# EulerCharacteristic

EulerCharacteristic[poly]

gives the Euler characteristic of a poly.

# Details

• EulerCharacteristic is also known as Euler number or EulerPoincaré 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

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## Basic Examples(1)

Euler characteristic of a polyhedron:

## Scope(4)

EulerCharacteristic works on polyhedrons:

Polyhedron with holes:

Polyhedrons with disconnected components:

EulerCharacteristic works on mesh regions:

## Properties & Relations(3)

Use EulerCharacteristic to compute PolyhedronGenus for a simple polyhedron:

Euler characteristic of a convex polyhedron equals 2:

Euler characteristic of UniformPolyhedron is 2:

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_2024_eulercharacteristic, author="Wolfram Research", title="{EulerCharacteristic}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/EulerCharacteristic.html}", note=[Accessed: 30-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_eulercharacteristic, organization={Wolfram Research}, title={EulerCharacteristic}, year={2019}, url={https://reference.wolfram.com/language/ref/EulerCharacteristic.html}, note=[Accessed: 30-May-2024 ]}