I have written before about the mathematical calendar formulas that may have been written into the Aztec Sun Stone. It has come into focus again because one of its key numbers may appear in the original measures of the Stonehenge ditch according to some of the data available, shortly after also appearing in the design of Hadrian’s Library complex. Now, it is also subject to focus because as we begin to look at the work of Susan Milbrath, we find references therein to a possible symbolic association between the Sun Stone and the Eclipse Year.
I might have stumbled over it before and forgotten but I can’t find reference to “Eclipse” or “346” in the first comprehensive post I tried to write on the subject. Whether or not I agree with Milbrath’s perspective on the question, there do indeed seem to be some interesting connections between the proportions proposed for the calendar stone and the Eclipse Year figure I call the “Best Eclipse Year Value” of 346.5939351 days.
Here is an interesting series that works all the way to the 5th power of the radius of the Sun Stone that at very least connects the Bat Palace Number (the internal ratio of Saturn’s cycles) to the Best Eclipse Year at the 4th power, if the inclusion of the Great Pyramid in the equation seems too out of place to anyone.
(Great Pyramid diagonal at proposed pavement level: 1067.077716 ft, adapted from raw data of (3018.110298 / 4) x (sqrt 2) = 1067.063129)
(1067.077716 x 2) / ((Sun Stone Diameter 3.747289674)^1) = 5.695197376 / 10^n = Bat Palace Number
(1067.077716 x 2) / ((Sun Stone Diameter 3.747289674)^2) = 151.9817754 = (30 / (Pi^2)) / 2; (30 / (Pi^2)) is part of prior calendar stone proceedings
(1067.077716 x 2) / ((Sun Stone Diameter 3.747289674)^3) = 40.55778673 = 162.2311470 / 4
(1067.077716 x 2) / ((Sun Stone Diameter 3.747289674)^4) = 10.82323232
(1067.077716 x 2) / ((Sun Stone Diameter 3.747289674)^5) = 2.888282803
The same digit three times starting at the second place like 2.888282803 is usually an indicator of seeing a simple fraction or multiple of the Best Eclipse Year.
Case in point, 2.888282803 x 120 = 346.5939364 Best Eclipse Year = ~346.62 days
I found this series quite interesting for a number of reasons in addition to those already mentioned. 10.82323232 is 1/4 of Jupiter Orbital Period / 4 (I noticed early on in Mayan studies that they seemed rather fond of 10.82323232 for some reason), but perhaps what is most interesting here is that in terms of diameter, the Aztec Sun Stone has a Mayan Twin at Yaxchilan according to the data.
I find it quite remarkable that the thickness given for the Yaxchilan altar by Morley (via Carolyn Tate) is .33 m = 1.082677165 ft, and I’m going to take a wild guess the intended figure was 10.82323232 / 10 ft.
It’s as if 3.747289674 really is a calendar number for all seasons and they knew that well.
(Meanwhile, in the next consecutively numbered structure at Yaxchilan after the one with the Sun Stone’s twin, there is described by Morley via Tate an altar with an already suggested diameter / thickness ratio of D/T R = 4.055778682).
There’s an odd little bonus that comes out of this, which is that 3.747289674 / 1.082323232 = 3.462264841. We’ve probably not much to do with that per se because we have numerous better things for approximations of the Eclipse Year, but somewhere out there someone might be suggesting that (sqrt 12) = 3.464101615 could be a suitable approximation of the Eclipse Year / 100.
My preference of Eclipse Year values is still “Best Eclipse Year” of 346.5939351 days, I really don’t pay much attention to the other ones unless they manage to impose themselves on something where 346.5939351 doesn’t fit but they haven’t had much luck with that so far. An even better approximation of the Eclipse Year can be had with valid numbers but it’s out of synch with the system a bit and lacks the resonance it needs to volunteer more often, so 346.5939351 frequently seems free to dominate.
I’d like to point out though that although the textbook figure for the Eclipse Year is 346.62 days, the semi-canonical translation of 1/19th of a Saros Cycle should be likely be 6585.3211 / 19 = 3.465958474. (6585 / 19 = 3.465789474). In other words, the figure I use is (once again) closer to the figure that may actually arise from traditional formulas.
