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PolyhedronGenus[poly]

gives the genus of the polyhedron poly.

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PolyhedronGenus

PolyhedronGenus[poly]

gives the genus of the polyhedron poly.

Details

  • PolyhedronGenus is also known as genus.
  • PolyhedronGenus gives the maximum number of cuttings along non-intersecting closed simple curves on the surface of the polyhedron without disconnecting the polyhedron.

Examples

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Basic Examples  (1)

Genus of a polyhedron:

Wolfram Language code: 𝒫 = Polyhedron[{{0.1875, 0.108253, 0.}, {0.205078, 0., 0.046875}, {0.21875, 0., 0.}, {0.205078, 0., -0.046875}, {0.175781, 0.101487, -0.046875}, {0.102539, 0.177603, -0.046875}, {0.109375, 0.189443, 0.}, {0.102539, 0.177603, 0.046875}, {0.175781, ... {38, 88, 45}, {8, 90, 89}, {14, 90, 15}, {50, 90, 91}, {44, 90, 51}, {14, 92, 91}, {20, 92, 21}, {56, 92, 93}, {50, 92, 57}, {20, 94, 93}, {26, 94, 27}, {62, 94, 95}, {56, 94, 63}, {26, 96, 95}, {32, 96, 33}, {68, 96, 87}, {62, 96, 69}}];
Wolfram Language code: PolyhedronGenus[𝒫]
Wolfram Language code: Region[𝒫]

Scope  (3)

PolyhedronGenus works on polyhedrons:

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

Tetrahedron:

Wolfram Language code: PolyhedronGenus[Tetrahedron[]]

Hexahedron:

Wolfram Language code: PolyhedronGenus[Hexahedron[]]

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: PolyhedronGenus[%]

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: PolyhedronGenus[%]

Properties & Relations  (3)

Use EulerCharacteristic to compute PolyhedronGenus for a simple polyhedron:

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]

Convex polyhedron has zero genus:

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: PolyhedronGenus[𝒫]

The genus of UniformPolyhedron is 0:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_polyhedrongenus, author="Wolfram Research", title="{PolyhedronGenus}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/PolyhedronGenus.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_polyhedrongenus, organization={Wolfram Research}, title={PolyhedronGenus}, year={2019}, url={https://reference.wolfram.com/language/ref/PolyhedronGenus.html}, note=[Accessed: 12-June-2026]}