## A remarkable discovery: All Solar system periods fit the Fibonacci series and the Golden Ratio

Posted: February 20, 2013 by tallbloke in Analysis, Astronomy, Astrophysics, climate, Cycles, data, Gravity, Natural Variation, Ocean dynamics, Solar physics, solar system dynamics, Tides
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Many other people have noticed Phi relationships in the solar system in the past, from Kepler onwards, and there are several websites which cover this interesting topic. But up until now, so far as I know,  no-one has been able to find a single simple scheme linking all the planets and the Sun into a harmonious whole system described by the basic Fibonacci series. A couple of weeks ago while I was on holiday, I had a few long ‘brainstorming sessions’ with Tim Cullen, and decided to roll my sleeves up and get the calculator hot to test my ideas. What I discovered is laid out below in the style of a simple ‘paper’. Encouraged by an opinion from a PhD astrophysicist that this is “a remarkable discovery”, I will be rewriting this for submission to a journal with the more speculative elements removed and some extra number theory added to give it a sporting chance of acceptance. For now, this post establishes the basics, but there is much more I have discovered, and I will be using some of that extra material in my presentation at the conference in September we are setting up to discuss Solar System Dynamics, the Solar-Planetary Theory, and Solar Terrestrial relations. There will be an update on the conference with booking details etc very soon.

Relations between the Fibonacci Series and Solar System Orbits

Roger Tattersall – February 13 2013

Abstract

The linear recurrence equation: an = an-1 + an-2 with the starting conditions: a1 = a2 = 1 generates the familiar Fibonacci series: 1,1,2,3,5,8,13… This paper will use the first twenty terms of the sequence to demonstrate a close match between the Fibonacci series and the dynamic relationships between all the planets, and two dwarf planets in the Solar System. The average error across the twenty eight data points is demonstrated to be under 2.75%. The scientific implication of the result is discussed.

Introduction

Since it was noticed that five synodic conjunctions occur as Earth orbits the Sun eight times while Venus orbits thirteen times, many attempts have been made to connect the Fibonacci series and it’s convergent ‘golden ratio’ of 1.618:1 to the structure of the solar system. Most of these attempts have concentrated on the radial distances or semi-major axes of the planet’s orbits, in the style of Bode’s Law, and have foundered in the inner solar system.

The present paper adopts a different approach, in order to simultaneously study the dynamic relations between planet pairs as signified by the frequencies of their synodic conjunctions in addition to their individual orbits. A static analysis of semi-major axes is inadequate to an understanding of a dynamic solar system in the same way that statically balancing a flywheel cannot reveal the out of balance forces which will cause vibration when it is rotated at high speed.

Method

The highest number in the series used (6765) is allowed to stand for the number of orbits of the Sun made by Mercury, the innermost planet. The number of orbits made by the other planets and dwarf planets during the same time period of ~1630 years taken by Mercury to complete 6765 orbits are calculated. Additionally, the number of synodic conjunctions between adjacent planet pairs made in the same period is calculated using the method derived by Nicolaus Copernicus:

Period = 1/((1/faster orbit)-(1/slower orbit))

Additionally, the harmonic periods associated with the Power Spectral Density (PSD) study made of the sunspot number by talkshop contributor ‘Bart’ and used in the subsequent posting on Jupiter and Saturn’s relationship with the solar cycle and independently confirmed by  Scafetta 2012a[7] are included. The results are then compared to the descending values of the Fibonacci series and the deviations from the series calculated.

Juno is selected as representative of the Asteroid Belt as it lies near the middle of the main core at a distance of 2.67 AU. By Kepler’s third law this object has an orbital period of: P=(SQR)2.673=4.36yr.

Results

Results are tabulated in table 1. The hypothetically vanished planet ‘Vulcan’ is shown in order to demonstrate the interesting phi relationships  which would have existed given its 2.67 year orbit.

