Sacred Number

After the last entry on Nineteen and the Lunar nodes, we should discuss the "lower register" of the Sun - Moon - Earth system, based on the solar year and the numbers 12 and 13.

Instead of 0.618 being attached to N [N=18 giving 18.618], 7/19 is attached (that is 0.36842) to N=12 to give 12.36842 lunar months per solar year, so that this fraction will resolve over the Metonic period of 19 years (see last posting). This is also quite closely proximate to 1/e (0.367879...), the reciprocal this time of the exponential constant 2.718281828...  rather than the golden mean.

19/7 is 2.7142857..., which whilst making it amenable to the Metonic also closely approximates the Megalithic yard of less than 2.72 - a unit of length recognised by Alexander Thom through statistical analysis of survey plans. One can see a relationship to the royal foot of 8/7 feet, its cubit of 12/7 feet and thence to a strange step of 19/7 feet rather than 20/7 [ a step usually being 2.5 feet). It must be the Astronomical Megalithic Yard that interests us here, a relevant formula for it being 19.008/7 feet, equally 19008/7000 which allows 19008 to provide prime number 11 in the numerator, and thence 11/7 making this a very fine Polar measure - it divides into the polar radius of the Earth, according to the ancient model of the Earth.

It was my brother's re-discovery of the Lunation Triangle with base 12, hypotenuse 13 and height 5 (the second Pythagorean triangle after 3-4-5), that revealed the ancient mechanism for harmonising the Sun and the Moon through an intermediate hypotenuse found at 3 high on the third side. Its length is the square root of 12² + 3² = 144 + 9 = 153 (fishes in a square root net, cast on the "right" side) 12.36931688... just slightly more than the 7/19 required for the Metonic.

This is the number of lunations in a solar year. It is close to but significantly not one third. The close thing gives you the rule of thumb, 37 lunations in three years, and interestingly 37 is a hexagonal number, 19 + 18. The number should be 365.242/29.53059 which equals 12.36825949, the underline being the signature of 7/19. The proximity to 1/e to 0.368 leads to the use of the megalithic yard (= e feet) to build the Lunation Triangle, whereupon the 0.368 approximation in the hypotenuse of 12.369, cancels with the 2.715428 feet approximation to e in the megalithic yard to yield exactly one foot at the end of 12 megalithic yards. The foot actually produced by this procedure is a Greek foot, 441/440 greater than the English but within that module.

This early link between metrology and astronomy through number appears very unlikely but was clearly intended and very useful.  It enables prediction of lunar phases and hence also of potential eclipses near the lunar nodes, also trackable using triangles (see previous) or using the Aubrey Hole simulator pioneered by Fred Hoyle and by inference, available as an early "clock" for astronomical events (see Robin Heath's descriptions of this).

I can now quote from an unpublished 1996 manuscript of a book written by Robin and myself.

Twelve and Thirteen

As we have seen, there are irrationals and their equivalent integers at work in the Moon's behaviour and it appears that the year is poised between the numbers twelve and thirteen, seen graphically in the lunation triangle.

Let us take the situation where the new moon began the year and twelve lunations [or lunar months] have passed. The Moon has returned to the Sun's position but 0.368 lunations remain in the year. As a part of the year, this period is 10.867 days long and therefore the Sun is 10.867 DAYS (of angle) from the year end on the Year Circle.

The Moon will travel 145.27 DAYS in that 0.368 lunations but 10.867 DAYS is travelled, by the Moon, in less than 20 hours and therefore [because the Moon reached year end in less than a day], in practical terms there are 13 orbits in [the time ]required for] 12 lunations. [In fact 12 lunations equals 12.9703 lunar orbits on average]

If we multiply 0.666 [2/3] by 12 over 13 we get 0.615 and so the ratio 12:13 is equivalent to 0.615/0.666 and, we think, a further veil has dropped away from the lunar number system. We can, in retrospect, see how the ratios equate like this: 0.666 is (2 over 3) which is (8 over 12/0.615) which is then 8/13 [because 8/13 = 0.615, a Fibonacci]

[If 12/13 = 0.615/0.666 = 8/13 times 3/2]

 The denominators are the creative forces and 12 evidently belongs to 0.666 [= 8/12] and 13 to 0.615 [=8/13], but in their relative size [one real fraction to the other], the other's denominator becomes the number of units relative to the other: [a fact perfectly expressed in the Lunation Triangle and ] a magnificent reciprocity based upon the field of pure number and expressed in the Moon's behaviour.

