Sierpiński curve

Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit ${\displaystyle n\to \infty }$ 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 ${\displaystyle n\to \infty }$) is ${\displaystyle 2}$.
The Euclidean length of the ${\displaystyle n}$^th iteration curve ${\displaystyle S_{n}}$ is

${\displaystyle l_{n}={2 \over 3}(1+{\sqrt {2}})2^{n}-{1 \over 3}(2-{\sqrt {2}}){1 \over 2^{n}},}$

i.e., it grows exponentially with ${\displaystyle n}$ beyond any limit, whereas the limit for ${\displaystyle n\to \infty }$ of the area enclosed by ${\displaystyle S_{n}}$ is ${\displaystyle 5/12\,}$ that of the square (in Euclidean metric).