OuterPolyhedron [poly]
gives the outer polyhedron of the polyhedron poly.
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
open all close allBasic Examples (1)
Get the outer polyhedron:
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}}];OuterPolyhedron[𝒫]Graphics3D[{Opacity[0.5], 𝒫}]Scope (3)
OuterPolyhedron works on polyhedrons:
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}}]]OuterPolyhedron[Tetrahedron[]]Cube :
OuterPolyhedron[Cube[]]Polyhedron with holes:
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}}}];OuterPolyhedron[𝒫]Polyhedrons with disconnected components:
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}}]OuterPolyhedron[𝒫]Properties & Relations (2)
OuterPolyhedron of a simple polyhedron is the polyhedron itself:
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}}];SimplePolyhedronQ[𝒫]OuterPolyhedron[𝒫]A simple polyhedron has the same polyhedron coordinates as its OuterPolyhedron :
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}}];SimplePolyhedronQ[𝒫]PolyhedronCoordinates[𝒫]PolyhedronCoordinates[OuterPolyhedron[𝒫]]Related Guides
History
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: 06-September-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: 06-September-2026]}