*** 

The square brackets are my attempt to make it more clear than mud. This remains an interesting insight linking again to a role for the Golden Mean, through Fibonacci approximation, in achieving near rationality within these two lunar time periods of orbit and synodic month. I went on (in Matrix of Creation) to find the reciprocal Golden Mean fraction of the lunation period, exactly 18.25 days, synchronous with Venus and the Practical Year of 365 days to a very high accuracy. The miracle is that all of these exact and approximate relationships coexist to create a rational environment at all, but this does require one to know the extent of the numerical network involved as well as its more sensational highlights.

To be able to form a hypothesis is a rare skill these days and this is exactly what is required to make sense of such an extended network of relationships.

"Knowledge gives nothing to a man until he gives everything to it"
(a Sufi aphorism through Idries Shah)

The Earth - Moon - Sun system is a numerical contruction based especially on the numbers 18 and 19. The first, eighteen, is a canonical or harmonic number because it only contains powers of 2 and 3, the first two primes. The second, 19, is a prime number divisible by no lower number. As a prime 19 has the cache of being associated with Creation and indeed the Moon manifests 19 very strongly in a number of ways.

Known to the (classical) Greeks as the Metonic period, in 19 solar years the Moon's phase and its location in the stars repeats to quite high accuracy. Since the Sun illuminates the Moon to generate its phase, then the Sun also must be in the same relative position to the Moon, nineteen years later.

Also known to the Greeks (but demonstrably known also in prehistory)  there is a period of just over 18 years in which an eclipse of a given sort will recur. Thus, after 18 years of an eclipse (solar or lunar) one can make an important prediction of another, similar one. This "just over 18 years" is in fact the time it takes for exactly 19 eclipse years - bringing 19 in again - a type of year (346.62 days long) mentioned in the previous article and associated with the movements of the lunar nodes - the crossing points at which all eclipses take place.

In the 1990's, probably 1993, an amazing "fact" arose for me. If one builds an 18:19 right angled triangle, the third side is almost 6 in length. Conversely, if 19 and 6 are used then the base becomes 18.027 solar years long and the Saros period is 18.030 years long. It seemed to me that this near-Pythagorean triangle of 18:19:6 had been incorporated in the relative lengths of the solar and eclipse year and hence in 19 times these.

The Near Pythagorean Triangle that occurs with 18:19 when the third side is whole,
to accurately approximate the Saros Period of 18.030 years or 19 eclipse years.

The simplest thing to realise about the Saros is that 223 (a prime) lunations occur within it and that in 19 years, 235 lunations complete the Metonic. The difference is 5 + 7 = 12 lunations, the length of the lunar year. This is true because such cycles exhaust their variety in a symmetrical fashion, so that there is, in a sense, a lunar year (of 12 lunar months) at both the beginning and the end of the Metonic. An eclipse one lunar year (12 lunations) into the 19 years will result in an eclipse at the end - and the whole thing is continuous whilst, for the sake of understanding, we must measure starting points. In reality, continuously, the Moon's position is a repetition of where it was 19 years ago as well as where it will be in 19 years.

The actual triangle has the Saros period as 223 lunations and the Metonic as 235 lunations long giving an 18.030 year Saros period. The "unit" becomes 12 lunations and the third side 6 times 12 = 72. 235 divided by 19 is, of course, 12 and 7/19ths of a lunation or 12.36842105 rather than the actual of 12.36826623 lunations per solar year, yet accurate to 99.998748% which is why the Metonic is an accurate 19 year synchronicity.

The Paradigm of Number Field as Cosmic Generative Principle

Something needs to change about how we view co-incidences between such a triangle and the reality found in time. The geometry represents a property found within the number field itself and this is the credible explanation for such a geometry "being used", that the number field can do very special things when these special relationships between numbers can be embodied within dynamical systems. Rather like the strange attractors of chaos theory, such arrangements allow far simpler time relationships to occur that seem mystical to us but are in fact artifices of number.

