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Graphics3D[GraphicsComplex[{{-1/2, 1/2 + 1/Sqrt[2], 1/2 + 1/Sqrt[2]},
   {-1/2, 1/2 + 1/Sqrt[2], (2 - 2*Sqrt[2])^(-1)}, {-1/2, (2 - 2*Sqrt[2])^(-1), 1/2 + 1/Sqrt[2]},
   {-1/2, (2 - 2*Sqrt[2])^(-1), (2 - 2*Sqrt[2])^(-1)}, {1/2, 1/2 + 1/Sqrt[2], 1/2 + 1/Sqrt[2]},
   {1/2, 1/2 + 1/Sqrt[2], (2 - 2*Sqrt[2])^(-1)}, {1/2, (2 - 2*Sqrt[2])^(-1), 1/2 + 1/Sqrt[2]},
   {1/2, (2 - 2*Sqrt[2])^(-1), (2 - 2*Sqrt[2])^(-1)}, {1/2 + 1/Sqrt[2], -1/2, 1/2 + 1/Sqrt[2]},
   {1/2 + 1/Sqrt[2], -1/2, (2 - 2*Sqrt[2])^(-1)}, {1/2 + 1/Sqrt[2], 1/2, 1/2 + 1/Sqrt[2]},
   {1/2 + 1/Sqrt[2], 1/2, (2 - 2*Sqrt[2])^(-1)}, {1/2 + 1/Sqrt[2], 1/2 + 1/Sqrt[2], -1/2},
   {1/2 + 1/Sqrt[2], 1/2 + 1/Sqrt[2], 1/2}, {1/2 + 1/Sqrt[2], (2 - 2*Sqrt[2])^(-1), -1/2},
   {1/2 + 1/Sqrt[2], (2 - 2*Sqrt[2])^(-1), 1/2}, {(2 - 2*Sqrt[2])^(-1), -1/2, 1/2 + 1/Sqrt[2]},
   {(2 - 2*Sqrt[2])^(-1), -1/2, (2 - 2*Sqrt[2])^(-1)}, {(2 - 2*Sqrt[2])^(-1), 1/2,
    1/2 + 1/Sqrt[2]}, {(2 - 2*Sqrt[2])^(-1), 1/2, (2 - 2*Sqrt[2])^(-1)},
   {(2 - 2*Sqrt[2])^(-1), 1/2 + 1/Sqrt[2], -1/2}, {(2 - 2*Sqrt[2])^(-1), 1/2 + 1/Sqrt[2], 1/2},
   {(2 - 2*Sqrt[2])^(-1), (2 - 2*Sqrt[2])^(-1), -1/2}, {(2 - 2*Sqrt[2])^(-1),
    (2 - 2*Sqrt[2])^(-1), 1/2}}, Polygon[{{6, 12, 10, 8, 4, 18, 20, 2},
    {1, 19, 17, 3, 7, 9, 11, 5}, {3, 24, 23, 4, 8, 15, 16, 7}, {5, 14, 13, 6, 2, 21, 22, 1},
    {9, 16, 15, 10, 12, 13, 14, 11}, {19, 22, 21, 20, 18, 23, 24, 17}, {16, 9, 7}, {5, 11, 14},
    {3, 17, 24}, {22, 19, 1}, {8, 10, 15}, {13, 12, 6}, {23, 18, 4}, {2, 20, 21}}]]]

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Archimedean Solid

Geometry