Riemann for Anti-Dummies Part 15

THE SOLAR SYSTEM'S HARMONIC TWIST

Significant insight can be obtained, for those wishing to master the art of changing one's own axioms, by re-living Kepler's transformation of his own thinking, from his initial hypothesis connecting the planetary orbits to the five Platonic solids, to the supersession of that hypothesis, under his concept of "World Harmony."

As presented in last week's installment, Kepler sought the reason underlying the ordering of the solar system by investigating, "why things were such and not otherwise: (namely) the number, size, and the motion of the circles." The anomaly between Mars and Jupiter, initially investigated in terms of the relationship of the distances between the planetary orbits, provoked Kepler to reject his initial attempts to answer this question using simple numerical values, or, geometrical relationships among plane figures. After much work, Kepler found that the underlying reason for the number and size of the planetary orbits corresponded to the ordering of five Platonic solids.

(An example of the this anomaly between Mars and Jupiter was recently pointed out by Jonathan Tennenbaum. If one extrapolates from the relationship of the distances between the orbits of Mars and Earth outward, one calculates two planetary orbits between Mars and Jupiter. On the other hand, extrapolating from the relationship of the distance between Jupiter and Saturn inward, only one planetary orbit between Mars and Jupiter is obtained.)

As to the motions of the planets, Kepler later discovered that his initial concept had to be superseded by a "more basic principle." Specifically, the non-uniform motion, i.e., eccentric, motion of the planets showed that the planetary orbits were not fixed circles, such as those found on spheres circumscribing and inscribing Platonic solids. Rather, the orbits were regions in which the planets moved non-uniformly, getting closer and farther from the Sun as they moved around it. Thus, the solids were not sufficient to account for these eccentricities.

But, what principle was determining these eccentricities? This presented a far different problem than determining merely the distances between the planetary orbits. Specifically, what is the appropriate measure of a non-uniform orbit? Kepler's equal area principle is such an appropriate measure within an orbit. But, what is an appropriate measure for determining intervals among eccentric orbits? Kepler's so-called third principle, (that the mean distance from the Sun equals the 3/2 power of the periodic time), is a first approximation of such a measure, but it doesn't express, "why these eccentricities and not others?"

Kepler recognized that the eccentricity of a planets' orbit is uniquely determined by its singularities, specifically its fastest speed at perihelion and its slowest speed a aphelion. These extreme motions reflect the intention of the planet's action in the intervening intervals. In other words, how much the planet speeds up, at each moment, from aphelion to perihelion, is a function of what it is to become at perihelion, and conversely, in the interval from perihelion back to aphelion. Thus, the principle determining these extremes, in turn determines the characteristic eccentricity of the whole orbit, which in turn, determines the distances between the planetary orbits.

As Kepler stated, "It was good, that for the formation of the distances the solid figures should give way to the harmonic relationships, and the greater harmonies between pairs of planets, to the universal harmonies of all, so far as this was necessary."

Since Kepler is almost unique among scientific discovers in presenting to us not only his discovery, but also the change in his own thinking which brought him to it, we quote at length from the concluding "envoi" of his "Harmonies of the World":

"For where there is a choice between different things which do not allow each other to have sole possession, in that case the higher are to be preferred, and the lower must give way, as far as is necessary, which the very word "cosmos," which means "decoration," seems to argue. But harmonic decoration is as far above the simple geometrical as life is above the body, or form above matter.

"For just as life completes the bodies of animate beings, because they were born to lead it, which follows from the archetype of the world, which is the actual divine essence, so motion measures out the regions allotted to the planets, to each its own, as a region has been assigned to a star so that it could move. But the five solid figures, in virtue of the word itself, relate to the spaces of the regions, and to the number of them and of the bodies; but the harmonies to the motions. Again, as matter is diffuse and unlimited in itself, but form is limited, unified, and itself the boundary of matter; so also the number of the geometrical proportions is infinite, the harmonies are few....Therefore, as matter strives for form, as a rough stone of the correct size indeed, strives for the Idea of the human form, so the geometrical proportions in the figures strive for harmonies; not so as to build and shape them, but because this matter fits more neatly to this form, this size of rock to this effigy, and also this proportion in a figure to this harmony, and therefore so that they may be built and shaped further, the matter in fact by its own form, the rock by the chisel into the appearance of an animate being, but the proportion of the spheres of the figures by its own, that is, by close and fitting harmony.

