Ratios of the Radii of Spheres
Circumscribed and Inscribed
Around Platonic Solids
Kepler recognized that the distances between planets corresponded to the ratios between circumscribed and inscribed spheres around each of the Platonic solids.
I have attempted to derive these ratios using geometry, and have succeeded in only three cases:
Cube:
If we choose a cube where each edge has a length of two, then the inscribed radius will be one, and the circumscribed radius can be found relatively quickly:
Octahedron:
Tetrahedron:
I resorted to Cartesian coordinates at this stage, which is anachronistic, but the following could easily be converted to geometric form:
