I know that I make frequent reference to the Great Pyramid’s “Missing Apex Section” – in fact, these passages from the very recent post “The Stars Built In” seem to make it particularly clear what my extended Munck model of the Great Pyramid implies in regard to the missing section:
I hope that by now, everyone understands what my model of the Great Pyramid actually represents. The origins of the extended model are simply that the height of the Great Pyramid as it currently stands was given by multiple sources as between about 452 and 454 feet; I originally proposed the value of 452.893421 feet purely because Carl Munck held this number in such high regard, referring to it as a “Holy of Holies” and I though this interpretation might make him happy.
It was only some time later that I realized myself more about what I was looking at, that this gesture had created a ratio of 1.718873385 x 10 – the value of 10 Morton Royal Cubits in Imperial feet – as the ratio between the whole pyramid and the part now missing.
In other words, I’ve been seeing this “unit value in Imperial measures as ratio” phenomenon since the very beginning of my Egyptological studies, and last saw it as recently as my last Egyptological study, where we find Remen in Imperial as ratio in the data from Mariette for one of the coffers in the Saqqara Serapeum, the ratio between the coffer’s inner and outer length, or even from the raw data, 3.85 m / 2.17 m = 1.214511041 = ~1.216733603.
How do we justify this assessment of the Great Pyramid’s missing section? If we look at the Great Pyramid in its present condition, there is almost 28 feet missing at the top, and yet from bottom to top we see it stripped only of the prized casing material, implying that there was a definite cut-off point some 28 below the projected apex with the pyramid above theis level made of a different material that supported the actual capstone. I would quite be surprised if the capstone itself exceeded 7 feet in height.
Frequent reference has also been made to how the numbers from this projection of the missing section turn up in a great many other places (including Stonehenge, for example in the mean proportions of the sarsen circle), and how astronomical data can easily be recovered from the missing apex section in spite of its absence.
In spite of all this, relatively little may have been offered recently in the way of demonstrations of this.
How about we begin at the beginning briefly?
If we take Munck’s height for the Great Pyramid (from the presumed pavement level) of 480.3471728 ft, and subtract from it the projected height at which the transition in materials is proposed to take place, 452.3894321 feet, we obtain the missing apex section’s height projection of
480.3471728 – 452.3894321 = 27.95783074 ft.
The interpretation of this figure is given as 27.94546572 ft, which is accurate then to 27.94546572 / 27.95783074 = .999557725. This is the first time the “Giza Standard” of accuracy on approximations of geometric necessity of .9995% or higher was seen in action.
If you find any of Michael Morton’s work on the Internet, or some of my older writing, you may see the number 27.94546572 referred to as the “LCS” constant. We were working with the idea that three serpent-themed monument including “Le Serpent Rouge” in France and a purported Scottish serpent effigy made of stone, were capable of holding a sort of “geomathematical conference” in which they were able to collectively generate this number, so we were aware of the number and some of its properties even before its rather astonishing appearance in the Great Pyramid.
Most of what has been learned about 27.94546572, however, has come after my realization of the ancient imperative to record astronomical data in monuments.
With a perimeter / height ratio for the Great Pyramid of 2 Pi, this height x 2 Pi = 27.94546572 x 2 Pi = perimeter 175.5865396 ft for the missing section.
Because this in itself sets up the ratio of 10 Royal Cubits in feet = 1.718873385 x 10 between the whole pyramid from the pavement and the missing section, no one thought to try converting these specifications for the missing apex into Royal Cubits – we’d already done that, it seemed – but that is another discovery from more recent years that we can use the Royal Cubit value that way as a mathematical probe, to retrieve some important numbers, including some of the “Mayan Wonder Numbers” from the Tikal temple pyramids in Guatemala.
So, even though the total projected height of the Great Pyramid from the pavement in Royal Cubits is already 480.3471728 / 1.718873385 = 279.4546573 Royal Cubits, we can still make a meaningful conversion of 27.94546573 feet to Royal Cubits, so that the height of the missing apex in Royal Cubits would be
So, even though the total projected height of the Great Pyramid from the pavement in Royal Cubits is already 480.3471728 / 1.718873385 = 279.4546573 Royal Cubits, we can still make a meaningful conversion of 27.94546573 feet to Royal Cubits, so that the height of the missing apex in Royal Cubits would be 27.94546573 / 1.718873385 = 16.25801294 Royal Cubits.
