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EulerCharacteristic[poly]

给出 poly 的欧拉示性数.

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范例  
基本范例  
范围  
属性和关系  
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EulerCharacteristic

EulerCharacteristic[poly]

给出 poly 的欧拉示性数.

更多信息

  • EulerCharacteristic 亦成为欧拉数或 Euler–Poincaré 示性数.
  • EulerCharacteristic 是一个描述多面体形状的拓扑不变量，不考虑它弯曲的方式.
  • 多面体的欧拉示性数 由 给出，其中 是顶点数， 为边数， 为面数.
  • 有 个洞和 个巷道的多面体满足 .
  • 网格区域的欧拉示性数由 χ=(-1)nMeshCellCount[poly,n] 给出.

范例

打开所有单元 关闭所有单元

基本范例  (1)

多面体的欧拉示性数：

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]

范围  (4)

EulerCharacteristic 适用于多面体：

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[]]

有洞的多面体：

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[%]

有断开组件的多面体：

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 适用于网格区域：

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

属性和关系  (3)

用 EulerCharacteristic 计算简单多面体的 PolyhedronGenus：

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]

凸多面体的欧拉示性数等于 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

UniformPolyhedron 的欧拉示性数为 2：

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

文本

Wolfram Research (2019)，EulerCharacteristic，Wolfram 语言函数，https://reference.wolfram.com/language/ref/EulerCharacteristic.html.

CMS

Wolfram 语言. 2019. "EulerCharacteristic." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/EulerCharacteristic.html.

APA

Wolfram 语言. (2019). EulerCharacteristic. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/EulerCharacteristic.html 年

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: 08-September-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: 08-September-2026]}