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HilbertCurve
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HilbertCurve

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

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gives the n^(th)-step Hilbert curve in dimension d.

Details and Options

  • HilbertCurve is also known as Hilbert space-filling curve.
  • HilbertCurve[n] returns a Line primitive corresponding to a path that starts at {0,0}, then joins all integer points in the 2n-1 by 2n-1 square, and ends at {2n-1,0}. »
  • HilbertCurve takes a DataRange option that can be used to specify the range the coordinates should be assumed to occupy.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

A 2D Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-e0oufp
Out[1]=1

Lengths of the approximations to the Hilbert curve:

In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-btyad4
Out[2]=2

The formula:

In[3]:=3
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https://wolfram.com/xid/0cpt0d0p0wqs3y-b7k7lx
Out[3]=3

Visualize the Hilbert curve in 2D with splines:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-pd44os
Out[1]=1

Scope  (8)Survey of the scope of standard use cases

Curve Specification  (4)

A 2D Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-hw51bs
Out[1]=1

A 3D Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-fdvkfs
Out[1]=1

An D Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-t1epq

The ^(th) approximation of the Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-dan8zd
Out[1]=1

Curve Styling  (4)

Hilbert curves with different thicknesses:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-bq85gl
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-ifj0kj
Out[2]=2

Thickness in scaled size:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-fh1qpo
Out[1]=1

Thickness in printer's points:

In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-kyrge4
Out[2]=2

Dashed curves:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-sjd36
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-bq2qbc
Out[2]=2

Colored curves:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-oxjl5
Out[1]=1

Generalizations & Extensions  (2)Generalized and extended use cases

Hilbert curve in 4D:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-4um6ja

Hilbert curve in D:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-zhpj02
Out[1]=1

Options  (1)Common values & functionality for each option

DataRange  (1)

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

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-ewpnb1
Out[1]=1

Specify a different range:

In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-fl0z4t
Out[2]=2

Applications  (4)Sample problems that can be solved with this function

HilbertCurve is constructed recursively by transforming cups into four cups linked together by lines:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-13fyon
In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-2gys50
Out[2]=2

Next iteration:

In[3]:=3
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https://wolfram.com/xid/0cpt0d0p0wqs3y-v6b49h
In[4]:=4
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https://wolfram.com/xid/0cpt0d0p0wqs3y-qi9fw9
Out[4]=4

Visualize the Hilbert curve in 2D:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-22pjxu
Out[1]=1

With splines:

In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-3jablb
Out[2]=2

In 3D:

In[3]:=3
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https://wolfram.com/xid/0cpt0d0p0wqs3y-qy8xih
Out[3]=3

With tubes:

In[4]:=4
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https://wolfram.com/xid/0cpt0d0p0wqs3y-o08fvf
Out[4]=4

Build a polygon from the Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-6z9a5
Out[1]=1

Apply a Hilbert curve texture to a surface:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-m00bp0
Out[1]=1

Properties & Relations  (3)Properties of the function, and connections to other functions

HilbertCurve consists of lines:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-mnr925
Out[1]=1

Find the perimeter of the 2D Hilbert curve:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-ji00t5
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-bac1jt
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cpt0d0p0wqs3y-ltfk9
Out[3]=3

3D Hilbert curve:

In[4]:=4
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https://wolfram.com/xid/0cpt0d0p0wqs3y-fmpnsn
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0cpt0d0p0wqs3y-gz6a76
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0cpt0d0p0wqs3y-san95
Out[6]=6

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

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-37ymx
Out[1]=1

Use RescalingTransform:

In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-dydlva
In[3]:=3
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https://wolfram.com/xid/0cpt0d0p0wqs3y-bu4y1v
Out[3]=3

Possible Issues  (2)Common pitfalls and unexpected behavior

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

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-dauf9b
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0cpt0d0p0wqs3y-sco19
Out[2]=2

HilbertCurve can be too large to generate:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-frq6rm
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

Traversal animations:

In[1]:=1
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https://wolfram.com/xid/0cpt0d0p0wqs3y-erb3pk
Out[1]=1
Wolfram Research (2017), HilbertCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertCurve.html.
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Wolfram Research (2017), HilbertCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertCurve.html.

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_hilbertcurve, author="Wolfram Research", title="{HilbertCurve}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertCurve.html}", note=[Accessed: 25-March-2025 ]}

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@misc{reference.wolfram_2025_hilbertcurve, author="Wolfram Research", title="{HilbertCurve}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertCurve.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hilbertcurve, organization={Wolfram Research}, title={HilbertCurve}, year={2017}, url={https://reference.wolfram.com/language/ref/HilbertCurve.html}, note=[Accessed: 25-March-2025 ]}

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@online{reference.wolfram_2025_hilbertcurve, organization={Wolfram Research}, title={HilbertCurve}, year={2017}, url={https://reference.wolfram.com/language/ref/HilbertCurve.html}, note=[Accessed: 25-March-2025 ]}