Five Easy Pieces

Practically the same idea (and code) as Give Me Some Space, just truncating the tetrahedron rather than rectifying it.

The code for the Manipulate is below; when exporting to a GIF I used "DisplayDurations" -> Prepend[Table[1/50, {199}], 1/2] inside Export to get the animation to pause on the original tetrahedron for half a second.

DynamicModule[{viewpoint = {Cos[2?/3], Sin[2?/3], 1/Sqrt[2]},
  g = .6, d = .2, n = 4,
  v = PolyhedronData["Tetrahedron", "VertexCoordinates"],
  e = {{2, 3, 4}, {1, 4, 3}, {4, 1, 2}, {3, 2, 1}},
  tt = PolyhedronData["TruncatedTetrahedron", "VertexCoordinates"],
  te = PolyhedronData["TruncatedTetrahedron", "Faces"][[2]],
  cols = RGBColor /@ {"#e43a19", "#f2f4f7", "#111f4d"},
  s, r},
 Manipulate[
  s = Haversine[? t];
  r = Haversine[2 ? t];
  Graphics3D[{Thickness[.004], EdgeForm[None],
    Table[
     {GraphicsComplex[
       (1/2 + r/2) v[[i]] + RotationTransform[2 ?/3 s, v[[i]]][#] & /@ (v/4),
       {cols[[1]], Polygon[e[[i]]], cols[[2]], Polygon[e[[Drop[Range[4], {i}]]]]}]},
     {i, 1, Length[v]}],
    GraphicsComplex[
     RotationTransform[0, {0, 0, 1}][RotationTransform[? s, {-Sqrt[3], -3, Sqrt[6]}][1/4 tt]],
     {cols[[1]], Polygon[te[[1, ;; 4]]], cols[[2]], Polygon[te[[1, 5 ;;]]]}]},
   Boxed -> False, ImageSize -> {540, 540}, PlotRange -> 2.5, 
   Background -> cols[[-1]], ViewPoint -> 10 viewpoint, 
   ViewAngle -> ?/125, ViewVertical -> {0, 0, 1}, 
   SphericalRegion -> True,
   Lighting -> {{"Ambient", GrayLevel[d]},
     {"Directional", GrayLevel[g], ImageScaled[{2, 0, 2}]},
     {"Directional", GrayLevel[g], ImageScaled[{-2, 2, 2}]},
     {"Directional", GrayLevel[g], ImageScaled[{0, -2, 2}]}}],
  {t, 0, 1}]
 ]