For a minute, we might puzzle over them not having managed 16.22311470 ft since this is one of the numbers that ancient architects seem to have wanted to build into everything, but what 16.25801294 is, is the reciprocal form of one the “Mayan Wonder Numbers”, and is in fact also 1/18th of another “Mayan Wonder Number” – none other than the dreaded “Real Mayan Annoyance”
292.64422329 / 18 = 16.25801294, so from the conversion to Royal Cubits, we only need to apply half of 360 (as in 360 / 2 Pi) in order to find the astronomically significant “Real Mayan Annoyance”
360 is one of the very first numbers that the Great Pyramid should make us think of, because of the role of 2 Pi in the geometry of circles.Virtually every number we can find in the Great Pyramid may respond to 360, 2 Pi, the Radian, or all of them, by releasing at least one stored significant data point. We also use 2 Pi to derive the same version of the Royal Cubit from whole numbers, that is embodied by the ratio between the missing apex section and the whole.
I very strongly agree with what Carl Munck said about the Great Pyramid – that its primary mathematical identify is as “a three-dimensional model of the number 2 Pi.”
For those who might be tuning in late and haven’t had the chance to read some of the older posts, the “Real Mayan Annoyance” was given such a preposterous name because it created considerable confusion at times whether we really seeing 2.926442329 or 2.920160646.
I had just found 2.920160646 at Tikal in proper context, and I thought, “Fantastic! The Maya knew this Egyptian number too. This will be easy then, because just like Egypt, every time we see “2.92-something” we will know it’s 2.920160646″ – and I of course was greatly annoyed when I first discovered that this was not so, and identifying “2.92-something” was going to be ten times as much work as I expected.
For those curious, let me take a minute to explain “proper context”.
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Munck’s own drawing of Tikal Temple I labelled with Maler’s data and conversions from meters to feet, from http://www.viewzone.com/bigpicture/bp112311.html.
Both the width and length of the Temple platform of Tikal Temple I are geodetic figures, and the width value could not be a more blatant one, using about 24.901 ft to represent 24901 miles equatorial circumference. We could think this just coincidence, but the length is an almost insanely clever second geodetic figure, and Maler’s data also shows us that the very same thing happened, only in reciprocal form, atop the El Castillo Pyramid at Chichen Itza.
We also have it on good authority that the width of Tikal Temple III is 890 cm = 29.19947507 feet.
Hence in this case at least we can be fairly certain that we are looking at 29.20160646 and not the dreaded “Real Mayan Annoyance” 2.926442329 x 10, because the situation is that while we have what is probably 24901.19743 displayed atop Temple I is as blatant a manner possible (and in “modern British feet” no less), atop Temple III is an equally blatant display of the very same number in cube root form!
29.20160646^3 = 24901.19742
From there, 2.926442329 is the “Real Mayan Annoyance” because there turned out to be a half dozen very similar numbers that exist within Munck’s mathematics that are apparently considerably less rewarding to get involved with.
Going back to the Great Pyramid’s missing apex section, if we convert its perimeter into Royal Cubits (27.94546572 x 2 Pi = perimeter 175.5865396 ft), 175.5865396 / 1.718873385 = 102.1521080 Royal Cubits, and 102.1521080 is another “Mayan Wonder Number” – also from Tikal, where we find it at the top of Temple IV. It belongs to what is still one of the most powerful and fruitful (Pi / 3) series I have ever seen, as was discovered because the Mayan architect actually seems to have combined it there with (Pi / 3) x 10 feet as another of the proportions of the same doorway.
So all of that comes in to play just in the conversion of the height and perimeter of the missing apex section into Royal Cubits!
The basic metrological unit of the height of the missing section is the Egyptian Mystery Unit phrased as 5 / Egyptian Mystery Unit 1.676727943 ft, which is to the Mystery Unit value what 1.067438159 is to the Egyptian Royal Foot – a frequently more useful form. 167.6727943 / 6 = missing section height 27.94546572 ft.