Discussion

This is a startling result. There is no currently accepted physical mechanism which can explain the clear and strong link between the Fibonacci sequence, the dynamic motion of the solar system, terrestrial cyclic phenomena at around 60 years and 205 years and solar activity levels. The underlying ratio is Phi, known as the golden section or ratio. This ratio does manifest itself elsewhere in nature. In plant biology, Phi is well known to appear in the spacing of leaf stems and the packing of seed heads. The leaf stem spacing maximises sunlight exposure and the seed packing maximises abundance[1]. In Geology, Phi relationships are evident in atomic, quasi-crystalline and other chemical structures[2].

Space has no crystalline structure. However it does have gravitational fields and electromagnetic fields permeating it. What kind of interaction of these fields with matter could bring about a situation whereby, approximately 4.5 billion years after the formation of the solar system, such close relationships to Phi are found to link every planet and two dwarf planets in the solar system? Evidently, harmonic and other periodic perturbations between planets and planet pairs have helped shape the system, and continue to maintain its internal relationships.

The average deviation from the Fibonacci series for the eight planets plus two dwarf planets orbits is 2.75%. This compares well with Bode’s Law which exhibits a 15% average deviation.  Solar activity cycles are represented by the inclusion of results from a Power Spectral Density (PSD) analysis which finds sunspot (SSN) activity peaks at 19.86 and 23.72 years, generating harmonics at 10.8 and 122 years[3]. This suggests that there is a link between planetary motion and solar activity levels.

Because the Sun’s gravity diminishes on an inverse square law, perturbation between Jovian planets will affect their orbits more strongly than the inner planets. Consequently, the Jovian planets excepting Saturn show a bigger deviation from the Fibonacci series than the three innermost planets.

It is suggested by Miles Mathis that inside Newton’s gravitational equation: F=GM1M2/r2 and Coulomb’s similar charge equation: F = kq1q2/r2  there is a unified field rather than two separate forces described by the two equations[4]. Mathis demonstrates that with a minimum of postulates, a fully mechanical ‘pool ball physics’ can be developed.  As well as providing a gravitational acceleration bringing extended bodies together, it also contains a repulsive electromagnetic force which although weak in our everyday experience, can become significant at the scale of astronomical bodies when they are in proximity. Importantly, the gravitational acceleration and the repulsive force scale differently as distance changes due to the different properties of the bodies they relate to.

This may explain why empty viable orbits are free of formation debris; the changing of the planetary orbits to create the most efficient order has over time traversed and swept the solar system clear of debris. The exception is the Asteroid Belt between Mars and Jupiter. Some evidence suggests its formation may be recent (3.2Ma).

Additionally, Mathis’ ‘foundational E/M field’ pervades space at varying densities (dependent on the proximity of emitting bodies), providing a ‘background’ against which the ratios of forces exerted by bodies will operate.  Mathis suggests that rather than trying to understand Phi in isolation, we can only appreciate the way that the two quantities which form the ratio can operate mechanically, by understanding the way in which they are relative to the ambient field in which they operate[5]. This is not a proposal for a ‘Universal Aether’, but for an interplanetary space which contains a density varying field of charge and spin bearing photons being constantly emitted and absorbed by matter.

Conclusion

The logical conclusion is that feedback is present via perturbations between the planets and Sun which arranges the planets into an order which minimises work done, enhances stability and maximises entropy. This calls to mind the constructal law, stated by Bejan in 1996 as follows: “For a finite-size system to persist in time (to live), it must evolve in such a way that it provides easier access to the imposed currents that flow through it.”[6]

A true system contains cybernetic feedback. The Phi relationships demonstrated here are evidence that the solar system truly is a system in the full sense of the word.

