In[]:=
Manipulate[Module[{},a1=(Translate[{Scale[s1,0.5*{1,1,1},{0,0,0}]},#1]&)/@(grow3*tetv);a2=(Translate[{Scale[{{RGBColor[0.3,1,0.2],s1},a1},0.5*{1,1,1},{0,0,0}]},#1]&)/@(grow2*tetv);a3=(Translate[{Scale[{{RGBColor[1,0.3,0.6],s1},a2},0.5*{1,1,1},{0,0,0}]},#1]&)/@(grow1*tetv);Graphics3D[{{Yellow,s1},a3,Scale[tet,tetgr*{1,1,1},{0,0,0}]},Boxed->False,SphericalRegion->True,ViewAngle->0.2,ViewPoint->{-1.5,-4,0.2},ImageSize->470,PlotRange->{{-0.4,0.6},{-0.56,0.6},{-0.3,0.65}}]],{{grow1,0.17,"stage 1"},0.17,0.5,ImageSize->Tiny},{{grow2,0.17,"stage 2"},0.17,0.5,ImageSize->Tiny},{{grow3,0.17,"stage 3"},0.17,0.5,ImageSize->Tiny},"","show tetrahedron",{{tetgr,0.3,""},0.3,1,ImageSize->Tiny},ControlPlacement->Left,TrackedSymbols:>{grow1,grow2,grow3,tetgr},Initialization:>(tv=N[1/(2*Sqrt[6])];oct=GraphicsComplex[PolyhedronData["Octahedron","VertexCoordinates"],Polygon[PolyhedronData["Octahedron","FaceIndices"]]];s1=Scale[Rotate[Rotate[oct,Pi/4,{0,0,1}],ArcTan[Sqrt[2]],{0,1,0}],0.5*{1,1,1}];tet=GraphicsComplex[PolyhedronData["Tetrahedron","VertexCoordinates"],Polygon[PolyhedronData["Tetrahedron","FaceIndices"]]];tetv=PolyhedronData["Tetrahedron","VertexCoordinates"];)]
Out[]=
