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Graphics3D[GraphicsComplex[{{0, (-1 - Sqrt[5])/2, 0}, {0, (1 + Sqrt[5])/2, 0},
   {-Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, Sqrt[(5 - Sqrt[5])/10]}, {-Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2,
    Sqrt[(5 - Sqrt[5])/10]}, {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, -Sqrt[(5 - Sqrt[5])/10]},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, -Sqrt[(5 - Sqrt[5])/10]}, {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0,
    -Sqrt[(5 - Sqrt[5])/10]}, {Sqrt[1/2 + 1/(2*Sqrt[5])], 0, Sqrt[(5 - Sqrt[5])/10]},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, -Sqrt[(5 - Sqrt[5])/10]},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, -Sqrt[(5 - Sqrt[5])/10]},
   {Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, Sqrt[(5 - Sqrt[5])/10]},
   {Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, Sqrt[(5 - Sqrt[5])/10]},
   {-Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, 0}, {-Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, 0},
   {Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, 0}, {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, 0},
   {-Sqrt[5/4 + Sqrt[5]/2], -1/2, 0}, {-Sqrt[5/4 + Sqrt[5]/2], 1/2, 0}, {Sqrt[5/4 + Sqrt[5]/2], -1/2, 0},
   {Sqrt[5/4 + Sqrt[5]/2], 1/2, 0}}, Polygon[{{12, 4, 3, 11, 8}, {4, 12, 2, 14}, {3, 4, 18, 17},
    {11, 3, 13, 1}, {8, 11, 15, 19}, {12, 8, 20, 16}, {16, 2, 12}, {14, 18, 4}, {17, 13, 3}, {1, 15, 11},
    {19, 20, 8}, {5, 9, 7, 10, 6}, {2, 16, 6, 10}, {18, 14, 10, 7}, {13, 17, 7, 9}, {15, 1, 9, 5},
    {20, 19, 5, 6}, {6, 16, 20}, {10, 14, 2}, {7, 17, 18}, {9, 1, 13}, {5, 19, 15}}]]]

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Johnson Polyhedra

Geometry