A-1 THE MERCURY Mt-BASED
EXPONENTIAL PLANETARY FRAMEWORK AND THE PHI-SERIES
Although the exponential function P(x) = Mt phix (x = -2 to 16, base Mt = 0.240842658 years ) generates successive mean sidereal and mean synodic periods, it is nevertheless the mean sidereal period of Mercury Mk 0 = Mt that provides the initial starting point for Mercury-based exponential period functions. In fact the latter constant is not only comparable to phi -3 = 0.236067978 years, the entire exponential function differs little from the Phi-Series for exponents x = -3 through 13, the one-year period and "synodic" position of Earth included. Thus there is a second and even simpler exponential planetary framework available that requires Phi alone, namely the Phi-Series itself, i.e., with respect to unity provided by the heliocentric position and motion of Earth.
Phi-Series Exponential Planetary Framework: Mean Periods, Distances,
In other words, the Phi-Series provides a complete theoretical planetary framework that incorporates mean values for the periods of revolution with odd-numbered exponents -3 through 13 and even-numbered exponents for the intermediate synodic cycles and thus corresponding mean heliocentric distances and mean orbital velocities.
PHI-SERIES AND THE SOLAR SYSTEM
Relations 5a and 5b. The Fundamental Period Constants
Relations 6a and 6b. The Fundamental Distance Constants
Thus with relation 6b we obtain a constant increase in mean planetary distances of 1.88995476295... as opposed to the ad hoc multiples of 2 that belong to the Titius-Bode relationship. But even so, this still leaves an unaccounted "gap" between Mars and Jupiter. For more on this complex topic see Figure 6 and part F below.
PHI-BASED EQUIANGULAR PERIOD SPIRAL
Although the above digressions and what follows next impinge on matters discussed in Part IV and later sections, returning to the technical side of the matter there seems little doubt that the phi-based exponential planetary frameworks can (and likely should) be considered in terms of equiangular period spirals based on relation 5b expressed in the form:
Relation 9. The Exponential Period Function and Equiangular Period Spiral
The resulting spiral (see Part IV) is
predicated on the
equiangular "square" dictated
by relation 5b, i.e., the Phi-squared increase in mean planetary
Thus for example, Figure 6c incorporates the Phi-Series mean sidereal and mean synodic periods from Mercury to Mars:
Delineated on the vertical
axis, the mean planetary periods increase by Phi squared per sidereal
of 360 degrees while the synodic periods occur at the 180-degree
points. Exactly the same configuration could be given for the
periods for Jupiter, Saturn and Uranus (or indeed any such segment of
Phi-Series) since the periods increase in the same manner, whereas a
(i.e., log-linear) representation necessarily requires logarithmic data
in addition, as shown in the inset. But there is far more to this
spiral, for although the above represents Solar System mean periods,
Time, it turns out that to produce corresponding equiangular
and velocity spirals would be entirely redundant, for both sets of
are already integral features of period spiral itself. The details are
discussed further in Part IV, but small wonder that Jacob Bernoulli
have called the equiangular spiral "Spira Mirabilis" and included it on
his tombstone, or that part of the title is retained here, albeit
with Archytas for reasons that will become apparent in the next few
On a more recent historical note, investigation reveals that research concerning the spiral form in related astronomical contexts includes the work of Lothar Komp1 in 1996 and William M. Malisoff in 1929.2 For the latter's inclusion of velocities, distances, periods and the logarithmic spiral see paragraph (7) in his 1929 letter to the editor of the Science ("Some New Laws of the Solar System".)
