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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]]), -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]]},
   {-(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]])}},
  Polygon[{{7, 9, 3, 1, 11}, {1, 3, 10, 8, 12}, {11, 1, 6, 8, 4}, {7, 5, 1, 12, 4},
    {1, 5, 9, 10, 6}, {8, 6, 3, 9, 2}, {7, 2, 10, 3, 5}, {4, 2, 9, 5, 11},
    {4, 12, 6, 10, 2}, {8, 2, 7, 11, 12}, {5, 3, 6, 12, 11}, {4, 8, 10, 9, 7}}]]]

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Kepler-Poinsot Polyhedra