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CantorMesh[n]

gives a mesh region representing the n^(th)-step Cantor set.

CantorMesh[n,d]

gives the n^(th)-step Cantor set in dimension d.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
DataRange  
MeshCellHighlight  
MeshCellLabel  
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MeshCellMarker  
MeshCellShapeFunction  
MeshCellStyle  
PlotTheme  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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CantorMesh

CantorMesh[n]

gives a mesh region representing the n^(th)-step Cantor set.

CantorMesh[n,d]

gives the n^(th)-step Cantor set in dimension d.

Details and Options

Examples

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Basic Examples  (2)

A 1D Cantor mesh:

Wolfram Language code: CantorMesh[2]

Lengths of the approximations to the Cantor mesh:

Wolfram Language code: Table[RegionMeasure[CantorMesh[n]], {n, 5}]//Rationalize

The formula:

Wolfram Language code: FindSequenceFunction[%, n]

A 2D Cantor mesh:

Wolfram Language code: CantorMesh[1, 2]

A 3D Cantor mesh:

Wolfram Language code: CantorMesh[1, 3]

Scope  (4)

A 1D Cantor mesh:

Wolfram Language code: CantorMesh[2]

A 2D Cantor mesh:

Wolfram Language code: CantorMesh[2, 2]

A 3D Cantor mesh:

Wolfram Language code: CantorMesh[2, 3]

The ^(th) approximation to the Cantor set:

Wolfram Language code: Table[CantorMesh[n], {n, 0, 5}]//Column

Options  (13)

DataRange  (1)

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

Wolfram Language code: CantorMesh[1, 2]
Wolfram Language code: RegionBounds[%]

Specify a different range:

Wolfram Language code: CantorMesh[1, 2, DataRange -> {{0, 1}, {0, 2}}]
Wolfram Language code: RegionBounds[%]

MeshCellHighlight  (2)

MeshCellHighlight allows you to specify highlighting for parts of a CantorMesh:

Wolfram Language code: CantorMesh[0, 2, MeshCellHighlight -> {{1, All} -> Red, {0, All} -> Black}]

Individual cells can be highlighted using their cell index:

Wolfram Language code: CantorMesh[0, 2, MeshCellHighlight -> {{1, 2} -> {Thick, Red}, {1, 3} -> {Dashed, Black}}]

Or by the cell itself:

Wolfram Language code: CantorMesh[0, 2, MeshCellHighlight -> {Line[{3, 4}] -> {Thick, Red}, Line[{2, 4}] -> {Dashed, Black}}]

MeshCellLabel  (2)

MeshCellLabel can be used to label parts of a CantorMesh:

Wolfram Language code: CantorMesh[0, 2, MeshCellLabel -> {0 -> "Index"}]

Individual cells can be labeled using their cell index:

Wolfram Language code: CantorMesh[0, 2, MeshCellLabel -> {{1, 1} -> "x", {1, 2} -> "y"}]

Or by the cell itself:

Wolfram Language code: CantorMesh[0, 2, MeshCellLabel -> {Line[{1, 3}] -> "x", Line[{3, 4}] -> "y"}]

MeshCellMarker  (1)

MeshCellMarker can be used to assign values to parts of a CantorMesh:

Wolfram Language code: CantorMesh[0, 2, MeshCellMarker -> {{0, 1} -> 1, {0, 2} -> 2, {0, 3} -> 3, {0, 4} -> 4}]

Use MeshCellLabel to show the markers:

Wolfram Language code: CantorMesh[0, 2, MeshCellMarker -> {{0, 1} -> 1, {0, 2} -> 2, {0, 3} -> 3, {0, 4} -> 4}, MeshCellLabel -> {0 -> "Marker"}]

MeshCellShapeFunction  (2)

MeshCellShapeFunction can be used to assign values to parts of a CantorMesh:

Wolfram Language code: CantorMesh[0, 3, MeshCellShapeFunction -> {1 -> (Tube[#1, .1]&)}]

Individual cells can be drawn using their cell index:

Wolfram Language code: CantorMesh[0, 3, MeshCellShapeFunction -> {{1, 2} -> (Tube[#1, .1]&)}]

Or by the cell itself:

Wolfram Language code: CantorMesh[0, 3, MeshCellShapeFunction -> {Line[{5, 7}] -> (Tube[#1, .1]&)}]

MeshCellStyle  (3)

MeshCellStyle allows you to specify styling for parts of a CantorMesh:

