EulerCharacteristic
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EulerCharacteristic
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 allclose allBasic Examples (1)Summary of the most common use cases
Scope (4)Survey of the scope of standard use cases
EulerCharacteristic works on polyhedrons:
https://wolfram.com/xid/0dqvplzptbm-cpw0pk
https://wolfram.com/xid/0dqvplzptbm-nc5g0k
https://wolfram.com/xid/0dqvplzptbm-jmbj0g
https://wolfram.com/xid/0dqvplzptbm-3o58n0
https://wolfram.com/xid/0dqvplzptbm-ruxc6r
https://wolfram.com/xid/0dqvplzptbm-x0euoi
https://wolfram.com/xid/0dqvplzptbm-ho2agf
Polyhedrons with disconnected components:
https://wolfram.com/xid/0dqvplzptbm-inu7wu
https://wolfram.com/xid/0dqvplzptbm-npcjle
EulerCharacteristic works on mesh regions:
https://wolfram.com/xid/0dqvplzptbm-ixfnxk
https://wolfram.com/xid/0dqvplzptbm-7j6wmc
Properties & Relations (3)Properties of the function, and connections to other functions
Use EulerCharacteristic to compute PolyhedronGenus for a simple polyhedron:
https://wolfram.com/xid/0dqvplzptbm-vvce7b
https://wolfram.com/xid/0dqvplzptbm-jfkzmb
https://wolfram.com/xid/0dqvplzptbm-bf9i3o
Euler characteristic of a convex polyhedron equals 2:
https://wolfram.com/xid/0dqvplzptbm-3uzite
https://wolfram.com/xid/0dqvplzptbm-z6nm5w
https://wolfram.com/xid/0dqvplzptbm-co5kpj
Euler characteristic of UniformPolyhedron is 2:
https://wolfram.com/xid/0dqvplzptbm-drl6fq
https://wolfram.com/xid/0dqvplzptbm-6kl0pb
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.
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.
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
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: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_eulercharacteristic, organization={Wolfram Research}, title={EulerCharacteristic}, year={2019}, url={https://reference.wolfram.com/language/ref/EulerCharacteristic.html}, note=[Accessed: 08-January-2025
]}