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

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Johnson Polyhedra

Geometry