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

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Kepler-Poinsot Polyhedra