Why Phi? – the Inex eclipse cycle, part 2

Posted: March 20, 2015 by oldbrew in Cycles, Fibonacci, modelling, moon, Phi, solar system dynamics
18 Inex cycles = 521 years [click to enlarge]

18 Inex cycles = 521 years
[click to enlarge]

In the wake of today’s solar eclipse and following an earlier post on the same topic, we have another perspective on the 521 year period that corresponds exactly to 18 Inex eclipse cycles.

An Inex corresponds to:
358 lunations (synodic months) = 28.94444 years
388.50011 draconic months
30.50011 eclipse years
Source: http://en.wikipedia.org/wiki/Inex

This means two Inex = 716 synodic months (358×2) and 777 draconic months (388.5×2).
This period will also be 61 eclipse or draconic years (777 – 716 or 30.5 x 2).

Each number in the diagram (below the top line) is derived from the numbers above it. Note that 18 Inex is the same period as 28 lunar nodal cycles. Both periods end at the lunar node they started at.

We can build on this, first by looking at data from a well-known science paper by Keeling & Whorf titled:
‘The 1,800-year oceanic tidal cycle: A possible cause of rapid climate change’

Figure 2 in the paper shows two long tidal cycle periods of 1823 years each:

By adding these two periods together, plus one year, we get 7 sets of 18 Inex cycles (3647 years = 521y x 7).

Returning briefly to Wikipedia it says re Inex cycles:
‘Their appearance and longitude are irregular due to a lack of synchronization with the anomalistic month (period of perigee). However, groupings of 3 inex cycles (˜87 years minus 2 months) comes close (˜1,151.02 anomalistic months), so eclipses are similar in these groupings.’

So we can re-define the 7 sets of 18 Inex as 21 sets of 3, x 2 (= 126 Inex).
We shall return to the 21 Inex figure shortly.

Turning to a new paper (H/T Talkshop contributor ren) by Zhao and Feng:
‘Correlation between solar activity and the local temperature of Antarctica during the past 11,000 years’ (2015)
we note a possible link between the long Inex period and sunspot numbers.
‘It is found that SSN and temperature have some common periodicities, such as the 208 (204) yr, 441 yr, 488 (483) yr, 521 (516) yr, 562 (568) yr and 962 yr cycles, during the interval of interest.’

The abstract of the paper notes:
‘We find that the variations of SSN and T have some common periodicities, such as the 208 year (yr), 521 yr, and ~1000 yr cycles. The correlations between SSN and T are strong for some intermittent periodicities.’ [T = local temperature at Vostok].

The 208 year period is approximately the well-known de Vries cycle.
With all this information we can put forward for discussion possible ratios between certain data items:

21 Inex = 55 average solar cycles (a ratio of 1:Phi² as 21 and 55 are Fibonacci numbers), or 1 Inex = Phi² solar cycles
5 de Vries = 2 Inex long periods of 521 years each

Since 35 Inex cycles = almost 51 Jupiter-Saturn(J-S) conjunctions:
210 Inex (35×6) = 306 J-S (51×6), or
21 x 10 Inex = 34 x 9 J-S, or
1 Inex = Phi x 0.9 J-S

That would also mean: 1 solar cycle = 0.9 J-S / Phi
That would make the Jose cycle of 9 (= 3²) J-S equal to (5 x 2 x Phi) solar cycles.

A post at the Talkshop by Roy Martin noted:
‘As a footnote: In the work on the tidal theory there is a very significant cycle of 110.3 years. It is interesting to note that 110.3 x 1.61803(phi) = 178.48 years, within 0.31 years of the nominal José cycle period as derived above from the Jupiter – Saturn synodic.’

Roy Martin: How do the planets affect the Sun – Updated

So here we have several interesting relationships between the Inex eclipse cycle, tides, Jupiter-Saturn synodics, sunspots, solar cycles and Fibonacci/Phi related harmonics.

