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 ${\displaystyle \left(\ln 8/\ln 4\ \right)=1.5=3/2}$. ^[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 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 (anti cross-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

• Self-avoiding walk
• Vicsek fractal

Notes

References

External links

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