"What has been said up to this point will be made clearer by the story of my discoveries. When, twenty four years ago I had engaged in this study, I first enquired whether the individual circles of the planets were separated by equal distances from each other (for in Copernicus the spheres, are separated, and do not mutually touch each other),. Of course, I acknowledged nothing as more splendid than the relationship of equality. However, it lacks a head and a tail, for this material equality provided no definite number for the moving bodies, no definite size for the distances. Therefore, I thought about the similarity of the distances to the spheres, that is about their proportion. But the same complaint followed, for although in fact distances between the spheres emerged which were certainly unequal, yet they were not unequally unequal, as Copernicus would have it, nor was the size of proportion nor the number of the spheres obtained. I moved on to the regular plane figures; they produced the distances in accordance with the ascription of their circles, but still in no definite number. I came to the five solids; in this case they revealed both the number of the bodies and nearly the right size for the intervals so much so that I appealed over the remaining discrepancy to the state of accuracy of astronomy. The accuracy of astronomy has been perfected in the course of twenty years;' and see! There was still a discrepancy between the distances and the solid figures, and the reasons for the very unequal distribution of the eccentricities among the planets were not yet apparent. Of course in this house of the cosmos I was looking for nothing but the stones of more elegant form, but of a form appropriate to stones not knowing that the Architect had shaped them into a fully detailed effigy of a living body. So little by little, especially in these last three years, I came to the harmonies, deserting the solid figures over fine details, both because the former were based on the parts of the form which the ultimate hand had impressed, but the figures from matter, which in the cosmos is the number of the bodies and the bare breadth of their spaces, and also because the former yielded the eccentricities, which the latter did not even promise. That is to say the former provided the nose and eyes and other limbs of the statue, for which these latter had only prescribed the external quantity of bare mass.

"Hence just as the bodies of animate beings have not been made, and a mass of stone is not usually made, according to the pure norm of some geometrical figure, but something is removed from the external round shape, however elegant (though the correct amount of bulk remains) so that the body can take on the organs necessary to life, and the stone the likeness of an animate being, similarly also the proportion which the solid figures were to prescribe for the planetary spheres, as lower, and having regard only in a body of a particular size and to matter, must have given way to the harmonies, as much as was necessary for the former as to be able to stand close and to adorn the motions of the globes."

A Still More Basic Principle

So, the geometrical proportions of the solids give way to the more basic principle of the harmonic principles. What principle, therefore, determines the harmonies?

Kepler himself stated that the harmonic proportions are determined by the ear, not numerical values. To what does the ear turn? To the universal principles of Classical artistic composition, as exemplified by J.S. Bach's well-tempered polyphonic compositions. It is in the domain of these compositions (Ideas) from which the values of the well-tempered intervals are derived, which, in turn, determine the harmonic proportions from which the planetary orbits derive their eccentricities.

With Piazzi's discovery of the asteroid Ceres, Gauss' subsequent determination of its orbit, and the follow up discoveries of the asteroids Pallas, Juno and Vesta, Kepler's principles were confirmed anew. The motion of each asteroid conformed to Keplerian principles, moving in elliptical paths with equal areas measuring equal portions of their periods, and, their mean distances from the Sun equaling the 3/2 power of their periodic times.

But, there was a twist. It now became possible to measure, in these orbits, cyclical changes in the eccentricities, that were occurring, but were hitherto beyond measurement, in all the planetary orbits. Furthermore, unlike the orbits of the major planets, which enclosed one another, the asteroid orbits intertwined. For example, at perihelion Pallas was closer to the Sun than Ceres, but at aphelion, Ceres was closer to the Sun than Pallas. This intertwining suggests the asteroids' orbits are both many individual Keplerian orbits, and one whole Keplerian orbit at the same time. What then, is the still more basic principle that governs the solar system which contains this new type of orbit represented by the asteroids?

