Harmony of the Spheres by Michæl Paukner on Flickr.

Harmony of the Spheres

Approximation of the intervals of the semi-minor axes of the planets in our solar system.

As the idea of the “harmony of the spheres” has been anchored in mankind for thousands of years, a coherence between geometry and the celestial relations has also been suspected for an equally long time. Plato associated the five regular solids named after him with the elements of fire, water, earth, air and a celestial-ethereal substance. He attributed the latter to the dodecahedron, a figure, that is enclosed by twelve pentagons. In geometrical regard it was again Johannes Kepler who 2000 years later developed the ancient ideas further. He started out on his search for order in the solar system by creating his well-known model which shows that the arrangement of the six planets, known in his time, is organized by the five Platonic solids. According to this the ratio of the radii of the inner and the outer sphere of the dodecahedron, for example, corresponds (if only very approximately) with that of the mean distances which Mars and Earth have from the Sun, or the semi-major axes of the elliptical orbits, repectively.

However, the structure of the whole system is determined by the semi-minor axes b, which already had a central importance in the harmonies of the velocities. What is most striking, is that the first and the fourth planet, counted from the inside as well as from the outside, are in a ratio of 4/1, relative to their semi-minor axes. The first and sixth planet, again calculated from the inside and the outside, show the proportion 25/1. The result is a clear higher structure, that is partitioned further by ratios of small integers. This order is illustrated in the image by circles. The differences from the real values amount to only a few thousandths, except for the intervals 8/3 and 3/2, where they are slightly more than one per cent.

source: www.keplerstern.com/Geometrical_Order/geometrical_order.html