Snub dodecahedron has 20 triangular faces that coincide with 20 faces of icosahedron. The icosahedron vertices are connected to points on the opposite sides which divide the sides in the ratio of 3/2. These lines run through the triangle sides of the sub dodecahedron.
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sd={GraphicsComplex[PolyhedronData["SnubDodecahedron","VertexCoordinates"],Polygon[PolyhedronData["SnubDodecahedron","FaceIndices"]]]};sdn={Scale[sd,0.364]};fi=1.618;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];perf=0;div=0.1666;ui=0.5Sqrt[3]/3;vE={-ui,-0.5,0};vF={2ui,0,0};vG={-ui,0.5,0};q1={vE,vF,vG,vE};q2=perf{vE,vF,vG,vE};tr1={Yellow,Translate[Polygon[Join[q1->q2]],{0,0,0}]};vH={-ui,div,0};i3={Red,Table[Rotate[Cylinder[{vH,vF},0.01],i2Pi/3,{0,0,1}],{i,3}]};i4=Rotate[Translate[{tr1,i3},{ui,0,0}],bet1,{0,1,0}];i5=Rotate[i4,Pi,{0,0,1}];i6=Table[Rotate[Rotate[Translate[{i4,i5},{0,0,fi/2}],-alf,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];i10a=Rotate[{i6},alf+bet1,{0,1,0}];i10b=Rotate[i10a,Pi,{0,1,0}];icolarge=Rotate[Scale[i10a,fi^2{1,1,1},{0,0,0}],-bet1,{0,1,0}];
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Graphics3D[{i10b,RGBColor[0.3,1,1],sdn},BoxedFalse,SphericalRegionTrue,ViewPoint{0,100,0},ViewAngle0.01,ImageSize250]
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