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. Platon associated the five regular bodies named after him with the elements fire, water, earth, air and a celestial-ethereal substance. The last one he attributed 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 the order in the solar system by creating his well-known model, that the arrangement of the six planets, known in his time, is organized by the five Platonic bodies. 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 semimajor axises of the elliptical orbits, repectively.
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However, the structure of the whole system, as explained in The Signature of the Spheres, is being determined by the semiminor 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 semiminor 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 per milles, 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 The Signature of the Spheres, page 23 ff).
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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 inner 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:
outer square: Pluto
circle: Neptune
middle square: Uranus
inner square: Saturn
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geometrical relations of areas: mean value 0,31%
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 semiminor axises. These ratios are like the ratios between areas of squares or circles, repectively. But why do the semiminor axes play the decisive part in the arrangement of the planetary orbits and why should their proportions be determined by the ratios of areas? ... ...