On “Justification”
Recently I have begun using the term “justification” to refer to an ever growing number of known instances where some of the otherwise questionable approximations I use turn out to be close approximations of actual numbers generated by from numbers known to have been used in ancient calendar systems.
The term is new but the phenomenon itself was first recognized several years ago, and quickly achieves greater prominence the deeper one gets into calendar studies.
One might be able to find some of the more classic examples of this by using “apologetic” as a search term, because I often feel apologetic over the B Value I use for the Half Venus Cycle, 18997.72194 days, which adds almost 18 days to the canonical, historical value of 18980 days. It leaves something to be desired for accuracy, certainly, yet not only does it seem to be what the mathematics dictate, but it falls close to things that may actually be a part of real-life ancient calculations.
For example, if we divide the Saros Cycle in contemporary form by the canonical HVC, we get
6585.3211 days / 18980 days = 346.9610095 / 10^n – not bad for an approximation of the 346.62 day Eclipse Year, but not that great either.
If we go to “correct” the formula, 6585.3211 / 346.62 = 18998.67607 / 10^n.
So, something very much like 18997.72194 and less like 18980 is an actual derivative of astronomical numbers, thus providing justification for an approximation like 18997.72194 in spite of the departure from 18980 that it represents.
We may also wish to note that 125 / Saros Cycle 6585.3211 = 18981.61048, which may provide some justification for the A value I use for the Half Venus Cycle.
These things may happen of course, because the natural cycles, in spite of many happy coincidences, do not necessarily fit together precisely as if by magic.
I also use approximation of the Earth year that tend to be somewhat higher than typical canonical figures, aside from one very awkward figure made from the Megalithic Foot and excessive exponential use of the number 12. The textbook values are in fact poorly represented by the numbers I use.
In classical terms, 18980 / 346.62 = 2 / 365.2476291 x 10^n, not bad for defining the Earth year – but go to make adjustments, and suddenly we may find ourselves looking at slightly higher figures like the ones I work with.
Half Venus Cycle 18980 / Anomalistic Month 27.55 = (27.55716878 / 4) x 10^n. It’s just not going to work out perfectly in all nice round numbers.
Long Count vs Calendar Round (Half Venus Cycle)
Unfortunately, it appears as if I may be habitually misusing the term “Long Count”. I’m not sure how that happened. It has all the earmarks of a rookie mistake on my part, but as a novice at dealing with Mayan calendar systems, I did try to humble myself and learn as much as I could from numerous sources and would not be using the term that way if it had not also been used that way by a number of works.
It may be that there is some frequent reticence to discuss the Long Count in greater or more specific detail that could contribute to such errors, but even then it’s hard to imagine such a glaring error going uncorrected for so long, even if there’s some possible contradiction at large. For example, compare:
The Mayans designed the Long Count calendar to last approximately 5,125.36 years, a time period they referred to as the Great Cycle [source: Jenkins]. [5125.36 x 365.25 = 1872037.74 days]
The Long Count is an astronomical calendar which is used to track longer periods of time. The Maya called it the “universal cycle.” Each such cycle is calculated to be 2,880,000 days long (about 7885 solar years) [7885 x 365.25 = 2879996.25 days]
An interesting example of Long Count number can be found in the introduction of the Venus table on page 24 of the Dresden Codex: the so-called Long Round number LR = 9.9.16.0.0 = 1366560 days, a whole multiple of the Tzolk’in, the Haab’, the Calendar Round, the Tun, Venus and Mars synodic periods [1366560 / 365.24 = 3659.301848 years]
At least if the first source quoted can be taken seriously, there probably isn’t a lot of actual harm done. This article might further mangle the distinctions, but may futher contribute to a consensus concerning the legitimacy of the first quoted figure
5,126 solar years x 365.25 = 1872271.5
The likely link between figures like this and the Calednar Round (Half Venus Cycle) would be (Pi^2), and the figure itself, 1872271.5, would seem to reflect Venus Orbital Period x 10^n / 12 (1872271.5 x 12 = 224.67258 x 10^n), something that is already reflected in using 1 / (12 x (Pi^2)) as the link between Venus Orbital Period and Calendar Round.
