Q:

Platonic solids and Kepler’s theory of planetary orbits

In the preface to the second edition of Classical Mathematical Physics: Classical Dynamical Systems (1988), Walter Thirring includes this tantalizing comment:

Even Kepler’s [orbital ratio] laws, which determine the radii of the planetary orbits, and which used to be passed over in silence as mystical nonsense, seem to point the way to a truth unattainable by superficial observation: the ratios of the radii of Platonic solids to the radii of inscribed Platonic solids are irrational, but satisfy algebraic equations of lower order. These irrational numbers are precisely the ones that are the least well approximated by rationals, and orbits with radii having these ratios are the most robust against each other’s perturbations, since they are the least affected by resonance effects.

This comment seems to imply that there may be some validity to Kepler’s claims in Mysterium Cosmographicum relating the planetary orbits to inscribed and circumscribed Platonic solids. Unfortunately, Thirring supplies no reference, and “Platonic solid” appears nowhere in the book’s index. Repeated efforts to find a reference online have also failed: with fewer keywords, the results are dominated by “debunkings” of Mysterium Cosmographicum; with more keywords, the results are dominated by Thirring’s preface itself.
Does anyone know of a source that discusses Thirring’s claim in more detail? If true, it’s a fascinating example of a supposedly discredited speculation turning out to be correct.

A:

Fortunately, we have good data on planetary orbits and can look at whether Kepler's theory of nested Platonic solids explains the ratios of their semimajor axes (given here in astronomical units (au) courtesy of Mathematica).
$$\begin{array}\\
\text{Mercury} && 0.38709893 \\
&& && 1.8685972 \\
\text{Venus} && 0.72333199 \\
&& && 1.3824912 \\
\text{Earth} && 1.00000011 \\
&& && 1.52366214 \\
\text{Mars} && 1.52366231 \\
&& && 3.41503690 \\
\text{Jupiter} && 5.20336301 \\
&& && 1.83286661 \\
\text{Saturn} && 9.53707032 \\
\end{array}$$
The ratios of the radii of a circumscribed sphere to an inscribed sphere for the five Platonic solids (in the order Kepler envisioned) are (also courtesy of Mathematica)
\begin{array}{ccc} \\
\text{Octahedron} && \sqrt3 && 1.732051 \\
\text{Icosahedron} && \sqrt{15-6\sqrt5} && 1.258409 \\
\text{Dodecahedron} && \sqrt{15-6\sqrt5} && 1.258409 \\
\text{Tetrahedron} && 3 && 3.000000 \\
\text{Cube} && \sqrt3 && 1.732051 \\
\end{array}
You can decide for yourself whether you think the fit is good. I think it is rather poor.
Added in response to John Dvorak's comment:
The ratio of Saturn's semimajor axis to Mercury's semimajor axis is predicted by this model to be $9(15-6\sqrt5)$ or $14.25233$. (This number is even in OEIS as “Johannes Kepler's polyhedron circumscribing constant”.) Unfortunately, this ratio is actually $24.637294$.
The Uranus:Saturn ratio is $2.01228085$ and the Neptune:Uranus ratio is $1.566804750$. Platonic values are not a good fit to either one.

