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Minkowski sausage

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

The Minkowski sausage^[3]orMinkowski 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 dimensionof${\displaystyle \left(\ln 8/\ln 4\ \right)=1.5=3/2}$.^[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 islandorMinkowski island/[snow]flake:

Island formed by a different generator^[5][6][7] with a dimension of ≈1.36521^[8] or 3/2^[5][b]

Anti-island (anticross-stitch curve), iterations 0-4^[b]

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]

See also[edit]

• Self-avoiding walk
• Vicsek fractal

Notes[edit]

References[edit]

External links[edit]

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