However, the same article may also contribute to a consensus regarding 2,880,000 days, which should also be taken into consideration. I find it interesting that if we go to see if there is a whole number of Eclipse Years in such a number we find 2,880,000 / 83 = 346.9879518 x 10, more similar to 6585.3211 days / 18980 days = 346.9610095 than to 346.62.
“Hidden” Components?
My latest attempts to work with canonical Mayan calendars have given me a renewed appreciation of the importance of the work of independent researchers like David Kenworthy or Bogna Krys, even if I cannot bring myself to agree wholesale with many of their contentions.
To David’s credit, his value of 346.6666666 does integrate neatly into canonical Mayan calendar systems, which attests to considerable legitimacy.
At the same time, I don’t know exactly what it is these researchers often have their hands on. Certainly there is the suggestion that they have made great contributions to our understanding of common ancient calendar systems, but whether they have successfully isolated and organized the elements of these system is another question.
I have already shown at the beginning of the post that these systems do not always necessarily break down into nice even numbers. It appears that even something as simple and straightforward as 346 and 2/3 may have escaped the radar of many authors on Mayan calendar system.
How the ancient dealt with equations that didn’t quite work out perfectly still seems like an open question, and penchant on the part of researchers for nice round numbers reminiscent of that seen with “Inductive Metrology” and metrology may have often distracted from their presence in canonical calendar systems.
As a result of such “imperfections” in ancient formulas, my most recent Mayan calendar studies leave me faced with a number of new approximations that beg for careful examination of each, and still leaves questions about the correct ways, and the best ways, for certain cycles to be represented.
A “nice round numbered” origin for the Saros Cycle can be made of 125 / 19 = 6578.947368 / 10^n, which may justify the approximation of the Saros as 13159.47254 / 2, but the net result of the variation between formulas may be variable values. We can also generate a Saros value as Eclipse Year 346.666666 x 19 = 6586.666666, which may in turn justify other approximations I’ve been working with.
18 / 273 = 6593.406593 as a possible Saros approximation is another matter in itself.
What does this say about the how the Maya felt about these these things, and how they responded to them, if not as I’ve been saying all along – that they realized the variations in figures resulting from different formulas, and attempted to embrace a more exact but more complicated system behind the scenes that would have acted like a safety net to catch these different possibilities and make them into mathematical truths? The question may be less about whether they did this, and more to what extent they did this.
In the meantime, we might note that 6585.3211 / 19 = 346.5958474, which may lend justification to the main approximation I use for the Eclipse Year, ((1.067438159^2) x 1.216733603) x 10^n / 4 = 346.5939351
Notes from Calendar Studies
I am currently attempting to table out – still informally at this point – the factors of multiples of common calendar values. If we assume that the ancients also had to go through such a process to determine what the best calendar systems and subsystems were, these sprawling tables may dramatically emphasize just what incredible ancient mathematical achievements these integrated calendars actually were.
I am still not at all certain what to make of all of the data, but there may be some surprises.
The Saturn Synodic Period of ~378 thus far seems to be trying to claim certain distinctions that may hint at greater importance than is usually suspected, given that the factors of 18980 x 378 = 7174440 include 360, 364, and 365, which has not been seen yet with any other calendar number product, although a few such products do have both 364 and 365 as factors.
The tables seem to be giving a lot of votes not only to 117 as Mercury Synodic Period, already a surprising departure, but they are also giving a good numbers of votes that 118 – believe it or not – may also have been used in ancient calendar systems as the Mercury Synodic Period, while at the same time very strangely I seem to be seeing few votes for anything representing the Mercury Orbital Period.
