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Graphics3D[GraphicsComplex[{{0, (-1 - Sqrt[5])/2, 0}, {0, (1 + Sqrt[5])/2, 0}, {Sqrt[1/8 - 1/(8*Sqrt[5])], (-1 - Sqrt[5])/4, -Sqrt[1 + 2/Sqrt[5]]},
   {Sqrt[1/8 - 1/(8*Sqrt[5])], (1 + Sqrt[5])/4, -Sqrt[1 + 2/Sqrt[5]]}, {Sqrt[1/8 + 1/(8*Sqrt[5])], (-3 - Sqrt[5])/4, Sqrt[(5 + Sqrt[5])/10]},
   {Sqrt[1/8 + 1/(8*Sqrt[5])], (3 + Sqrt[5])/4, Sqrt[(5 + Sqrt[5])/10]}, {Sqrt[1/4 + 1/(2*Sqrt[5])], -1/2, Sqrt[1 + 2/Sqrt[5]]},
   {Sqrt[1/4 + 1/(2*Sqrt[5])], 1/2, Sqrt[1 + 2/Sqrt[5]]}, {Sqrt[5/8 + 11/(8*Sqrt[5])], (-1 - Sqrt[5])/4, Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0]},
   {Sqrt[5/8 + 11/(8*Sqrt[5])], (1 + Sqrt[5])/4, Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0]}, {-Sqrt[1 + 2/Sqrt[5]], 0, Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0]},
   {-Sqrt[1 + 2/Sqrt[5]]/2, -1/2, -Sqrt[1 + 2/Sqrt[5]]}, {-Sqrt[1 + 2/Sqrt[5]]/2, 1/2, -Sqrt[1 + 2/Sqrt[5]]}, {Sqrt[1 + 2/Sqrt[5]], 0, Sqrt[(5 + Sqrt[5])/10]},
   {Sqrt[5/8 + Sqrt[5]/8], -(1 + Sqrt[5])^2/8, 0}, {Sqrt[5/8 + Sqrt[5]/8], (3 + Sqrt[5])/4, 0}, {Sqrt[(5 + Sqrt[5])/10], 0, -Sqrt[1 + 2/Sqrt[5]]},
   {-Sqrt[(5 + Sqrt[5])/2]/2, -(1 + Sqrt[5])^2/8, 0}, {-Sqrt[(5 + Sqrt[5])/2]/2, (3 + Sqrt[5])/4, 0}, {-Sqrt[5 + 2*Sqrt[5]]/2, -1/2, 0}, {-Sqrt[5 + 2*Sqrt[5]]/2, 1/2, 0},
   {Sqrt[5 + 2*Sqrt[5]]/2, -1/2, 0}, {Sqrt[5 + 2*Sqrt[5]]/2, 1/2, 0}, {Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0], 0, Sqrt[1 + 2/Sqrt[5]]},
   {Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0], (-1 - Sqrt[5])/4, Sqrt[(5 + Sqrt[5])/10]}, {Root[1 - 100*#1^2 + 80*#1^4 & , 1, 0], (1 + Sqrt[5])/4, Sqrt[(5 + Sqrt[5])/10]},
   {Root[1 - 20*#1^2 + 80*#1^4 & , 1, 0], (-3 - Sqrt[5])/4, Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0]}, {Root[1 - 20*#1^2 + 80*#1^4 & , 1, 0], (3 + Sqrt[5])/4,
    Root[1 - 5*#1^2 + 5*#1^4 & , 1, 0]}, {Root[1 - 20*#1^2 + 80*#1^4 & , 2, 0], (-1 - Sqrt[5])/4, Sqrt[1 + 2/Sqrt[5]]},
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    {12, 13, 4, 17, 3}, {3, 17, 9}, {17, 4, 10}, {4, 13, 28}, {13, 12, 11}, {12, 3, 27}, {27, 1, 18}, {9, 22, 15}, {10, 16, 23}, {28, 19, 2}, {11, 20, 21}, {27, 3, 9, 15, 1},
    {9, 17, 10, 23, 22}, {10, 4, 28, 2, 16}, {28, 13, 11, 21, 19}, {11, 12, 27, 18, 20}}]]]

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Archimedean Solid

Geometry