Three different kinds of truncation of RT are studied here.
Truncating the five-fold and three-fold faces simultaneously
Truncating the five-fold and three-fold faces simultaneously
Such truncated RT is the shape of the nodes of Zometool construction set. See: https://community.wolfram.com/groups/-/m/t/2608558
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Needs["PolyhedronOperations`"];Manipulate[rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtn=Rotate[rt,Pi,{0,0,1}];rtman=Truncate[rt,trunc];Graphics3D[{rtman},BoxedFalse,SphericalRegionTrue,ImageSize250,PlotRange->1.6,ViewPoint{100,100,100},ViewAngle0.006],{{trunc,0.5,"truncate"},0,0.5},TrackedSymbolstrunc,SaveDefinitions->True]
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Truncating the five-fold faces
Truncating the five-fold faces
This truncation leads to dodecahedron. It can also be considered as the edge truncation of dodecahedron.
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Manipulate[dod=PolyhedronData["Dodecahedron","Faces"];vA={x,0,0};vB={0,y,0};vC={-x,0,0};vD={0,-y,0};vE={x/2+a/fi,y/2-a,0};vF={-x/2-a/fi,y/2-a,0};vG={-x/2-a/fi,-y/2+a,0};vH={x/2+a/fi,-y/2+a,0};r1=Polygon[{vA,vB,vC,vD}];hex11=Polygon[{vA,vE,vF,vC,vG,vH}];r2=Translate[hex11,{0,0,fiy}];r3=Translate[hex11,{0,0,-fiy}];r4=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,0,1}];r5=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,1,0}];trdod=Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];dodnn=Scale[Rotate[dod,Pi,{0,0,1}],1.0456{1,1,1},{0,0,0}];Graphics3D[{dodnn,Cyan,trdod},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->200,PlotRange->1.7,ViewAngle->0.01],{{a,-0.1,"truncate"},0.425,-0.425},TrackedSymbols:>{a}]
The geometry of C80 fullerene can be studied at an intermediate stage of truncation. For this purpose the hexagon is constructed from a golden rhombus of RT.
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rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtin=Scale[rt,0.999{1,1,1},{0,0,0}];dod=PolyhedronData["Dodecahedron","Faces"];dodv=dod[[1]];ico=PolyhedronData["Icosahedron","Faces"];fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];m=fi^2Sin[alf];x=Cos[alf];y=Sin[alf];vA={x,0,0};vB={0,y,0};vC={-x,0,0};vD={0,-y,0};vE={x/2,y/2,0};vF={-x/2,y/2,0};vG={-x/2,-y/2,0};vH={x/2,-y/2,0};r1=Polygon[{vA,vB,vC,vD}];hex1=Polygon[{vA,vE,vF,vC,vG,vH}];Graphics3D[{r1,Cyan,Translate[hex1,{0,0,0.001}]},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->200]
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The sphere size is determined.
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spx=Sphere[{0,0,0},x/2];spxa=Translate[spx,{x/2,y/2,0}];spxb=Translate[spx,{-x/2,y/2,0}];Graphics3D[{spxa,spxb,r1,Cyan,hex1},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->200]
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r=N[(x/2)/Sin[Pi/5]];p5v=Table[r{Sin[i2Pi/5],Cos[i2Pi/5],0},{i,5}];p5=Polygon[p5v];sp5=Rotate[Map[Translate[spx,#]&,p5v],0Pi,{0,0,1}];Graphics3D[{p5,sp5},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->200]
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RT is assembled from 30 golden rhombuses.
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rtf=Scale[Rotate[rt,-alf,{0,1,0}],0.99];r2=Translate[r1,{0,0,fiy}];r3=Translate[r1,{0,0,-fiy}];r4=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,0,1}];r5=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,1,0}];rta=Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];Graphics3D[{rta},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->200]
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A polyhedron with 42 faces and 80 vertices is assembled from 30 hexagons and 12 pentagons. The relationship with RT, DOD, and ICO are shown.
80 spheres are placed at the vertices to approximate the geometry of C80 fullerene. See: https://demonstrations.wolfram.com/C80Fullerene/
Dodecahedra are placed at the vertices with face to face connections.
Truncating the three-fold faces
Truncating the three-fold faces
This leads to icosahedron.
Another arrangement of 80 spheres or RTs at the vertices of truncated icosahedron
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Truncated rhombic triacontahedron (RT)
by Sandor Kabai
Wolfram Community, STAFF PICKS, October 4, 2023
https://community.wolfram.com/groups/-/m/t/3026266
by Sandor Kabai
Wolfram Community, STAFF PICKS, October 4, 2023
https://community.wolfram.com/groups/-/m/t/3026266