I recently mentioned the figure of 411.8202744 in regard to Giza subsidiary pyramid G1a, the northernmost of the three (actually four including G1d) small pyramids on the east side of the Great Pyramid.
“The duration of one full moon cycle is: 411.78443029 days (411 days 18 hours 49 minutes 35 seconds)” https://en.wikipedia.org/wiki/Year
1.067438159 / 25920 = 411.8202744
This figure is still under investigation. Its mathematical properties are somewhat curious – it shows considerable “resonance” and functionality, and yet it may leave some things to be desired.
One reason for this may be the slight mismatching in trying to relate 411.78443029 to other figures. For example, the “textbook” value for the Saro Cycle is 6585.3211 days, and
6585.3211 / 16 = 411.5825688
411.78443029 x 16 = 6588.550885
Not quite a perfect fit, so what are we intended to do with that exactly?
This may be one of the reasons why multiple ancient calendar values seem to exist, which may be one way of resolving it.
Formulas relating the Full Moon Cycle to other calendar values may otherwise have yet to be developed yet. Without them there doesn’t seem to be much certainty whether 411.8202744 would be the A, B, or C value for the Full Moon Cycle, or perhaps even another.
Things are still less than perfectly clear because adapting some possible formulas doesn’t necessarily provide resolution.
6584.495235 is proposed as a likely representation of 6585.3211.
6584.495235 / 16 = 411.5309522 = 411.8202744 / 1.000723277. That much is sensible enough, but 411.8202744 x 16 = 6589.257649, which is currently suggested as the F value for the Saros Cycle based on 6584.495235 as the A value, which still seems a bit awkward.
410.9362960 has also been proposed as a possible representation of the textbook figure of 411.78443029. By percentage, it’s probably not a bad approximation, and we can tell it has considerable resonance just because 500 / 1.216733603 = 410.9362960, but whether 410.9362960 really fits into the calendar tables and if so where, remains to be to be seen.
Some of what we do know about 411.8202744: It’s a good responder to (Pi / 3) – good enough that one wonders why this wasn’t already found at Tikal – and quite confusingly a poor responder to 2 / 1.622311470 and barely responsive to 2 Pi – while being almost profoundly responsive to square root of 60 (the most powerful mathematical probe yet known for these numbers).
Here we see sqrt 60 working at least to the 20th power in combination with 411.8202744.
24901.19742 / ((sqrt 60)^20) = 10 x 411.8202744
This may endow 411.8202744 with considerable resonance since it can be found from any point in that long and important series simply by applying sqrt 60 as a probe like that.
One of the numbers found that way is one of the very first numbers found at Tikal, from the doorway of Temple II.
(1 / (7.396853331 x 1.177245771)) / ((sqrt 60)^5) = 411.8202744 / 10^n. Other important numbers in this remarkable series include 1.067438159 / 2, 4.134170221, 3.202314484, 19.21388690, and 1152.833214 (the lowest few numbers in the series are unusual but may have just been endorsed as important by their participation in this series).
Thus 411.8202744 can be readily found from many points in Egypt, including the King’s Chamber (Height 19.21388690, diagonal 19.21388690 x 2?) and coffer of the Great Pyramid (Width 3.202314484?).
It can also be found at Stonehenge simply by applying (Pi / 3) to the “Indicidental” Megalithic Yard of 2.719256444 ft which is one of the values for the Meg Yard that’s applicable there.
2.719256444 x ((Pi / 3)^9) = 411.8202787 / 100.
Along the way, the series passes 2 / 584.032129 and the proposed outer lintel circle diameter of Stonehenge, 327.0127142, which gives another way of readily finding it there. More ways will likely follow.
Even so, I don’t think there’s any getting around that in spite of some very impressive pedigree (and again 10674381.59 / 25920 = 411.8202774), 411.8202744 is still in ways a rather strange number that may take some time to fully understand and categorize.
–Luke Piwalker
Luke Piwalker 
Hi.
Perhaps that number, 411.8202744, or 411.78443029, doesn’t fit because it isn’t completely correct, in terms of ancient calendars. Re the apparent discrepancy between the full moon cycle and the Saros cycle, I have been told that you have to be careful about lunar month lengths and solar year lengths that you find from astronomical sites like NASA, which always quote these lengths in terms of international atomic time, whereas for calendrical purposes one always needs these lengths in terms of mean solar days, information that isn’t available from such astronomical sources, because of uncertainty in the slowing of the Earth rotation rate (accounted for by “Delta T”) as one reckons further into the remote past or distant future. The mean solar day depends on earth’s rotations, and fluctuates. According to Wikipedia, “Earth’s mean solar day is now slightly longer than it was during the 19th century due to tidal friction”. This makes it difficult to quantify the moon’s full cycle over thousands of years. (What’s more, should we be even be thinking of measuring the moon in terms of the stellar day and the sidereal day, which are shorter than the mean solar day by about 3 minutes 56 seconds?)
Confusingly, it’s hard to know why cycles are given slightly different lengths sometimes.
According to https://calendars.wikia.org/wiki/Full_moon_cycle, the average duration of the anomalistic month is:
AM = 27.55454988 days (see Meeus (1991) eq. 48.1)
The synodic month has an average duration of:
SM = 29.530588853 days (see Meeus (1991) eq. 47.1)
The full moon cycle is the beat period of these two, and has a duration of:
FC=(SM×AM)/(SM−AM)=411.78443d
However, the Nasa website has: Synodic Month (New Moon to New Moon) = 29.530589 days
Anomalistic Month (perigee to perigee) = 27.554550 days
These figures give a full cycle of 411.7842205269 days, according to the equation above.
