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:

SimplePolyhedronQ:

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_2023_uniformpolyhedron, author="Wolfram Research", title="{UniformPolyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/UniformPolyhedron.html}", note=[Accessed: 19-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_uniformpolyhedron, organization={Wolfram Research}, title={UniformPolyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/UniformPolyhedron.html}, note=[Accessed: 19-April-2024 ]}