PHYLLOTAXIS AND THE EXPONENTIAL PLANETARY CONSTANTS
Although the present treatment has concentrated on Time, it now becomes necessary to consider the results in terms of the relationship between Phi, the Fibonacci Series and natural growth. In other words, physical considerations concerning growth itself, with time, "distance" and speed (i.e., rate of growth) integral components. So far the generated exponential planetary frameworks have largerly concerned the mean periods, i.e., time, but as understood from the outset, this was to obtain more data and introduce a more productive approach to the structure of the Solar System. Carried though all this, however, were still the inter-relationships between Time, Distance and Velocity provided by the velocity expansions to the third law of planetary motion and the third law itself. Moreover, as can be seen in Figure 6c, the obvious complexities of the Phi Series in this specific astronomical context reveal that the exact values for the mean periods also occur elsewhere in the table among the mean Velocities (e.g., the mean sidereal period of Mars and the mean velocity of Mercury; see also Table 1) in a complex, if not distinctly "ourobotic" context that will be discussed in later Sections. As for the occurrence of the Phi Series in the present context, those unfamilar with the subject might wish to bear in mind that Phi, the Fibonacci, Lucas and related series, far from being confined to plant and animal growth alone, occur in numerous diverse contexts over an enormous range that extends from the structure of quasi-crystals out to the very structure of spiral galaxies. And this being so, should there really be any great surprise if Phi should also prove to be an underlying element in the structure of planetary systems? It has long been recognized that although Phi and the Fibonacci Series are intimately related to the subject of natural growth that they are hardly limited to these two fields alone. Remaining with the Phi-Series, Jay Kappraff 3 points out that the French architext Le Corbusier "developed a linear scale of lengths based on the irrational number (phi), the golden mean, through the double geometric and Fibonacci (phi) series" for his Modular System. The latter's interest in the topic is explained further in the following informative passage from Jay Kappraff's CONNECTIONS: The Geometric Bridge between Art and Science:
As a young man, Le Coubusier studied the elaborate spiral patterns of stalks, or paristiches as they are called, on the surface of pine cones, sunflowers, pineapples, and other plants. This led him to make certain observations about plant growth that have been known to botanists for over a century.A great deal of additional information concerning this complex topic is obtainable from the above works and the other references, but for the present it is sufficient to return to the ongoing line of inquiry, noting from the various examples cited that actual phyllotaxic ratios in nature do not necessarily produce Phi itself--the limiting value of Fibonacci and Lucas ratios--but rather numbers obtained from ratios closer to the commencing sequence: 1,1,2,3,5,8,13,21,... For example, the ratios 8:5 = 1.6, 13:8 = 1.625, and somewhat closer to Phi, the ratio 89:55 that results in 1.6181818...
Plants, such as sunflowers, grow by laying down leaves or stalks on an approximately planar surface. The stalks are placed successively around the periphery of the surface. Other plants such as pineapples or pinecones lay down their stalks on the surface of a distorted cylinder. Each stalk is displaced from the preceding stalk by a constant angle as measured from the base of the plant, coupled with a radial motion either inward or outward from the center for the case of the sunflower [see Figure 3.21 (b)] or up a spiral ramp as on the surface of the pineapple. The angular displacement is called the divergence angle and is related to the golden mean. The radial or vertical motion is measured by the pitch h. The dynamics of plant growth can be described by and h; we will explore this further in Section 6.9 [Coxeter, 1953].
Each stalk lies on two nearly orthogonally intersecting logarithmic spirals, one clockwise and the other counterclockwise. The numbers of counterclockwise and clockwise spirals on the surface of the plants are generally successive numbers from the F series, but for some species of plants they are successive numbers from other Fibonacci series such as the Lucas series. These successive numbers are called the phyllotaxis numbers of the plant. For example, there are 55 clockwise and 89 counterclockwise spirals lying on the surface of the sunflower; thus sunflowers are said to have 55, 89 phyllotaxis. On the other hand, pineapples are examples of 5, 8 phyllotaxis (although, since 13 counterclockwise spirals are also evident on the surface of a pineapple, it is sometimes referred to as 5, 8, 13 phyllotaxis). We will analyze the surface structure of the pineapple in greater detail in Section 6.9.
