Sándor Kabai
​Space filling structure around rhombic triacontahedron (RT)​
Rhombic polyhedra are used for building a space fillings structure around an RT.
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fi=GoldenRatio;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​bet1=ArcTan[1/fi^2];​​rtx=Cos[alf];​​rty=Sin[alf];​​rh=GraphicsComplex[PolyhedronData["RhombicHexecontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicHexecontahedron","FaceIndices"]]];​​rhn=Rotate[rh,Pi,{0,0,1}];​​rt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];​​rtn=Rotate[rt,Pi,{0,0,1}];​​rtf=Rotate[rt,bet,{0,1,0}];​​rtr=Rotate[rt,Pi/2,{0,0,1}];​​y=Sin[alf];​​m=2yfi;​​m1=mfi^2;​​ri=GraphicsComplex[PolyhedronData["RhombicIcosahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicIcosahedron","FaceIndices"]]];​​icov=PolyhedronData["Icosahedron","VertexCoordinates"];

Bilinski

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y=N[Sin[alf]];​​vA={1/fi,0,-1};​​vB={0,1,-1};​​vC={-1/fi,0,-1};​​vD={0,-1,-1};​​b1=Polygon[{vA,vB,vC,vD}];​​vE={1/fi,1/fi,0};​​vF={1/fi,0,1};​​vG={1/fi,-1/fi,0};​​b2=Polygon[{vA,vE,vF,vG}];​​vI={0,-1,1};​​vH={0,-fi,0};​​b3=Polygon[{vA,vG,vH,vD}];​​b4=Polygon[{vG,vF,vI,vH}];​​b5=Rotate[{b3,b4},Pi,{1,0,0}];​​b6=Rotate[{b1,b2,b3,b4,b5},Pi,{0,1,0}];​​bil0={b1,b2,b3,b4,b5,b6};​​bil=Scale[bil0,y];​​bil1=Translate[bil,{0,0,yfi+y}];​​bil2=Rotate[bil1,alf,{0,1,0}];​​bil5=Table[Rotate[bil2,i2Pi/5,{0,0,1}],{i,5}];​​b2=Rotate[bil1,Pi,{1,0,0}];​​b3=Rotate[Rotate[{b2,bil1},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,0,1}];​​b4=Rotate[Rotate[{bil1,b2,b3},Pi/2,{0,1,0}],Pi/2,{0,0,1}];​​b30=Table[Rotate[Rotate[{bil1,b2,b3,b4},bet,{1,0,0}],i2Pi/5+2Pi/2+Pi/10,{0,0,1}],{i,5}];​​Graphics3D[{rt,bil5},BoxedFalse,SphericalRegionTrue,ImageSize300]​​
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(*Step1:Place30BilinskionRT*)​​Graphics3D[{rt,b30},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint{0,0,100}]