Both of these estimates are, I think, referring to atomic time. It is believed the Babylonians expressed all periods in synodic months, probably because they used a lunisolar calendar. They had:
223 (synodic) months = 239 returns in anomaly (anomalistic month) = 242 returns in latitude (draconic month). This is now known as the saros period which is very useful for predicting eclipses. (again, Wikipedia)
Perhaps that’s all we can be certain of.
Thanks
Melissa
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Hi, Melissa
Many thanks for the fascinating feedback. I think you raise a number of very good points.
It is true that I’m working with contemporary values that can not only vary slightly according to different sources (or even according to which NASA page one is looking at), but they can also vary over time, so there may be some risk of introducing values that might not have been known to the ancients.
To make things more difficult, I have some mistrust of software applications that are allegedly capable of projecting backwards to see what values the ancients would have seen for these cycles, not only because of the earlier controversies over Giza alignments where different people using different software models would give markedly different results, but also because Michael Morton’s concept of a “Archaeo-Sky Matrix” eventually seemed to fall prey to significant variations in data between different release numbers of the same software, giving unrepeatable results.
I did try to back-project several cycles using the available data and it didn’t seem as if there were significant changes at least in the short term.
Thankfully, I think there is some leeway built into the accuracy – or even a lot of leeway depending on how one looks at it – and I think that is due in part to the particular calendar math that is being sourced, starting with the loss of about 1/4 day when the Solar Year of about 365.25 is represented as 365 or the loss of about 1/3 of a day when the Lunar Year of about 354.36 is represented as 354, and so forth.
I think with a number of proposed ancient systems including mine, the absolute accuracy comes into the picture once we have determined how things are best approximated – we have some freedom to choose our own rules, but once we do we’d better stick to them, to so speak. Even within that framework there may be some leeway because of the ancients running different approximations for the same value – 365.0200808 and 365.47562611, for example.
The utility and even the possible origins and antiquity of 365.0200808 (300 Remens of 1.216733603) are hopefully self-evident, but 365.47562611 also has considerable utility in representing the calendar year as well.
Teobert Maler’s data for the El Castillo suggests that the architects used a symmetrical set of relationship between 6 x 10^n and Pi to generate 365.47562611 as a square root, i.e., with the pyramid having sides of
Maler 5821 cm = 190.9776903 ft = ~190.9859317 (600 / Pi) ft = sqrt 36475.62611
Maler 5747 cm = 188.5498688 ft = ~188.4955592 (60 Pi) ft = sqrt 35530.57584
It looks like they may have used this scheme with the lovely mathematical symmetry of combining 600 / Pi with 60 x Pi because it’s able to generate valid square roots for a Solar calendar year of 364.7562611 and a Lunar leap year of 355.3057584 (and thanks again to Jim Wakefield for point out that because the Lunar Year is actually somewhat larger than ~354, a Lunar leap yet would be required to keep calendars on track, just as a Solar leap year is required with a year of about ~365 days rather than ~365.25 days.
These variations may also happen because of the metrology, i.e., 365.0200808 = 300 Remens is seemingly the best representation of the calendar year, and we seem to see this in the diagonals of many less major Egyptian pyramids whose bases appear to be measured out in simple numbers of Royal Cubits, but in this situation the true Remen at a 45* angle to the Royal Cubit is slightly shorter.
1.718873385 x sqrt 2 = 1.215427027 x 2, which is a Remen closer to (12 / (Pi^2)) = 1.215854204 ft = 364.7562611 / 300, than it is to 1.216733603 = 365.200808 / 300.
Here this seems less negotiable than with relatively “abstract” mathematical formulas for calendar functions, because it’s referring to the geometric reality of actual existing architecture.
As a result, it seems likely the ancient Egyptians also observed 364.7562611 as well as 365.2008080 and other possible approximations of the calendar year.
This may also get into geodesy along with getting into metrology, since the ratio between 1.215854204 and 1.216733603 is the same as the ratio between 364.756261 and 365.2008080, the ubiquitous fine ratio 1.000723277, which is also the ratio between the consensus mean circumference of the world in miles, 24883.2, and the ideal equatorial circumference formed as the cube of 24 Remens of 1.216733603:
24901.19742 / 24883.2 = 1.000723277
As always, with planetary or lunar cycles there can be considerable initial difficulty can be choosing the right pair that is separated by the required ratio. 411.8202787 certainly seems to belong, although its strange enough that it probably requires the assistance of additional values to achieve all of its tasks, but it remains to be seen whether it will end up a primary value for the Full Moon Cycle in the scheme of things.
It has some great pedigree, which often helps to identify primary approximations, but whether or not it’s separated from other useful variations by 1.000723277 as the main “A” and “B” calendar values are, remains to be seen. Technically, 411.8202787 x 1.000723277 = 412.1271453 and 411.8202787 / 1.000723277 = 411.5316273.
Both of those departures actually seem a little surprising, and possibly excessive. It seems like there’s a valid 411.7-something figure that should quite possibly be in there somewhere, and the reciprocal of the double Remen at 410.9362960 is another very worthy and well-pedigreed candidate that seems to have been excluded there too somehow.
For the aforementioned reasons I’m not terribly worried about the accuracy of 411.8202787 as a valid approximation of the Full Moon Cycle, but I do still have numerous concerns about the accuracy of its labeling. It has the utility or prominence to suggest it belongs to the “A” or “B” group of calendar numbers, but not necessarily the relationships to other candidate approximations to confirm this or to illuminate the structure of the rest of the Full Moon Cycle values that would have been recognized by what the formulas mandate from there.
Anyway, thanks again for the thought-provoking comments. I hope that you and other readers are enjoying my blog and will stop in often.
Cheers!
–Luke Piwalker
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