References

[6] Bejan, Adrian (1997). “Advanced Engineering Thermodynamics,” (2nd ed.). New York: Wiley.

http://en.wikipedia.org/wiki/Constructal_theory

[7] Scafetta, Nicola (2012a) “Multi-scale harmonic model for solar and climate cyclical variation
throughout the Holocene based on Jupiter–Saturn tidal frequencies plus the
11-year solar dynamo cycle”  JASTP

1. oldbrew says:

Guys: think of a starting point of 89/8

Half the assumed 178 year cycle is 89 years
89 year period = 8 x 11.125 years = 8 solar magnetic reversals aka sunspot cycles
11.125 = 11 and one eighths = 89 / 8 (again)

Also,
11.125 years = 133 and a half months = 267 / 2 = 89 x 3 / 2 = 89 x 1.5 months
8 x 1 .5 months (89 /8 formula) = 1 year

Not sure yet if that can be taken further in the ‘micro’ direction, but note that:
1.5 months = 45.65625 days (365.25 / 8) and also 45 degrees of orbital movement (360 / 8),
and that 89 / 2 = 44.5, so any ideas welcome.

On the ‘macro’ side, 89 x 52 = 4628. The significance of 52 (2x2x13) years is not clear, but consider 8 x 52 (89/8 formula). That is also 2 x 4 x 52 = 2 x 208 = 2 de Vries cycles?

‘a ~ 208 year-cyclicity, named de Vries or Suess cycle (Damon and Sonett, 1991, Stuiver and Braziunas, 1993 and Wagner et al., 2001), is documented in various Holocene records (e.g. Schimmelmann et al., 2003, Raspopov et al., 2008, Taricco et al., 2009, Incarbona et al., 2010 and Di Rita, 2011). It might as well be present from historical sunspot observations (Ma and Vaquero, 2009). Its influence on several climatic parameters has been discussed by Raspopov et al. (2007), who document a non-linear response of the climate system in various geographic regions.’

http://www.sciencedirect.com/science/article/pii/S003101821200096X

Btw re the 4628 cycle: 2 x 2 x 34 x 34 = 4624 (the next 4-year [2x2] period up to 4628 is not completed)

2. oldbrew says:

TB says: ‘Mars at 47, lies at a golden section within the golden section’

The square root of 47 is 6.85565~
Phi^4 = 6.8541~

3. tallbloke says:

Excellent!

4. Gray says:

The rest of the series…

Mercury orbits = 739.061
Venus orbits = 288
Synodic Venus = 449

incidentally tallbloke 6785 / 739.061 = 9.180

Jupiter orbits 15
Saturn orbits = 6.042
JS Synodic – 9

Saturn orbits = 6.042
Synodic Uranus = 3.92414235995
Uranus Orbit = 2.11878024916

Uranus orbit: 2.11878024916
Synodic Neptune : 1.03488372093
Neptune orbit = 1.08019540614

http://www.jupitersdance.com/PlanetRelationships2.jpg

5. Gray says:

Correction : 6765 / 739.061 = 9.15350695004

6. oldbrew says:

Re. ‘starting point of 89/8′:
360 degrees / 89 = 4.044944 x 11.125 = 44.9 recurring = 45 degrees

Which proves that 89 / 8, i.e. Fibonacci, directly links to degrees of orbit and rotation (45 x 8 = 360 of course).

Re. ‘The significance of 52 (2x2x13) years is not clear’:
52 = 5² + 3³

89 x 5² = 2225
89 x 3³ = 2403
2403 – 2225 = 178 = 89 x 2

If anyone wants to talk about a 1000 year solar cycle, we can offer:
89 years x 11.125 = 990.125 years

That would be a one-eighth part of a 7921 (89²) cycle.

7. Gray says:

Mercury figure 6765 divided by 449 Mercury/Venus Synodic periods

6765 / 449 = 15.066

So 15 and 9 scales to:

Jupiter orbits = 15
Saturn orbits = 6.042
JS Synodic = 9

8. oldbrew says:

Using 89 as a base gave:
4628 / 11.125 (i.e. 89 /8) = 416 (solar cycles in 4628 years)

However:
4628 / Phi = 2860.26
2860.26 / 16 = 178.75
178.75 / 16 = 11.171875
(16 = number of solar cycles in the 178~ year period)

These results are closer to the numbers we actually know of, e.g. Jose 178.7 year cycle.
Gray’s website refers to ‘the recognised sunspot frequency of 11.171 years’.

