The dodecahedron – 3

Doing the math


The geometry

  1. Start with a square with corners A, B, C, D.
  2. Make E be the midpoint of AB, F the midpoint of CD, and G the midpoint of EF.
  3. Proportion EG using the golden ratio to find H, so that GH is the longer part.
  4. Proportion GF in the same way to find J.
  5. Draw line HK perpendicular to the face of square, and make its length = GH.
  6. Draw a similar line JL. KL is the ridge of the roof.
  7. Draw line HM, perpendicular to EF.
  8. Take the right triangle HKM, and solve for angle K.
  9. Make AB = 10, so EB = HM = 5 = k.
  10. If k = 5, then GR/5 = 1/m, so m = 3.09016994374948.
    The golden ratio (GR:1) = 1/2(sqrt[5]+1):1 = 1.61803398874989:1.
  11. k² + m² = h², so h = 5.87785252292473.
  12. Cosine(K) = m/h = 0.525731112119135, so K = 58.28 degrees
    (the cosine of an angle = adjacent/hypotenuse).
  13. 2(K)= 116.56 or 116° 34' = the dihedral angle.

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