A couple of methods are introduced to depict all polyhedra. Polyhedra can be set with “nn” from 1 to 215
1. Frame all polyhedra with cylinders and spheres
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Manipulate[​​name=PolyhedronData[][[nn]];​​frame=GraphicsComplex[PolyhedronData[name,"VertexCoordinates"],Tube[PolyhedronData[name,"EdgeIndices"],tubedia]];​​poly=GraphicsComplex[PolyhedronData[name,"VertexCoordinates"],Polygon[PolyhedronData[name,"FaceIndices"]]];​​vert=poly[[1]];​​sp=Sphere[{0,0,0},sps];​​spall=Map[Translate[sp,#]&,vert];​​Graphics3D[{RGBColor[1,0.6,0.9],poly,RGBColor[0.9,1,0],frame,Cyan,spall},Boxed->False,SphericalRegion->True,ImageSize->300],{{nn,164,"number"},1,255,1},{{sps,0.07,"sphere size"},0.02,1},​​{{tubedia,0.05,"did"},0.01,1},​​TrackedSymbols:>{nn,cyls,cyls,sps,tubedia}]
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number
sphere size
did
2. Another way of creating cylinders along the edges
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nn=175;​​rd=PolyhedronData[PolyhedronData[][[nn]],"Faces"];​​rdedge=PolyhedronData[PolyhedronData[][[nn]],"EdgeIndices"];​​rdv=PolyhedronData[PolyhedronData[][[nn]],"VertexCoordinates"];​​rdfr=Map[Cylinder[rdv[[#]],0.05]&,rdedge];​​sp=Sphere[{0,0,0},0.1];​​sprd=Map[Translate[sp,#]&,rdv];​​Graphics3D[{rd,Cyan,rdfr,Yellow,sprd},Boxed->False,SphericalRegion->True,ImageSize->300]
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Another possibility is using stellate from 0.
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Needs["PolyhedronOperations`"];​​nn=171;​​st=1.2;​​pol=PolyhedronData[PolyhedronData[][[nn]],"Faces"];​​rd=Stellate[pol,1.2];​​rdedge=PolyhedronData[PolyhedronData[][[nn]],"EdgeIndices"];​​rdv=PolyhedronData[PolyhedronData[][[nn]],"VertexCoordinates"];​​rdfr=Map[Cylinder[rdv[[#]],0.07]&,rdedge];​​sp=Sphere[{0,0,0},0.2];​​sprd=Map[Translate[sp,#]&,rdv];​​Graphics3D[{rd,Cyan,rdfr,Yellow,sprd},Boxed->False,SphericalRegion->True,ImageSize->300]
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Graphics3D[{Cyan,rdfr,Yellow,sprd},Boxed->False,SphericalRegion->True,ImageSize->300]
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Graphics3D[{rd,Yellow,sprd},Boxed->False,SphericalRegion->True,ImageSize->300]
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3. Framing by perforating and placing cylinders on the edges
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nn=200;​​p1=0.2;​​tubedia=0.02;​​base=PolyhedronData[PolyhedronData[][[nn]],"Faces"];​​ee=Normal[base]/.p:Polygon[__]:>(p/.pts:{{_?NumericQ,_,_},__}:>Module[{cent,innervtx},cent=Mean[pts];​​innervtx={1-p1,p1}.{#,cent}&/@pts;​​Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;​​name=PolyhedronData[][[nn]];​​base=PolyhedronData[PolyhedronData[][[nn]],"Faces"];​​pt=GraphicsComplex[PolyhedronData[name,"VertexCoordinates"],Tube[PolyhedronData[name,"EdgeIndices"],tubedia]];​​Graphics3D[{Purple,Scale[base,0.99],White,pt,FaceForm[Green,Yellow],ee},Boxed->False,ImageSize->300,SphericalRegion->True]
4. Using geodesate and truncate, as well as glow
5. Using perforate with geodesate and truncate
6. Using shrink

CITE THIS NOTEBOOK

Showing all polyhedra in various ways​
by Sándor Kabai​
Wolfram Community, STAFF PICKS, November 14, 2023
​https://community.wolfram.com/groups/-/m/t/3064990