Why Phi? – the Inex eclipse cycle, Fibonacci and Lucas

Posted: January 27, 2015 by oldbrew in Fibonacci, moon, Phi, solar system dynamics

"Lunar eclipse at sunrise Minneapolis October 2014" by Tomruen - Own work. [via Wikimedia Commons]

“Lunar eclipse at sunrise Minneapolis October 2014” by Tomruen – Own work. [via Wikimedia Commons]

It’s well-known that Fibonacci numbers and the Lucas series are closely related, as shown here.

There’s a lunar connection – the Inex cycle is the cousin of the shorter Saros and lasts nearly 29 years, such that:
18 Inex = 521 years
18 and 521 are both Lucas numbers

What does this tell us?

‘The significance of the inex cycle is not in the prediction, but in the organization of eclipses: any eclipse cycle, and indeed the interval between any two eclipses, can be expressed as a combination of saros and inex intervals. Also when a saros series has terminated, then often one inex after the last eclipse of that saros series, the first eclipse of a new saros series occurs. This in-coming and ex-iting of saros series separated by an interval of 29 years suggested the name for this cycle.’ – Wikipedia [bold added]

Every Lucas number is the sum of two Fibonacci numbers:
18 = 13 + 5
521 = 377 + 144

The number of sidereal months (lunar orbits) in 521 years is 6965.
6965 = 199 x 35
199 is a Lucas number (sum of two Fibonacci numbers: 144 + 55)
521y / 199 = 2.6180904 (~Phi²) = 35 sidereal months

The question then is: why do 18 Inex last 521 years, which is 199 x 35 sidereal months – where 18, 199 and 521 are Lucas numbers and 35 sidereal months is Phi² years?

The chances of this arrangement happening ‘at random’ must be remote, statistically speaking.
Therefore it appears the system is organising itself in this way.

The 521 year period has been called the ‘Hyper Saros’ and even been described as the ‘basic period’ of the solar system by a Belgian Professor.

NB 29 (see earlier quote) is also a Lucas number (sum of Fibonacci 21 + 8).
The Inex cycle is not exactly 29 years because 18 x 29 = 522, not 521.
Actual Inex = 521 / 18 (the two Lucas numbers) = 28.9444~ years.
Nearest Fibonacci equations: 377 / 13 = 29 exactly, 610 / 21 = 29.0476

Wikipedia also notes:
‘Although the inex series lasts much longer than the saros, it is not unbroken: at the beginning and end of a series, eclipses may fail to occur. However once settled down, inex series are very stable and run for many thousands of years.’

Another lunar cycle is the Metonic cycle (~19 years) which has a lesser-known cousin the Callipic cycle (~76 years = 4 x 19). The Greeks used it in their famous Antikythera mechanism. 76 is another Lucas number.

‘This was a more accurate approximation, obtained by taking one day away from every fourth of Meton’s cycles, so creating a 76-year cycle with a mean year of exactly 365.25 days.’

521 / 76 is almost equivalent to the Fibonacci equation 377 / 55 (about 99.98%).
Both are very close to the fourth power of phi i.e. 6.8541.

Numbers fans can pursue all this further here, or at various other sites discussing Lucas and Fibonacci numbers.

  1. Brett Keane says:

    I am able to see that there are potential planetary solar effects via gravitational mechanics and fluid dynamics. What might be the physical effects of Inex and similar? Brett

  2. oldbrew says:

    Hi Brett – I’m not aware of anything specific to do with the Inex cycle, but if any other readers are, please send a comment.

    Lunar eclipses in general can cause spring tides (i.e. higher than normal).

  3. oldbrew says:

    Jupiter, Venus and Mercury have their own Lucas-related synodic relationship.

    11 J-V = 18 V-Me = 29 J-Me (within a few days)
    11, 18 and 29 all belong to the Lucas series.

    That period of around 7.13 years is also close to six Chandler wobbles.


    Venus and Mercury rotations are also linked in a 7:29 ratio, both numbers being in the Lucas series.
    Mercury and Earth orbit periods are nearly 29:7 as well.

  4. tallbloke says:

    OB: The chances of this arrangement happening ‘at random’ must be remote, statistically speaking. Therefore it appears the system is organising itself in this way.

    I don’t know how to calculate it, but the odds of this being random must be astronomical. (sorry)

    So we have a Moon which has phi relationships with Earth in terms of density, diameter, spin rate, precession of lines of apse and nodes…

    Why Phi?

  5. oldbrew says:

    We also found: 34 Saros = 55 x 149 sidereal months = 34 x 18, +1 years (34 and 55 are Fibonacci numbers)


    So eclipse cycles and Phi are closely linked. A sidereal month is also the Moon’s rotation period.

    Also 20 Inex = 610 ‘draconic years’ (= ‘eclipse years’).
    610 is a Fibonacci number.

    Eclipse year: ‘the time for the Sun (as seen from the Earth) to complete one revolution with respect to the same lunar node (a point where the Moon’s orbit intersects the ecliptic). This period is associated with eclipses: these occur only when both the Sun and the Moon are near these nodes; so eclipses occur within about a month of every half eclipse year. Hence there are two eclipse seasons every eclipse year.’ [wikia.com]

  6. oldbrew says:

    Another one: 383 Saros = 371 lunar nodal cycles (383 is a prime number)
    383 + 371 = 754 = 377 x 2 (377 is a Fibonacci number, as is 2)

    383 x 2 = 766 Saros which is the basis of lunar chart 2 here:


  7. […] 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 […]