# Polyhedron

Polyhedron[{f1,,fn}]

represents a filled polyhedron inside the closed surfaces with polygon faces fi.

Polyhedron[{f1,,fn}{{g1,,gm},}]

represents a polyhedron with voids {g1,,gm},.

Polyhedron[{poly1,poly2,}]

represents a collection of polyhedra polyi.

Polyhedron[{p1,,pn},data]

represents a polyhedron in which coordinates given as integers i in data are taken to be pi.

# Details and Options

• Polyhedron can be used a geometric region and a graphics primitive.
• Polyhedron[{f1,,fn}] is a volume region, representing all the points inside the closed surface with polygon faces fi.
• A point is an element of the polyhedron if a ray from the point in any direction crosses the boundary polygon faces an odd number of times.
• Polyhedron[{f1,,fn}{{g1,,gm},}] specifies a polyhedron with voids consisting of an outer polyhedron Polyhedron[{f1,,fn}] and one or several inner polyhedra Polyhedron[{g1,,gm}],.
• A point p is an element of the polyhedron if it is in the outer polyhedron but not in any inner polyhedron.
• Polyhedron[{poly1,poly2,}] is a collection of polyhedra polyi with or without voids and is treated as a union of polyi for geometric computations.
• Polyhedron[{p1,,pn},data] effectively replaces integers i that appear as coordinates in data by the corresponding pi.
•  Polyhedron[{p1,…,pn},{f1,…,fn}] polyhedron boundary faces fi with points {po1,…,pok} Polyhedron[{p1,…,pn},{{f1,…,fk}{{g1,…,gl},…}] outer polyhedron boundary faces fi with points {po1,…,pok} and inner polyhedron boundary faces gj with points {pv1,…,pvl} etc. Polyhedron[{p1,…,pn},{{b1,…,bn},{f1,…,fk}{{g1,…,gl},…},…}] a collection of several polyhedra
• As a geometric region, the polygon faces fi can have any embedding dimension, but must all be simple polygons and have the same embedding dimension.
• In a graphics, the points of the polygon faces fi can be Scaled and Dynamic expressions.
• Graphics renderings is affected by directives such as FaceForm, EdgeForm, Texture, Specularity, Opacity and color.
• The following options and settings can be used in graphics:
•  VertexColors Automatic vertex colors to be interpolated VertexNormals Automatic effective vertex normals for shading VertexTextureCoordinates None coordinates for textures

# Examples

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## Basic Examples(1)

A polyhedron:

Its graphic image:

Its volume:

## Scope(11)

### Graphics(8)

#### Specification(2)

Polyhedra:

Polyhedra with voids:

#### Styling(5)

Color directives specify the face colors of polyhedra:

Texture can be used to specify a texture to be used on the faces of polyhedra:

Texture can work together with a different Opacity:

Texture can work together with different Lighting:

FaceForm and EdgeForm can be used to specify the styles of the edges and faces:

Colors can be specified at vertices using VertexColors:

Normals can be specified at vertices using VertexNormals for polyhedra:

#### Coordinates(1)

Use Scaled coordinates:

### Regions(3)

Embedding dimension:

Geometric dimension:

Volume:

Centroid:

A polyhedron is bounded:

Find its range:

## Possible Issues(1)

Degenerate polyhedra are not valid geometric regions:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_polyhedron, author="Wolfram Research", title="{Polyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Polyhedron.html}", note=[Accessed: 14-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_polyhedron, organization={Wolfram Research}, title={Polyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/Polyhedron.html}, note=[Accessed: 14-July-2024 ]}