At any rate, if you caught the (sqrt 12) trick, you probably thought of trying 12 / 3.462264841 = 346.5939364 / 100. In other words, 3.462264841 IS 346.5939364, it’s just written backwards.
I should point out that the diameter of the Sun Stone does enjoy good interactivity with the Best Eclipse Year value. For the Sun Stone, Radian / Pi is the double short Solar Year, and the speculative model of the Sun Stone allows for both long and short Solar Year, so that Radian / Pi probably can be considered a specifically indicated number.
(Radian / Pi) / ((Sun Stone Diameter 3.747289674)^1) = 4.866934409 = 1/10th inner sarsen circle radius Stonehenge
(Radian / Pi) / ((Sun Stone Diameter 3.747289674)^2) = 1.298787879 = 1/80th outer sarsen circle diameter Stonehenge
(Radian / Pi) / ((Sun Stone Diameter 3.747289674)^3) = (Best Eclipse Year 346.5939366) / 10^n
(Radian / Pi) x ((Sun Stone Diameter 3.747289674)^1) = 2 / Real Mayan Annoyance
Also for the Sun Stone, circumference / Best Eclipse Year = 6793.227761 / 2 / 10^n, where 6793.227761 is trying valiantly to earn its stripes as a legitimate version of the Nodal Cycle.
11.77245771 / 346.5939366 = 6793.227732 / 2 / 10^n
Combined with the A form of the Calendar Round, 11.77245771 / Pi = 3.747289674 may perform another vital function linking the Calendar Round to the fourth root of the Saros (if the Saros generated can be verified as legitimate)
Calendar Round A 18983.99126 days / 3.747289674 = 5.066059181 x 10^n; 5.066059181^4 = 6586.899474 / 10; Saros Cycle textbook value 6585.3211 days
We also get another Saros figure by multiplying Calendar Round 18983.99126 x Best Eclipse Year 346.5939366 = 6579.736262, which may have more substantiation than the figure generated from a 4th root; 6579.736262 seems to be part of a premiere pair of approximations characterized by having the 1.000723277 relationship; 6579.736262 x 1.000723277 = 6584.495234, what might be called the “Best Value” for the Saros, although the greater resonance and solvency probably belongs to 6579.736262.
I think sooner or later it’s going to come up that a reason that some of the preferred approximations for longer Lunar cycles like these may still be unsolved is because there may be multiple representative sets – suppose the Saros, Nodal, Metonic or other Lunar cycles were such an important business that they granted them two table rows and six different variations instead of one row and three variations, that might explain quite a lot but I”ll believe it when I see it.
Certainly with the variances exemplified by actual Mayan Codices or implied by obvious operations (Mayan Calendar Number 819 / Venus Orbital Period 225 = 364 / 10^n not 365 / 10^n), the Solar Year along with the Synodic Periods of Venus and Mercury have been afforded multiplicity.
It’s also interesting to wonder if the same may apply to the Lunar Month / Lunar Year, and whether this might at last allow the Squared Royal Cubit to be a legitimate Lunar Month. So far it has been excluded by the very systematics involved themselves, but a separate situation might be permissive.
In some of the work I’ve done lately there’s been emphasis on the difference between the Eclipse Year and a simple fraction of the Metonic Cycle. In canonical numbers one might be able to get away with “Jupiter Orbital Period x 8) / 10^n = Eclipse Year and Eclipse Year x 2 x 10 = Metonic Cycle” but something a little more intricate and versatile seems to happen with the numbers I use, where again, we seem to be able to actually distinguish between the Eclipse Year and the most outstanding candidate so far for the main value of the Metonic Cycle.
The main thing that got me looking at Morley’s work again is because I have a number of quotes of altar proportions from a multivolume source not indicated in my notes (my first guess is the multivolume Inscriptions of the Peten). In Volume 3 of the mysterious source work, there is an altar associated with a Stelae 27 (likely a sizeable site?) given dimensions of Diameter .56 m = 1.837270341 ft, Thickness .71 = 2.329396325 ft.
.56 / .71 = .7887323944 = 1 / 1.267857143.
