Great Rhombic Triacontahedron (GRT) explained
Sándor Kabai
GRT is one of the stellations of rhombic triacontahedron (RT).
GRT is one of the stellations of rhombic triacontahedron (RT).
1. GRT is produced by moving the faces of RT towards the centre
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Manipulate[fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];rtsize=Sqrt[2]/fi/2/2/Cos[alf];rtx=Cos[alf];rty=Sin[alf];rtvA={rtx,0,0};rtvB={0,rty,0};rtvC={-rtx,0,0};rtvD={0,-rty,0};rts1={rtvA,rtvB,rtvC,rtvD,rtvA};(*rts2=0.8{rtvA,rtvB,rtvC,rtvD,rtvA};r1=Polygon[Join[rts1->rts2]];*)r1=Polygon[rts1];r2=Translate[r1,{0,0,firty/comp}];r3=Translate[r1,{0,0,-firty/comp}];r4=Rotate[Rotate[{r2,r3},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,1,0}];r5=Rotate[Rotate[{r2,r3},Pi/2+0.001,{0,1,0}],Pi/2+0.001,{1,0,0}];rtp=Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];rt0={Rotate[Rotate[Scale[rt,rtsize],-alf+bet1,{0,1,0}],Pi/8,{0,0,1.3}]};rt=PolyhedronData["RhombicTriacontahedron","Faces"];Graphics3D[{Cyan,rtp},BoxedFalse,SphericalRegionTrue,PlotRange1.6,ViewPoint{0,0,100},ViewAngle0.01],{{comp,1,"compress"},1,4.236},TrackedSymbols->comp]
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The same with RT frame. Notice the solid RT at the centre.
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Manipulate[rtsize=Sqrt[2]/fi/2/2/Cos[alf];rtx=Cos[alf];rty=Sin[alf];rtvA={rtx,0,0};rtvB={0,rty,0};rtvC={-rtx,0,0};rtvD={0,-rty,0};rts1={rtvA,rtvB,rtvC,rtvD,rtvA};rts2=0.9{rtvA,rtvB,rtvC,rtvD,rtvA};r1=Polygon[Join[rts1->rts2]];r2=Translate[r1,{0,0,firty/comp}];r3=Translate[r1,{0,0,-firty/comp}];r4=Rotate[Rotate[{r2,r3},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,1,0}];r5=Rotate[Rotate[{r2,r3},Pi/2+0.001,{0,1,0}],Pi/2+0.001,{1,0,0}];rtp=Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];rt0={Rotate[Rotate[Scale[rt,rtsize],-alf+bet1,{0,1,0}],Pi/8,{0,0,1.3}]};rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtsm=Scale[rt,0.97/fi^2];Graphics3D[{Pink,rtsm,Cyan,rtp},BoxedFalse,SphericalRegionTrue,PlotRange1.6,ViewPoint{0,0,100},ViewAngle0.01],{{comp,1,"compress"},1,4.236},TrackedSymbols->comp]
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2. GRT is produced by enlarging faces of RT in their own planes.
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shrink[t_,Polygon[x_List,opts___]]:=Module[{c=Plus@@x/Length[x]},Polygon[Map[(c+(1-t)(#-c))&,x],opts]]Manipulate[rtpol=PolyhedronData["RhombicTriacontahedron","Faces","Polygon"];rtsr={rtpol/.y_Polygonshrink[exp,y]};Graphics3D[{rtpol,Yellow,rtsr},BoxedFalse,SphericalRegionTrue,ImageSize300,PlotRange3.6,ViewPoint{0,0,100},ViewAngle0.01],{{exp,0,"expand"},-3.23,0.5},TrackedSymbols->exp]
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3. Components of a Great Rhombic Triacontahedron
The great rhombic triacontahedron (GRT) can be considered to be a composition of 30 golden rhombuses (a stellation of the rhombic triacontahedron). This Demonstration shows how the protruding parts (stars) can also be interpreted as parts of a concave rhombic hexecontahedron.
