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Graphics3D[GraphicsComplex[{{0, -(1/Sqrt[2]), -(1/Sqrt[2])}, {0, -(1/Sqrt[2]), 1/Sqrt[2]},
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   {-(1/Sqrt[2]), 0, 1/Sqrt[2]}, {1/Sqrt[2], 0, -(1/Sqrt[2])}, {1/Sqrt[2], 0, 1/Sqrt[2]},
   {1/2, (1 + Sqrt[2])/2, 0}, {-1/2, (1 + Sqrt[2])/2, 0}, {(-1 - Sqrt[2])/2, 1/2, 0},
   {(-1 - Sqrt[2])/2, -1/2, 0}, {-1/2, (-1 - Sqrt[2])/2, 0}, {1/2, (-1 - Sqrt[2])/2, 0},
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    {13, 12, 5, 1}, {15, 14, 1, 7}, {9, 16, 7, 3}, {3, 10, 9}, {5, 12, 11}, {1, 14, 13},
    {7, 16, 15}}]]]

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Johnson Polyhedra

Geometry