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 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.
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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,ImageSize350,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},SaveDefinitions->True]
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