3.7.2 Nature responds to a physical constraint After more than 100 years of study, just what causes plants to grow in accord with the dictates of Fibonacci series and the golden mean remains a mystery. However, recent studies suggest some promising hypotheses as to why such patterns occur [Jean, 1984], [Marzec and Kappraff, 1983], [Erickson, 1983].
A model of plant growth developed by Alan Turing states that the elaborate patterns observed on the surface of plants are the consequence of a simple growth principle, namely, that new growth occurs in places "where there is the most room," and some kind of as-yet undiscovered growth hormone orchestrates this process. However, Roger Jean suggests that a phenomenological explanation based on diffusion is not necessary to explain phyllotaxis. Rather, the particular geometry observed in plants may be the result of minimizing an entropy functionsuch as he introduces in his paper .
Actual measurements and theoretical considerations indicate that both Turing's diffusion model and Jean's entropy model are best satisfied when successive stalks are laid down at regular intervals of 2Pi /Phi^ 2 radians, or 137.5 degrees about a growth center, as Figure 3.22 illustrates for a celery plant. The centers of gravity of several stalks conform to this principle. One clockwise and one counterclockwise logarithmic spiral wind through the stalks giving an example of 1,1 phyllotaxis.
The points representing the centers of gravity are projected onto the circumference of a circle in Figure 3.23, and points corresponding to the sequence of successive iterations of the divergence angle, 2Pi n/Phi^ 2, are shown for values of n from 1 to 10 placed in 10 equal sectors of the circle. Notice how the corresponding stalks are placed so that only one stalk occurs in each sector. This is a consequence of the following spacing theorem that is used by computer scientists for efficient parsing schemes [Knuth, 1980].Theorem 3.3 Let x be any irrational number. When the points [x] f, [2x] f, [3x] f,..., [nx] f are placed on the line segment [0,1], the n + 1 resulting line segments have at most three different lengths.Here clock arithmetic based on the unit interval, or mod 1 as mathematicians refer to it, is used, as shown in Figure 3.24, in place of the interval mod 2pi around the plant stem. It turns out that segments of various lengths are created and destroyed in a first-in-first-out manner. Of course, some irrational numbers are better than others at spacing intervals evenly. For example, an irrational that is near 0 or I will start out with many small intervals and one large one. Marzec and Kappraff  have shown that the two numbers 1/Phi and 1/Phi^2 lead to the "most uniformly distributed" sequence among all numbers between Phi and 1. These numbers section the largest interval into the golden mean ratio,Phi :l, much as the blue series breaks the intervals of the red series in the golden ratio.
Moreover, [(n + 1)x] f will fall into one of the largest existing segments. ( [ ] f means "fractional part of ").
Thus nature provides a system for proportioning the growth of plants that satisfies the three canons of architecture (see Section 1.1). All modules (stalks) are isotropic (identical) and they are related to the whole structure of the plant through self-similar spirals proportioned by the golden mean. As the plant responds to the unpredictable elements of wind, rain, etc., enough variation is built into the patterns to make the outward appearance aesthetically appealing (nonmonotonous). This may also explain why Le Corbusier was inspired by plant growth to recreate some of its aspects as part of the Modulor system. (Jay Kappraff, Chapter 3.7. The Golden Mean and Patterns of Plant Growth, CONNECTIONS : The Geometric Bridge between Art and Science, McGraw-Hill, Inc., New York, 1991:89-96, bold emphases supplied.)
