I’m continuing to try to round up further data on ancient American “altars” – or calendar stones as I prefer to think of them. I’m perhaps a bit too busy trying to think of them as advanced calculators for intricate calendar cycles to think of them as “altars”, but obviously in spite of any myths commemorating the exploits of the planets, the ancients knew the cycles of the planets well enough to know they had the predictability of inanimate objects, which is probably enough to make them more the ancient version of NASA much more than it is something that earmarks them as any kind of cult.
Finding (and organizing) data is often a time consuming and relatively unrewarding process, that can exclude being able to get deeper into the mathematical work, but I am quite intrigued with the subject once again, especially since a review of the data gathered thus far suggests a possible pattern of relationships between many of these “altars” and classic examples of their kind such as the Aztec Sun Stone or Tizoc Stone.
If there is very much overlap between their measurement values, it may make our job understanding them much easier.
When pointing out the occurrence of one of the Sun Stone numbers in the design of Hadrian’s Library recently, I had the occasion to mention how generally rare it’s been to encounter some of these numbers, even in Egypt thus far, yet suddenly there seems to be more and more of these rare numbers showing through the assembled data on circular “altars” upon closer inspection, which implies there is indeed significant overlap.
There are other patterns showing through the data – if I wanted a case in point that the number 1.0823232323 seems to have meant a lot to ancient American astronomer-architects, I probably needn’t look any further. We already know this number as “standard approximation Jupiter Orbital Period 4329.2929292 days / 10^n”, but there are apparently other reasons for it to have found favor in ancient astronomical formulas, and the same may be true for other numbers that have started to become familiar from examining ancient American calendar stones (“altars”).
Also, there have been discoveries that highlight the importance of closer examination of the planetary cycles themselves – quite notably, the discovery that the “Bat Palace Number” turns out to function as twice the ratio between Saturn Orbital Period and Saturn Synodic Period.
It comes to my attention that there may be more along such lines to be inferred from closer observation of the prograde and retrograde cycles of the planets.
While 1.0823232323 could be considered emblematic of Jupiter by way of being a simple fraction of its Orbital Period (divided by 10^n), given whole numbers for retrograde data (for which we will be “assured” that the simple minded ancients could only work with whole numbers) the ratio between Mars’ Synodic Period and Days in Retrograde calculates as 780 / 72 = 10.83333333.
What we are seeing may relate an often absurd amount of iconographic blurring or other related impacts on mythology brought about by changes in what is happening in terms of astronomy. In terms of the sort of mythographic studies I’ve pointed out like Brendan Crawford’s efforts to clarify the association of the planet Saturn with various ancient Egyptian mythological characters or “deities”
Even when there a very strong association might be made, there tend to be things that appear as if anomalies; as sensible as an association between Thoth and Saturn may be, there are ancient textual references that curiously involve references to the Sun, the Moon, or Mars.
To me, what we are seeing take place in the mythology may have meaningful parallels in what is taking place in the mathematics; there are ancient references to Saturn as “a better Sun” – an apparent Solar reference that may not make a great deal of sense outside of a connotation such as that Saturn’s Synodic Period divided by its raw Retrograde Period gives the inverse of the Solar Year. There’s something that’s decided “Solar” about something as “not Solar” as Saturn.
What we have just seen for Mars, 780 / 72 = 10.83333333 = Jupiter Orbital Period 4329.292929 / 4 / 100, may help to explain how Mars found its way into the discussion of Saturn, and etc.
I believe I have recently pointed out a number of facts about a certain set of numbers, including their exponential merits. We have a number of meter-like figures to consider, but when we go to ask which if any might have been afforded “primacy” on the basis of exponential merit, it seems to be 3.289868134 that comes out as one of the leaders, if not the leader.
In theory, while a meter based on the squared Radian would find usefulness in many equations because the squared Radian finds usefulness in many equations, the exponential use of such a “meter” would still be expected to have limitation. We don’t usually have that much expectation of getting exponential use out of the Radian itself. Sometimes we can find the Radian working as a data retrieval tool up to the 7th power or so, usually we’re probably doing good if we’re getting the Radian to work for us up to the 3rd or 4th power in general. A meter based on Radian^2 if itself squared then already equals Radian^4, so we probably shouldn’t expect a “squared Radian meter” to have exponential value past the second power of itself most of the time.
