Contentment by design - Playing with the icosahedron

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The icosahedron – doing the math

The only way I can see to figure the dihedral angle is to work from the 2-fold symmetry:

Calculating the dihedral angle from the icosahedron 2-fold symmetry view

Start with a golden rectangle with corners A, B, C, D.
Golden ratio (GR:1) = 1/2(sqrt[5]+1):1 = 1.6180.
Assume AB = 12, then AD = 19.4164.
Bisect the first golden rectangle with another golden rectangle with endpoints EF so that G splits AD into two.
AG = AD/2 = 9.7082.
EF - AB = x = 7.4164.
x/2 = GF = Adjacent = 3.7082.
GF² + AG² = AF² = 13.7507 + 94.2492 = 107.9999.
AF = Hypotenuse = 10.3923.
Cosine = Adjacent/Hypotenuse = 0.3568.
AFG = 69.0948.
2(AFG) = AFD = Dihedral angle = 138.19, or 138º 11'.

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