For more on this topic see also Dr. Ron Knott's extensive treatment The Fibonacci Numbers and the Golden Section, the latter's related links and the The Phyllotaxis Home Page of Smith University)
With respect to the Phi-series and the exponential planetary frameworks under consideration, accepting (a) that an exponential component does exist in the structure of the Solar System, and (b) that the inverse-velocity relationships are indeed an integral feature of the latter, then it becomes possible to consider phyllotaxis in this explicit context, especially since the spiral form can be considered to be operating here also. At which point it may be recalled that in seeking to reduce the common minor deviations in the inverse-velocity relationships in the Phi-based planetary frameworks a substitute base period for Mercury (Mt3 = 0.2395640) years was applied and Phi retained as the constant of linearity. However, although the determination of the new base period Mt3 was necessary in terms of the initial framework, there was nevertheless another way that the common deviations could have been reduced to zero, namely the substitution of a slightly different value for the major constant Phi itself. Or, if one wishes, the establishment of a practical ratio similar to those discussed above that nevertheless reduced all inverse-velocity errors to zero. This requirement is readily achieved by back-solving, resulting in the retention of the present day estimate for the mean sidereal period of Mercury (Mt = 0.240827 years) as the base period but the substitution of a a new, slightly lower value of 1.6171413367027 for the constant of linearity. With this substitution the minor deviations in the inverse-velocity relationships are still reduced to zero while the resulting exponential planetary framework is found to differ only marginally from the other three (see Table 3 below).
The question that now arises
is of considerable interest, for how does this new constant of
compare with the Fibonacci and Lucas ratios discussed above in
with natural growth? Although not entirely comparable, it turns out
the zeroing constant is indeed close to some of the phyllotaxic ratios
in question, slightly lower, in fact, than the Sunflower's 89:55
In other words, the value in question -1.617141336703 - is
closest to the Lucas Series ratio 76
/ 47 followed by the Fibonacci
Series ratio of
55 / 34.
The occurrence of the Lucas ratio in this context is perhaps the least
surprising given the well-known relationship that exists between the
Series and the Lucas Series, namely that the difference between the two
is the value obtained from reciprocal exponent of the generating power
applied in the former. For example, in the Phi-Series exponential
framework the theoretical mean sidereal period of Uranus
years) is generated by Phi raised to the ninth power, while Lucas
76 is less than this by exactly
Phi to the minus ninth power,
0.0131556174.., and the same applies in the case of the eighth powers
the 47-year period, and so on. But is it pure coincidence that the
76 and 47-year periods correspond to the respective Phi
periods for Uranus and the Saturn-Uranus synodic? And does the Lucas
predominate here, or is there a Fibonacci component as suggested by
of the 55:34 ratio? Either way, there is little variance
the new exponential periods of the Lucas-Fibonacci (MtLF)
and those provided by Mt3 and the two
frameworks as shown in Table 3, which features the modern estimate for
the mean sidereal period of Mercury for the initial exponential
framework (Mt-based) and also the last variant that employs the
constant of linearity. Noteworthy in the MtLF-based data (but possibly
coincidental) is the unforced correlation between the value for the
sidereal period of Saturn of 29.45867 years in the latter and the
estimate of 29.45252 years.
As explained above, the
MtLF exponential planetary framework also provides error-free
relationships, which perhaps suggests that it should provide the
planetary framework. The following log-linear representation of the
as the diagonal reference line is applied to compress of the range of
periods and facilitate the comparison between the exponential
and Solar system parameters. Here with the diagonal providing the
frame, deviations above and below the line represent longer and shorter
periods respectively and thus also deviations in heliocentric distance,
i.e., the greater distance above the line the further out for from the
Sun, and below the line, the closer in with respect to the frame of
Thus the expected deviations for Pluto, Neptune, Mars and to a lesser
Uranus are all evident, as is the suggested location of Earth in the
position between Venus and Mars.