I could talk more about 3.289868134, including its origins in Munck’s work; 3.289868134 = Pi x (PI / 3) and 3.289868134 = 10.33542556 / Pi, but if anyone happens to be familiar with Carl Munck’s work, they may know that his equations and calculations told him that 10.33542556 was of major importance. I’m still quite impressed that his work managed to retrieve those facts even out from under the shadow of the Royal Cubit, which is so similar when presented as 6 Royal Cubits = 6 x 1.718873385 ft = 10.31324031 ft. To my knowledge, none of us have ever written the book about how to keep the two 100% sorted out, but they are obviously both of considerable importance.
I could also talk more about 1.315947254. It still appears in my notes abbreviated as “CHOL” because Munck’s extensive mathematical explorations seemed to have showed him that this number was important enough to make sense to us a his “geomathematical” coordinate for no less than the mighty Cholula pyramid. Even setting aside his potentially contentious “geomathematical” concepts, this number has great merit and very meaningful mathematical properties, and would have easily mattered to an ancient mathematicians who were aware of it.
Some of my work with Andrews’ data suggests that we may have at least one example where an ancient architect may have thought enough of the number to display it both forward and backward in the same room by deliberately choosing a set of overall proportions that probably don’t make sense any other way and that may discourage us from trying to resolve the puzzle with a geodetic expression.
Simply put, 13.15947254 / 4 = 3.289868134, which has earned 3.289868134 a tentative nomination as a “Cholula Meter” before although I’m allowing the term to fall into disuse because I’m still not sure that we should try to think of 3.289868134 as a meter. As tempting as the idea of ancient American metric may be, or for as many as may have been tempted by the prospect, there are likely still considerable merits in any anti-metric arguments that address the possibility of common use of metric in archaeology being able to obscure important metrological or astronomical data.
At any rate, in case no one’s tried it at home yet, a consequence of some of the preceding facts is that
3.289868134^2 = 10.823232323
The subject of the set of number consisting of these and related figures sometimes falls by the wayside, but another consideration that keeps trying to move the discussion to the front burner, so to speak, is that the most solvent approximation of the Saros Cycle may not be the most accurate one.
Rather, the most useful and relevant approximation of the Saros Cycle of approx. 6585.3211 days may actually be 13159.47254 / 2 = 6579.736270, with the accuracy being comparable to that involving the Half Venus Cycle and its approximations, minimum accuracy being very roughly 1 in 1000 days.
3.289868134 aspires to re-enter the discussion which has lately begun to turn toward the recognition of Jupiter and Saturn cycles by the Maya, because using rounded numbers, Jupiter Synodic Period 399 days / Jupiter Days in Retrograde 121 = 3.297520661, something that would likely benefit from being approximated as 3.289868134.
An alternate of course might be to demarcate this as equal to 3 Indus Feet, which should not be overlooked (or go untested) but 3.289868134 may be most worthy of examination for being part of a general trend being observed here, particularly if an argument ends up resting on exponential value.
Surprisingly, even though I consider Indus Units as probably directly derivative of Megalithic Feet, I do not associate Indus Units with exponential value, let alone the sort of exponential feats that the putative Megalithic Foot value of 1.177245771 is often capable of. I have to do no more than simply square the standard Indus Foot value to come up with something I have no idea whatsoever what to do with. The likelihood is probably that ancient people preferred to think of it – and work with it – as the Megalithic Foot rather than the Indus Foot or its derivatives.
In talking about the early efforts with the calendar stones, I’ve mentioned that there seemed to be considerable relevance to be afforded to a figure of 30 / (Pi^2) = 3.039635509 in some of the calendar calculations.
Are we all that surprised to discover that 20 / 3.039635509 = 6.579736270 and 40 / 3.039635509 = 1.315947254?
For all this (and more) we should probably expect to see a good deal of 1.315947254 and related numbers appearing at various sites to help afford them with much greater functionality for astronomical calculations and data storage and retrieval.
