Dodecahedron is planar

pts = {{-Sqrt[1 + 2/Sqrt[5]], 0,
    Root[1 - 20 #1^2 + 80 #1^4 &, 3]}, {Sqrt[1 + 2/Sqrt[5]], 0,
    Root[1 - 20 #1^2 + 80 #1^4 &, 2]}, {Root[1 - 20 #1^2 + 80 #1^4 &,
     1], 1/4 (-3 - Sqrt[5]),
    Root[1 - 20 #1^2 + 80 #1^4 &, 3]}, {Root[1 - 20 #1^2 + 80 #1^4 &,
     1], 1/4 (3 + Sqrt[5]), Root[1 - 20 #1^2 + 80 #1^4 &, 3]}, {Sqrt[
    5/8 + 11/(8 Sqrt[5])], 1/4 (-1 - Sqrt[5]),
    Root[1 - 20 #1^2 + 80 #1^4 &, 3]}, {Sqrt[5/8 + 11/(8 Sqrt[5])],
    1/4 (1 + Sqrt[5]),
    Root[1 - 20 #1^2 + 80 #1^4 &, 3]}, {Root[1 - 20 #1^2 + 80 #1^4 &,
     2], 1/4 (-1 - Sqrt[5]), Sqrt[
    5/8 + 11/(8 Sqrt[5])]}, {Root[1 - 20 #1^2 + 80 #1^4 &, 2],
    1/4 (1 + Sqrt[5]), Sqrt[
    5/8 + 11/(8 Sqrt[5])]}, {-(1/2) Sqrt[1 + 2/Sqrt[5]], -(1/2),
    Root[1 - 100 #1^2 + 80 #1^4 &, 1]}, {-(1/2) Sqrt[1 + 2/Sqrt[5]],
    1/2, Root[1 - 100 #1^2 + 80 #1^4 &, 1]}, {Sqrt[
    1/4 + 1/(2 Sqrt[5])], -(1/2), Sqrt[5/8 + 11/(8 Sqrt[5])]}, {Sqrt[
    1/4 + 1/(2 Sqrt[5])], 1/2, Sqrt[5/8 + 11/(8 Sqrt[5])]}, {Sqrt[
    1/10 (5 + Sqrt[5])], 0,
    Root[1 - 100 #1^2 + 80 #1^4 &, 1]}, {Root[
     1 - 100 #1^2 + 80 #1^4 &, 1], 1/4 (-1 - Sqrt[5]),
    Root[1 - 20 #1^2 + 80 #1^4 &, 2]}, {Root[1 - 100 #1^2 + 80 #1^4 &,
      1], 1/4 (1 + Sqrt[5]),
    Root[1 - 20 #1^2 + 80 #1^4 &, 2]}, {Root[1 - 5 #1^2 + 5 #1^4 &,
     1], 0, Sqrt[
    5/8 + 11/(8 Sqrt[5])]}, {Root[1 - 20 #1^2 + 80 #1^4 &, 3],
    1/4 (-1 - Sqrt[5]),
    Root[1 - 100 #1^2 + 80 #1^4 &, 1]}, {Root[1 - 20 #1^2 + 80 #1^4 &,
      3], 1/4 (1 + Sqrt[5]),
    Root[1 - 100 #1^2 + 80 #1^4 &, 1]}, {Sqrt[1/8 + 1/(8 Sqrt[5])],
    1/4 (-3 - Sqrt[5]), Root[1 - 20 #1^2 + 80 #1^4 &, 2]}, {Sqrt[
    1/8 + 1/(8 Sqrt[5])], 1/4 (3 + Sqrt[5]),
    Root[1 - 20 #1^2 + 80 #1^4 &, 2]}};
edges = {{1, 14}, {1, 15}, {1, 16}, {2, 5}, {2, 6}, {2, 13}, {3,
    7}, {3, 14}, {3, 19}, {4, 8}, {4, 15}, {4, 20}, {5, 11}, {5,
    19}, {6, 12}, {6, 20}, {7, 11}, {7, 16}, {8, 12}, {8, 16}, {9,
    10}, {9, 14}, {9, 17}, {10, 15}, {10, 18}, {11, 12}, {13,
    17}, {13, 18}, {17, 19}, {18, 20}};

projection[p_, t_, r_] := Abs[1/(1 + t p[[3]])] ( {
      {1, 0, 0.17},
      {0, 1, 0.17}
     } ).(RotationMatrix[r, {0, -2, 1.5}].p);

Manipulate[
 Graphics[GraphicsComplex[Map[projection[#, t, r] &, pts],
   Line[edges]]], {t, 0, 0.7}, {{r, 0.132}, 0, Pi}]