D. SIMILARITIES AND
A visual comparison between the twelve mean periods of the Mt3-based planetary framework and Solar System mean data is provided in Figure 5 (for data concerning Neptune see Table 3). The next and outermost theoretical planetary position (period: 526.8669 years; mean distance: 65.233 A.U.) provides the inverse-velocity data for IMO although there is no known planet in the region. However, it is relevant to note that Clyde W. Tombaugh (the discoverer of Pluto) wrote in 1980 that the search for a tenth Solar System planet occasioned a number of reports, mostly arising from observed irregularities in the orbits of known objects. Although it remained unconfirmed, a planet with a mean distance of 65.5 A.U. was in fact proposed by Joseph L. Brady of the Lawrence Livermore Laboratory, University of California in 1972.4 Since that time further proposals concerning a possible planet in the outer regions have been made by Van Flandern and Harrington, (50 A.U.-100 AU.), Whitmire and Matese (80 A.U.), Anderson (78-100 A.U.) and Powell (60.8 A.U., later modified to 39.8 A.U.) To date no tenth planet has been found, but most of these proposals require planets with highly inclined orbits, large eccentricities, and relatively long intervals between returns, all of which complicate confirmation, especially for small objects. However, further delineation on a wider scale may eventually be forthcoming from the gravity-based analyses of Aleksandr N. Timofeev, Vladimir A. Timofeev and Lubov G. Timofeeva;4 see also Aleksandr Timofeev's: Two fundamental laws of nature in the gravity field.
In terms of
departures from the norm perhaps the most
to accept is that Earth may currently be occupying a resonant synodic
location between Venus and Mars. The establishment of the
heliocentric concept notwithstanding, it would still appear
difficult to perceive the position of Earth as anything other than an
and unquestioned constant. However, the relatively recently advent of
Theory, its application in astronomical contexts and the investigations
carried out by Sussman,5
Laskar,9, 10, 11, 12 and others have now
changed matters irrevocably. The Solar System can
no longer be wound backwards or forwards indefinitely like some
and well understood mechanical device, as Ivars Petersen13 has
Newton's Clock: Chaos in the Solar system.
can the positions of any of its various members be considered
not even that of Earth.
Whether Earth has always been
in the synodic location between Venus and Mars is uncertain, but the
zone of habitability is generally
by the orbits of the latter pair of planets, and it is an open question
whether life would necessarily have developed at either extremity, or
it had, whether it would have necessarily flourished, given the
periodic extinctions which appear to have taken place at Earth's more
central synodic location. This even suggests that a fortuitous element
may have played a role in the continuance, if not the very development
of life here on Earth, and that while life may still abound in the
it may not be quite as common-place as previously supposed. Whether
has a direct bearing on the negative results obtained over the last
decades by the Search for Extra-Terrestrial Intelligence (SETI) is,
another matter altogether. Past, Present or Future, this is hardly a
simple issue, as the extensive avenues explored by Gerry Zeitlin in OPEN
For present purposes it may be noted that deviations exist between the Solar System and the exponential planetary frameworks, and that depending on the degree of confidence assigned to the latter, it may be feasible to quantify these anomalies in terms of planetary masses, mean distances, and the conservation of angular momentum, etc. This still leaves the anomalous position of Neptune, but it is possible to suggest a number of scenarios based on mass-distance changes that might include a further belt of asteroids and/or cometary material at approximately 65 astronomical units from the sun periodically perturbed by an object or objects in a eccentric polar orbit, etc. Apart from the exponential framework itself, very little of this is actually new, though scenarios based on total angular momentum might well remain problematic owing to uncertainties concerning the complete inventory and total mass of the Solar System itself. On the other hand, new avenues and new insights concerning the structure of the Solar System have already begun to surface; e.g., the Fibonacci-related paper by Aleksandr N Timofeev entitled: "Sprouts of New Gravitation Without Mathematical Chimeras of XX Century."
