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Graphics3D[GraphicsComplex[{{0, 0, Sqrt[5/8 + Sqrt[5]/8] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, -Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {Sqrt[1/2 + 1/(2*Sqrt[5])], 0, Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, -Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, -Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {-Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {-Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, -Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, -Sqrt[1/8 + 1/(8*Sqrt[5])] - Sqrt[(85 + 31*Sqrt[5])/10]/44}},
  Polygon[{{10, 4, 2, 5, 11}, {11, 7, 3}, {5, 9, 7}, {2, 8, 9}, {4, 6, 8}, {10, 3, 6}, {11, 5, 7}, {5, 2, 9},
    {2, 4, 8}, {4, 10, 6}, {10, 11, 3}, {7, 9, 1}, {9, 8, 1}, {8, 6, 1}, {6, 3, 1}, {3, 7, 1}}]]]

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Johnson Polyhedra

Geometry