The icosahedron – 7

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

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

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