IN THE ASTEROID BELT/MARS-JUPITER GAP
E. LUCAS AND FIBONACCI RESONANCES IN THE SOLAR SYSTEM
Further out among the anomalous planets Neptune and Pluto there are additional resonances, and with respect to the former it is also known that both Earth and Neptune are locked in similar resonant relationships. Denoting synodic periods by Ts, inner and outer mean sidereal periods by T1 and T2 and resonant relationships by: T1 : Ts : T2, both planets are in fact in 2:1:1 resonant relationships with adjacent bodies (Earth with Mars; Neptune with Uranus) while Neptune is also locked in a further 3:2:1 resonant relationship with Pluto. The latter's mean period produces poor results throughout as a base parameter for the exponential frameworks, but in comparison to its neighbors the gas giants, this small planet is already anomalous on a number of counts. Undoubtedly problems exist with the location of Pluto in the present context, but it is nevertheless still Neptune that represents the major discrepancy in the outer regions of the Solar System. Whether resonances among the four major superior planets will shed any light on the matter remains to be seen, but there is far more to this whole matter than mean motion resonances in any case, since real-time resonances in the Solar System must also be addressed. The question that now arises is how best to investigate these resonances on one hand and display them effectively on the other.
RESONANCES: THE INFERIOR PLANETS
For this purpose the methodology of Bretagnon and Simon14 adapted to time-series analysis is particularly useful, especially the power series data and formulas for deriving heliocentric distances. The adaptation (related formulas and methodology are provided in Times Series Analysis) will be explained in more detail later, but for present purposes it is sufficient to note that for any part of a planet's orbit at any point in time the instantaneous value of the radius vector can be treated as the mean value of an equivalent mean distance orbit and consequently also provide corresponding periods and velocities for the same. In other words, each planetary orbit may be considered in terms of successive mean motion orbits extending outwards from the shortest distance at perihelion to the longest at aphelion. In this way not only the varying distances, but also the velocities and periods may be treated as continuous functions over successive intervals. Instantaneous values of successive radius vectors may then be used to generate corresponding periods that serve to illustrate some of the better-known resonances among the inferior and superior planets. With respect to the former, particularly the adjacent planets Venus, Earth and Mars there seems little doubt that from a dynamic viewpoint Earth's location between Venus and Mars is highly complex. In addition to the resonances listed by Lothar Komp it may be noted that although the Venus-Mars mean synodic period is 0.914224 years, in practice the elliptical nature of the orbits of the three planets cause the instantaneous sidereal and synodic velocities to vary widely and also periodically coincide. But Earth is not only locked in a 2:1:1 resonance with Mars, but also in a 13:5:8 resonant relationship with Venus, which is itself linked to Mars by a further 3:2:1 resonance. Moreover, a plot of the true varying sidereal and synodic motion in the form of time-series data reveals the existence of even more complex resonant relationships as seen in see Figure 7 below:
This actual example
computed a number of years ago remains part of a
relatively inconclusive but not entirely negative investigation of
resonances and their possible inter-relationship with solar activity.
that time even the more obvious feature--that all the numbers involved
belong to the Lucas Series, i.e., 1, 3, 4, 7, 11 was not noted;
were the other resonances encountered examined in terms of the
Series per se. The present example (which repeats after almost
years) is however but one of a number of approaches that can be applied
to the problem. It may be further noted here that in addition to
a resonant intermediate synodic location between Venus and Mars, that
corresponding inverse-velocity function for Earth may also defined in
of the inverse-velocities of the three adjacent gas giants (the
and the Saturn-Jupiter synodics respectively) which are in turn subject
to real-time periodic variations of their own.
But there still remains the unexplained occurrence of the Lucas 76:47 and Fibonacci 55:34 ratios and why the former gives the better correction for the inverse-velocity functions in question. On the other hand, there is the apparent linkage between the major superior and the terrestrial planets provided by the inverse-velocity functions and the undoubted Fibonacci relationships that exist among the more massive group of planets, Jupiter and Saturn especially.
RESONANCES: THE MAJOR SUPERIOR PLANETS
Perhaps the best known resonance in the Solar System involves the relative motion of Jupiter with respect to Saturn. But before examining this example in detail it is necessarily to emphasize the predominance of this pair of planets above all others, including the adjacent major superior planets Uranus and Neptune. Alone Jupiter accounts for 71% of the planetary mass in the Solar System and more than half of the total angular momentum. Saturn comes next with 21% of the mass and and 25% of the angular momentum; taken together Jupiter and Saturn thus account for 92% of the mass and more than 85% of the angular momentum. The further inclusion of Uranus lifts the totals to 95% and 92% respectively, while all four major superior planets account for more than 99% of both the planetary mass and the total angular momentum of the entire Solar System.
