Taking a Dual

scale[shape_, t_] := Module[{faces = PolyhedronData[shape, "Faces"]},
   faces[[1]] = Map[t, faces[[1]]];
   faces
   ];
scaleV[shape_, t_] :=
  Module[{coords = PolyhedronData[shape, "VertexCoordinates"]},
   Map[t, coords]
   ];
fade[a_, t_] := (Tanh[10 (t - a)] + 1)/2;
frame[t_] :=
 Graphics3D[{
   Opacity[If[t < 0.45, 1 - fade[0.2, t]/2, 1 - fade[0.45, t]]], Red,
   If[t < 0.77, scale["Cube", # &], Opacity[1]],
   Opacity[1 - fade[0.5, t]/2], Blue,
   If[t < 0.8, Sphere[{0, 0, 0}, 1/2], Opacity[1]],
   Opacity[fade[0.4, t]], Green,
   If[t < 1.1, Point[scaleV["Octahedron", #/Sqrt[2] &]], Opacity[1]],
   Opacity[If[t < 1.1, 1 - fade[0.8, t]/2, 1 - fade[1.1, t]]],
   If[t < 1.2, scale["Octahedron", #/Sqrt[2] &], Opacity[1]],
   Opacity[fade[1, t]], Blue,
   Opacity[1 - fade[1.4, t]],
   Sphere[{0, 0, 0}, 1/(2 Sqrt[3])],
   Opacity[1],
   Red,
   Opacity[fade[0.84, t]],
   Point[scaleV["Cube", #/3 &]],
   Opacity[1],
   scale["Cube", #*
      If[t < 1.6, 1/3, If[t < 2, 1/3 + 5/3 (t - 1.6), 1]] &]
   }
  , Boxed -> False,
  PlotRange -> {{-0.6, 0.6}, {-0.6, 0.6}, {-0.6, 0.6}},
  ViewPoint -> {3 Sin[t Pi/4], 3 Cos[t Pi/4], 2}];
Manipulate[frame[t], {t, 0, 2}]