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SimplePolyhedronQ[poly]

gives True if the polyhedron poly is simple and False otherwise.

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SimplePolyhedronQ

SimplePolyhedronQ[poly]

gives True if the polyhedron poly is simple and False otherwise.

Details

  • A polyhedron is simple if it has no voids and non-intersecting boundary faces.
  • A simple polyhedron is topologically equivalent to a ball.

Examples

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Basic Examples  (2)

Test whether a polyhedron is simple:

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: SimplePolyhedronQ[𝒫]
Wolfram Language code: Graphics3D[𝒫]

SimplePolyhedronQ gives False for non-simple polyhedrons:

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: Graphics3D[𝒫]

Scope  (3)

SimplePolyhedronQ works on polyhedrons:

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: SimplePolyhedronQ[𝒫]
Wolfram Language code: Region[𝒫]

Cube:

Wolfram Language code: SimplePolyhedronQ[Cube[]]

Dodecahedron:

Wolfram Language code: SimplePolyhedronQ[Dodecahedron[]]

Tetrahedron:

Wolfram Language code: SimplePolyhedronQ[Tetrahedron[]]

Polyhedron with holes:

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: SimplePolyhedronQ[%]

Polyhedrons with disconnected components:

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: SimplePolyhedronQ[%]

Properties & Relations  (3)

A convex polyhedron is simple:

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: SimplePolyhedronQ[𝒫]

The OuterPolyhedron of a simple polyhedron is simple:

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: SimplePolyhedronQ[OuterPolyhedron[𝒫]]

Simple polyhedrons do not have holes:

Wolfram Language code: InnerPolyhedron[𝒫]

The UniformPolyhedron is simple:

Wolfram Language code: UniformPolyhedron[{3, 5}]
Wolfram Language code: SimplePolyhedronQ[%]

Possible Issues  (1)

For nonconstant polyhedrons, SimplePolyhedronQ returns False:

Wolfram Language code: poly = Polyhedron[{{a, b, c}, {d, e, f}, {0, 1, 0}, {0, 0, 1}}, {{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}}];
Wolfram Language code: ConstantRegionQ[poly]
Wolfram Language code: SimplePolyhedronQ[poly]
Wolfram Research (2019), SimplePolyhedronQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SimplePolyhedronQ.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_simplepolyhedronq, author="Wolfram Research", title="{SimplePolyhedronQ}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/SimplePolyhedronQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_simplepolyhedronq, organization={Wolfram Research}, title={SimplePolyhedronQ}, year={2019}, url={https://reference.wolfram.com/language/ref/SimplePolyhedronQ.html}, note=[Accessed: 13-June-2026]}