Of the four major planets, the heliocentric positions of the first three not only compare to successive positions on the exponential planetary frameworks, they also permit the generation of the three inverse-velocity relationships discussed in Part II. But there are other considerations to be factored into this complex equation, for Jupiter is not only the largest planet by far in terms of size and mass, it is also the swiftest moving major planet, followed in due order by Saturn (the next most massive) and then Uranus. Neptune at present represents an anomaly though it obviously cannot be ignored. But if one is going to concentrate on the major planets then it would be logical to expect that the influence of Jupiter and Saturn would predominate, followed next by Uranus. In other words, the three adjacent planets that belong to the five successive sidereal and synodic periods from Jupiter out to Uranus from the original log-linear segment. But since the sidereal and synodic relationships between Jupiter, Saturn and Uranus have long been known and to some extent researched, whatever it is that remains to be determined must be more than this alone, or even perhaps entirely different. Then again, perhaps it is something relatively simple but difficult to check exhaustively. Now at least the exponential planetary frameworks provide a basis for comparison, as do the inverse-velocity relationships. Finally, the phyllotaxic Fibonacci/Lucas ratios at least permit the narrowing of the inquiry to an investigation of real-time resonances among the the four most massive objects in the Solar System.
E4. THE JUPITER-SATURN
CYCLES AND URANUS
As mentioned earlier, the present methods were first adapted a decade or more ago to generate real-time data to investigate the possible influence of planetary motion on Solar Activity cycles--an investigation that included resonances, but not exhaustively. Here the same methods can be directed towards more specific goals, though it is as well to be aware of the complexities in attempting to come to terms with interactions that involve multiple elliptical orbits and varying motion. A real-time period function for Jupiter will vary on either side of the mean sidereal period by the range permitted by the planet's eccentricity, in this case approximately 11 to 12.75 years, and a similar situation prevails in the case of Saturn (i.e., approximately 27 to 32 years). The more complex Jupiter-Saturn synodic cycle on the other hand has a somewhat wider theoretical range (approximately 17 to 24 years) with corresponding data derived from the synodic formula and periods obtained from the Jupiter-Saturn radius vectors. Time-series results in this case provide sinusoidal period functions that follow the variations of the respective radius vectors over time. Thus over approximately 59 years the 5:3:2 resonances of Jupiter and Saturn will be displayed as five sinusoidal waveforms for the former (i.e., 5 sidereal cycles), two sinusoidal cycles for Saturn, plus a three-cycle synodic waveform that maps the relative but varying motion of Jupiter with respect to Saturn over the same interval. "Resonances" occur when all three waveforms coincide -- three times in the present example. But before proceeding there are two further matters that require explanation and emphasis. The first is that as long as the basic 5:3:2 relationship for Jupiter and Saturn holds, multiplications need not stop at the approximate 59 year period; nor for that matter, need the well-known 1:1:2 resonance of Uranus with respect to Neptune necessarily remain with unity (the latter provided by the mean period of Neptune), i.e.,
At this juncture the matter begins to focus more firmly on the Fibonacci and Lucas Series, for in seeking to embrace the latter it seems that while it is still necessary to concentrate on the relative motion of Jupiter with respect to Saturn, the relative motion of Jupiter with respect to Uranus also has a significant role to play. The mean value of this period is readily obtained from the the mean sidereal periods of Jupiter and Uranus by way of the general synodic formula. Given to the sixth decimal the mean synodic period of Jupiter with respect to Uranus thus turns out to be 13.820371 years. What follows next is perhaps surprising, for in dealing with multiple harmonics -- which is essentially what