SierpinskiCurve

SierpinskiCurve[n]

gives the line segments representing the n-step Sierpiński curve.

Details and Options

SierpinskiCurve is also known as Sierpiński space-filling curve.
SierpinskiCurve returns a Line primitive corresponding to a closed path around {0,0}.

SierpinskiCurve takes a DataRange option that can be used to specify the range the coordinates should be assumed to occupy.

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 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:

DataRangerange 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:

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SierpinskiCurve

Details and Options

Examples

Basic Examples (2)