Circles and loops around icosidodecahedron (ID)
Sándor Kabai
The intersecting points of six great circles of the sphere determine the vertices of icosidodecahedron. If the circles are replaced with waves, then various loops can be produced around ID.
The intersecting points of six great circles of the sphere determine the vertices of icosidodecahedron. If the circles are replaced with waves, then various loops can be produced around ID.
In[]:=
(*SixcirclespassthroughtheverticesofID*)fi=GoldenRatio;w1=ParametricPlot3D[{fiSin[x],fiCos[x],0Sin[nx]},{x,0,2Pi},PlotStyle{Cyan,Tube[0.03]}][[1]];w2=Rotate[w1,2ArcTan[1/fi],{0,1,0}];w5=Table[Rotate[w2,i2Pi/5-Pi/5,{0,0,1}],{i,5}];id=PolyhedronData["Icosidodecahedron","Faces"];Graphics3D[{id,w1,Yellow,w5},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
Out[]=
In[]:=
(*Thesecerclesarethegreastcirclesofasphere*)sp=Sphere[{0,0,0},fi];Graphics3D[{sp,id,w1,w5},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
Out[]=
In[]:=
sp1=Sphere[{0,0,0},0.1];vid=PolyhedronData["Icosidodecahedron","VertexCoordinates"];sp30=Map[Translate[sp1,#]&,vid];Graphics3D[{id,w1,w5,Green,sp30},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
Out[]=
In[]:=
dv=Table[fi{Sin[i2Pi/10],Cos[i2Pi/10],0},{i,11}];dec=Polygon[dv];dect=Tube[dv,0.06];Graphics3D[{Yellow,dect,w1,Green,dec},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
Out[]=
In[]:=
(*TendecagonscoincidewiththeedgesofID*)w2=Rotate[dect,2ArcTan[1/fi],{0,1,0}];w5=Table[Rotate[w2,i2Pi/5-Pi/5,{0,0,1}],{i,5}];id=PolyhedronData["Icosidodecahedron","Faces"];Graphics3D[{dec,Pink,dect,Green,w5},BoxedFalse,SphericalRegionTrue,ImageSize300]
Out[]=
In[]:=
n=10;m1=ParametricPlot3D[{(2n+Cos[nx])Cos[x]/n,(2n+Cos[nx])Sin[x]/n,0},{x,0,2Pi},PlotStyle{Yellow,Tube[0.06]}][[1]];Graphics3D[{id,m1},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
Out[]=
In[]:=
w2=Rotate[m1,-2ArcTan[1/fi],{0,1,0}];w5=Table[Rotate[w2,i2Pi/5,{0,0,1}],{i,5}];id=PolyhedronData["Icosidodecahedron","Faces"];Graphics3D[{Cyan,id,m1,w1,w5},BoxedFalse,SphericalRegionTrue,ImageSize300,ViewPoint->{0,0,100}]
Out[]=
