Description of Kepler's 3D model

To Kepler, the six planets (Copernican paradigm) orbited crystalline spheres. The thickness of a sphere was the minimum to contain the elliptical orbit of its planet. The five gaps between these six concentric spheres were spaced by the Platonic solids.
  • Gap #1: Saturn/Jupiter, the cube
    harmonic ratio = sqrt(3) ~= 1.732
  • Gap #2: Jupiter/Mars, the tetrahedron
    harmonic ratio = 3
  • Gap #3: Mars/Earth, the dodecahedron
    harmonic ratio ~= 1.258
  • Gap #4: Earth/Venus, the icosahedron
    harmonic ratio ~= 1.258
  • Gap #5: Venus/Mercury, the octahedron
    harmonic ratio = sqrt(3) ~= 1.732
  • The cube and the octahedron, having the same harmonic ratio, might be interchanged. Ditto the dodecahedron and icosahedron. However, Kepler argued that the order listed above was the correct order.

    The sympathy of the cube and octahedron (that is, sharing the same harmonic ratio) is elementary. The sympathy of the dodecahedron and icosahedron is not elementary, but will be familiar to students of Euclid (are there any still alive?) as the second proposition of Hypsicles, author of the work sometimes known as "Euclid Book XIV". Apparently this result is due to Aristaeus the Elder, ca 320 BCE, in his work, Comparison of the five regular solids. See (Heath, 1960; v.3, p. 438).


    Rev'd 19 jan 2002 by ralph abraham