Final Results

Here are the gap ratios, based on modern data from (Moore/Hunt, 1983), compared with the harmonic ratios of Kepler's model with the min/max convention, with errors.

The five inner planets?

Gap Planet data Fraction Value Kepler Error
1 Saturn R-min/Jupiter R-max 1347/815.7 1.6513 1.7321 5%
2 Jupiter R-min/Mars R-max 740.9/249 2.9755 3.0000 1%
3 Mars R-min/Earth R-max 206/152 1.3553 1.2584 7%
4 Earth R-min/Venus R-max 147/109 1.3486 1.2548 7%
5 Venus R-min/Mercury R-max 107.4/69.7 1.5409 1.7321 13%
Not too bad! Mercury should be closer to the Sun perhaps?

What about the outer planets?

Here we take the liberty of guessing Kepler's reaction to the discovery of two or three new planets, two centuries after his death. We choose to reuse the cube or octahedron (harmonic ratio, 1.7321) for the Gap #A (between Saturn and Uranus), the dodecahedron or icosahedron (harmonic ratio, 1.2548) for Gap #B (between Uranus and Neptune), and also for GAp #C (between Neptune and Pluto), just because that gives the best fit. The results:
Gap Planet data Fraction Value Kepler Error
A Uranus R-min/Saturn R-max 2735/1501 1.7149 1.7321 1%
B Neptune R-min/Uranus R-max 4456/3004 1.1834 1.2548 6%
C Pluto R-min/Neptune R-max 4425/4537 0.9753 1.2584 28%
Excepting Pluto, which truly does not fit Kepler's scheme because its R-min is actually less than Neptune's R-max (thus, there is no gap, and the min/max ratio is less than one) these results are surprisingly good.

Revised 21 jan 2002 by ralph abraham