CITE THIS NOTEBOOK: Fifteen great circles circumscribing an icosahedron by Sándor Kabai. Wolfram Community 14 OCT 2022.
Three golden rectangles at right angles to each other determine the vertices of an icosahedron. There are 15 different circles through the pairs of opposite edges of the icosahedron. These circles are the great circles of a sphere circumscribing the icosahedron, since they are in the plane of the golden rectangles, and by one definition, a great circle is the intersection of a sphere and a plane passing through the center of the sphere. The intersections of the circles determine the vertices of a dodecahedron and an icosidodecahedron. The 30 edges of the icosahedron coincide with the longer diagonals of the golden rhombus bordering the rhombic triacontahedron.
Three golden rectangle with one great circle. Vertices are emphasised by spheres.
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az={Red,Cylinder[{{0,0,-1.3},{0,0,1.3}},0.03]};ax={Green,Cylinder[{{-1.3,0,0},{1.3,0,0}},0.03]};ay={Blue,Cylinder[{{0,-1.3,0},{0,1.3,0}},0.03]};spc=Sphere[{0,0,0},0.951];ico=PolyhedronData["Icosahedron","Faces"];icot=ico/.PolygonTube;icovc=PolyhedronData["Icosahedron","VertexCoordinates"];id0=PolyhedronData["Icosidodecahedron","Faces"];id=Rotate[Scale[id0,1/1.7],Pi,{0,0,1}];idv=PolyhedronData["Icosidodecahedron","VertexCoordinates"];dod0=GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]];dod=Scale[dod0,0.6787{1,1,1},{0,0,0}];dodt=dod/.PolygonTube;dodv=PolyhedronData["Dodecahedron","VertexCoordinates"];fi=GoldenRatio;ϕ=0.618;w=0.5;v=0.5*(1/fi);z=fi/2;alf=ArcTan[fi];bet=ArcTan[1/fi];vA={fi/2,0.01,0};vB={0,0.01,fi^3/2};vC={-fi/2,0.01,0};vD={0,0.01,-fi^3/2};cub=PolyhedronData["Cube","Faces"];cuba={Rotate[Scale[cub,1.1*{1,1,1},{0,0,0}],bet,{0,1,0}]};gr1=Polygon[{{0.5,0,-fi/2},{0.5,0,fi/2},{-0.5,0,fi/2},{-0.5,0,-fi/2}}];gr2=Rotate[Rotate[{gr1},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,1,0}];gr3=Rotate[Rotate[gr1,Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,0,1}];grn=Rotate[{gr1,gr2,gr3},-bet,{0,0,1}];c1=ParametricPlot3D[0.951*{Sin[x],0,Cos[x]},{x,0,2*Pi}][[1]];c3=Rotate[Rotate[c1,Pi/2+0.001,{1,0,0}],bet,{0,1,0}];c2=Rotate[Rotate[c1,Pi/2+0.001,{1,0,0}],bet+Pi/2+0.001,{0,1,0}];c15=Table[Rotate[c1,i2Pi/5,{0,0,1}],{i,5}];c25=Table[Rotate[c2,i2Pi/5,{0,0,1}],{i,5}];c35=Table[Rotate[c3,i2Pi/5,{0,0,1}],{i,5}];sp=Sphere[{0,0,0},0.06];icoball=Map[Translate[sp,#]&,icovc];idball=Rotate[Map[Translate[sp,#]&,idv/1.7],Pi,{0,0,1}];dodball=Map[Translate[sp,#]&,0.6787dodv];dod=Scale[dod0,0.6787{1,1,1},{0,0,0}];Graphics3D[{c1,c2,c3,Pink,dod,Yellow,ico,Cyan,id},Boxed->False,SphericalRegion->True,ImageSize200]
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Graphics3D[{dod,Cyan,dodball},Boxed->False,SphericalRegion->True,ImageSize200]
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Graphics3D[{id,Red,idball},Boxed->False,SphericalRegion->True,ImageSize200]
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grn=Rotate[{gr1,gr2,gr3},bet,{0,1,0}];Graphics3D[{c1,c2,c3,{Yellow,icoball},grn},Boxed->False,SphericalRegion->True,ViewPoint{0,0,100}]
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Three golden rectangles with three great circles and axes.
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Graphics3D[{ax,ay,az,c1,c2,c3,{Yellow,icoball},grn},Boxed->False,SphericalRegion->True,ViewPoint{0,100,0}]
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Vertices are connected to from an ICO
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Graphics3D[{{Blue,icot},ax,ay,az,{Yellow,icoball},grn},Boxed->False,SphericalRegion->True,ViewPoint{0,100,0}]
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15 circles with obtained by rotating the 3 circles five times. The intersections correspond to the vertices of ICO (and other polyhedra with icosahedral symmetry, such as RT, RH), DOD , ID.