I’m far from sure I have any idea what these numbers mean yet, but there is at least one valid number that resembles it. 1.264223085 is one of them.
A curious thing about this is that from the raw data 1.83270341 x 2.329396325 = 1 / 2.342418159 but the resemblance to its own reciprocal becomes more pronounced when I begin to try to “fine tune” the values. Probably alternately, 2.342418159 / 2 = 1.17120906 = ~117 / 100 so the long “Dresden” versions of Mercury (117) and Venus (585) may be riding along there?
I’m starting to think that 1.837270341 may mean 1.838737943 for the diameter, with the resulting circumference being diameter 1.838737943 x Pi = circumference 5.776565613.
What that would probably do is land the number near a busy intersection of equations; also there seem to be other altars that share some of these proportions (see Chinkultic and Tzimin Kax?)
1.838737943 is probably a new Wonder Number. I’ve been seeing this number and began to realize it was probably something important, but I hadn’t quite guessed much more. One thing this number is, is the long “585” day version of the Venus Synodic Period, times Pi.
5.776565613 / 4 = 2.888282807
A little while back in this same post, we saw
“(1067.077716 x 2) / ((Sun Stone Diameter 3.747289674)^5) = 2.888282803
The same digit three times starting at the second place like 2.888282803 is usually an indicator of seeing a simple fraction or multiple of the Best Eclipse Year.
Case in point, 2.888282803 x 120 = 346.5939364 Best Eclipse Year = ~346.62 days”
Hence, 5.776565613 = 346.5939368 / 60
Possibly what they did with the a x b = ~1 / b factor is set it up to juggle both versions of the Venus Synodic Period, although this remains to be seen.
Interestingly, Morley seems to describe another altar associated with the same stela (Stella 27) at the same unspecified site with a diameter of .33 m = 1.082677165 = ~1.082323232. The thickness is the same (.71 m).
.71 / .33 = 2.151515152 / 10
.33 / .71 = 4.647887324 / 10
These rather resemble the sqrt of 2160 and its reciprocal: sqrt 2160 = 4.647580015 x 10 and (1 / 4.647580015) x 10 = 2.151657415, which in reverse gives one of the already suspected figures for the meaning of “.71 m”:
1.082323232 x 2.151657415 = 2.328788808. I believe the possibility of this number has been reported in the vicinity of Tikal before (nominated as probable width of stairway of Topoxte Structure C, data from Pinto & Noriega 1995).
In the same vein, in pouring over my data tables for circular altars it came to my attention that there was no data present for the “Coyolxauhqui stone“, one of the more famous Aztec altars.
In looking to amend this, I stumbled over a blog post by Steve Minor that purports that a museum label from the Templo Mayor Museum that states “…its circumference is irregular, with a maximum diameter of 3.25 m (10’8″) and a minimum of 3.04 (10′); its thickness is 30 centimeters (about 1′) and it weighs about 8 tons.”
I’m suprised this hasn’t been mentioned before because the value 3.25 was already in my tables where it had been recognized as 10 “Hashimi Cubits” = 10.67438159 ft, but perhaps I was cautious because of the lack of data. I suspect that what happened is that I might have obtained only the diameter value of 325 from a paper by Emily Umberger?
The dataset quoted above gives as basic parameters
Major diameter 3.25 m = 10.66272966 ft
Minor diameter 3.04 m = 9.973753281 ft
Thickness 30 cm = .3 m = .9842519685 ft
Major / minor diameter ratio 3.25 / 3.04 = 1.069078947
Major diameter / thickness 3.25 / .3 = 10.83333333
Minor diameter / thickness 3.04 / .3 = 10.13333333
The last of these figures of course resembles the Greek Foot value, and the one just before it of 10.83333333 much resembles the suggested figure of 10.823232323 that seems to be recurring through these artifacts, which is after all a simple fraction of the standard primary Jupiter Orbital Period approximation of 4329.2929292 days for 4332.59, which I believe is good accuracy especially for such a large figure.
These “altars” thus may have a possible common theme of Jupiter running through them.
