In this context a golden polyhedron is defined as a solid having faces related to the golden ratio. Five golden rhombic polyhedra are introduced in this study:
1. Acute golden rhombohedron (prGR)
2. Obtuse golden rhombohedron (obGR)
3. Golden rhombic dodecahedron (Bilinski 1961 or RD2: 2 prGR + 2 obGR )
4. Rhombic icosahedron (RI, Fedorov 1887 : 5 prGR + 5 obGR )
5. Rhombic triacontahedron (RT, Kepler 1610: 10 prGR + 10 obGR)
(Notice the difference between the years 1961, 1887 and 1610)
Each is bordered by golden rhombuses (6 - prGR and obGR, 12 - BIL, 20 - IR, 30 - RT, 60 RH). Each rhombic polyhedron (RP) can be constructed from prGR and obGR units, which otherwise can be used to fill space (as a spatial version of Penrose tiling). A version of RT is also shown, i.e. the great RT (GRT) together with a kind of relationship between RT and RH.
1. Acute golden rhombohedron (prGR)
2. Obtuse golden rhombohedron (obGR)
3. Golden rhombic dodecahedron (Bilinski 1961 or RD2: 2 prGR + 2 obGR )
4. Rhombic icosahedron (RI, Fedorov 1887 : 5 prGR + 5 obGR )
5. Rhombic triacontahedron (RT, Kepler 1610: 10 prGR + 10 obGR)
(Notice the difference between the years 1961, 1887 and 1610)
Each is bordered by golden rhombuses (6 - prGR and obGR, 12 - BIL, 20 - IR, 30 - RT, 60 RH). Each rhombic polyhedron (RP) can be constructed from prGR and obGR units, which otherwise can be used to fill space (as a spatial version of Penrose tiling). A version of RT is also shown, i.e. the great RT (GRT) together with a kind of relationship between RT and RH.
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fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];rtx=Cos[alf];rty=Sin[alf];y=Sin[alf];diap=N
3+
;diao=N6
5
3-
;prGR=PolyhedronData["AcuteGoldenRhombohedron","Faces"];obGR=PolyhedronData["ObtuseGoldenRhombohedron","Faces"];ri=PolyhedronData["RhombicIcosahedron","Faces"];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rh=PolyhedronData["RhombicHexecontahedron","Faces"];(*Bilinski*)p1b=Translate[prGR,{0,0,diap/2}];p2b=Rotate[Rotate[p1b,Pi,{0,0,1}],2bet1,{0,1,0}];prolate=Translate[Rotate[Rotate[{p1b,p2b},Pi/2+0.001-bet1,{0,1,0}],Pi/2+0.001,{1,0,0}],{-2rty,0,0}];o1=Translate[obGR,{0,0,diao/2}];o2=Rotate[o1,-bet1,{0,-1,0}];o3=Rotate[o2,Pi,{1,0,0}];oblate={o2,o3};bil={prolate,oblate};Graphics3D[{bil}];6
5
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Row[Graphics3D[#,Boxed->False,ImageSize->140,ViewPoint->{3,6,3}]&/@{prGR,obGR,bil,ri,rt,rh}]
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Relationship of prGR and obGR
Relationship of prGR and obGR
It is shown how prGR is compressed to obtain obGR.
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Manipulate[red=diap/diao;prGRt=prGR/.Polygon->Tube;prGRc=Scale[{prGR,Yellow,prGRt},{wide,wide,height},{0,0,0}];Graphics3D[{prGRc,Cyan,obGR},ImageSize260,BoxedFalse,SphericalRegionTrue,ViewPoint{0,10,3},PlotRange1.3,ViewAngle0.1],{{wide,1,"widen"},1,1.618},{{height,1,"compress"},1,0.236},TrackedSymbols{wide,height},SaveDefinitions->True]
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Rhombic dodecahedron of the second kind (Bilinski) consisting of 2 prGR and 2 obGR
Rhombic dodecahedron of the second kind (Bilinski) consisting of 2 prGR and 2 obGR
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Manipulate[cyl=Cylinder[{{-2.2,0,0},{1.6,0,0}},0.03];y=Sin[alf];sp=Sphere[{0,0,0},0.1];p1b=Translate[prGR,{0,0,epdiap/2}];p2b=Rotate[Rotate[p1b,Pi,{0,0,1}],2bet1,{0,1,0}];prolate=Translate[Rotate[Rotate[{p1b,p2b},Pi/2+0.001-bet1,{0,1,0}],Pi/2+0.001,{1,0,0}],{-2yep,0,0}];o1=Translate[obGR,{0,0,eodiao/2}];o2=Rotate[o1,-bet1,{0,-1,0}];o3=Rotate[o2,Pi,{1,0,0}];oblate={o2,o3};Graphics3D[{sp,{Red,cyl},Yellow,oblate,Cyan,prolate},BoxedFalse,SphericalRegionTrue,ImageSize{300,200},PlotRangeAll,ViewAngle0.2],{{eo,1,"fly oblate"},1,2},{{ep,1,"fly prolate"},1,1.7},TrackedSymbols{eo,ep},SaveDefinitions->True]
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RT is assembled from one RI, five prGR and five obGR
RT is assembled from one RI, five prGR and five obGR
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Manipulate[p=N[rty/Cos[2Pi/20]];s=N[pSin[2Pi/20]];d=N[rty/Sin[2Pi/10]/2];h=Sqrt[1-p^2];eps={bet-bet1};p1=Translate[prGR,{0,0,diap/2}];ob1=Rotate[Translate[obGR,{0,0,-diao/2}],-bet+bet1-Pi/2,{0,1,0}];ob5=Table[Rotate[Translate[ob1,{-fly2+(-p-2s),0,1.344}],i2Pi/5,{0,0,1}],{i,5}];riup=Translate[ri,{0,0,2diao+1+fly1}];p5top=Table[Rotate[Rotate[p1,alf-bet1,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];Graphics3D[{{Pink,riup},cyl,Cyan,p5top,sp,Yellow,ob5},ImageSize300,BoxedFalse,SphericalRegionTrue,ViewPoint{0,10,4},PlotRange{{-2.6,2.6},{-2.6,2.6},{-0.2,4.8}},ViewAngle0.1],{{fly1,1.6,"fly RI"},0,1.6},{{fly2,0,"fly obGR"},0,1.2},TrackedSymbols{fly1,fly2},SaveDefinitions->True]
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RI is further divided
RI is further divided
Rhombic hexecontahedron (RH) consisting of 20 prGR
Rhombic hexecontahedron (RH) consisting of 20 prGR
RH is divided into two symmetrical parts.
RH is divided into four symmetrical parts
Cube and RT relationship
Cube and RT relationship
Great rhombic triacontahedron (GRT) from RT
Great rhombic triacontahedron (GRT) from RT
Each face of RT is moved in the direction of the origin until the edges meet again and form a GRT. The difference in distances is fi^4 = 4.236
RT frame, just for the record
RH from RT
RH from RT
Rhombuses are grown in the planes of RT faces until their edges meet to form an RH.