Cube and RT relationship
Cube and RT relationship
Sándor Kabai
A cube and six halves of Bilinski polyhedron (rhombic dodecahedron of the second kind) makes a rhombic triacontahedron (RT). If the edge of cube is two, then the volume of RT is 8 + (6/2) 4 = 20.
A cube and six halves of Bilinski polyhedron (rhombic dodecahedron of the second kind) makes a rhombic triacontahedron (RT). If the edge of cube is two, then the volume of RT is 8 + (6/2) 4 = 20.
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Manipulate[ϕ=GoldenRatio;cub=PolyhedronData["Cube","Faces"];vA={1/ϕ,0,ϕ};vB={0,1,ϕ};vC={-1/ϕ,0,ϕ};vD={0,-1,ϕ};vE={1,-1,1};vJ={ϕ,0,1};vF={1,1,1};vG={-1,1,1};vH={-1,-1,1};vI={-ϕ,0,1};r1=Polygon[{vA,vB,vC,vD}];r11=Polygon[{vA,vD,vE,vJ}];r12=Polygon[{vA,vJ,vF,vB}];r13=Polygon[{vC,vB,vG,vI}];r14=Polygon[{vC,vI,vH,vD}];tri1=Polygon[{vH,vD,vE}];tri2=Polygon[{vB,vF,vG}];sq=Polygon[{vE,vJ,vF,vG,vI,vH}];cyl={Green,Cylinder[{{0,0,-1.5},{0,0,1}},0.2]};b0={cyl,sq,tri1,tri2,r1,r11,r12,r13,r14};cuba=Scale[cub,2];b1=Translate[b0,{0,0,w}];b2=Rotate[b1,Pi,{1,0,0}];b3=Rotate[Rotate[b1,Pi/2+0.001,{0,0,1}],Pi/2+0.001,{1,0,0}];b4=Rotate[Rotate[b1,Pi/2+0.001,{0,0,1}],-Pi/2+0.001,{1,0,0}];b5=Rotate[Rotate[b1,-Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,0,1}];b6=Rotate[Rotate[b1,-Pi/2+0.001,{1,0,0}],-Pi/2+0.001,{0,0,1}];bilhalf={b1,b2,b3,b4,b5,b6};Graphics3D[{Pink,cuba,Yellow,bilhalf},PlotRange2.6,SphericalRegionTrue,BoxedFalse,ImageSize300,ViewPoint{2,-5,2},ViewAngle0.2],{{w,0.808,"elevate half RDs"},0,1},TrackedSymbols{w}]
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