Currently one of the possible Saros values I am attempting to experiment with is 6588. It does seem to generate some significant factors, especially when multiplied by 224 as Venus Orbital Period, or by 18980.
There are a few numbers I’ve found it interesting to experiment with – 312, 315, 320, 325, 336, 338, 676 and 767. I don’t know if I should call them “helpers” or “facilitators” or I should call them anything at this point. They may all turn to be helpful in generating products with significant factors because they are related to actual calendar numbers by simple ratios. 336 for example is 1.5 times Venus Orbital Period written as 224 (which like 364 may eventually be more or less a fact of life with integrated calendars).
It’s interesting that David Kenworthy mentioned the possibility of a second Aubrey Circle at Stonehenge with 52 holes, since the well known Aubrey Circle has 56, and my tables may be telling me that a 364 day year may be most compatable with 52 and 56 in combination.
At this point, it’s still rather mysterious to me whether for example the optimal representation of the Mercury Orbital Period might have been 87 days, and there 4332 or 4333.333333, or perhaps something else, might have been the most ideal representation of Jupiter’s Orbital Period. I likewise have little certainty about the best representation in whole numbers of of the Saturn Orbital Period of ~10759 days.
87 is a factor of the products of 6588 x 116, 6939 x 10759, 6939 x 377, 6939 x 116, 10759 x 312, 10759 x 315, 10759 x 336, 10759 x 585, 10759 x 378, 10759 x 366, 10759 x 354, 10759 x 117, 10759 x 225, 6585 x 377, 4332 x 377, 4332 x 27.55, 354 x 27.55, 819 x 377, 819 x 27.55 and etc, so we can see some good possibilities even while additional possible candidates for the Mercury Orbital Period such as 8784 / 100 begin to assert themselves also (we may wish to note some of the successes of a decimal number, 27.55 as Anomalistic Month, seen here as well).
What else might be be learned by attempting go through the same pains as the originators of these ancient calendars most likely also had to do (presumably without a handy online factor calculator to assist them)? Potentially a great deal, but only time will really tell.
I may have a lot of do-overs ahead of me for taking the concept of “factors” too literally and not going full throttle by omitting the decimal place.
To give an example, 819 is the specified Mayan calendar number, yet the factors of 819 are 1, 3, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819.
We have to add more zeroes before it shows us more, the factors of 81900 being 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 18, 20, 21, 25, 26, 28, 30, 35, 36, 39, 42, 45, 50, 52, 60, 63, 65, 70, 75, 78, 84, 90, 91, 100, 105, 117, 126, 130, 140, 150, 156, 175, 180, 182, 195, 210, 225, 234, 252, 260, 273, 300, 315, 325, 350, 364, 390, 420, 450, 455, 468, 525, 546, 585, 630, 650, 700, 780, 819, 900, 910, 975, 1050, 1092, 1170, 1260, 1300, 1365, 1575, 1638, 1820, 1950, 2100, 2275, 2340, 2730, 2925, 3150, 3276, 3900, 4095, 4550, 5460, 5850, 6300, 6825, 8190, 9100, 11700, 13650, 16380, 20475, 27300, 40950, 81900.
One more reason we here at the Pi Jedi academy have the motto, “We don’t care where the decimal goes unless there’s a root function involved“.
To think of all the discussions and even arguments I’ve been in about whether ancient people knew what a decimal point was when all they had to was multiply everything by a million to get it out of the way and get on with business as usual!
I’ve underscored 315, mentioned earlier in this post, and 156 in the list above. 156 is something that I seem to be seeing a lot of in the tables, and not that surprisingly it is 312 / 2 and 156 / 2 = 780 / 10, so until I get better at spotting simple multiple and fractions, that is another challenge, and yet in spite of that obstacle and the obstacles imposed by correct decimal placement with the factor calculator, things nonetheless seem to be going well – better than I yet realize, in fact.
–Luke Piwalker
Luke Piwalker 