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The Minkowski sausage or Minkowski curve is a fractal first proposed by and named for Hermann Minkowski as well as its casual resemblance to a sausage or sausage links. The initiator is a line segment and the generator is a broken line of eight parts one fourth the length.

The Sausage has a Hausdorff dimension of ${\displaystyle \left(\ln 8/\ln 4\ \right)=1.5=3/2}$. It is therefore often chosen when studying the physical properties of non-integer fractal objects. It is strictly self-similar. It never intersects itself. It is continuous everywhere, but differentiable nowhere. It is not rectifiable. It has a Lebesgue measure of 0. The type 1 curve has a dimension of ⁠ln 5/ln 3⁠ ≈ 1.46.

Multiple Minkowski Sausages may be arranged in a four sided polygon or square to create a quadratic Koch island or Minkowski island/flake:

See also

• Self-avoiding walk
• Vicsek fractal

Notes

1. ^ Quadratic Koch curve type 2
2. ^ Quadratic Koch curve type 1
3. ^ Neither type 1 nor 2
4. This has been called the "zig-zag quadratic Koch snowflake".

References

1. Cohen, Nathan (Summer 1995). "Fractal antennas Part 1". Communication Quarterly: 7–23.
2. Ghosh, Basudeb; Sinha, Sachendra N.; and Kartikeyan, M. V. (2014). Fractal Apertures in Waveguides, Conducting Screens and Cavities: Analysis and Design, p. 88. Volume 187 of Springer Series in Optical Sciences. ISBN 9783319065359.
3. Lauwerier, Hans (1991). Fractals: Endlessly Repeated Geometrical Figures. Translated by Gill-Hoffstädt, Sophia. Princeton University Press. p. 37. ISBN 0-691-02445-6. The so-called Minkowski sausage. Mandelbrot gave it this name to honor the friend and colleague of Einstein who died so untimely (1864-1909).
4. ^ Addison, Paul (1997). Fractals and Chaos: An illustrated course, p. 19. CRC Press. ISBN 0849384435.
5. ^ Weisstein, Eric W. (1999). "Minkowski Sausage", archive.lib.msu.edu. Accessed: 21 September 2019.
6. ^ Pamfilos, Paris. "Minkowski Sausage", user.math.uoc.gr/~pamfilos/. Accessed: 21 September 2019.
7. ^ Weisstein, Eric W. "Minkowski Sausage". MathWorld. Retrieved 22 September 2019.
8. Mandelbrot, B. B. (1983). The Fractal Geometry of Nature, p. 48. New York: W. H. Freeman. ISBN 9780716711865. Cited in Weisstein MathWorld.
9. Schmidt, Jack (2011). "The Koch snowflake worksheet II", p. 3, UK MA111 Spring 2011, ms.uky.edu. Accessed: 22 September 2019.

External links

• "Square Koch Fractal Curves". Wolfram Demonstrations Project. Retrieved 23 September 2019.

• De Rham curves
• L-systems
• Hermann Minkowski