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EulerCharacteristic
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EulerCharacteristic

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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 allclose all

Basic Examples  (1)Summary of the most common use cases

Euler characteristic of a polyhedron:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-b6vqy4
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-2e6img
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqvplzptbm-8ezgj0
Out[3]=3

Scope  (4)Survey of the scope of standard use cases

EulerCharacteristic works on polyhedrons:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-cpw0pk
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-nc5g0k
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqvplzptbm-jmbj0g
Out[3]=3

Tetrahedron:

In[4]:=4
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https://wolfram.com/xid/0dqvplzptbm-3o58n0
Out[4]=4

Hexahedron:

In[5]:=5
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https://wolfram.com/xid/0dqvplzptbm-ruxc6r
Out[5]=5

Polyhedron with holes:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-x0euoi
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-ho2agf
Out[2]=2

Polyhedrons with disconnected components:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-inu7wu
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-npcjle
Out[2]=2

EulerCharacteristic works on mesh regions:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-ixfnxk
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-7j6wmc
Out[2]=2

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

Use EulerCharacteristic to compute PolyhedronGenus for a simple polyhedron:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-vvce7b
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-jfkzmb
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqvplzptbm-bf9i3o
Out[3]=3

Euler characteristic of a convex polyhedron equals 2:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-3uzite
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-z6nm5w
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dqvplzptbm-co5kpj
Out[3]=3

Euler characteristic of UniformPolyhedron is 2:

In[1]:=1
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https://wolfram.com/xid/0dqvplzptbm-drl6fq
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dqvplzptbm-6kl0pb
Out[2]=2
Wolfram Research (2019), EulerCharacteristic, Wolfram Language function, https://reference.wolfram.com/language/ref/EulerCharacteristic.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_eulercharacteristic, author="Wolfram Research", title="{EulerCharacteristic}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/EulerCharacteristic.html}", note=[Accessed: 16-May-2025 ]}

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@misc{reference.wolfram_2025_eulercharacteristic, author="Wolfram Research", title="{EulerCharacteristic}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/EulerCharacteristic.html}", note=[Accessed: 16-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_eulercharacteristic, organization={Wolfram Research}, title={EulerCharacteristic}, year={2019}, url={https://reference.wolfram.com/language/ref/EulerCharacteristic.html}, note=[Accessed: 16-May-2025 ]}

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@online{reference.wolfram_2025_eulercharacteristic, organization={Wolfram Research}, title={EulerCharacteristic}, year={2019}, url={https://reference.wolfram.com/language/ref/EulerCharacteristic.html}, note=[Accessed: 16-May-2025 ]}