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DualPolyhedron[poly]

多面体 poly の双対多面体を与える．

詳細とオプション
Details and Options Details and Options
例題  
例  
スコープ  
アプリケーション  
基本的なアプリケーション  
多面体の操作  
特性と関係  
考えられる問題  
関連項目
関連するガイド
履歴
引用書式

DualPolyhedron

DualPolyhedron[poly]

多面体 poly の双対多面体を与える．

詳細とオプション

例題

すべて開く すべて閉じる

例  (2)

12面体の双対多面体：

Wolfram Language code: DualPolyhedron[Dodecahedron[]]
Wolfram Language code: Graphics3D[%]

スペースシャトルの双対多面体を求める：

Wolfram Language code: 𝒫 = DualPolyhedron[Polyhedron[{{-4.999492168426514, -0.6817100048065186, 0.569242000579834}, {-4.999759197235107, -0.4911530017852783, 0.8052060008049011}, {-5.349475860595703, -0.47093498706817627, 0.5660619735717773}, {-4.999759197235107, 0.491153001785278 ... }, {291, 218, 220}, {211, 259, 258}, {280, 206, 218}, {212, 258, 288}, {225, 187, 219}, {245, 197, 196}, {200, 236, 235}, {263, 196, 207}, {274, 205, 193}, {282, 210, 205}, {268, 193, 188}, {226, 219, 210}, {269, 188, 187}, {215, 288, 287}}]];

可視化する：

Wolfram Language code: Graphics3D[𝒫, Boxed -> False]

スコープ  (3)

DualPolyhedron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: DualPolyhedron[𝒫]
Wolfram Language code: Graphics3D[{Opacity[0.2], 𝒫, %}]

プラトンの立体のDualPolyhedronにはTetrahedronが含まれる：

Wolfram Language code: DualPolyhedron[Tetrahedron[1]]

Cube：

Wolfram Language code: DualPolyhedron[Cube[1]]
Wolfram Language code: Graphics3D[%]

Dodecahedron：

Wolfram Language code: DualPolyhedron[Dodecahedron[1]]

Octahedron：

Wolfram Language code: DualPolyhedron[Octahedron[1]]

Icosahedron：

Wolfram Language code: DualPolyhedron[Icosahedron[1]]

特別な多面体の表現：

Wolfram Language code: 𝒫 = ExampleData[{"Geometry3D", "SpaceShuttle"}, "BoundaryMeshRegion"]
Wolfram Language code: DualPolyhedron[𝒫]
Wolfram Language code: Graphics3D[{Opacity[0.2], 𝒫, %}]

アプリケーション  (9)

基本的なアプリケーション  (3)

プラトンの立体とその双対：

Wolfram Language code: Grid[Table[{Graphics3D[f[1], Boxed -> False], [image], Graphics3D[DualPolyhedron[f[1]], Boxed -> False]}, {f, {Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron}}]]

アルキメデスの立体とその双対：

Wolfram Language code: Multicolumn[Table[Row[{Graphics3D[f, Boxed -> False, ImageSize -> 50], [image], Graphics3D[DualPolyhedron[f], Boxed -> False, ImageSize -> 50]}], {f, PolyhedronData["Archimedean", "Polyhedron"]}], 2, Spacings -> 3]

プラトンの立体の双対複合多面体：

Wolfram Language code: Table[Graphics3D[{Opacity[0.5], DualPolyhedron[f[1]], f[1]}, Boxed -> False], {f, {Tetrahedron, Cube, Dodecahedron}}]

アルキメデスの立体：

Wolfram Language code: Table[Graphics3D[{Opacity[0.5], f, DualPolyhedron[f]}, Boxed -> False], {f, PolyhedronData["Archimedean", "Polyhedron"]}]

多面体の操作  (6)

DualPolyhedronを使って，針操作のような多面体の操作を計算する：

Wolfram Language code: needle[poly_] := DualPolyhedron[TruncatedPolyhedron[poly]]
Wolfram Language code: needle[Tetrahedron[1]]
Wolfram Language code: Graphics3D[%]

メタ操作：

Wolfram Language code: meta[poly_] := DualPolyhedron[BeveledPolyhedron[poly]]
Wolfram Language code: meta[Tetrahedron[1]]
Wolfram Language code: Graphics3D[%]