  1. oldbrew says:

    The Zhao & Feng paper abstract also says:

    ‘The millennial variation of SSN leads that of T by 30–40 years, and the anti-phase relation between them keeps stable nearly over the whole 11,000 years of the past. As a contrast, the correlations between CO2 and T are neither strong nor stable.’ [bold added]

  2. p.g.sharrow says:

    @Oldbrew; nice post and the comment “‘The millennial variation of SSN leads that of T by 30–40 years, and the anti-phase relation between them keeps stable nearly over the whole 11,000 years of the past.”

    That is the money shot that I have been looking for, thank you. pg

  3. tallbloke says:

    Very nice work Stuart.

    I think the anti-phase relation is not SSN and T but 10Be and T. The authors appear to have forgotten that cosmic ray incidence is anti-phase with solar activity. Unless they mean that the millennial lag of 30 yrs makes it anti-phase due to a 60 yr cycle?

  4. oldbrew says:

    Thanks pg and TB. I realise some people may find it a bit turgid looking at all the numbers, but the key is understanding how they fit together.

    Btw the 441 year period recorded by Z & F looks a lot like the Saturn-Mars-Earth cycle of ~442 years:
    15 Saturn = 441.71 years, 235 Mars = 442 years less a few hours.
    Also 441 = 21², and 441y x 4 = 21 Uranus.
    442 years x 4 is 60 Saturn, 149 Jupiter and 89 J-S conjunctions.

  5. oldbrew says:

    Note in the diagram that synodic months is shown as 6444 (537 x 12).
    That was to show that it’s 537 lunar years.

    6444 SM is also 716 x 9, 716 being the number of synodic months in 2 Inex, as mentioned in the post.

  6. Roger,
    Try to remember that the illusionary linear time T, while like the illusionary entropy (s) must always increase, must also always have (“please accept”) the congratulates, -1/(time), “frequency” and -1/(entropy), “temperature”. Please also remember that -1/l(linear) is never “linear” Those guys constructing that what was, now only the remaining “is”, were restricted from giving “any” clue to the possible critters.

  7. TLMango says:

    Hi Oldbrew

    Nice conversation!
    when J = 11.862242 and S = 29.457784
    J/4 x S/10 / (J/4 – S/10) = 441.6054951
    441.6054951 x 5 = 2208.027476 years ‘ the Hallstatt short cycle
    441.6054951 x 10/3 = 1472.018317 ‘the Bond cycle

  8. oldbrew says:

    Thanks TLM. I recommend this site for planetary data:


  9. TLMango says:

    Thanks for the link Oldbrew
    Some incredible work has come from Nicola Scafetta and also Hans Jelbring.
    I use orbital periods J = 11.862242 and S = 29.457784 because these are the values that
    Scafetta and Jelbring use. I simply don’t want to transpose frequency values when I read
    their papers. Also, the astronomer John F.W. Herschel used the values J = 11.862 and
    S = 29.457 when he estimated the Jupiter/Saturn great conjunction to be ~883 years.

    Nobody knows the actual orbital periods of Jupiter and Saturn to any significant number of
    decimal places because they are changing. This is a very interesting phenomenon known
    as the Jupiter/Saturn ‘Great Inequality’. When Jupiter approaches Saturn there is a mutual
    attraction between the two. This causes a slight acceleration for Jupiter and a slight
    deceleration for Saturn. When Jupiter pulls away from Saturn, there is a slight deceleration
    for Jupiter and a slight acceleration for Saturn. Over long time scales, the orbital periods
    of Jupiter and Saturn fluctuate inversely. When Jupiter’s period increases, Saturn’s decreases
    and vice versa.

    (883 x 5/2 = 2208) and (883 x 5/3 = 1472)

  10. TLMango says:

    Oh no!
    Sorry, should read (883.2 x 5/2 = 2208) and (883.2 x 5/3 = 1472)

  11. oldbrew says:

    TLM: 60 Saturn = 149 Jupiter, more or less, which is 89 conjunctions (149 – 60).
    The period of 89 J-S is around 1768 years , half that is 884 years (cf. your 883 years).
    Which is twice the 441.7y period you mentioned earlier.