prGR

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ySqrt[3]
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(*LongerdiameterofprGR*)​​dia=N
3+
6
5
;​​(*DiameterofcubeinRT*)​​sqdia=ySqrt[3];​​prGR=GraphicsComplex[PolyhedronData["AcuteGoldenRhombohedron","VertexCoordinates"],Polygon[PolyhedronData["AcuteGoldenRhombohedron","FaceIndices"]]];​​(*prGRwithtipattheorigion*)​​prGRup=Translate[prGR,{0,0,dia/2}];​​prGRtilt=Table[Rotate[Rotate[Rotate[prGRup,Pi,{0,0,1}],2bet1,{0,1,0}],i2Pi/3,{0,0,1}],{i,3}];​​u1={prGRup};​​u2=Translate[u1,{0,0,sqdia}];​​u3=Rotate[Rotate[u2,Pi,{0,0,1}],2bet1,{0,1,0}];​​u4=Table[Rotate[Rotate[{u2,u3},alf-bet1,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​u5=Rotate[u4,Pi,{0,1,0}];​​
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(*Step2:Place30prGRonRT*)​​Graphics3D[{rt,Pink,u4,u5},ViewPoint{0,100,0},ImageSize300,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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(*Step3:CombineStep1andStep2*)​​Graphics3D[{b30,Pink,u4,u5},ViewPoint{0,100,0},ImageSize300,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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(*Step4:InstallseriesofprGR,whichwillbethebackbonesofthestructure*)​​prGRup1=Translate[prGR,{0,0,dia/2-dia+sqdia}];​​seru2=Table[{Hue[0.8^i],Translate[prGRup1,{0,0,idia}]},{i,3}];​​seru3=Rotate[Rotate[seru2,Pi,{0,0,1}],2bet1,{0,1,0}];​​seru4=Table[Rotate[Rotate[{seru2,seru3},alf-bet1,{0,1,0}],i2Pi/5,{0,0,1}],{i,6}];​​seru5=Rotate[seru4,Pi,{0,1,0}];​​Graphics3D[{rt,seru4,seru5},ViewPoint{0,100,0},ImageSize350,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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Graphics3D[{b30,seru4,seru5},ViewPoint{0,0,100},ImageSize350,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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​​(*Aframeforlateruse*)​​vv1={prGRup,prGRtilt};​​vv2=Translate[vv1,{0,0,dia}];​​vv3=Rotate[Rotate[vv2,Pi,{0,0,1}],2bet1,{0,1,0}];​​vv4=Table[Rotate[Rotate[{vv2,vv3},alf-bet1,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​vv5=Rotate[vv4,Pi,{0,1,0}];​​Graphics3D[{rt,vv4,Pink,vv5},ViewPoint{0,100,0},ImageSize300,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]

Parts of RH made from 20 prGR

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p1=Rotate[Rotate[Translate[prGR,{0,0,dia/2}],(Pi/2-bet1),{0,1,0}],Pi,{0,0,1}];​​p2=Rotate[Rotate[p1,Pi,{1,0,0}],0bet,{0,1,0}];​​top5=Table[Rotate[Rotate[{p1},bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​middlet=Table[Rotate[Rotate[{p2},bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​bottom5=Rotate[top5,Pi,{0,1,0}];​​middleb=Rotate[middlet,Pi,{0,1,0}];​​Graphics3D[{top5,bottom5,middlet,middleb,Pink,u4},ViewPoint{0,100,0},ImageSize300,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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(*Step5:InstallpartsofRH*)​​top1=Rotate[Translate[{top5,middlet},{0,0,fi^2}],Pi,{0,0,1}];​​Graphics3D[{u4,Green,b30,Yellow,top1},ViewPoint{0,100,0},ImageSize340,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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top2=Table[Rotate[Rotate[Rotate[top1,Pi,{0,0,1}],-2bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];​​Graphics3D[{Green,b30,Yellow,top1,top2},ViewPoint{0,0,100},ImageSize340,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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Graphics3D[{seru4,Green,b30,Yellow,top1,top2},ViewPoint{0,0,100},ImageSize340,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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(*Step6:Clusterof20RT*)​​idv=PolyhedronData["Icosidodecahedron","VertexCoordinates"];​​rt30id=Rotate[Map[Translate[rt,#]&,m*idv],0Pi,{0,0,1}];​​Graphics3D[{rt30id},ViewPoint{0,0,100},ImageSize340,ViewAngle0.01,BoxedFalse,SphericalRegionTrue];
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Graphics3D[{seru4,rt30id,Green,b30,Yellow,top1,top2},ViewPoint{0,0,100},ImageSize340,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
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​​(*Step7:Place12RTaticovertices*)​​icov=PolyhedronData["Icosahedron","VertexCoordinates"];​​ico12=Rotate[Map[Translate[rtn,#]&,1.22fim*icov],Pi,{0,0,1}];​​Graphics3D[{seru4,rt30id,Cyan,ico12},ViewPoint{0,0,100},ImageSize340,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]