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, as explained in Signature of the Celestial Spheres, 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 next Figure 4.2 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 (the exact values can be found in Signature of the Celestial Spheres, page 17).
The evolving arrangement is so dazzlingly clear and simple, that we have to wonder once again, why, at least to the authors knowledge, it has not been mentioned anywhere. What we see is something like a reflection in Jupiter, the greatest member of the planetary community, even though Venus and Neptune are not yet integrated into the represented order. We achieve this by using the most simple regular figures: circle, square and triangle. The proportions 2/1 and 4/1 can be derived from the ratios of the areas of the incircle and the circumcircle of a quadrangle respectively a triangle. Thus the circle cuts off proportions in the form 4/p, p/2 etc. As an example we obtain the following arrangement for the four outer planets:
It may be surprising at first to find three of the orbits as quadrangles. What we are getting at, however, is the representation of the principle behind the distances, i.e. the semi-minor axes. These ratios are like the ratios between areas of squares or circles, repectively. But why do the semi-minor axes play the decisive part in the arrangement of the planetary orbits and why should their proportions be determined by the ratios of areas?