Calculating in the other direction:
11.171 x 16 x 16 x Phi = 4627.215~
4627.215 / 11.171 = 414.22~ (solar cycles in 4627~ years)

Previous discussions on the Talkshop refer to a 4627 year cycle (not 4628), e.g.

9. oldbrew says:

More triangles for the schematic list?

Mercury 369, Venus 144, 369-144 = 225 = 15×15 (Jupiter 178 /11.86 = 15)

Earth 89, Mars 47.3, Jupiter 11.86² (=141 approx = 3.45% above exact match) OR
Earth 89, Mars 47.3, Venus 144 (5.6% above exact match)

Also: Me – (E + V) = 369 – 233 = 136 = E + Ma (136.3)

10. oldbrew says:

‘Mercury 369, Venus 144, 369-144 = 225 = 15×15 (Jupiter 178 /11.86 = 15)’

The Venus sidereal year is also 225 (224.7 days).

The creationists have some interesting stats here (just sticking to the figures):
http://creationwiki.org/File:Phi_and_planets_graph.jpg
http://creationwiki.org/Golden_ratio_in_the_planets

11. Max™ says:

Something which may help make sense of the Fibonacci resemblance:

For Earth, every 97,000 miles we travel around the Sun, we travel 447,000 miles roughly towards Vega.

http://calgary.rasc.ca/images/howfast_solsys18.gif

Fibonacci sequences naturally describe spirals, the planets follow spirals as they orbit the Sun.

12. oldbrew says:

This paper: ‘The ∼ 2400-year cycle in atmospheric radiocarbon concentration: bispectrum of 14C data over the last 8000 years’…
http://www.ann-geophys.net/20/115/2002/angeo-20-115-2002.pdf

…reports that:
‘A principle feature of the time series is the long period of ∼ 2400 years, which is well known. The
lines with periods of 710, 420 and 210 years are found to be the primary secular components of power spectrum.’

The 2400 year period is 89 x 27 = 2403 years (27 = 3³)

89 x 8 = 712 (cf. 710) = 4 Jose cycles of 178 years (= 2 x 89 years)

210 has already been identified in this thread as the de Vries cycle (208), and 420 = 2 x 210.

13. oldbrew says:

More on the 89-year period: re. the reported 160-minute solar oscillation.

Take a notional solar cycle of 11.125 years ( = 89 years / 8) and convert it to hours.
11.125 x 365.25 x 24 = 97521.748 hours

NASA data says the sidereal rotation period of the Sun is 609.12 hours
97521.748 / 609.12 = 160.1 hours (160.10268)

160.10268 x 60 = 9606.1608 minutes

9606.1608 / 60 = 160.10268 minutes

http://tallbloke.wordpress.com/2012/04/23/evidence-for-a-160-minute-oscillation-affecting-galaxies-and-the-solar-system/

http://ray.tomes.biz/160min.html

14. tallbloke says:

160.10268 x 60 = 9606.1608 minutes

9606.1608 / 60 = 160.10268 minutes

Hmmm, this doesn’t seem all that informative at first glance.

15. oldbrew says:

Sorry, it should have been the other way round because there’s no reason to divide by 60 on the last line. If we assume the 160.01 and use it as a divisor:

9606.1608 / 160.01 = 60.03475~

(The other calculation returns 160.1 not 160.01)

There are some interesting comments here about the Sun, Jupiter and Kotov’s wave, as they call it. The link was in the last comment on the ’160-minute’ thread mentioned above.

‘Strange Coincidences between the Sun, Jupiter and OJ 287′
http://lempel.pagesperso-orange.fr/coincidences_etranges_02_uk.html

See: Relations with the 160 mn wave of Kotov.

