({{, (RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron])["name"]
gives the uniform polyhedron with the given name.}, {, (RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron])[{n,m}]
gives the uniform polyhedron with n sides of each face and m faces meeting at each vertex point.}, {, (RefLink[UniformPolyhedron, paclet:ref/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.}, {, (RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron])[{x,y,z},{r,θ,ϕ},…]
centers the uniform polyhedron at {x,y,z}.}})

OpenersNotesSection(Spacer1)Details

(RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron]) is also known as Platonic solid, Archimedean solid or regular star polyhedron.

(RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron]) is typically used to generate base shapes for 3D modeling and as geometric regions.

(RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron]) generates a (RefLink[Polyhedron, paclet:ref/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:

OpenersPrimaryExamplesSection(Spacer1)Examples  (18)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

A dodecahedron:

poly=UniformPolyhedron["Dodecahedron"]

Graphics3D[poly]

Get a list of uniform polyhedra:

EntityList[LinguisticAssistant]

Graphics3D[UniformPolyhedron[#]] & /@ Take[%,6]

OpenersExampleSection1(Spacer1)Scope  (9)

OpenersExampleSubsection0(Spacer1)Basic Uses  (6)

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

Table[Graphics3D[UniformPolyhedron[{3,m}]],{m,3,5}]

Color directives specify the face colors of uniform polyhedrons:

Table[Graphics3D[{c,UniformPolyhedron[{3,5}]}],{c,{Red,Green,Blue,Yellow}}]

(RefLink[FaceForm, paclet:ref/FaceForm]) and (RefLink[EdgeForm, paclet:ref/EdgeForm]) can be used to specify the styles of the interior and boundary:

Graphics3D[{FaceForm[Pink],EdgeForm[Directive[Dashed,Thick,Blue]],UniformPolyhedron[{3,5}]}]

Uniform polyhedra are 3-dimensional geometric regions:

RegionQ[UniformPolyhedron[{3,4}]]

Geometric dimension:

RegionDimension[UniformPolyhedron[{3,4}]]

Find the geometric properties of a uniform polyhedron:

Volume[UniformPolyhedron["Cube"]]

Centroid:

RegionCentroid[UniformPolyhedron["Cube"]]

Surface area:

SurfaceArea[UniformPolyhedron["Cube"]]

Modify the orientation of a uniform polyhedron:

UniformPolyhedron[{1,Pi/3,Pi/4},"Cube"]

Graphics3D[%]

Translate it:

UniformPolyhedron[{1,2,3},{1,Pi/3,Pi/4},"Cube"]

OpenersExampleSubsection1(Spacer1)Specifications  (3)

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

UniformPolyhedron["Cube"]

Graphics3D[%]

Uniform polyhedra can also be specified in Schläfli notation:

UniformPolyhedron[{4,3}]

Wenninger numbers:

UniformPolyhedron[3]

Entities:

UniformPolyhedron[LinguisticAssistant]

Alternate polyhedron numbering notations include Wenninger numbers:

UniformPolyhedron[{"Wenninger",4}]

Uniform numbers:

UniformPolyhedron[{"Uniform",22}]

Kale numbers:

UniformPolyhedron[{"Kaleido",27}]

Coxeter numbers:

UniformPolyhedron[{"Coxeter",25}]

OpenersExampleSection4(Spacer1)Properties & Relations  (4)

Using (RefLink[PolyhedronData, paclet:ref/PolyhedronData]) to get a uniform polyhedron:

PolyhedronData["Dodecahedron"]

Graphics3D[UniformPolyhedron["Dodecahedron"]]

Use (RefLink[ConvexPolyhedronQ, paclet:ref/ConvexPolyhedronQ]) to check the property of a uniform polyhedron:

poly=UniformPolyhedron["Dodecahedron"]

ConvexPolyhedronQ[poly]

(RefLink[SimplePolyhedronQ, paclet:ref/SimplePolyhedronQ]):

SimplePolyhedronQ[poly]

All faces of uniform polyhedrons are uniform polygons:

poly=UniformPolyhedron["Dodecahedron"]

faces = poly[[2]]

Graphics3D[Polygon[PolyhedronCoordinates[poly][[#]]]]&/@faces

A uniform polyhedron is bounded:

poly=UniformPolyhedron[{3,5}];

BoundedRegionQ[poly]

Get its range:

rr=RegionBounds[poly]

Graphics3D[{Red,poly,{EdgeForm[{Dashed,Red}],Opacity[0.1,Yellow],Cuboid@@Transpose[rr]}}]

OpenersExampleSection5(Spacer1)Possible Issues  (1)

(RefLink[UniformPolyhedron, paclet:ref/UniformPolyhedron]) gives polyhedra with exact coordinates:

poly=UniformPolyhedron["GreatStellatedDodecahedron"]

PolyhedronCoordinates[poly]//Short

Convert to approximate numbers for fast computation:

Nest[BeveledPolyhedron[#,1/3]&,N[poly],2]//AbsoluteTiming

Compare to:

Nest[BeveledPolyhedron[#,1/3]&,poly,2]//AbsoluteTiming

OpenersExampleSection6(Spacer1)Neat Examples  (2)

Generate cubes of varying radii:

polys=Table[UniformPolyhedron[r,{4,3}],{r,1,4}];

Graphics3D[{Opacity[0.2],polys}]

Generate cubes of varying starting angles:

polys=Table[UniformPolyhedron[{1,t Pi/6, Pi/3},{4,3}],{t,4}];

Graphics3D[{Opacity[0.2],polys}]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[Cube, paclet:ref/Cube])  ▪  (RefLink[Dodecahedron, paclet:ref/Dodecahedron])  ▪  (RefLink[Icosahedron, paclet:ref/Icosahedron])  ▪  (RefLink[Octahedron, paclet:ref/Octahedron])  ▪  (RefLink[Tetrahedron, paclet:ref/Tetrahedron])  ▪  (RefLink[Polyhedron, paclet:ref/Polyhedron])  ▪  (RefLink[PolyhedronData, paclet:ref/PolyhedronData])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Basic Geometric Regions, paclet:guide/GeometricSpecialRegions])}, {▪(RefLinkPlain[Polyhedra, paclet:guide/Polyhedra])}})}})