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 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]
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]
^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.
^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).
^ abAddison, Paul (1997). Fractals and Chaos: An illustrated course, p. 19. CRC Press. ISBN 0849384435.
^ abWeisstein, Eric W. (1999). "Minkowski Sausage", archive.lib.msu.edu. Accessed: 21 September 2019.
^ abPamfilos, Paris. "Minkowski Sausage", user.math.uoc.gr/~pamfilos/. Accessed: 21 September 2019.