Minkowski Sausage

The Minkowski Sausage is the "quadratic type 2" variant of the Koch Snowflake curve, named after Hermann Minkowski. Where the Koch curve uses triangles, this one uses only right angles. Every line is cut into 4 equal parts and replaced with 8 segments that step up, across, down, down, across and back up again, like a square wave. Repeat that on every segment and the line turns into a thick, wriggling band: the sausage.

Turn on the "Minkowski island" toggle to put four of these curves on the sides of a square. Because every bump outward is matched by a bump inward of the same size, the island always encloses exactly the area of the square you started with, while its boundary grows longer with every step.

Each step replaces a line with 8 copies, each $\frac{1}{4}$ as long. The Hausdorff dimension is therefore

This is a little higher than the Quadratic Snowflake, which uses the "quadratic type 1" curve with 5 segments of $\frac{1}{3}$ (dimension $\ln 5 / \ln 3 \approx 1.46$).

The page draws it as an L-System with a turn angle of $90°$ and these rules:

const SAUSAGE_AXIOM = "F";
const ISLAND_AXIOM = "F+F+F+F";
const RULES = { F: "F+F-F-FF+F+F-F" };

For more curves from the same family, see the Quadratic Koch Island and the Cesàro Fractal.