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Sierpiński Curve

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There are several fractal curves associated with Sierpiński.

SierpinskiCross

The area for the first Sierpiński curve illustrated above (Sierpiński 1912) is

A=1/3(7-4sqrt(2)).

The curve is called the Sierpiński curve by Cundy and Rollett (1989, pp. 67-68), the Sierpiński's square snowflake by Wells (1991, p. 229), and is pictured but not named by Steinhaus (1999, pp. 102-103). The nth iteration of the first Sierpiński curve is implemented in the Wolfram Language as SierpinskiCurve[n].

SierpinskiCurve

The limit of the second Sierpiński's curve illustrated above has area

A=5/(12).

The Sierpiński arrowhead curve is another Sierpiński curve.

See also

Exterior Snowflake, Gosper Island, Hilbert Curve, Koch Antisnowflake, Koch Snowflake, Peano Curve, Peano-Gosper Curve, Sierpiński Arrowhead Curve

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References

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.Dickau, R. M. "Two-Dimensional L-Systems." http://mathforum.org/advanced/robertd/lsys2d.html.Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, p. 34, 1989.Sierpiński, W. "Sur une nouvelle courbe continue qui remplit toute une aire plane." Bull. l'Acad. des Sciences Cracovie A, 462-478, 1912.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wagon, S. Mathematica in Action. New York: W. H. Freeman, p. 207, 1991.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 229, 1991.

Referenced on Wolfram|Alpha

Sierpiński Curve

Cite this as:

Weisstein, Eric W. "Sierpiński Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SierpinskiCurve.html

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