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EulerCharacteristic[poly]

poly のオイラー(Euler)標数を与える．

詳細
Details and Options Details and Options
例題  
例  
スコープ  
特性と関係  
関連項目
関連するガイド
履歴
引用書式

EulerCharacteristic

EulerCharacteristic[poly]

poly のオイラー(Euler)標数を与える．

詳細

  • EulerCharacteristicは，オイラー数あるいはオイラー・ポアンカレ(Poincaré)標数としても知られている．
  • EulerCharacteristicは，多面体の形状をその曲げられ方とは無関係に説明する，位相不変量である．
  • 多面体のオイラー標数 は で与えられる．ただし， は頂点数， は辺の数， は面の数である．
  • 個の空隙と 個のトンネルを持つ多面体は を満足する．
  • メッシュ領域のオイラー標数は χ=(-1)nMeshCellCount[poly,n]で与えられる．

例題

すべて開く すべて閉じる

例  (1)

多面体のオイラー標数：

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]

スコープ  (4)

EulerCharacteristicは多面体に使うことができる：

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[]]

穴のある多面体：

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[%]

不連続成分がある多面体：

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はメッシュ領域に使うことができる：

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

特性と関係  (3)

EulerCharacteristicを使って単純な多面体のPolyhedronGenusを計算する：

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]

凸多面体のオイラー標数は2である：

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のオイラー標数は2である：

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

テキスト

Wolfram Research (2019), EulerCharacteristic, Wolfram言語関数， 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: 05-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: 05-September-2026]}