SierpinskiCurve

SierpinskiCurve[n]

gives the line segments representing the n^(th)-step Sierpiński curve.

Details and Options

Examples

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Basic Examples  (2)

A 2D Sierpiński curve:

Lengths of the approximations to the Sierpiński curve:

The formula:

Visualize the Sierpiński curve in 2D with splines:

Scope  (6)

Curve Specification  (2)

A 2D Sierpiński curve:

The ^(th) approximation of the Sierpiński curve:

Curve Styling  (4)

Sierpiński curves with different thicknesses:

Thickness in scaled size:

Thickness in printer's points:

Dashed curves:

Colored curves:

Options  (1)

DataRange  (1)

DataRange allows you to specify the range of mesh coordinates to generate:

Specify a different range:

Applications  (4)

SierpinskiCurve is constructed recursively by transforming segments into curves linked together by lines:

Next iteration:

Visualize the Sierpiński curve in 2D:

With splines:

Build a polygon:

Apply a Sierpiński curve texture to a surface:

Properties & Relations  (3)

SierpinskiCurve consists of lines:

Find the perimeter of the 2D Sierpiński curve:

DataRangerange is equivalent to using RescalingTransform[{…},range]:

Use RescalingTransform:

Possible Issues  (2)

By default, the coordinates of the Sierpiński curve are not in the unit square:

Using DataRange to generate the Sierpiński curve in the unit square:

SierpinskiCurve can be too large to generate:

Neat Examples  (1)

Traversal animations:

Wolfram Research (2017), SierpinskiCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/SierpinskiCurve.html.

Text

Wolfram Research (2017), SierpinskiCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/SierpinskiCurve.html.

CMS

Wolfram Language. 2017. "SierpinskiCurve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SierpinskiCurve.html.

APA

Wolfram Language. (2017). SierpinskiCurve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SierpinskiCurve.html

BibTeX

@misc{reference.wolfram_2025_sierpinskicurve, author="Wolfram Research", title="{SierpinskiCurve}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/SierpinskiCurve.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_sierpinskicurve, organization={Wolfram Research}, title={SierpinskiCurve}, year={2017}, url={https://reference.wolfram.com/language/ref/SierpinskiCurve.html}, note=[Accessed: 22-February-2025 ]}