Medial rhombic triacontahedron (MRT) explained​
​By Sándor Kabai
MRT is a stellation of rhombic triacontahedron (RT). I worked out a couple of methods for its presentation.
​https://demonstrations.wolfram.com/TwoStellationsOfTheRhombicTriacontahedron/​
​https://demonstrations.wolfram.com/CubesOnARhombicHexecontahedron/​
​https://mathworld.wolfram.com/MedialRhombicTriacontahedron.html
Assembled from deltahedron faces
fi=GoldenRatio;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​bet1=ArcTan[1/fi^2];​​​​vE={-1/fi^2,0,0};​​vF={0,-1/fi,0};​​vG={fi,0,0};​​vH={0,1/fi,0};​​​​vI={fi^2,0,0};​​vJ={0,1,0};​​vK={-fi^2,0,0};​​vL={0,-1,0};​​fi2=Translate[Polygon[{vI,vJ,vK,vL}],{0,0,1.618}];​​​​w1={vE,vF,vG,vH};​​w2=0.5{vE,vF,vG,vH};​​w=Polygon[Join[w1->w2]];​​p1=Translate[w,{1,0,fi}];​​p2=Rotate[p1,Pi,{0,0,1}];​​sp=Sphere[{0,0,0},0.1];​​(*AfaceofMRTispartofarhombushavingratioofdiagonalequalto"golden ratio squared"*)​​Graphics3D[{sp,fi2,Yellow,p1,p2},BoxedFalse,ViewPoint{0,0,100},ViewAngle0.01,SphericalRegionTrue]
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p3=Rotate[{p1,p2},Pi,{0,1,0}];​​p4=Rotate[Rotate[{p1,p2,p3},Pi/2,{0,1,0}],Pi/2,{0,0,1}];​​p5=Rotate[Rotate[{p1,p2,p3},Pi/2,{1,0,0}],Pi/2,{0,0,1}];​​pmrt=Table[Rotate[Rotate[{p1,p2,p3,p4,p5},-bet,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];​​Graphics3D[{pmrt},BoxedFalse,ViewPoint{0,0,100},ViewAngle0.01,SphericalRegionTrue]
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w=Polygon[w1];​​p1=Translate[w,{1,0,fi}];​​p2=Rotate[p1,Pi,{0,0,1}];​​p3=Rotate[{p1,p2},Pi,{0,1,0}];​​p4=Rotate[Rotate[{p1,p2,p3},Pi/2,{0,1,0}],Pi/2,{0,0,1}];​​p5=Rotate[Rotate[{p1,p2,p3},Pi/2,{1,0,0}],Pi/2,{0,0,1}];​​cub=PolyhedronData["Cube","Faces"];​​cubn=Scale[cub,0.999×2fi];​​(*Thisisonefifthofmrt,asifplacedonacubesized2fi*)​​fifth={p1,p2,p3,p4,p5};​​Graphics3D[{fifth,Yellow,cubn},BoxedFalse,SphericalRegionTrue]
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Rotate the fifth part until the tips of two faces are in a vertical line
axis=Cylinder[{{0,0,-4},{0,0,4}},0.1];​​tiltfifth=Rotate[{fifth},bet,{1,0,0}];​​Graphics3D[{axis,tiltfifth},BoxedFalse,SphericalRegionTrue,ViewPoint{4,2,2}]
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tiltcub=Rotate[{cubn},bet,{1,0,0}];​​Graphics3D[{tiltcub,axis,Red,tiltfifth},BoxedFalse,SphericalRegionTrue,ViewPoint{4,2,2}]
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Rotate the one gifth five times to get mrt
Combining mrt and compound of five cubes
Whaterver ICO and RT can do, MRT can do better in clusters and in rings
MRT in RT
m = distance of two parallel faces
icon = central ico, s1,s2,s3 = connecting ico corresonding to s5 geometry
This is the unit (icon, s1, s2, s3) which is repeated fives times around an r circle
Bring icon to the centre, and rotate ring five times