displaystyle A_n$ $\textstyle =$ $\displaystyle {L_n}^2N_n = ({\textstyle{5\over 9}})^n.$ (3)

The Capacity Dimension is therefore
$\displaystyle d_{\rm cap}$ $\textstyle =$ $\displaystyle -\lim_{n\to \infty}{\ln N_n\over\ln L_n} = -\lim_{n\to\infty}{\ln(5^n)\over\ln(3^{-n})}$  
  $\textstyle =$ $\displaystyle {\ln 5\over\ln 3} \approx 1.464973521.$ (4)

See also Box Fractal, Sierpinski Carpet, Sierpinski Sieve


References

Dickau, R. M. ``Cantor Dust.'' http://forum.swarthmore.edu/advanced/robertd/cantor.html.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 103-104, 1993.

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




© 1996-9 Eric W. Weisstein
1999-05-26