Sierpiński curve
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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 completely fill the unit square: thus their limit curve, also called the Sierpiński curve, is an example of a space-filling curve.
This article is missing information about other Sierpiński curves, see Sierpiński Curve on Wolfram MathWorld. (January 2019) |
"Sierpinski square snowflake" redirects here. For other uses, see Sierpinski carpet.
Because the Sierpiński curve is space-filling, its Hausdorff dimension (in the limit ) is .
The Euclidean length of the th iteration curve is
i.e., it grows exponentially with beyond any limit, whereas the limit for of the area enclosed by is that of the square (in Euclidean metric).