Some of my works related to stellation of polyhedra are summarized here.
Cube to rhombic dodecahedron (RD )
Cube to rhombic dodecahedron (RD )
Stellation degree is 2.
Stellation degree is 2.
In[]:=
Needs["PolyhedronOperations`"];Manipulate[cub=PolyhedronData["Cube","Faces"];cubs=Stellate[cub,st];cub1=Scale[cub,2];Graphics3D[{cub,{Cyan,cubs},Opacity[0.6],cub1},Boxed->False,SphericalRegion->True,ImageSize->200],{{st,0.9,"stellate"},0,2},TrackedSymbols->{st},SaveDefinitions->True]
Out[]=
Building RD on cube from square slabs. Vary “pcs”. See also:
In[]:=
pcs=12;r=0.5/pcs;s=2r;cub=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Polygon[PolyhedronData["Cube","FaceIndices"]]];base=Translate[Scale[cub,{1,1,r},{0,0,0}],{0,0,r/2}];ped=Table[Translate[Scale[base,{(1-is),(1-is),1},{0,0,0}],{0,0,ir}],{i,0.5/r}];rd1=Translate[{ped},{0,0,0.5-r}];rd2=Table[Rotate[rd1,i2Pi/4+0.001,{0,1,0}],{i,4}];rd3=Table[Rotate[rd2,i2Pi/2+Pi/2+0.001,{1,0,0}],{i,2}];rdt=GraphicsComplex[PolyhedronData["RhombicDodecahedron","VertexCoordinates"],Tube[PolyhedronData["RhombicDodecahedron","FaceIndices"],0.03]];rdtn=Scale[Rotate[rdt,Pi/4,{0,0,1}],0.85{1,1,1},{0,0,0}];sp=Sphere[{0,0,0},0.71];Graphics3D[{sp,Cyan,rdtn,Green,cub,LightBlue,rd2,rd3},SphericalRegionTrue,BoxedFalse,ImageSize->200]
Out[]=
Tetrahedron to Cube
Tetrahedron to Cube
Stellation degree is 3.
In[]:=
Manipulate[tet=PolyhedronData["Tetrahedron","Faces"];tets=Stellate[tet,st];cub2=Rotate[Rotate[Rotate[Scale[cub,0.71],Pi/4,{0,0,1}],ArcTan[Sqrt[2]],{1,0,0}],-Pi/6,{0,0,1}];Graphics3D[{tet,tets,Yellow,Opacity[0.1],cub2},Boxed->False,SphericalRegion->True,ViewPoint->{5,4,3},ImageSize->200],{{st,0.7,"stellate"},0,3,Appearance->"Labeled"},TrackedSymbols->{st},SaveDefinitions->True]
Out[]=
Octahedron to rhombic dodecahedron (RD)
Octahedron to rhombic dodecahedron (RD)
With stellation degree 1.5 and to stella octangula (STO) with stellation degree 3. See also: https://demonstrations.wolfram.com/StellaOctangula/
In[]:=
Manipulate[oct=PolyhedronData["Octahedron","Faces"];octs=Stellate[oct,st];cub3=Scale[cub,1.41];Graphics3D[{oct,Pink,octs,Yellow,Opacity[0.1],cub3},Boxed->False,SphericalRegion->True,ImageSize->200],{{st,1.5,"stellate"},0,3,Appearance->"Labeled"},TrackedSymbols->{st},SaveDefinitions->True]
Out[]=
RD to STO
RD to STO
Stellation degree is 2.
In[]:=
Manipulate[rd=PolyhedronData["RhombicDodecahedron","Faces"];rds=Stellate[rd,st];cub4=Rotate[Scale[cub,2.3],Pi/4,{0,0,1}];Graphics3D[{rd,Yellow,rds,Yellow,Opacity[0.1],cub4},Boxed->False,SphericalRegion->True,ImageSize->200],{{st,0.9,"stellate"},0,2},TrackedSymbols->{st}]
Out[]=
In[]:=
Manipulate[sto=PolyhedronData["StellaOctangula","Faces"];stos=Stellate[sto,st];cub2a=Rotate[Rotate[Rotate[Scale[cub,1.41],Pi/4,{0,0,1}],ArcTan[Sqrt[2]],{1,0,0}],-Pi/6,{0,0,1}];cub2at=cub2a/.Polygon->Tube;Graphics3D[{sto,stos,Yellow,cub2at},Boxed->False,SphericalRegion->True,ImageSize->200],{{st,1.5,"stellate"},0,1.5},TrackedSymbols->{st}]
Dodecahedron
Dodecahedron
Icosahedron
Icosahedron
Mathematica Spikey. See: https://demonstrations.wolfram.com/StellatingAnIcosahedronWithTriangularSlabs/
Stellating the stella octangula
Stellating the stella octangula
Stellation degree is 1.5 to produce a structure fitted into a cube.
Compound of 5 cubes is a stellation of RT. See: https://community.wolfram.com/groups/-/m/t/3027031 and https://demonstrations.wolfram.com/CompoundOf5Cubes/
Rhombic hexecontahedron (RH) is a stellation of RT. See: https://demonstrations.wolfram.com/HowFivePetalFlowersGrowOnARhombicTriacontahedronToBecomeARho/
Many of my demos deal with RH, as well as my book “Rhombic structures”: https://www.amazon.com/Rhombic-Structures-Geometry-Modeling-Crystals-ebook/dp/B00XABWJ78
Many of my demos deal with RH, as well as my book “Rhombic structures”: https://www.amazon.com/Rhombic-Structures-Geometry-Modeling-Crystals-ebook/dp/B00XABWJ78
Two addition major stellations of RT are the great rhombic triacontahedron (GRT) and the median rhombic triacontahedron (MRT). See: https://demonstrations.wolfram.com/TwoStellationsOfTheRhombicTriacontahedron/
Great rhombic triacontahedron (GRT) as a stellation of RT. See also: https://community.wolfram.com/groups/-/m/t/2823046
GRT can be produced from the faces of RT. For this purpose the faces are moved towards the origin, or enlarged in their own planes. The latter approach is used below. The size is fi^3. Part of GRT coincides with rhombic hexecontahedron (RH).
GRT can be considered as a composition of perforated RH and a concave 120 faced polyhedron (RT120) (which is created by stellating RT inwards by fi/2).
The RT120 can also be created from parts built on median rhombic triacontahedron (MRT) . The diagram shown in this demo is a composition of polygons, 60 of which comprise the assembly . See: https://demonstrations.wolfram.com/ComponentsOfAGreatRhombicTriacontahedron/
S-shaped parts of the faces of GRT bypass one another. This is a basic idea behind producing various geometrical sculptures. See: https://demonstrations.wolfram.com/GreatRhombicTriacontahedronSculpture/ and https://demonstrations.wolfram.com/BallsAndSticksGeometricalSculpture/
Lamp fixtures often made in the for of stellated polyhedron. See: https://community.wolfram.com/groups/-/m/t/3017635 and https://community.wolfram.com/groups/-/m/t/3018026
Towers of icosahedra fit into a stellated icosahedron. see: https://demonstrations.wolfram.com/IcosahedronTowers/
Stellate all polyhedra
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Miscellaneous stellated polyhedra
by Sándor Kabai
Wolfram Community, STAFF PICKS, October 8, 2023
https://community.wolfram.com/groups/-/m/t/3030327
by Sándor Kabai
Wolfram Community, STAFF PICKS, October 8, 2023
https://community.wolfram.com/groups/-/m/t/3030327