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OuterPolyhedron[poly]

gives the outer polyhedron of the polyhedron poly.

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Basic Examples  
Scope  
Properties & Relations  
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OuterPolyhedron

OuterPolyhedron[poly]

gives the outer polyhedron of the polyhedron poly.

Details

  • OuterPolyhedron is also known as polyhedron outer shell.
  • Typically used to decompose a polyhedron as a difference of simple polyhedra, even when the original construction of the polyhedron was using crossing curves etc.
  • OuterPolyhedron is defined by the canonicalization performed in CanonicalizePolyhedron.
  • OuterPolyhedron gives a polyhedron of the form Polyhedron[{p1,p2,…},{{f1,f2,…},…}], where pk are explicit coordinates, and fk are integer lists referring to polygon faces.
  • In general, the result will be a Polyhedron object representing a disjoint union of simple polygons.

Examples

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Basic Examples  (1)

Get the outer polyhedron:

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 ... }, {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}, {22, 23, 24}, {23, 22, 21}, {24, 21, 22}, {21, 24, 23}}];
Wolfram Language code: OuterPolyhedron[𝒫]
Wolfram Language code: Graphics3D[{Opacity[0.5], 𝒫}]

Scope  (3)

OuterPolyhedron works on polyhedrons:

Wolfram Language code: OuterPolyhedron[Polyhedron[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {1, 0, 1}}, {{2, 3, 4, 1}, {1, 4, 8, 5}, {4, 3, 7, 8}, {3, 2, 6, 7}, {2, 1, 5, 6}, {5, 8, 7, 6}}]]

Tetrahedron:

Wolfram Language code: OuterPolyhedron[Tetrahedron[]]

Cube:

Wolfram Language code: OuterPolyhedron[Cube[]]

Polyhedron with holes:

Wolfram Language code: 𝒫 = 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: OuterPolyhedron[𝒫]

Polyhedrons with disconnected components:

Wolfram Language code: 𝒫 = 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: OuterPolyhedron[𝒫]

Properties & Relations  (2)

OuterPolyhedron of a simple polyhedron is the polyhedron itself:

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: OuterPolyhedron[𝒫]

A simple polyhedron has the same polyhedron coordinates as its OuterPolyhedron:

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: PolyhedronCoordinates[𝒫]
Wolfram Language code: PolyhedronCoordinates[OuterPolyhedron[𝒫]]
Wolfram Research (2019), OuterPolyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/OuterPolyhedron.html.

Text

Wolfram Research (2019), OuterPolyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/OuterPolyhedron.html.

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_outerpolyhedron, author="Wolfram Research", title="{OuterPolyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/OuterPolyhedron.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_outerpolyhedron, organization={Wolfram Research}, title={OuterPolyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/OuterPolyhedron.html}, note=[Accessed: 13-June-2026]}