Sierpinski curve

1st order Sierpiński curve

Sierpiński curves
1st and 2nd order

Sierpiński curves
1st to 3rd order

The Sierpiński curves are a recursively defined sequence of continuous closed fractal curves . The Sierpiński curve is an example of a space-filling curve that completely fills the unit square in the transition . They were defined in 1912 by the Polish mathematician Wacław Sierpiński . ${\displaystyle n\rightarrow \infty }$

properties

• The limit of the area enclosed by the Sierpiński curve is (in Euclidean metric). ${\displaystyle 5 \over 12}$
• The Euclidean length of the curve is growing exponentially with : . ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle l_{n}={\frac {2}{3}}(1+{\sqrt {2}})2^{n}-{\frac {1}{3}}(2-{\sqrt {2}}){\frac {1}{2^{n}}}}$
• Since the curve is space-filling, it has the Hausdorff dimension in the limit value . ${\displaystyle 2}$

Web links

Commons : Sierpinski curve  - album with pictures, videos and audio files

• Eric W. Weisstein : Sierpinski curve . In: MathWorld (English).
• Interactive demonstration of the Sierpinski curve.