Number 32 in my demonstrations

Sándor Kabai
A couple of examples of structures are introduced where the number 32 occurs.
RT 12 - RH20 = 32 items
In[]:=
fi=GoldenRatio;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​bet1=ArcTan[1/fi^2];​​ce=N[2Sin[alf]];​​d3=ceSqrt[3];​​(*fd=RTfacesdistance*)​​fd=fice;​​d5=Sqrt[ce^2+fd^2];​​icov=PolyhedronData["Icosahedron","VertexCoordinates"];​​dodv=PolyhedronData["Dodecahedron","VertexCoordinates"];​​​​rt=PolyhedronData["RhombicTriacontahedron","Faces"];​​rtn=Rotate[rt,2Pi/10,{0,0,1}];​​​​rh=PolyhedronData["RhombicHexecontahedron","Faces"];​​rhc=Scale[rh,1/fi{1,1,1},{0,0,0}];​​rhn=Rotate[rh,2Pi/10,{0,0,1}];

RT12 + RT20 = 32 items

In[]:=
rt12=Map[Translate[rtn,#]&,fdicov];​​rt20=Map[Translate[rtn,#]&,fddodv];​​Graphics3D[{{Green,rt12},rt20},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ImageSize240]​​
Out[]=
There are overlaps between RT12 and RT20 as can be seen from this image of RT12
In[]:=
Graphics3D[{{Green,rt12}},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ImageSize240]
Out[]=

RT12 + RH20 = 32 items

In[]:=
rt12=Map[Translate[rtn,#]&,fifdicov];​​rh20=Map[Translate[rhn,#]&,fifddodv];​​Graphics3D[{Cyan,rt12,Yellow,rh20},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ImageSize240]

RH12 + RH20 = 32 items

In[]:=
rh12=Map[Translate[rhn,#]&,6.2icov];​​rh20=Map[Translate[rhn,#]&,fifddodv];​​Graphics3D[{Yellow,rh12,Cyan,rh20},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ImageSize240]
Out[]=

RH12 + RT20 = 32 items

In[]:=
rh12=Map[Translate[rhn,#]&,fifdicov];​​rt20=Map[Translate[rtn,#]&,fddodv];​​Graphics3D[{Yellow,rh12,Pink,rt20},SphericalRegionTrue,BoxedFalse,ViewPoint{0,0,100},ImageSize240]
Out[]=

32 RTs at the vertices of a large RT

In[]:=
fi=GoldenRatio;​​alf=ArcTan[fi];​​ce=2Sin[alf];​​fd=fice;​​d5=Sqrt[ce^2+fd^2];​​rt=PolyhedronData["RhombicTriacontahedron","Faces"];​​rtv=PolyhedronData["RhombicTriacontahedron","VertexCoordinates"];​​rtrt=Map[Translate[rt,#]&,d5rtv];​​rtbig=Scale[rt,0.999fi^3{1,1,1},{0,0,0}];​​Graphics3D[{Cyan,rtrt,{Yellow,Opacity[0.6],rtbig},Red,Scale[rt,fi^2{1,1,1},{0,0,0}]},SphericalRegionTrue,BoxedFalse,ViewPoint->{0,0,100},ImageSize300]
Out[]=

Spherical segments at RT vertices

In[]:=
​​sp=Sphere[{0,0,0},0.99];​​segm1=SphericalPlot3D[{1},{θ,0,0.5},{ϕ,0,2Pi},PlotStyleGreen,PlotPoints10,Mesh->5,BoundaryStyleBlack][[1]];​​segm2=SphericalPlot3D[{1},{θ,0,0.5},{ϕ,0,2Pi},PlotStyleYellow,PlotPoints10,Mesh->6][[1]];​​tico=PolyhedronData["TruncatedIcosahedron","Faces"];​​Graphics3D[{sp,segm1},BoxedFalse,SphericalRegionTrue,ImageSize150]​​
Out[]=

20 + 12 interwoven loops