Harmonic ratios of the Platonic solids

Five exercizes in Euclidean solid geometry:
These values are computed with the "bc" program of the UNIX operating system. The approximate values in parens are taken from Kepler, Mysterium Cosmographicum, 2nd edn. (Duncan, 1981; p. 28). By the way, the term minor refers to a root of a certain type of fourth order algebraic equation, as defined in Euclid, Book X, Prop. 76 (Heath, 1960; v.3, pp. 163-164).

Note: Although there are five Platonic solids, there are only three harmonic ratios. Kepler knew this, but choose not emphasize it in his books.


Rev'd 19 jan 2002 by ralph abraham