結合操作：

Wolfram Language code: join[poly_] := DualPolyhedron[TruncatedPolyhedron[poly]]
Wolfram Language code: join[Tetrahedron[1]]
Wolfram Language code: Graphics3D[%]

圧縮操作：

Wolfram Language code: zip[poly_] := DualPolyhedron[AugmentedPolyhedron[poly]]
Wolfram Language code: zip[Tetrahedron[1]]
Wolfram Language code: Graphics3D[%]

オルソ操作：

Wolfram Language code: ortho[poly_] := DualPolyhedron[TruncatedPolyhedron[TruncatedPolyhedron[poly]]]
Wolfram Language code: ortho[Tetrahedron[1]]
Wolfram Language code: Graphics3D[%]

拡張操作：

Wolfram Language code: expand[poly_] := DualPolyhedron[TruncatedPolyhedron[TruncatedPolyhedron[poly, 0.5]]]
Wolfram Language code: expand[Tetrahedron[1]]
Wolfram Language code: Graphics3D[%]

特性と関係  (4)

単純多面体の双対は単純多面体である：

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: SimplePolyhedronQ[#]& /@ {𝒫, DualPolyhedron[𝒫]}

プラトンの立体の双対はプラトンの立体である：

Wolfram Language code: 𝒫 = Cube[1];
Wolfram Language code: DualPolyhedron[𝒫]
Wolfram Language code: {Graphics3D[𝒫], Graphics3D[%]}

双対の双対はもとの多面体である：

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: DualPolyhedron[DualPolyhedron[𝒫]];
Wolfram Language code: {Graphics3D[𝒫], Graphics3D[%]}

錐体は自己双対多面体である：

Wolfram Language code: DualPolyhedron[Pyramid[]]
Wolfram Language code: Graphics3D[%]

考えられる問題  (3)

DualPolyhedronがサポートするのは単純多面体のみである：

Wolfram Language code: 𝒫 = Polyhedron[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 0}, {2, 0, 0}, {1, 1, 0}, {1, 0, 1}}, {{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: SimplePolyhedronQ[𝒫]
Wolfram Language code: DualPolyhedron[𝒫]

多面体の双対は位相的双対である：

Wolfram Language code: dual = DualPolyhedron[Tetrahedron[1]]

双対の双対：

Wolfram Language code: DualPolyhedron[dual]

DualPolyhedronは退化した多面体を返すことがある：

Wolfram Language code: DualPolyhedron[Polyhedron[{{-1, 0, 0}, {-1/2, -1/2, -(1/Sqrt[2])}, {-1/2, -1/2, 1/Sqrt[2]}, {-1/2, 1/2, -(1/Sqrt[2])}, {-1/2, 1/2, 1/Sqrt[2]}, {0, -1, 0}, {0, 1, 0}, {1/2, -1/2, -(1/Sqrt[2])}, {1/2, -1/2, 1/Sqrt[2]}, {1/2, 1/2, -(1/Sqrt[2])}, {1/2, 1/2, 1/Sqrt[2]}, {1, 0, 0}}, {{4, 10, 8, 2}, {3, 9, 11, 5}, {9, 6, 8, 12}, {3, 1, 2, 6}, {5, 7, 4, 1}, {11, 12, 10, 7}, {12, 11, 9}, {3, 5, 1}, {6, 9, 3}, {5, 11, 7}, {8, 10, 12}, {1, 4, 2}, {2, 8, 6}, {7, 10, 4}}]]
Wolfram Language code: RegionQ[%]
Wolfram Research (2019), DualPolyhedron, Wolfram言語関数， https://reference.wolfram.com/language/ref/DualPolyhedron.html.

テキスト

Wolfram Research (2019), DualPolyhedron, Wolfram言語関数， https://reference.wolfram.com/language/ref/DualPolyhedron.html.

CMS

Wolfram Language. 2019. "DualPolyhedron." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DualPolyhedron.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_dualpolyhedron, author="Wolfram Research", title="{DualPolyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/DualPolyhedron.html}", note=[Accessed: 13-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_dualpolyhedron, organization={Wolfram Research}, title={DualPolyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/DualPolyhedron.html}, note=[Accessed: 13-September-2026]}