WOLFRAM

Wolfram Language & System Documentation Center

CanonicalizePolyhedron[poly]

gives a canonical representation of the polyhedron poly with shared coordinates and with inner and outer boundaries.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
Cite this Page

CanonicalizePolyhedron

CanonicalizePolyhedron[poly]

gives a canonical representation of the polyhedron poly with shared coordinates and with inner and outer boundaries.

Details

  • CanonicalizePolyhedron is used to get a simple standard representation of a polyhedron from various representations and descriptions.
  • CanonicalizePolyhedron converts a polyhedron to an optimized standard form Polyhedron[{p1,p2,…},{outer1,outer2inner2,…}].
  • The points pi are the vertex points of non-intersecting polygonal faces and sorted into Sort order.
  • An outer boundary outeri is a closed surface with polygonal faces {fi1,fi2,…}, possibly touching at edges.
  • An inner boundary inneri is a closed surface with polygonal faces {fj1,fj2,…}, possibly touching at edges.

Examples

open all close all

Basic Examples  (1)

Find the canonical form of a Polyhedron:

Wolfram Language code: 𝒫 = Polyhedron[{{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}, {{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}}];
Wolfram Language code: CanonicalizePolyhedron[𝒫]
Wolfram Language code: Graphics3D[%]

Scope  (3)

CanonicalizePolyhedron works on polyhedra:

Wolfram Language code: Polyhedron[{{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}, {{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}}];
Wolfram Language code: CanonicalizePolyhedron[%]

Tetrahedron:

Wolfram Language code: CanonicalizePolyhedron[Tetrahedron[]]

Octahedron:

Wolfram Language code: CanonicalizePolyhedron[Octahedron[]]

Polyhedron with voids:

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: CanonicalizePolyhedron[%]
Wolfram Language code: Graphics3D[{Opacity[0.5], %}]

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: CanonicalizePolyhedron[%]

Applications  (1)

Set up a graphics complex with shared coordinates:

Wolfram Language code: 𝒞𝒫 = CanonicalizePolyhedron[Polyhedron[{{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}, {{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}}]]
Wolfram Language code: Graphics3D[{GraphicsComplex[𝒞𝒫[[1]], Polyhedron[𝒞𝒫[[2]]]]}]

Properties & Relations  (5)

Using CanonicalizePolyhedron to get PolyhedronCoordinates:

Wolfram Language code: 𝒫 = 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}}];
Wolfram Language code: First[CanonicalizePolyhedron[𝒫]]
Wolfram Language code: PolyhedronCoordinates[𝒫]

The CanonicalizePolyhedron of a platonic solid is a polyhedron:

Wolfram Language code: CanonicalizePolyhedron[Icosahedron[]]

The CanonicalizePolyhedron of simple polyhedra preserve the number of polyhedron coordinates:

Wolfram Language code: 𝒫 = Polyhedron[{{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{0, 1, 0}, {1, 0, 0}, {0, 0, 0}}, {{0, 0, 1}, {0, 0, 0}, {1, 0, 0}}, {{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}}];
Wolfram Language code: SimplePolyhedronQ[𝒫]
Wolfram Language code: Length[PolyhedronCoordinates[#]]& /@ {𝒫, CanonicalizePolyhedron[𝒫]}

OuterPolyhedron gives the canonical representation of the outer polyhedron:

Wolfram Language code: OuterPolyhedron[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}}]]

InnerPolyhedron gives the canonical representation of the inner polyhedron:

Wolfram Language code: InnerPolyhedron[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}}]]

Possible Issues  (1)

CanonicalizePoyhedron works only on geometric regions:

Wolfram Language code: 𝒫 = Polyhedron[{{0, 0, 0}, {1, 1, 0}, {0, 1, 0}, {0, 0, 1}}, {{2, 3, 4}, {3, 2, 1}}];
Wolfram Language code: {CanonicalizePolyhedron[𝒫], RegionQ[𝒫]}
Wolfram Research (2019), CanonicalizePolyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/CanonicalizePolyhedron.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_canonicalizepolyhedron, author="Wolfram Research", title="{CanonicalizePolyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/CanonicalizePolyhedron.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_canonicalizepolyhedron, organization={Wolfram Research}, title={CanonicalizePolyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/CanonicalizePolyhedron.html}, note=[Accessed: 13-June-2026]}