WOLFRAM

Wolfram Language & System Documentation Center

EulerCharacteristic[poly]

gives the Euler characteristic of a poly.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

EulerCharacteristic

EulerCharacteristic[poly]

gives the Euler characteristic of a poly.

Details

  • EulerCharacteristic is also known as Euler number or Euler–Poincaré characteristic.
  • EulerCharacteristic is a topological invariant that describes the shape of the polyhedron, regardless of the way it is bent.
  • The Euler characteristic for a polyhedron is given by , where is the number of vertices, the number of edges and the number of faces.
  • A polyhedron with voids and tunnels satisfies .
  • The Euler characteristic for a mesh region is given by χ=(-1)nMeshCellCount[poly,n].

Examples

open all close all

Basic Examples  (1)

Euler characteristic of a polyhedron:

Wolfram Language code: 𝒫 = Polyhedron[{{-Sqrt[1 + 2/Sqrt[5]], 0, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, {Sqrt[1 + 2/Sqrt[5]], 0, Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0]}, {Root[1 - 20*#1^2 + 80*#1^4 & , 1, 0], (-3 - Sqrt[5])/4, Root[1 - 20*#1^2 + 80*#1^4 & , 3, 0]}, {R ... {{15, 10, 9, 14, 1}, {2, 6, 12, 11, 5}, {5, 11, 7, 3, 19}, {11, 12, 8, 16, 7}, {12, 6, 20, 4, 8}, {6, 2, 13, 18, 20}, {2, 5, 19, 17, 13}, {4, 20, 18, 10, 15}, {18, 13, 17, 9, 10}, {17, 19, 3, 14, 9}, {3, 7, 16, 1, 14}, {16, 8, 4, 15, 1}}];
Wolfram Language code: EulerCharacteristic[𝒫]
Wolfram Language code: Region[𝒫]

Scope  (4)

EulerCharacteristic works on polyhedrons:

Wolfram Language code: 𝒫 = Polyhedron[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}];
Wolfram Language code: EulerCharacteristic[𝒫]
Wolfram Language code: Region[𝒫]

Tetrahedron:

Wolfram Language code: EulerCharacteristic[Tetrahedron[]]

Hexahedron:

Wolfram Language code: EulerCharacteristic[Hexahedron[]]

Polyhedron with holes:

Wolfram Language code: Polyhedron[{{0, 0, 0}, {0, 3, 0}, {3, 3, 0}, {3, 0, 0}, {0, 0, 3}, {0, 3, 3}, {3, 3, 3}, {3, 0, 3}, {1, 1, 1}, {1, 2, 1}, {2, 2, 1}, {2, 1, 1}, {1, 1, 2}, {1, 2, 2}, {2, 2, 2}, {2, 1, 2}}, {{{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}} -> {{{10, 11, 12, 9}, {9, 12, 16, 13}, {12, 11, 15, 16}, {11, 10, 14, 15}, {10, 9, 13, 14}, {13, 16, 15, 14}}}}];
Wolfram Language code: EulerCharacteristic[%]

Polyhedrons with disconnected components:

Wolfram Language code: Polyhedron[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 1, 1}, {2, 1, 1}, {1, 2, 1}, {1, 1, 2}}, {{{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}, {{5, 6, 7}, {5, 6, 8}, {6, 7, 8}, {5, 7, 8}}}]
Wolfram Language code: EulerCharacteristic[%]

EulerCharacteristic works on mesh regions:

Wolfram Language code: MengerMesh[1, 3]
Wolfram Language code: EulerCharacteristic[%]

Properties & Relations  (3)

Use EulerCharacteristic to compute PolyhedronGenus for a simple polyhedron:

Wolfram Language code: 𝒫 = Polyhedron[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}];
Wolfram Language code: SimplePolyhedronQ[𝒫]
Wolfram Language code: 1 - EulerCharacteristic[𝒫] / 2 == PolyhedronGenus[𝒫]

Euler characteristic of a convex polyhedron equals 2:

Wolfram Language code: 𝒫 = Polyhedron[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}];
Wolfram Language code: ConvexPolyhedronQ[𝒫]
Wolfram Language code: EulerCharacteristic[𝒫] == 2

Euler characteristic of UniformPolyhedron is 2:

Wolfram Language code: UniformPolyhedron[{3, 5}]
Wolfram Language code: EulerCharacteristic[%]
Wolfram Research (2019), EulerCharacteristic, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerCharacteristic.html.

Text

Wolfram Research (2019), EulerCharacteristic, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerCharacteristic.html.

CMS

Wolfram Language. 2019. "EulerCharacteristic." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/EulerCharacteristic.html.

APA

Wolfram Language. (2019). EulerCharacteristic. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EulerCharacteristic.html

BibTeX

@misc{reference.wolfram_2026_eulercharacteristic, author="Wolfram Research", title="{EulerCharacteristic}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/EulerCharacteristic.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_eulercharacteristic, organization={Wolfram Research}, title={EulerCharacteristic}, year={2019}, url={https://reference.wolfram.com/language/ref/EulerCharacteristic.html}, note=[Accessed: 13-June-2026]}