I am not saying that this is a physical explanation (yet) for how the Moon could have come to have such relationships based around 19. Rather, that these relationships do work in the same sense that a machine allows things to be achieved that are otherwise impossible. Were such work associated with the number field, just as harmony is, then such structures in time echo to the fundamental structure of the universe prior to any Big Bang. In this sense, they might function like natural laws based on number rather than on physics, as in gravitation and the like.

The Golden Mean is already a perfect example of such an edifice since, it is the only real number proportion to one where the reciprocal and the square have the same fractional part (0.6180 = the reciprocal and 2.6180... the square). This artifice is found broadly in life but astronomically in the orbit of Venus through the Fibonacci series (see Matrix of Creation and Sacred Number on this).  Venus manifests phi just as the Moon manifests the properties of the number field relating to 19. Jupiter and Saturn manifest 18:19 in their synodic periods to give us the seven day week.

Mystifying numerical relationships, that appear regressive to modern science, probably belong to the number field, a field that few study in depth or with practical application. If these relationships are found, as I and others have shown, in the practical dynamics of the Earth's environment; then the association with prehistorical knowledge and then later mysticism has given a false impression of superstitious thinking when the facts speak for themselves - of a valid numerical not physical knowledge.

How this works can be shown using the triangle of the previous entry involving 18.618 and 19.618:

 The third side is quite interesting, apparently physically meaningless*, being just over 10 times the reciprocal of phi. I would like to work out an intermediate hypotenuse that is 19 long - the Metonic in years whilst the Draconic is 18.618 years (year of the lunar nodes) which period is 19.618 eclipse years long.

*The actual length of the third side is 6.1835265 which was rounded up in the diagram to 6.184. However, the number of lunations in half a year is 6.18413, two numbers within 99.99% of each other. There are 6800.076 days in a Draconic period of 18.618 solar years and 19.618 eclipse years in the same period making the form of the triangle correspond to the same period and the division of it by different units, the solar and eclipse years.

19² - (18.618)² = 361 - 346.62 = 14.38 and the sqrt is 3.792. If we divide the third side above (6.1835) by this the result is 1.631. This is nearly 1.618 as if, with N:N+1 and N=N.618, the whole number hypotenuse is found near the golden mean point on the third side, as below:

 In nature, such a working fact could generate an integer relationship where it would otherwise be highly unlikely.

The Moon has been a primary influence in the Earth's evolution as a planet since it was created by an unusual collision with the Earth. These influences include:

1. Kneading the crust when the Moon formed orbiting only tens of thousands of miles above its surface.
2. Tilting the Earth to create our familiar phenomena of weather, seasons and large temperate zones.
   
3. Stabilising the tilt which otherwise would have oscillated wildly.
4. Continuing to regulate seismic and volcanic activity.
5. Protecting Earth from many bombardments, as does Jupiter.
6. Producing tidal influences in more recent times that evolve shorelines and large tidal environments.

During its existence the Moon's orbit has receded and the orbital period has therefore lengthened, as the energy from tidal interactions has slowed the rotation of the Earth, transferring energy to the Moon. The current orbit is on average 27.32166 days and the month 29.53059 days, the time it takes the Moon to again catch up with the Sun.

It's orbit is about 5.1 degrees relative to the plane of the solar system, effectively to the Sun's path through the year. The Sun's path, because of the tilt of the earth, is tilted and about 23 degrees to the celestial equator, which is the Equator of the Earth projected up into the sphere of the fixed stars.

This all means that the Moon does not follow the same path as the Sun in the sky. It travels above and below this path and only crosses at two points, an ascending node and a descending node where the Moon moves to the north of the Sun and to the south of the Sun respectively.

The positions of these nodes is moving, in the opposite sense to the movement of the Sun. Also, where these nodes are, the Moon can come in front of the Sun to cause some sort of solar eclipse, or the Moon can be opposite to the Sun in which case a lunar eclipse of the Moon by the Earth's shadow can occur.

