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Graphics3D[GraphicsComplex[{{(-1 - Sqrt[5])/2, 0, Root[1 + 2*#1^2 - 11*#1^4 + 8*#1^6 + #1^8 & , 4, 0]/2},
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{Sqrt[5 + 2*Sqrt[5]]/2, 1/2, Root[1 + 2*#1^2 - 11*#1^4 + 8*#1^6 + #1^8 & , 1, 0]/2},
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Polygon[{{1, 2, 4}, {1, 3, 2}, {1, 5, 3}, {1, 8, 5}, {2, 6, 4}, {2, 9, 6}, {3, 5, 7}, {3, 7, 10},
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{12, 8, 1, 4}, {9, 2, 3, 10}, {15, 13, 5, 8}, {14, 6, 9, 17}, {10, 7, 16, 18}, {19, 20, 12, 11},
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{17, 9, 10, 18, 23}}]]]
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Johnson Polyhedra
Geometry