Cantor Square Fractal

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

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

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

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

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

The first three steps are illustrated above.

The size of the unit element after the th iteration is

$\begin{displaymath} L_n=\left({1\over 3}\right)^n \end{displaymath}$

and the number of elements is given by the Recurrence Relation

$\begin{displaymath} N_n=4N_{n-1}+5(9^n) \end{displaymath}$

where $N_1\equiv 5$ , and the first few numbers of elements are 5, 65, 665, 6305, .... Expanding out gives

$\begin{displaymath} N_n=5\sum_{k=0}^n 4^{n-k}9^{k-1}=9^n-4^n. \end{displaymath}$

The Capacity Dimension is therefore

$\displaystyle D$	$\textstyle =$	$\displaystyle -\lim_{n\to\infty} {\ln N_n\over \ln L_n} = -\lim_{n\to\infty} {\ln(9^n-4^n)\over\ln(3^{-n})}$
	$\textstyle =$	$\displaystyle -\lim_{n\to\infty}{\ln(9^n)\over\ln(3^{-n})} = {\ln 9\over \ln 3}={2\ln 3\over\ln 3}=2.$

Since the Dimension of the filled part is 2 (i.e., the Square is completely filled), Cantor's square fractal is not a true Fractal.

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 82-83, 1991.

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