However there may still be more to the story, speaking of exponential functions. It’s rare enough that we get valid figures as the square roots of important numbers; far more often these square roots are things that need to to be discarded because they don’t belong to the system in question, although there is still some question about what all the ancients would have done to compensate for that.
The standard figure for the Jupiter Orbital Period is made more attractive by having one of these relatively rare valid square roots, which is
sqrt 4329.292929 = 65.79736263 — it’s managed to drift a bit at the last several places from 20 / 3.039635509 = 6.579736270, as these figures often do when working to only 10 digits, but it’s quite obvious what it is.
Perhaps even more surprisingly, 65.79736263 also has a valid square root, which is
sqrt 65.79736263 = 8.111557349 = 16.22311470 / 2
Thus the ancient Mayans have managed to intrigue us with their references and calculations until we have learned yet another remarkable thing about the 1.622311470, which I insist that it seems is one of a small handful of essential numbers that the ancient architects seemed eager to try to incorporate into virtually everything, 1.177245771 being another such number.
Of course, we have also been seeing a remarkable amount of the Hashimi Cubit of 1.067438159 ft (see Egyptian Royal Foot, Sothic Cycle, Hipparchic Cycle) in the calendar stones, is what it looks like. Even though we have found it through completely unrelated functions, are we completely surprised to find that besides getting the Jupiter Orbital Period approximation from 8.111557349 = 16.22311470 / 2 at the fourth power, and a probable Saros Cycle approximation by squaring 8.111557349 = 16.22311470 / 2, that if we cube 8.111557349 = 16.22311470 / 2, we get 8.111557349^3 = 533.7190804 = 1067.438159 / 2.
I didn’t even know that 1067.438159 / 2 had a valid cube root, but apparently they did the way they have grouped these numbers together so enthusiastically, and as it increasingly appears, consistently.
Now, if you’ve been able to follow along with some of my posts this year, going back to the return to the legend of the Lake Moeris pyramids in Egypt, and the Faiyum Oasis region where they were reported and its architecture, you hopefully already know that a “Faiyum Number” was recognized as seemingly being recurrent in the ancient architecture of the Faiyum Oasis, and then eventually I returned to Tikal to try again to tackle the “Bat Palace”, only to find an apparent solution in the form of the “Bat Palace Number” which soon proved to be double the “Faiyum Number”.
I’ve been throwing out “maybe we shouldn’t be too surprised” left and right in this post, but let’s all go ahead and be however surprised as we like about this one, because that’s what I’m going to do.
So we have seen (1.622311470 / 2) giving off astronomical data at the second, third and fourth power, which is so remarkable that surely our luck has run out by now, right?
(16.22311470 / 2)^5 = 35.11730789
Hopefully everyone knows what that is – it’s twice the projected perimeter of the Great Pyramid’s missing apex section (divided by 10^n).
That was the next thing I was going to talk about, is how we usually don’t get far after finding a new Mayan Wonder Number before we being to recognize its presence in the Great Pyramid or at Stonehenge (what a remarkable thing that is in itself, and we are still generally using so few of the numbers present at either one, that’s almost all I ever talk about at Stonehenge is just the sarsen circle).
We did already know this was here because back when the Faiyum Number was recognized, we did find out that it traced all the way back to Egypt’s 4th Dynasty at least because of its presence in the model of the Great Pyramid; I guess it’s just the drama of the presentation now that we know what the Bat Palace may be trying to say
Perimeter Great Pyramid Missing Apex Section 175.5865396 ft
(351.1730789 x 10) / 2 = 175.5865396)
So (16.22311470 / 2)^5 = 35.11730789 gives us not only another very clever way that 16.2231147 was incorporated into the Great Pyramid, but all the astronomical data that comes with 16.22311470 / 2, but in observing what may be “vicarious” occurrence of the same number between ancient Egyptians and ancient Americans,
1 / 175.5865396 = 5.695197379 / 10^n.
The reciprocal of the perimeter of the Great Pyramid’s missing section is the Tikal “Bat Palace” Number.