It eludes me at the moment whether the Greek Foot is to be expected to have astronomical significance besides being a simple Solar Year fraction (365/360 = 1.0138888888), but given its relationship to the Remen it’s probably fair to credit it with more.
I suspect that 3.25 / 3.04 might be intended to represent 1.069734371. It may often keep a low profile so it doesn’t collide with 1.067438159, but it is a Wonder Number in its own right and it belongs to one of the original Wonder Number series located at Tikal.
We start by dividing the orbital period of Venus in half, then dividing it by (Pi / 3)
(VOP 224.8373808 / 2 = 112.4186904) / ((Pi / 3)^2) = Wonder Number 1.0251355300 x 10^n
(VOP 224.8373808 / 2 = 112.4186904) / ((Pi / 3)^3) = (1 / Wonder Number 1.021521078) x 10^n
(VOP 224.8373808 / 2 = 112.4186904) / ((Pi / 3)^4) = (1 / Wonder Number 1.069734371) x 10^n
All this and more makes 1.069734371 a valuable piece of data because of the data we can easily extract from it, and the resultant series may well be THE most profound series ever discovered, because it also contains 1.177245771 and BOTH of the two most powerful mathematical probes ever discovered, 2 / 1.622311470 and 10 x (1 / (sqrt 60)).
If we accept 10.67438159 as the diameter and 1.069734371 as the Major/Minor diameter ratio, the minor diameter becomes 10.67438159 / 1.069734371 = 9.978534746
I’m not certain what to think of that; it may be something of an exotic number and possibly an awkward one, but it does happen to be a simple fraction of one of the figures currently used for Jupiter’s Synodic Period
9.978534746 x 400 = 399.141390 the “A Value” of Jupiter Synodic Period (“textbook” value 398.88 days; “B value” is thought to be 399.4300799).
Because the artifact is an ellipse rather than a regular circle, even though it is not exactly a regular ellipse in shape either, we may be able to more closely approach what may be its original symmetrical design by entering the major and minor diameter into calculations for the perimeter of an ellipse
The linear eccentricity may get away with being something like perhaps 16 Megalithic Feet (?) = 16 x 1.177245771 = 1.883593234 x 10^n.
It may be almost unavoidable unless these calculations are upset by adjustments to the major or minor diameter, that the perimeter is going to turn out to be 16.22311470 x 2 = 32.44622940, which may have a great deal of reason for being present if this can be demonstrated by further explorations.
Tentatively – and we may want to take another quick look at the equations above that involve the Great Pyramid’s projected 1067.077716 base diagonal at the proposed pavement level and its exponential interactions with the diameter of the Aztec Sun Stone, before we observe
32.44622940 / 10.67438159 = 3.039635517, there’s 30 / (Pi^2) again, that we’ve seen repeatedly in the proceedings of even very few calendar stones
32.44622940 / (10.67438159^2) = 2.847598703, the Double Faiyum Number = the Bat Palace Number / 2.
32.44622940 / (10.67438159^3) = Great Pyramid diagonal at pavement level / 4 / 10^n
The two can be multiplied at least one power as well, so here we see 1.067438159 not only functioning up to the FOURTH power, but interactive with the Great Pyramid diagonal in the process.
I find all four of those equations rather encouraging.
Before I forget, there is something important I should point out lest someone overlook it, which is quite easy to do when there is so much to consider, but the suggested diameter of 10.67438159 is intimately related to the value 11.77245771 / Pi that seems to recur through a number of cited calendar stones (“altars”), including the Sun Stone and it’s partial “twin” from Yaxchilan
11.77245771 / Pi = 3.747289670 = 4 / 1.067438159
As a length measurement, 3.747289670 ft would be a value in Hashimi Cubits / Egyptian Royal Feet, but one important reason of many that this value was probably settled upon for the diameter of the Aztec Sun Stone is because 37.47289670 is a simple fraction — 1/6th — of the Venus Orbital Period of 224.8373808. 224.8373808 / 6 = 37.47289670.
We have more to learn about the Coyolxauhqui Stone, but suffice it for now that it might well be equipped to be on the very same wavelength as other major calendar stones.
–Luke Piwalker
Luke Piwalker 