  12. TLMango says:

    What is really incredible is that because the Jupiter/Saturn orbital periods fluctuate inversely,
    ratios and frequencies that are near Phi will eventually be equal to Phi. This is also true about

  13. oldbrew says:

    Near Phi or ‘a Fibonacci equivalent’ probably, e.g. Jupiter-Saturn 60:49 is almost a 2:5 ratio (= 60:150).
    As we all know Venus-Earth is almost 13:8.

    Phi is known to be the ‘most irrational number’, and its Fibonacci derivatives can be inferred by studying the data.

    “The golden ratio is the slowest of all continued fractions to converge.” Just what a planet needs 😉


  14. TLMango says:


    The 1768 year conjunction is a good one. But…. recently I’ve discovered the grand-daddy
    of commensurabilities. Try this one out !!! [796656.31047 years]

    J = 11.862242 ——– S = 29.457784
    S / 7 x J / 5 / (S / 7 – J / 5) = 5.438409623
    S / 5 x J / 4 / (S / 5 – J / 4) = 5.971220172
    J x S / (J + S) = 8.456804035
    S / 4 x J / 3 / (S / 4 – J / 3) = 8.538561328
    S / 7 x J / 4 / (S / 7 – J / 4) = 10.0425614
    J x S / 5 / (J – S / 5) = 11.70503387
    S / 5 x J / 3 / (S / 5 – J / 3) = 12.02373048
    J / 2 x S / 7 / (J / 2 – S / 7) = 14.48729425
    S / 3 x J / 2 / (S / 3 – J / 2) = 14.97868444
    S x J / (S – J) = 19.85931224
    J x S / 3 / (J – S / 3) = 57.01397771
    S / 2 x J / (S / 2 – J) = 60.94838271
    S / 7 x J / 3 / (S / 7 – J / 3) =65.46604571
    J / 2 x S / 5 / (J / 2 – S / 5) = 883.2109902

    5.438409623 x 146487 = 796656.3105
    5.971220172 x 133416 = 796656.3105
    8.456804035 x 94203 = 796656.3105
    8.538561328 x 93301 = 796656.3105
    10.0425614 x 79328 = 796656.3107
    11.70503387 x 68061 = 796656.3102
    11.862242 x 67159 = 796656.3105
    12.02373048 x 66257 = 796656.3104
    14.48729425 x 54990 = 797656.3108
    14.97868444 x 53186 = 796656.3106
    19.85931224 x 40115 = 796656.3105
    29.457784 x 27044 = 796656.3105
    57.01397771 x 13973 = 796656.3105
    60.94838271 x 13071 = 796656.3105
    65.46604571 x 12169 = 796656.3102
    883.2109902 x 902 = 796656.3132

    Plus lunar cycles:
    18.03124147 x 44182 = 796656.3105
    18.6043369 x 42821 = 796656.3105

    The 796656.31047 year commensurability has 49791.01941 and 99582.03881 year cycles
    built in. These 50000/100000 year waves dominate when all commensurabilities are graphed.

  15. These 50000/100000 year waves dominate when all commensurabilities are graphed.
    Nice work! does your word commensurabilities have anything to do with commensurate or incommensurate your use of suffix “abilities” suggest fantasy, never a measurable, at any scale.

  16. oldbrew says:

    Abstract: ‘Celestial commensurabilities: some special cases’ by H. Jelbring


    @ TLM: I prefer the JPL data tbh.

  17. oldbrew says:.March 27, 2015 at 1:26 pm

    “Abstract: ‘Celestial commensurabilities: some special cases’ by H. Jelbring..
    @ TLM: I prefer the JPL data tbh.”

    Thanks OB.
    Does this mean that PI can become a rational 22/7 in your world?
    I like that incommensurate is “that” with respect any rational. but also incommensurate
    with any other independent irrational. Phi/PI is never a ratio, nor rational! Such is but an independent irrational, that may become useful for earthling understanding of what “may be likely”.

