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]}

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 $\left(\ln 8/\ln 4\ \right)=1.5=3/2$.^{[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. ISBN9783319065359.

^Lauwerier, Hans (1991). Fractals: Endlessly Repeated Geometrical Figures. Translated by Gill-Hoffstädt, Sophia. Princeton University Press. p. 37. ISBN0-691-02445-6. The so-called Minkowski sausage. Mandelbrot gave it this name to honor the friend and colleague of Einstein who died so untimely (1864-1909).

^ ^{a}^{b}Addison, Paul (1997). Fractals and Chaos: An illustrated course, p. 19. CRC Press. ISBN0849384435.

^ ^{a}^{b}Weisstein, Eric W. (1999). "Minkowski Sausage", archive.lib.msu.edu. Accessed: 21 September 2019.

^ ^{a}^{b}Pamfilos, Paris. "Minkowski Sausage", user.math.uoc.gr/~pamfilos/. Accessed: 21 September 2019.