Wolfram Language code: CantorMesh[0, 2, MeshCellStyle -> {{1, All} -> Red, {0, All} -> Black}]

Individual cells can be highlighted using their cell index:

Wolfram Language code: CantorMesh[0, 2, MeshCellStyle -> {{1, 1} -> {Thick, Red}, {1, 2} -> {Dashed, Black}}]

Or by the cell itself:

Wolfram Language code: CantorMesh[0, 2, MeshCellStyle -> {Line[{1, 3}] -> {Thick, Red}, Line[{3, 4}] -> {Dashed, Black}}]

Give explicit color directives to specify colors for individual cells:

Wolfram Language code: CantorMesh[1, 2, MeshCellStyle -> {{2, All} -> Black, {2, 1} -> Red, {2, {2, 3}} -> Pink}]

PlotTheme  (2)

Use a theme with grid lines and a legend:

Wolfram Language code: CantorMesh[1, 2, PlotTheme -> "Detailed"]

Use a theme to draw a wireframe:

Wolfram Language code: CantorMesh[1, 2, PlotTheme -> "Lines"]

Applications  (4)

The Cantor set is generated from the unit interval by repeatedly removing the middle third of the cells:

Wolfram Language code: Column[{CantorMesh[0], CantorMesh[1], CantorMesh[2], CantorMesh[3]}]

In 2D:

Wolfram Language code: Row[{CantorMesh[0, 2], CantorMesh[1, 2], CantorMesh[2, 2]}, Spacer[60]]

In 3D:

Wolfram Language code: Row[{CantorMesh[0, 3], CantorMesh[1, 3], CantorMesh[2, 3]}, Spacer[60]]

Find the length of the Cantor mesh:

Wolfram Language code: Table[RegionMeasure[CantorMesh[k]], {k, 5}]//Rationalize

The general formula:

Wolfram Language code: FindSequenceFunction[%, n]

Find the measure of the 2D Cantor mesh:

Wolfram Language code: Table[RegionMeasure[CantorMesh[k, 2]], {k, 5}]//Rationalize

The general formula:

Wolfram Language code: FindSequenceFunction[%, n]

Find the measure of the 3D Cantor mesh:

Wolfram Language code: Table[{k, RegionMeasure[CantorMesh[k, 3]]}, {k, 0, 3}]//Rationalize

The general formula:

Wolfram Language code: FindSequenceFunction[%, n]

Properties & Relations  (4)

The output of CantorMesh is always a full-dimensional MeshRegion:

Wolfram Language code: CantorMesh[3, 2]
Wolfram Language code: {MeshRegionQ[%], RegionDimension[%]}

CantorMesh consists of intervals in 1D:

Wolfram Language code: CantorMesh[2]
Wolfram Language code: MeshCells[%, RegionDimension[%]]

Rectangles in 2D:

Wolfram Language code: CantorMesh[1, 2]
Wolfram Language code: MeshCells[%, RegionDimension[%]]

Hexahedrons in 3D:

Wolfram Language code: CantorMesh[0, 3]
Wolfram Language code: MeshCells[%, RegionDimension[%]]

The total length removed of the Cantor set is 1:

Wolfram Language code: Table[RegionMeasure[CantorMesh[n]] / 2, {n, 5}]//Rationalize

Guess the general formula:

Wolfram Language code: FindSequenceFunction[%, i]
Wolfram Language code: Limit[Sum[%, {i, 1, n}], n -> Infinity]

DataRange->range is equivalent to using RescalingTransform[{…},range]:

Wolfram Language code: CantorMesh[1, 2, DataRange -> {{1, 2}, {1, 3}}, Frame -> True]

Use RescalingTransform:

Wolfram Language code: box = TransformedRegion[CantorMesh[1, 2], RescalingTransform[{{0, 1}, {0, 1}}, {{1, 2}, {1, 3}}]];
Wolfram Language code: MeshRegion[box, Frame -> True]

Possible Issues  (1)

CantorMesh can be too large to generate:

Wolfram Language code: CantorMesh[20, 3]
Wolfram Research (2017), CantorMesh, Wolfram Language function, https://reference.wolfram.com/language/ref/CantorMesh.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_cantormesh, author="Wolfram Research", title="{CantorMesh}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/CantorMesh.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_cantormesh, organization={Wolfram Research}, title={CantorMesh}, year={2017}, url={https://reference.wolfram.com/language/ref/CantorMesh.html}, note=[Accessed: 13-June-2026]}