Sierpinski curve

Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit ${\displaystyle n\rightarrow \infty }$ completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.

Because the Sierpiński curve is space-filling, its Hausdorff dimension (in the limit ${\displaystyle n\rightarrow \infty }$) is ${\displaystyle 2}$.
The Euclidean length of

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