rolling dodecahedron

vs = N[PolyhedronData["Dodecahedron", "VertexCoordinates"]];
faces = (List @@ PolyhedronData["Dodecahedron", "Faces"][[2]])[[1]];
d = Norm[vs[[1]]];
Manipulate[
 Module[{vsP, facesP},
  vsP = (vs.RotationMatrix[2 Pi t, {0, 0, 1}]).RotationMatrix[
     4 Pi t, {0, 1, 0}];
  facesP =
   Map[#[[;; 2]] &,
    SortBy[Table[vsP[[face]], {face, faces}], -Max[Last /@ #] &], {2}];
  Graphics[{
    {EdgeForm[Black], FaceForm[Directive[Opacity[0.9], White]],
     Table[Polygon[face], {face, facesP}]},
    Circle[{0, 0}, d]
    }
   ]
  ]
 , {t, 0, 1}]