The internal parts are revealed as the shells are removed:
1. the concave rhombic hexecontahedron (CRH)
2. the relationship to the median rhombic triacontahedron (MRT)
3. the rhombic triacontahedron (RT)
The building can be started with the stellation diagram.
The great rhombic triacontahedron (GRT) can be considered to be a composition of 30 golden rhombuses (a stellation of the rhombic triacontahedron). This Demonstration shows how the protruding parts (stars) can also be interpreted as parts of a concave rhombic hexecontahedron.
The internal parts are revealed as the shells are removed:
1. the concave rhombic hexecontahedron (CRH)
2. the relationship to the median rhombic triacontahedron (MRT)
3. the rhombic triacontahedron (RT)
The building can be started with the stellation diagram.
In[]:=
fi=GoldenRatio;alf=ArcTan[fi];bet=ArcTan[1/fi];vA={fi,0,fi^2};vB={0,1,fi^2};vC={-fi,0,fi^2};vD={0,-1,fi^2};vE={fi+fi^2,-fi,fi^2};vF={fi+2fi^2,0,fi^2};vG={fi+fi^2,fi,fi^2};vH={fi+1,1/fi,fi^2};vI={fi+2,0,fi^2};vJ={fi+1,-1/fi,fi^2};vK={0,-1-2fi,fi^2};vL={fi,(-1-2fi)+1,fi^2};vM={1/fi,(-1-2fi)+fi,fi^2};vO={-1,-fi,fi^2};vP={1,-fi,fi^2};vR={fi^2,-(1+fi),fi^2};vQ={-fi-2fi^2,0,fi^2};w={Yellow,Polygon[{vA,vB,vC,vD}]};z1={RGBColor[0.8,1,0.6],Polygon[{vK,vP,vD,vO}]};z2=Rotate[z1,Pi,{0,0,1}];u1={RGBColor[1,0.4,0.7],Polygon[{vK,vL,vM}]};u2=Rotate[u1,Pi,{0,0,1}];u3=Rotate[{u1,u2},Pi,{1,0,0},{0,0,fi^2}];r1={RGBColor[0,1,1],Polygon[{vL,vR,vP,vM}]};r2=Rotate[r1,Pi,{0,0,1}];r3=Rotate[{r1,r2},Pi,{1,0,0},{0,0,fi^2}];v1=Polygon[{vJ,vE,vF,vG,vH,vI}];v2=Rotate[v1,Pi,{0,0,1}];Manipulate[Module[{p0,p1,p2,p3,p4,p5,mrt},p0=Rotate[{w,z1,z2,u1,u2,u3,r1,r2,r3,v1,v2},-bet,{0,1,0}];p1={If[sw,w,{}],If[z,{z1,z2},{}],If[u,{u1,u2,u3},{}],If[r,{r1,r2,r3},{}],If[v,{v1,v2},{}]};p2=Rotate[p1,Pi,{0,1,0}];p3=Rotate[{p1,p2},Pi,{0,1,0}];p4=Rotate[Rotate[{p1,p2,p3},Pi/2+0.001,{0,1,0}],Pi/2+0.001,{0,0,1}];p5=Rotate[Rotate[{p1,p2,p3},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,0,1}];mrt=Table[Rotate[Rotate[{p1,p2,p3,p4,p5},-bet,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];Graphics3D[{p0,mrt},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewAngle0.1,ViewPoint{2,2,10},PlotRange7.2]],{{sw,False,"rhombic triacontahedron"},{True,False},ControlPlacementLeft,ImageSizeTiny},{{z,False,"median rhombic triacontahedron"},{True,False},ControlPlacementLeft,ImageSizeTiny},{{r,False,"concave rhombic hexecontahedron"},{True,False},ControlPlacementLeft,ImageSizeTiny},{{u,False,"star"},{True,False},ControlPlacementLeft,ImageSizeTiny},{{v,False,"great rhombic triacontahedron"},{True,False},ControlPlacementLeft,ImageSizeTiny},Delimiter,TrackedSymbols{v,r,u,z,sw}]
4. Pages from the book Rhombic Structures