# UniformPolyhedron

UniformPolyhedron["name"]

gives the uniform polyhedron with the given name.

UniformPolyhedron[{n,m}]

gives the uniform polyhedron with n sides of each face and m faces meeting at each vertex point.

UniformPolyhedron[{r,θ,ϕ},]

rescales the uniform polyhedron by a factor r and rotates by an angle θ with respect to the z axis and angle ϕ with respect to the y axis.

UniformPolyhedron[{x,y,z},{r,θ,ϕ},]

centers the uniform polyhedron at {x,y,z}.

# Details

• UniformPolyhedron is also known as Platonic solid, Archimedean solid or regular star polyhedron.
• UniformPolyhedron is typically used to generate base shapes for 3D modeling and as geometric regions.
• UniformPolyhedron generates a Polyhedron centered at the origin with unit edge length.
• Uniform polyhedrons can be specified with their standard names, Schläfli symbols {n,m} or Wenninger numbers, including:
•  {4,3} "Cube" {5,3} "Dodecahedron" {3,5} "Icosahedron" {3,4} "Octahedron" {3,3} "Tetrahedron"
•  {{5,2},5} "SmallStellatedDodecahedron" {{5,2},3} "GreatStellatedDodecahedron" {3,{5,2}} "GreatIcosahedron" {5,{5,2}} "GreatDodecahedron"

# Examples

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## Basic Examples(2)

A dodecahedron:

Get a list of uniform polyhedra:

## Scope(9)

### Basic Uses(6)

Generate an equilateral tetrahedron, octahedron, icosahedron, etc.:

Color directives specify the face colors of uniform polyhedrons:

FaceForm and EdgeForm can be used to specify the styles of the interior and boundary:

Uniform polyhedra are three-dimensional geometric regions:

Geometric dimension:

Find the geometric properties of a uniform polyhedron:

Centroid:

Surface area:

Modify the orientation of a uniform polyhedron:

Translate it:

### Specifications(3)

A uniform polyhedron can be specified by its standard Wolfram Language name:

Uniform polyhedra can also be specified in Schläfli notation:

Wenninger numbers:

Entities:

Alternate polyhedron numbering notations include Wenninger numbers:

Uniform numbers:

Kaleido numbers:

Coxeter numbers:

## Properties & Relations(4)

Using PolyhedronData to get a uniform polyhedron:

Use ConvexPolyhedronQ to check the property of a uniform polyhedron:

All faces of uniform polyhedrons are uniform polygons:

A uniform polyhedron is bounded:

Get its range:

## Neat Examples(2)

Generate cubes of varying radii:

Generate cubes of varying starting angles:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_uniformpolyhedron, author="Wolfram Research", title="{UniformPolyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/UniformPolyhedron.html}", note=[Accessed: 18-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_uniformpolyhedron, organization={Wolfram Research}, title={UniformPolyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/UniformPolyhedron.html}, note=[Accessed: 18-July-2024 ]}