Currently, the Moon is in such an orbit that it can just completely cover the Sun in part of its orbit (a total solar eclipse) though in other parts it is further away and smaller (causing an annular eclipse).

Many people think that the phenomenon of totality in solar eclipses represents an unlikely fact, indicative of some possibility that the Moon seems designed to do this. Certainly the likelihood of a technical civilisation co-inciding with this condition is theoretically an unlikely natural event.

However, the deeper one looks into the time periods associated with the Moon, the larger a network of co-incidences emerge and here I am going to take one area and show how ridiculously unlikely it seems. The subject here is the movement of the Moon's nodes relative to the year and the nodal or Draconic year of 18.618 years. (Other information can be found in Matrix of Creation or Sun, Moon and Earth by Robin Heath)

Because the Sun moves along the ecliptic in the opposite direction to the lunar nodes, the Sun will cross a node where the Moon might be in less than half a year. The Sun will recross a node after less than a year, in fact in 346.62 days on average, a time period called the Eclipse Year.

If 346.62 is subtracted from 365.242, the solar year, then the difference is 18.622 days. The square of 18.622 is 346.779 days, very similar to the length of the eclipse year. The square root of 364.62 is in fact 18.618 days and if this is divided into the solar year, then 365.242/18.618 = 19.618. That is, the eclipse year and the solar year appear to have a natural divisor, the period of which is 18.618 days long.

Using the Megalithic technology of right angled triangles, the two years can be represented as an N:N+1 triangle as below. A right angled triangle can always be represented in this form, where the difference between the hypotenuse and the base is divided into each of these to normalise the triangle. Technically this is called a superparticular triangle and it represents the essence of the triangle and the relationship between the two longer sides.

 The logic of this triangle is quite clear once you can read it. The base line of 18.618 must be one less than the hypotenuse of 19.618 (units of 18.618 days) because the sun moves 18.618 DAYS on the ecliptic "year circle" whilst the lunar nodes move away from the Sun just one DAY. In moving away from the Sun, a node is moving towards an earlier reunion with the Sun, around the far side of the year circle.

In the micro world, the creator of the relationship is the relative motions of the Sun and the Nodes. If this was a different ratio than 18.618 to 1, then another N:N+1 triangle would emerge but, because all right angled triangles can be normalised in this way, another interesting numerical relation would be available. However this triangle is special because the unit underlying it of 18.618 days also figures in the N of the N:N+1 triangle, making the eclipse year the unit squared.

Were the unit larger, what would happen? If the nodes moved more quickly then the triangle will get steeper with N reducing in size. This means that the excess of the solar year over the eclipse year is becoming a larger proportion of the solar year, i.e more than 18.618 days. Thus, the units of the triangle and the N implicit within its N:N+1 form diverge at any value other than they are.

Only in the present relative motions of the lunar nodes, and solar motion on the ecliptic, can the triangle be normalised. This is extraordinary enough. However, it is also true that the length of a day further divides up the excess of 18.618 days into the number 18.618 that also belongs to the N:N+1 essence of the triangle. This cannot be possible unless the rotation of the Earth is, by design, perfectly in time with this relationship.

Finally, in the macro world, the 19.618 of the hypotenuse represents also the number of eclipse years in the Draconic period of 18.618 years, the latter being the meaning of the base of 18.618 years. In that time the nodes have completed one complete revolution of the ecliptic, represented here by the number 1, which is how this normalisation of triangles manifests outside the "surface" of the solar year - all beyond the year is the reciprocal of all found within the year. This is a natural consequence of recurrence since each divisor of a period also manifests in longer periodicities that are numerically self-similar but reciprocal.

Every day, we can say, that the Moon's nodes move 1/18.618 of a DAY [on average] on the ecliptic

The above relationship is very hard to explain and since 1994, when it began to become clearer to Robin [Heath] and myself, it has remained a curiosity awaiting and explanation that would reach an audience. Within lies just one of the great co-incidences outside of science that the Moon provides. [In Sacred Number I use a John Michell anomoly relating the Earth, Moon, Great Pyramid and Pi of 22/7.]