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Graphics3D[GraphicsComplex[{{0, (-1 - Sqrt[5])/2, -Sqrt[701/2 - 131/(2*Sqrt[5])]/88},
   {0, (1 + Sqrt[5])/2, -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}, {Sqrt[1/2 + 1/(2*Sqrt[5])], 0,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88 + Sqrt[(5 - Sqrt[5])/10]}, {Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88 + Sqrt[(5 - Sqrt[5])/10]}, {Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88 + Sqrt[(5 - Sqrt[5])/10]}, {-Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}, {-Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}, {Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}, {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}, {-Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88 + Sqrt[(5 - Sqrt[5])/10]}, {-Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88 + Sqrt[(5 - Sqrt[5])/10]}, {-Sqrt[5/4 + Sqrt[5]/2], -1/2,
    -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}, {-Sqrt[5/4 + Sqrt[5]/2], 1/2, -Sqrt[701/2 - 131/(2*Sqrt[5])]/88},
   {Sqrt[5/4 + Sqrt[5]/2], -1/2, -Sqrt[701/2 - 131/(2*Sqrt[5])]/88},
   {Sqrt[5/4 + Sqrt[5]/2], 1/2, -Sqrt[701/2 - 131/(2*Sqrt[5])]/88}},
  Polygon[{{5, 11, 10, 4, 3}, {15, 14, 8, 1, 6, 12, 13, 7, 2, 9}, {11, 5, 2, 7}, {10, 11, 13, 12},
    {4, 10, 6, 1}, {3, 4, 8, 14}, {5, 3, 15, 9}, {9, 2, 5}, {7, 13, 11}, {12, 6, 10}, {1, 8, 4},
    {14, 15, 3}}]]]

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Johnson Polyhedra

Geometry