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Minkowski sausage

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Not to be confused with Minkowski cover.

The Minkowski sausage[3] 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.[4]

First iterations of the quadratic type 2 Koch curve, the Minkowski sausage[a]
First iterations of the quadratic type 1 Koch curve[b]
Alternative generator with dimension of ⁠ln 18/ln 6⁠ ≈ 1.61[c]
Higher iteration of type 2[a]
Example of a fractal antenna: a space-filling curve called a "Minkowski Island"[1] or "Minkowski fractal"[2][b]
Generator
island[c]

The Sausage has a Hausdorff dimension of .[a] It is therefore often chosen when studying the physical properties of non-integer fractal objects. It is strictly self-similar.[4] 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.[b]

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

Islands
Island formed by a different generator[5][6][7] with a dimension of ≈1.36521[8] or 3/2[5][b]
Island formed by using the Sausage as the generator[a][d]
Anti-island (anticross-stitch curve), iterations 0-4[b]
Anti-island: the generator's symmetry results in the island mirrored[a]
Same island as the first formed from a different generator ,[6] which forms 2 right triangles with side lengths in ratio: 1:2:√5[7][b]
Quadratic island formed using curves with a different generator[c]

See also

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Notes

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  1. ^ a b c d e Quadratic Koch curve type 2
  2. ^ a b c d e f Quadratic Koch curve type 1
  3. ^ a b c Neither type 1 nor 2
  4. ^ This has been called the "zig-zag quadratic Koch snowflake".[9]

References

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  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. ^ a b Addison, Paul (1997). Fractals and Chaos: An illustrated course, p. 19. CRC Press. ISBN 0849384435.
  5. ^ a b Weisstein, Eric W. (1999). "Minkowski Sausage", archive.lib.msu.edu. Accessed: 21 September 2019.
  6. ^ a b Pamfilos, Paris. "Minkowski Sausage", user.math.uoc.gr/~pamfilos/. Accessed: 21 September 2019.
  7. ^ a b 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.
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