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Graphics3D[GraphicsComplex[{{0, (-1 - Sqrt[5])/2, -1/2}, {0, (-1 - Sqrt[5])/2, 1/2},
   {0, (1 + Sqrt[5])/2, -1/2}, {0, (1 + Sqrt[5])/2, 1/2}, {-Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4,
    (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4,
    (-5 - Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4,
    (5 + Sqrt[10*(5 + Sqrt[5])])/10}, {-Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2,
    (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10}, {-Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2,
    (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10}, {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2,
    (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10}, {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2,
    (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10}, {-Sqrt[1/2 + 1/(2*Sqrt[5])], 0, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[1/2 + 1/(2*Sqrt[5])], 0, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4, (5 + Sqrt[10*(5 + Sqrt[5])])/10},
   {-Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4, (5 + Sqrt[10*(5 + Sqrt[5])])/10},
   {Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4, (-5 - Sqrt[10*(5 + Sqrt[5])])/10},
   {Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4, (-5 - Sqrt[10*(5 + Sqrt[5])])/10},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, (5 + 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[(5 - Sqrt[5])/10]/2, (-1 - Sqrt[5])/4, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {Sqrt[(5 - Sqrt[5])/10]/2, (1 + Sqrt[5])/4, (-5 - 2*Sqrt[5*(5 + 2*Sqrt[5])])/10},
   {-Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, -1/2}, {-Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, 1/2},
   {-Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, -1/2}, {-Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, 1/2},
   {Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, -1/2}, {Sqrt[5/8 + Sqrt[5]/8], (-3 - Sqrt[5])/4, 1/2},
   {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, -1/2}, {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, 1/2},
   {-Sqrt[5/4 + Sqrt[5]/2], -1/2, -1/2}, {-Sqrt[5/4 + Sqrt[5]/2], -1/2, 1/2},
   {-Sqrt[5/4 + Sqrt[5]/2], 1/2, -1/2}, {-Sqrt[5/4 + Sqrt[5]/2], 1/2, 1/2},
   {Sqrt[5/4 + Sqrt[5]/2], -1/2, -1/2}, {Sqrt[5/4 + Sqrt[5]/2], -1/2, 1/2},
   {Sqrt[5/4 + Sqrt[5]/2], 1/2, -1/2}, {Sqrt[5/4 + Sqrt[5]/2], 1/2, 1/2},
   {-Sqrt[(5 + 2*Sqrt[5])/5], 0, (-5 - Sqrt[10*(5 + Sqrt[5])])/10},
   {Sqrt[(5 + 2*Sqrt[5])/5], 0, (5 + Sqrt[10*(5 + Sqrt[5])])/10}},
  Polygon[{{20, 13, 19, 11, 12}, {16, 13, 20}, {15, 19, 13}, {7, 11, 19}, {40, 12, 11}, {8, 20, 12},
    {30, 4, 8}, {26, 34, 16}, {32, 24, 15}, {2, 28, 7}, {36, 38, 40}, {4, 26, 16, 20, 8},
    {34, 32, 15, 13, 16}, {24, 2, 7, 19, 15}, {28, 36, 40, 11, 7}, {38, 30, 8, 12, 40}, {14, 21, 9, 10, 22},
    {22, 10, 6}, {10, 9, 39}, {9, 21, 5}, {21, 14, 17}, {14, 22, 18}, {18, 29, 37}, {6, 25, 3}, {39, 31, 33},
    {5, 1, 23}, {17, 35, 27}, {18, 22, 6, 3, 29}, {6, 10, 39, 33, 25}, {39, 9, 5, 23, 31},
    {5, 21, 17, 27, 1}, {17, 14, 18, 37, 35}, {29, 3, 4, 30}, {3, 25, 26, 4}, {25, 33, 34, 26},
    {33, 31, 32, 34}, {31, 23, 24, 32}, {23, 1, 2, 24}, {1, 27, 28, 2}, {27, 35, 36, 28}, {35, 37, 38, 36},
    {37, 29, 30, 38}}]]]

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Johnson Polyhedra