Minkowski sausage

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

• Self-avoiding walk
• Vicsek fractal

Notes

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