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Graphics3D[GraphicsComplex[{{0, 0, -5/Sqrt[50 - 10*Sqrt[5]]}, {0, 0, 5/Sqrt[50 - 10*Sqrt[5]]},
   {-Sqrt[2/(5 - Sqrt[5])], 0, -(1/Sqrt[10 - 2*Sqrt[5]])}, {Sqrt[2/(5 - Sqrt[5])], 0,
    1/Sqrt[10 - 2*Sqrt[5]]}, {(1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), -1/2, -(1/Sqrt[10 - 2*Sqrt[5]])},
   {(1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), 1/2, -(1/Sqrt[10 - 2*Sqrt[5]])},
   {-(1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), -1/2, 1/Sqrt[10 - 2*Sqrt[5]]},
   {-(1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), 1/2, 1/Sqrt[10 - 2*Sqrt[5]]},
   {-(-1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), -Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])]/2,
    -(1/Sqrt[10 - 2*Sqrt[5]])}, {-(-1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]),
    Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])]/2, -(1/Sqrt[10 - 2*Sqrt[5]])},
   {(-1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), -Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])]/2, 1/Sqrt[10 - 2*Sqrt[5]]},
   {(-1 + Sqrt[5])/(2*Sqrt[10 - 2*Sqrt[5]]), Sqrt[(5 + Sqrt[5])/(5 - Sqrt[5])]/2, 1/Sqrt[10 - 2*Sqrt[5]]}},
  Polygon[{{2, 12, 8}, {2, 8, 7}, {2, 7, 11}, {2, 11, 4}, {2, 4, 12}, {5, 9, 1}, {6, 5, 1}, {10, 6, 1},
    {3, 10, 1}, {9, 3, 1}, {12, 10, 8}, {8, 3, 7}, {7, 9, 11}, {11, 5, 4}, {4, 6, 12}, {5, 11, 9}, {6, 4, 5},
    {10, 12, 6}, {3, 8, 10}, {9, 7, 3}}]]]

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Platonic Polyhedra

Geometry