So, (16.22311470 / 2)^5 = (2 / Bat Palace Number) x 10^n
At (16.22311470 / 2)^6 = 2.848560569 x 10^n
Now has our luck run out? Well, no, because 2.848560569 / 2 = 1.424280285, which is one of the original Tikal “Wonder Numbers” = 1.676727943 / 1.177245771, which was later found at Stonehenge and at Giza — but that’s got to already be much more than we can rightfully expect from exponential function here, obviously.
Yet there is still another round.
At (16.22311470 / 2)^7 = 2.310822428 x 10^n, we are still on a familiar track an in the realm of what is useful. 2.310822428 = AEMY Megalithic Yard 2.720174976 / Megalithic Foot 1.177245771, which is all we need to do generate simple fractions of the “Best Eclipse Year” value.
At (16.22311470 / 2)^8 = 1.874277728 x 10^n, we no longer seem to have functional fractions of the Venus Orbital Period (i.e., 224.8373808 / 12 = 1.873644840 and etc), and yet we still haven’t run out relevance.
2 / 1.874277728 = 1.067077718, which is 1/1000th of the Great Pyramid’s projected base diagonal length at the proposed pavement level (i.e., the proposed diagonal for Munck’s own model of the Great Pyramid)
We can actually go at least as high as (16.22311470 / 2)^9 without running out of things we can recognize, although we may be truly pushing the limits of usefulness now – we are about at the point now where Stonehenge’s reach may begin to exceed its grasp, and perhaps several particularly attentive readers will be able to think of exact what I mean by that.
If it helps to have a reminder, the Hashimi Cubit is almost the cube root of the Remen – almost. Standard Remen = 1.216733603 ft, Hashimi Cubit cubed = 1.067438159^3 = 1.216264895. What do we do about that, and what did the ancients do about it exactly?
We have certainly learned about new data we can extract from the base of the Great Pyramid by using the numbers we can find at the top. Do the Great Pyramid’s sometimes somewhat odd proportions make more sense now? They do to me!
We have also learned that for as much accumulated incentive as there may already be for the ancients trying to include 1.622311470 into all of their permanent architecture, there was still more to the story, just as there is probably still more to the story even now, just as there is still more to the story of the related astronomical calculations.
As I begin to look more closely at “altars” from Tikal and elsewhere including the “Altar de Sacrificios” site in Guatemala, for several of the altars from this site, using data from both Maler and Morley, I don’t know if it’s the right answer, but I should note that for Altar 1, the total diameter of the glyph band should be (82 + 25 + 25 = 132 cm) = 4.330708661 ft = 346.4566929 / 8 / 10^n = ~346.62 / 8 / 10^n and for Altar 3 the diameter inside the glyph band should be 108 cm = 3.543307087 ft, so there we have what look like the Eclipse Year and Lunar Year both associated with the glyph band diameters of these artifacts.
One of the altars in question at Altar de Sacrificios in Guatemala, as photographed by Teobert Maler, showing the band of glyphs on the top surrounding a smooth center.
A little bit more that has been observed in pouring over the data for the “altars”:
Interestingly, when we bring Lunar Year into conjunction with Saturn’s primary values, the Orbital and Synodic Periods
Synodic Period 378 / 354 = ~1.067438159
Orbital Period 10759 / 354 = ~30 / (Pi^2)
These equations work precisely with established values.
So when we are seeing 1.067438159 or 30 / (Pi^2) in these altars, one of “x” number of things they represent are links between the Lunar Year and Saturn’s Cycles, meaning they are things one would use to expand a statement of Lunar Year data into a statement of Saturn data as well.
We being to see some of what looks like (24 Remens x Pi) / 10^n = 9.173955233 = 1 / 1.090042380
Why would we see this? In context,
Saturn Synodic Period 377.8016633 / 9.173955233 = 411.8198244 thus far the preferred form of the Full Moon Cycle of “411.78443029” days, and
Saturn Synodic Period 377.8016633 x 9.173955233 = 346.5935546 thus far the preferred form of the Eclipse Year of ~346.62 days
It does not appear to be so conversant with Saturn’s Orbital Period (?) but thus 9.173955233 distinguishes itself as a particularly important link between key Lunar Values and the Saturn Synodic Period.
–Luke Piwalker
Luke Piwalker 