12 polar zonohedron

https://demonstrations.wolfram.com/PolarZonohedronBuiltOnADodecahedron/
A triangle determined by an edge and the center of a dodecahedron is half of a rhombus, with diagonal ratio equal to the square of the golden ratio. Based on this relationship, polar zonohedra can be built on each face of the dodecahedron.
The diagonal ratio of the second rhombus of the polar zonohedron is

2

.
This assembly can be built with Zometool using yellow struts.

In[]:=

Manipulate[op=1;ϕ=1.618;α=ArcTan[ϕ];β=ArcTan[1/ϕ];β1=ArcTan[1/ϕ^2];h=2*ϕ*Sin[α];b=2*ϕ*Cos[α];d=b/Cos[Pi/5];f=1/Tan[Pi/5];vO={0,0,0};vP={d/2,0,h/2};vQ={f,1,h};vR={f,-1,h};vS={2*f-d/2,0,3*(h/2)};vT={(d/2)*Sin[Pi/10],(d/2)*Cos[Pi/10],h/2};pol1={RGBColor[0.3,1,0.7],Polygon[{vO,vP,vQ,vT}]};pol2=Polygon[{vP,vQ,vS,vR}];pz1=Translate[Table[Rotate[{pol1,pol2},i*2*(Pi/5),{0,0,1}],{i,5}],{0,0,up1}];pz2={Specularity[0.6],RGBColor[0.5,0.3,0],Rotate[{pz1},Pi,{0,1,0},{0,0,5*(h/4)+up1+up2}]};pz5=Table[Rotate[Rotate[Rotate[{pz1,pz2},Pi,{0,0,1}],-2*β,{0,1,0}],i*2*(Pi/5),{0,0,1}],{i,n}];pz6=Rotate[{pz1,pz2,pz5},Pi,{0,1,0}];dod={Yellow,Opacity[op],Scale[Rotate[GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]],Pi/5,{0,0,1}],aff*(2/ϕ)*{1,1,1},{0,0,0}]};Graphics3D[{pz1,pz2,pz5,If[pz6s,pz6,{}],dod},Boxed->False,SphericalRegion->True,ViewAngle->0.045,ViewPoint->{7,20,0},ImageSize->300,PlotRange->8],{{up1,0,"move polar zonohedron"},0,1.4},{{up2,0,"move half polar zonohedron"},0,0.8},{{aff,1,"dodecahedron size"},1,3},{{n,5,"show lateral polar zonohedra"},0,5,1,ControlType->RadioButton},{{pz6s,False,"show lower half"},{False,True}},TrackedSymbols->Manipulate]