The author claims to have found proof that, just as the Moon is very gradually moving away from the Earth, so for the same reasons Jupiter is moving away from the Sun.

16. tallbloke says:

Earth-Moon masss ratio ~80:1 Lunar orbital distance ~60 Earth radii

Sun-Jupiter Mass ratio ~1000:1 Jovian orbital distance ~1000 solar radii

17. oldbrew says:

The 609.12 hours figure is based on an arbitrary decision to measure at 26 degrees from the equator, ‘approximately the point where we see most of the sunspots’

So with a very slight tweak the 160.01 division could return 60 exactly. Obviously we don’t know what that indicates but it doesn’t do the 11.125 solar/Fibonacci hypothesis any harm.

18. oldbrew says:

Let’s try it another way.

11.125 x 365.25 = 4063.4062 Earth days
609.12 hours = 25.38 Earth days = 1 Sun rotation
4063.4062 / 25.38 = 160.1 solar rotations

So there are approximately 160.1 (could be 160.01) solar rotations in a solar cycle.

If the incoming ‘wave’ acting on the Sun occurs every 160.01 Earth minutes, and Earth minutes derive from the time taken to orbit the Sun, the evidence for a ‘solar clock’ seems to be there.

19. oldbrew says:

Using Roy Martin’s harmonics list…
http://tallbloke.files.wordpress.com/2013/03/table2_synodic_periods_table_01.jpg?w=614

…the total of all Jupiter synodics is:
9+276+164+728+80+13+14 = 1284

One solar rotation takes 25.38 days (NASA figure). Using Roy’s J/S base period of 178.732 years:
(178.732 x 365.25) / 25.38 = 2572.94

2572.94 / 1284 = 2.00385~

That equates to an average of 1 Jupiter synodics per 2 solar rotationsi.e. a 1:2 ratio.

20. oldbrew says:

One solar rotation takes 25.38 days.

25.38³ = 16348.384 days
16348.384 days / 365.25 = 44.76 years
44.76 x 4 = 179.04 years (Jose cycle)

So the cube of the solar rotation period = a quarter of the Jose cycle.

21. tallbloke says:

Good one!
I had some thought s on solar rotation related to planetary periodicity a while back. I got panned for it, but I still think the inverse relationships are real somehow.
http://tallbloke.wordpress.com/2011/11/19/solar-planetary-spin-orbit-coupling-more-evidence/

22. Gray says:

The final triangle in the series is:

Sun rotation 0.069 years 178 / 0.069 = 2,579.7
Mercury orbit .240846 years 178 / .240846 = 739.06147496741
Sun synodic Mercury .096739 years = 178/.096739 = 1,840.002

23. oldbrew says:

Found something in the link that might be of interest. You said:

‘So for the orbits of Jupiter(11.86 years) and Saturn(29.46 years) we find that the squares (multiplication by itself) of the orbital periods are 140.67 and 867.3. The cube roots of these values are 5.2 and 9.54′

Compare those cube root values with the ‘mean distance from Sun (AU)’ figures for J and S here:
http://www.windows2universe.org/our_solar_system/planets_table.html

Any comment?

24. oldbrew says:

Re. the cube root values (as per Newton / Kepler, ignore previous post) – there seems to be a correlation between planetary pairs.

Earth:Mars = 1:1.52
Uranus:Neptune = 1:1.56

Mercury:Venus = 1:1.86
Jupiter:Saturn = 1:1.83

Looking at the links between these pairings, we have:

Venus:Earth = 1:1.385 (1.382 = 2 – 0.618)
Saturn:Uranus = 1:2.01 (2)

For the outer planets, the cube root ratios and the ‘distance from Sun’ ratios are identical.

For the inner planets, the exception is Earth:Mars, where the distance ratio is 1:1.88 (similar to M:V and J:S) but the cube root ratio of 1:1.52 is like U:N, as shown. That may well relate to the fact that Mars is the last inner planet before Jupiter, and/or the existence of the asteroid belt between the two.

