({{, (RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron])[poly] 
gives the augmented polyhedron poly by replacing each face by a pyramid. }, {, (RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron])[poly,h]
gives the augmented polyhedron with a pyramid of height h.}})

OpenersNotesSection(Spacer1)Details and Options

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) is also known as stellated polyhedron, kiss polyhedron or pyramid augmented polyhedron.

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) generates a (RefLink[Polyhedron, paclet:ref/Polyhedron]) consisting of faces obtained by replacing each face of poly with a pyramid of height h.

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) takes the same options as (RefLink[Polyhedron, paclet:ref/Polyhedron]).

OpenersPrimaryExamplesSection(Spacer1)Examples  (13)

OpenersExampleSection0(Spacer1)Basic Examples  (2)

Augmented polyhedron of a dodecahedron:

AugmentedPolyhedron[Dodecahedron[]]

Graphics3D[%]

Find the augmented polyhedron of the space shuttle:

𝒫=AugmentedPolyhedron[];

Visualization:

Graphics3D[𝒫,Boxed-> False]

OpenersExampleSection1(Spacer1)Scope  (4)

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) works on polyhedra:

𝒫 = Polyhedron[{{0,0,0},{1,0,0},{0,1,0},{0,0,1}},{{1,2,3},{1,2,4},{2,3,4},{1,3,4}}];

AugmentedPolyhedron[𝒫]

Graphics3D[%]

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) of Platonic solids includes (RefLink[Tetrahedron, paclet:ref/Tetrahedron]):

AugmentedPolyhedron[Tetrahedron[1]]

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

AugmentedPolyhedron[Cube[1]]

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

AugmentedPolyhedron[Dodecahedron[1]]

Graphics3D[%]

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

AugmentedPolyhedron[Octahedron[1]]

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

AugmentedPolyhedron[Icosahedron[1]]

Support polyhedron meshes:

𝒫=ExampleData[{"Geometry3D","SpaceShuttle"},"BoundaryMeshRegion"]

AugmentedPolyhedron[𝒫]

Graphics3D[%]

Augment the polyhedron by different heights:

𝒫 = Polyhedron[{{0,0,0},{1,0,0},{0,1,0},{0,0,1}},{{1,2,3},{1,2,4},{2,3,4},{1,3,4}}];

Table[Graphics3D[AugmentedPolyhedron[𝒫, h]],{h,{0.1,0.2,0.3,0.4}}]

OpenersExampleSection3(Spacer1)Applications  (4)

OpenersExampleSubsection0(Spacer1)Basic Applications  (3)

Gallery of Platonic solids and their augmented polyhedra:

Grid[Table[{Graphics3D[f[1],Boxed-> False],,Graphics3D[AugmentedPolyhedron[f[1]],Boxed-> False]},{f,{Tetrahedron,Cube,Octahedron,Dodecahedron,Icosahedron}}]]

Gallery of Archimedean solids and their augmented polyhedra:

Multicolumn[Table[Row[{Graphics3D[f,Boxed-> False,ImageSize-> 50],,Graphics3D[AugmentedPolyhedron[f],Boxed-> False,ImageSize-> 50]}],{f,PolyhedronData["Archimedean","Polyhedron"]}],2,Spacings->3]

Augmented compounds of Platonic solids:

Table[Graphics3D[{Opacity[0.5],AugmentedPolyhedron[f[1]],f[1]},Boxed-> False],{f,{Tetrahedron,Cube,Dodecahedron}}]

Archimedean solids:

Table[Graphics3D[{Opacity[0.5],f,AugmentedPolyhedron[f]},Boxed-> False],{f,PolyhedronData["Archimedean","Polyhedron"]}]

OpenersExampleSubsection1(Spacer1)Polyhedron Operations  (1)

Use (RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) to compute the polyhedron operations such as zip operation:

zip[poly_]:=DualPolyhedron[AugmentedPolyhedron[poly]]

zip[Tetrahedron[1]]

Graphics3D[%]

OpenersExampleSection4(Spacer1)Properties & Relations  (1)

The augmented simple polyhedron is simple:

𝒫  = Polyhedron[{{0,0,0},{1,0,0},{0,1,0},{0,0,1}},{{1,2,3},{1,2,4},{2,3,4},{1,3,4}}];

SimplePolyhedronQ[#]&/@{𝒫, AugmentedPolyhedron[𝒫]}

OpenersExampleSection5(Spacer1)Possible Issues  (2)

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) only supports simple polyhedra:

𝒫=;

SimplePolyhedronQ[𝒫]

AugmentedPolyhedron[𝒫]

(RefLink[AugmentedPolyhedron, paclet:ref/AugmentedPolyhedron]) can give degenerate polyhedra:

AugmentedPolyhedron[]

RegionQ[%]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[BeveledPolyhedron, paclet:ref/BeveledPolyhedron])  ▪  (RefLink[DualPolyhedron, paclet:ref/DualPolyhedron])  ▪  (RefLink[TruncatedPolyhedron, paclet:ref/TruncatedPolyhedron])  ▪  (RefLink[Polyhedron, paclet:ref/Polyhedron])}})}})

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