Box Fractal

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

A Fractal which can be constructed using String Rewriting by creating a matrix with 3 times as many entries as 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 be the number of black boxes, the length of a side of a white box, and the fractional Area of black boxes after the 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

$\displaystyle d_{\rm cap}$	$\textstyle =$	$\displaystyle -\lim_{n\to \infty}{\ln N_n\over\ln L_n} = -\lim_{n\to\infty}{\ln(5^n)\over\ln(3^{-n})}$
	$\textstyle =$	$\displaystyle {\ln 5\over\ln 3} = 1.464973521\ldots.$	(4)

References

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