25. oldbrew says:

Correction – the Earth:Mars distance ratio IS 1:1.52, so it’s not an exception.
(The actual Mars orbital period is 1.88 years, copied in error)

Extending the analysis to dwarf planet Ceres (between Mars and Jupiter), we get:

Mars:Ceres 1:1.81 (very close to Jupiter:Saturn = 1:1.83)
Ceres:Jupiter 1:1.88 (very close to Mercury:Venus = 1:1.86)

Looking at it as a whole, we have:

1.86 — 1.385 — 1.52 — 1.81 — 1.88 — 1.83 — 2 — 1.56

The planetary pairs with the lowest ratios (Mars/Earth, Neptune/Uranus*) orbit either side of the Jupiter/Saturn ‘axis’ (ignoring the dwarf planet Ceres).

*Quoting myself (March 4 @ 8:36am):
‘Consider the other ‘half’ of the system: imagine we fold it over so all the planets are back-to-back.
Now, Mars/Earth = Neptune/Uranus, and so on.’

26. oldbrew says:

@ Gray: improvements to the ‘speculation’ here…

Saturn’s orbit period cubed (sopc) = 29.457³ = 25560.276 years
Jupiter/Saturn synodic 19.86 / 2 is 9.93 (conjunctions+oppositions)
25560.276 / 9.93 = 2574.05 (J/S synodics in sopc)

Other results from that:
25560.276 / 178.74 = 143.00255 (no. of Jose cycles in sopc)
25560.276 / (143 x 16) = 11.1715 (one solar cycle, 16 per Jose cycle)
25560.276 / 19.86 = 1287.02 = 143 x 9

9.93 x 365.25 = 3626.9325 days
3626.9325 / 25.38 = 142.9 (no. of solar rotations in 9.93 years)

Solar rotations = (2574 x 25.38 days) / 365.25 = 178.86
2574 / 18 = 143 rotations per J-S half-synodic (confirms previous calc.)
18 x 9.93 periods per Jose cycle = 178.74

(21 x 9.93 = 208.53 = one de Vries cycle?)

Another view:
143 x 19.86 (9.93 x 2) = 2840
25560.276 / 2840 = 9

(1) no. of s.r. per Jose cycle = no. of J-S conjns. + opposns. in Saturn s.o.p.³

(2) no. of s.r. in 9.93y (J/S conjns. + opposns.) and
no. of J-S conjns. + opposns. in Jose cycle* =
no. of Jose cycles in Saturn s.o.p.³

[(1) = 2574, (2) = 143]

NB 9.93 x 9 = 89 (89.37)
(143 cf. Fibonacci 144 ?)

(* 2:1 s.r. to all J synodic conjunctions already identified)

Conclusions at this stage:

The cube of the orbit period seems significant.
Definite link between s.r. and orbit period.
S.r. / Jupiter synodic conjunctions link confirmed.

Number of solar rotations per (ideal) Jose should be 2574 (143×18)

J-S half-synodic x 18 = 9.93 x 18 = 178.74
(Jose cycle – J-S figure : Sun figure = 1:1.0006713)

Now we turn to the cube of the orbit period for the other ‘gas giants’.

Jupiter s.o.p.

11.86³ = 1668.2228 years
1668.2228 / 9.93 (J-S/2) = 168 = 21 x 8
(21 x 9.93 = 208.53 = de Vries cycle?)

No. of solar rotations in 1668.2228 years:
(1668.2228 x 365.25) / 25.38 = 24007.816
24007.816 / 168 = 142.904 (143)

NB 1668.2228 / 3 = 556.0743 years
One Eris orbit = 557 years approx.

Uranus s.o.p.

84.01³ = 592915.67 years
592915.67 / 143² = 29

Not sure if this means anything, but:
9.93 x 29 = 287.97 (9.93 = J-S half-synodic period)
287.97 / 2 = 143.985

Neptune s.o.p.

