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Sierpiński curves are a recursively defined sequence of continuous closed plane fractal curves discovered by Wacław Sierpiński, which in the limit n → ∞ {\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 is 2 {\displaystyle 2}. The Euclidean length of the n {\displaystyle n} iteration curve S n {\displaystyle S_{n}} is
i.e., it grows exponentially with n {\displaystyle n} beyond any limit, whereas the limit for n → ∞ {\displaystyle n\to \infty } of the area enclosed by S n {\displaystyle S_{n}} is 5 / 12 {\displaystyle 5/12\,} that of the square.