  18. oldbrew says:

    Square root of 5: ‘As of December 2013, its numerical value in decimal has been computed to at least ten billion digits’

    ‘This number appears in the fractional expression for the golden ratio’


    That should be enough digits 😉

  19. TLMango says:

    Will & OB
    I’m a layman, so I am sometimes prone to the misuse of terminology.
    By commensurate I mean that 67159 x J = 27044 x S.
    But the 796656.31047 year period is more than commensurate, it is a grand confluence
    of all the solar system’s major frequencies.
    Over long time scales, the orbital periods of Jupiter and Saturn fluctuate inversely.
    Jupiter and Saturn regulate the solar system’s frequencies with J : S ratio fluctuation..
    There are many confluences that exist within the range of J/S orbital fluctuation.
    A good analogy would be: Someone has His hand on the radio dial and and is rolling back
    and forth. Each radio station would be analogous to a confluence of the solar systems
    current set of frequencies.
    As for Phi, Pi and E ?? These constants are always going to show up in nature. They bring
    beauty to mathematics.

  20. oldbrew says:

    Notrickszone features the Zhao and Feng paper (see post above) under the heading:
    ‘The sun drives the climate’ – By Dr. Sebastian Lüning and Prof. Fritz Vahrenholt


    Another paper, this time by Volobuev is also quoted:
    ‘Here we solve an inverse heat conductivity problem: calculate the boundary heat flux density (HFD) from known evolution of temperature. Using meteorological temperature record during (1958–2011) we calculated the HFD variation about 0.2–0.3 W/m2 in phase with solar activity cycle. This HFD variation is derived from 0.5 to 1 °C temperature variation and shows relatively high climate sensitivity per 0.1 % of solar radiation change. This effect can be due to the polar amplification phenomenon, which predicts a similar response 0.3–0.8 °C/0.1 %.’

  21. Chaeremon says:

    @oldbrew, let’s take a few empirical observables, here for lunar<->planet conjunctions restricted in the following, but not limited to, Jupiter and Saturn, so that opponents of cycle research can [perhaps] see what they object due to ignorance and instilled-into handwaving rituals like academic shamans.

    From a theoretical view we can expect a window of opportunity = ±½ × 27.3~ days for the repetition of lunar<->planet conjunctions, and depart from that point to investigation of a more narrow window size (as said for just Jupiter and Saturn).

    We find that lady luna, together with her tellurian siamese twin, has indeed ladylike preferences (aka. attractive physical bias), since she doesn’t need nor bother this much theoretical time in her window: the apparent physically necessary window size is about ±5 days when observing these conjunction intervals (in the 600ce to 2100ce time-frame) and that means a ~30% segment (dynamic view!) of the lunar orbit Does Not Participate in recurrence of (these) conjunctions. This is what I mean by physical bias on things which have measurable impact, like amplitude of oceanic and atmospheric tides (and LOD, etc).

    From this attraction result we go further curiously (keeping in mind that 100% Earth [also means: barycenter] is always involved in these conjunctions) and investigate: where is Jupiter during the lunar<->Saturn conjunction window and where is Saturn during the lunar<->Jupiter conjunction window:

    - the lunar<->Jupiter conjunctions repeat with Saturn period (@ 10750.5~ days)
    - the lunar<->Saturn conjunctions repeat with Jupiter period (@  4327.8~ days)

    both inside same window …

    This secular view is that each jovian appears physically fixed at heliocentric longitude (how can I say, in 100% of the solar system), fixed during the other jovian’s lunar+tellurian conjunction. Thank you, thank you very much Mr. Newton, you gave us a grand scientific point of view.

    Disclaimer: the conjunctions above are all (in contradistinction to cherry-picked) conjunctions during the 600ce to 2100ce time-frame, IINM, with 1 × conjunction at the begin of the respective interval and for same planet 1 × conjunction at the end of the respective interval, both within one-and-the-same measured window of opportunity; this in constrast to cycles on pure mathematical fictions by which empirically unobservable matching temporal distances are guestimated.