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Minkowski Sausage


MinkowskiSausage
MinkowskiMotif

A fractal curve created from the base curve and motif illustrated above (Lauwerier 1991, p. 37).

MinkowskiSausageLengths

As illustrated above, the number of segments after the nth iteration is

 N_n=4·3^n,
(1)

and the length of each segment is given by

 epsilon_n=(1/(sqrt(5)))^n,
(2)

so the capacity dimension is

D=-lim_(n->infty)(lnN_n)/(lnepsilon_n)
(3)
=(2ln3)/(ln5)
(4)
=log_59
(5)
 approx 1.36521
(6)

(Mandelbrot 1983, p. 48).

The term Minkowski sausage is also used to refer to the Minkowski cover of a curve.


See also

Minkowski-Bouligand Dimension, Minkowski Cover

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References

Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 37-38 and 42, 1991.Mandelbrot, B. B. The Fractal Geometry of Nature. New York: W. H. Freeman, pp. 32 and 48-49, 1983.Peitgen, H.-O. and Saupe, D. (Eds.). The Science of Fractal Images. New York: Springer-Verlag, p. 283, 1988.Trott, M. "The Mathematica Guidebooks Additional Material: Vibrating Koch Drum." http://www.mathematicaguidebooks.org/additions.shtml#N_1_07.

Referenced on Wolfram|Alpha

Minkowski Sausage

Cite this as:

Weisstein, Eric W. "Minkowski Sausage." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MinkowskiSausage.html

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