The initial work on this was done by Gauss, whose investigation into the changing eccentricities and the intertwinings provoked his creation of new mathematical metaphors, which, like Leibniz' calculus, had applications far beyond the original paradoxes that gave rise to them. The changing eccentricities provoked Gauss to conceive of the orbits as elliptical rings in which the mass of the planet, or asteroid, was distributed in the ring according to Kepler's equal area principle. (For a more complete treatment of Gauss' concept, see the pedagogical series, "Dance With the Planets" 98406bmd002; 98416bmd001; 98426bmd001 )

Gauss also considered the implications of the intertwining of the asteroid orbits, for the geometrical characteristics of the solar system as a whole. Gauss took this up in a preliminary way in an 1804 paper, "On the Determination of the Geocentric Positions of the Planets." Here Gauss considered the inverse of the problem he confronted in the determination of the Ceres orbit. In that case, Gauss had a few geocentric positions of Ceres, from which he had to determine its heliocentric orbit. Now he considered the inversion. What characteristics of a heliocentric orbit govern the geometry of its geocentric positions. For this, he explicitly turned to Leibniz' and Carnot's "Geometry of Position."

Each planet or asteroid makes a circuit through the zodiac. But, since the Earth is also moving, the zodiac changes its position with respect to an observer on the Earth. Consequently, the locus of all geocentric positions of a planet or asteroid form a zone on the celestial sphere, that Gauss called its "zodiacus." The determination of the boundaries of that zone required the construction of function that mapped the changes of the heliocentric positions of the planet and the Earth, onto the celestial sphere. While Gauss was able to calculate specific values for this function, more importantly, he investigated its general characteristics. He showed that the nature of that zodiacus depends on the relationship of the planet's orbit to the Earth's. Either the planets' orbit is completely inside the Earth's, completely outside, or, it overlaps. Gauss showed that the first two situations determined a zodiacus with definite boundaries, but, in the third case those boundaries were indeterminable. He noted, ironically, that, the implications of this paradox had, until then, been avoided, because none of the known planets or asteroids, had ever appeared in strange places, such as near the poles of the ecliptic! Nevertheless, Gauss was pointing out a crucial principle on which Riemann would later rely. Specifically, that orbits that completely enclose one another defined completely different geometrical characteristics than those that overlapped.

(If you want to have some fun, take two rings, one bigger than the other. Study the relationship between positions on the two rings when the smaller ring is inside the larger. Now, compare these relationships with two rings that are interlinked. I leave it to the reader to discover the difference on your own.)

Finally, think of the implications of these intertwining asteroid orbits for Kepler's harmonic proportions. Preliminary calculations performed by this author for 10 asteroids show that when the extreme speeds of each asteroid are individually compared with the extreme speeds of Jupiter and Mars, similar harmonic proportions to those Kepler found for the major planets occur. The diverging and converging intervals each asteroid makes with Jupiter correspond to intervals Kepler would consider consonant. With Mars, the diverging intervals are consonant, while the converging intervals correspond to the deisis that Kepler found between Jupiter and Mars.

The twist comes up in forming intervals among the asteroids themselves. Since their orbits overlap the very meaning of converging and diverging intervals is different than in the intervals between the major planets. For example, when is Ceres converging towards Pallas or diverging away from it? When both are moving away from the Sun, Ceres is getting closer to Pallas, while, when both are moving closer to the Sun, they become divergent. Unlike the major planets, however, the point of divergence and convergence does not occur at the extreme positions. And since the eccentricities of the asteroids's orbits are changing, where these orbits cross over from diverging to converging is itself changing. Now, think of the connectivity involved when thinking of this relationship among many asteroids, not just two, as in this example!

This braided, overlapping characteristic is not limited to orbits within the range of the asteroid belt. In fact, the solar system is filled with orbits that similarly overlap, including asteroids whose orbits overlap the Earth's. Such overlapping orbits, suggests a new set of harmonic relationships, akin to the transformation of Bach's well-tempered polyphony by Beethoven in his late quartets.