is under consideration here -- it is one thing to invoke Fibonacci variants of the basic 5:3:2 resonant relationship between Jupiter and Saturn, and quite another to expect that the Jupiter-Uranus harmonics would also relate to the Lucas Series in this precise context, especially in an opposite sense. Nor for that matter is it likely that one would anticipate that while it is necessary to reverse the order of the fibonacci triples to maintain the resonant relationship between Jupiter and Saturn (i.e., 5:3:2 to obtain 5 cycles of Jupiter, 3 Synodics and 2 cycles of Saturn in approximately 59 years, and so on), that the Lucas harmonic expansion would follow its normal order, i.e., 4, 7, 11, 18, 29, ... etc. But this said, we are at least familiar with the Phi-Series planetary frameworks, the relationship between the latter and the Lucas Series and we are already dealing with the mean periods of revolution and synodic cycles expressed in years in both contexts. Again, however, bearing in mind the variance that results from the true orbital motions of the three planets in question, the relationship between the reversed Fibonacci triples and the Lucas harmonics is still not immediately apparent. One of the main reasons for this is that it only becomes clear after the multiple periods of the Jupiter-Saturn triples are averaged, and then only with the longer intervals is the relationship easily detectable. For example, based on a mean sidereal period of 11.869237 years for Jupiter, a corresponding mean synodic period 19.881324 years and mean sidereal period for Saturn of 29.452520 years, the fifth, third and second multiples (i.e., the 5:3:2 resonance) occur after 59.346 years, 59.644 years and 58.905 years respectively, whereas the average for all three products is 59.298 years. The fourth (4) Lucas augmentation of the Jupiter-Uranus mean synodic period on the other hand occurs after 55.282 years--a loose correlation easily dismissed as a chance occurrence. However, further investigation reveals that the 5:3:2 Jupiter-Saturn and Jupiter-Uranus Lucas multiple 4 are seemingly co-associated, for the next Lucas number (7) is similarly associated with the next reversed Fibonacci triple after 5:3:2, and as the two sets both proceed to their larger numbers, the difference between the averages of the Fibonacci triples and the Lucas multipliers becomes increasingly less. Thus by the time the 89:55:34 Fibonacci triple is reached the average of 1050.41 years is more closely approximated by the 1050.38 years obtained from the 76th multiple of the mean Jupiter-Uranus synodic cycle. In other words, the Fibonacci and Lucas assignments proceed sequentially, side-by-side in strict order. Thus the harmonic Fibonacci triples of the Jupiter-Saturn triad are related to the Lucas harmonics of the Jupiter-Uranus synodic cycle in the following manner for the given periods (rounded here to the nearest year for clarity and convenience):
3 -Lucas 4 ( 59 Years )
5 -Lucas 7 ( 94 Years )
8 -Lucas 11 ( 153 Years )
13 -Lucas 18 ( 248 Years )
21 -Lucas 29 ( 401 Years )
34 -Lucas 47 ( 649 Years )
55 -Lucas 76 ( 1050 Years )
Returning to the matter at hand, however, we have now arrived at the 76:47 Lucas ratio in true consort with the 55:34 Fibonacci ratio, with Lucas harmonics always occupying the position between the highest and next highest values in the associated Fibonacci triple. And here, as can be seen in Figure 10 -- real-time 89:55:34 multiples of the Jupiter-Saturn cycles and the 76th Jupiter-Uranus cycle -- the latter component also moves towards the nexus of the Jupiter-Saturn cycles, and this increasingly so with time.
As a first approximation it therefore seems that the relative motions of Jupiter, Saturn and Uranus, and predominantly that of the first (the largest, swiftest and most massive of the three) are intimately associated with the Golden Ratio. Not included here, yet likely also involved are the relative motions of Saturn with respect to Uranus, the motion of Saturn with respect to Neptune, and additional complications arising out the dominance provided by Jupiter with respect to all three. Nevertheless, the situation may be summarised at this initial stage in terms of the relative motions of the three major superior planets Jupiter, Saturn and Uranus as follows:
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