Minkowski Sausage

$\begin{figure}\begin{center}\BoxedEPSF{Minkowski_Sausage.epsf scaled 680}\end{center}\end{figure}$

A Fractal created from the base curve and motif illustrated below.

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

The number of segments after the th iteration is

$\begin{displaymath} N_n=8^n, \end{displaymath}$

and

$\begin{displaymath} \epsilon_n=\left({1\over 4}\right)^n, \end{displaymath}$

so the Capacity Dimension is

$\begin{displaymath} D\equiv-\lim_{n\to\infty} {\ln N_n\over \ln\epsilon_n} = -\l... ...^n} = {\ln 8\over \ln 4} = {3\ln 2\over 2\ln 2} = {3\over 2}. \end{displaymath}$

References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 37-38 and 42, 1991.

Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 283, 1988.

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