164.7885³ = 4474872.7 years (nearly 4.5 million years)
4474872.7 / (143 x 175) = 178.8161 years = Jose cycle

4474872.7 / 12.78279 (J-N synodic) = 350070.1 conjunctions
350070.1 / 2574 (143 x 18) = 136.00236 (136 = 34 x 2 x 2)

——-
All the gas giants plus the Sun have clear links to multiples of 143.

143 solar rotations take 9.936 years (based on 25.38 days each).
One J-S half-synodic takes 9.93 years.
These two figures may in reality be exactly the same period.
(The difference between them is around 0.0006%)

27. Gray says:

Hi Oldbrew

http://www.jupitersdance.com/solution.jpg

28. oldbrew says:

Hi Gray

Sorry, I’m having trouble joining the dots there. Where does the 55 come from, for example?

More speculation probably…

Jupiter-Saturn synodic period = Jupiter orbit period³ / Uranus orbit period
19.859 = 1668.223 / 84.01 (19.857)

29. Gray says:

Hi OldBrew.

I’m sorry, It’s an abstract concept which I’m trying to convey as logically as I can.

I’ll work some more on it…

30. oldbrew says:

I’ve expanded the data for the constant that Ulric Lyons showed earlier.
This is where the ratios of the synodic periods of 3 contiguous planets are compared.

Mercury — Venus — Earth (Me-V 0.3958, V-E 1.5987, Me-E 0.31726)

V-E / Me-E = 5.0391
V-E / Me-V = 4.0391

Venus — Earth — Mars (V-E 1.5987, E-Ma 2.1353, V-Ma 0.91423)

E-Ma / Ma-V = 2.3356
E-Ma / E-V = 1.3356

Jupiter — Saturn — Uranus (J-S 19.859, J-U 13.812, S-U 45.363)

S-U / J-U = 3.2843
S-U / J-S = 2.2843

Saturn — Uranus — Neptune (S-U 45.363, U-N 171.39, S-N 35.869)

U-N / S-N = 4.7782
U-N / S-U = 3.7782

Orbit periods from NASA used to calculate synodic periods here:
http://www.1728.org/synodic.htm

The only rule is that the largest figure is divided by the two smaller ones.
The difference between the two results is always 1.

31. oldbrew says:

In order to make more sense of the available data, there may be a case for ‘superimposing’ a template on it. Take the Jose cycle:

The synodic period figure is 178.72 years, so let’s suppose a template figure of 180.
The difference is: 180 / 178.72 = 1.00716 (180 is a multiple of 2,3 and 5 in Fibonacci)

Jupiter-Saturn half-synodic = 9.93 x 1.007 = 10 exactly (2 x 5 in Fibonacci)
Full J-S synodic 19.86 x 1.007 = 20 exactly (9 x 20 = 180)

9.93 = 143 x 25.38 day solar rotations
143 x 1.007 = 144 exactly (Fibonacci)

Jupiter orbital period = 11.862 years
11.862 x 1.007 = 11.95 (12 = square root of 144 in Fibonacci)

Saturn orbital period = 29.46 years
29.46 x 1.007 = 29.666 = 89 / 3 (Fibonacci, 144 / 89 = Phi)

32. tallbloke says:

Oldbrew: Nice job, and starts to tie in with Gray’s Jose period approach too. This is looking really good. I had a chat with Gray on the phone last night and agreed a way forward. I will review his comments and tie them in with his diagrams as best I can, then email him my summary for him to annotate and return. We’ll include you in the email loop for comments too. Then I’ll write a summary post, run it by you both for final comments, amend and publish.

33. oldbrew says:

I’m sending a triangle diagram by e-mail, re this:

‘Since it was noticed that five synodic conjunctions occur as Earth orbits the Sun eight times while Venus orbits thirteen times, many attempts have been made to connect the Fibonacci series and it’s convergent ‘golden ratio’ of 1.618:1 to the structure of the solar system.’