As it turns out even with little if any a priori knowledge of astonomy or ancient calendars at the time it was first proposed, if we follow along with the logic that because there Giza is dominated by a pyramid that is a massive advertisement for 2 Pi, that we should try throwing 2 Pi at every number we find there, creating a pyramid with an upper section of this height turns out to be quite a desirable goal for anyone wanting to express important numbers from astronomy (or geodesy).
Let’s now being the roll call…
Temporarily ignoring correct decimal placement, which is inconsequential here
27.94546572 / (2 Pi^1) = 4.447659006 = 1 / 224.8373808
At the very first power of 2 Pi, the projected missing height of the apex has responded with the Venus Orbital Period, in its primary form – its most useful and important form.
27.94546572 / (2 Pi^2) = 7.078669160 = 353.9334580 / 200
At the second power of 2 Pi, the projected missing height has given us twice the standard value for the Lunar Calendar Year, with 353.9334580 approximating “354” days. This is still the most useful representation of the Lunar Year that I’m aware of.
This downward progression with useful numbers (some in reciprocal form) as output continues down to
27.94546572 / (2 Pi^7) = 7.228568039, which is the reciprocal form of the revised perimeter of Menkaure’s pyramid, “G3”, itself a measurement with both geodetic and astronomical function because the very important Hashimi Cubit is the root of it in sensu Petrie (whole number of units): 360 x 360 x 1.067438159 = 1383.399858 revised G3 perimeter = 1 / 7.228568039. I wouldn’t know 7.228568039 on sight myself but a simple reciprocal check of 1 / “x” reveals what it is.
At 27.94546572 / (2 Pi^9), we find the reciprocal of 54614.43732, which is 2 x 360 x 360 Palestinian Feet. It doesn’t seem to be often used but its significance is recognizable.
Beyond that things get a bit strange so we will demarcate that as as territory to be avoided.
In the upward direction 27.94546572 x (2 Pi^2) = 110.3247266, which is our best Eclipse Year divided by Pi.
110.3247266 x Pi = 346.5939368
That is at least ten important pieces of data recovered, including the Venus Orbital Period, Lunar Year, and Eclipse Year, just by applying the ratio 2 Pi to 27.94546572 value actually found atop a 2 Pi pyramid.
If you run the progression yourself, on the way down you will meet what looks like half of the “Not Venus” number. Pay no mind to that, as we’ve already found the true Venus Orbital Period at the start of the downward series. What the number really is, is the reciprocal of the square root of the generic volume of a sphere.
The reciprocal of Munck’s Giza Vector Number is in there too, which is also an astonomy-related number, including that is square was apparently used in ancient reckoning of time.
360 will pull 3 more important pieces of data from 27.94546572, and the Radian value 57.29577951, at least four more. Now at 17 pieces of important data just from putting 27.94546572 atop a 2 Pi pyramid. That of course isn’t the crown jewel of the pyramid, that’s just what the crown jewel rests on.
None of this yet touches on the fact that the projected missing section was also found to have significant base diagonal, vertical edge, and slope length. The base diagonal length as determined by ratio to the whole (that is, the base diagonal of the whole pyramid divided by 10 Royal Cubits in feet) is a geodetic figure related to the polar circumference in miles by 360 as a ratio.
Were it only that we could make the corresponding projection for Chephren’s pyramid, I presume we might see something equally splendid, but sadly I still no have reliable data on the current height of “G2”, and while there is some data from Maragioglio and Rinaldi that may even include corroboration of a platform height established well below the base of the capstone just as we would want to predict, I have not been able to obtain the vital parameter of either height of the present platform or the length of its sides, in order to perform the calculations.
Unfortunately, I also have little faith at present in historical sources (there is more than one) that amazingly, claim to offer dimensions of the Great Pyramid’s original capstone.
Likewise, Petrie’s data avails us little corroboration of the Great Pyramid’s present height. His course heights are different at different corners, and even if we somehow miraculously added them all up without generating egregious cumulative error, there is still the matter that the surviving top course may represent merely the protrusion that fit into a formed socket in the base of the missing section, meaning that its height cannot be expected to assuredly be at the same height on the pyramid as the base of of the section that’s missing.
These specifications for the missing section have now stood for at least 15 years without anything prompting a revision – even when I ask myself on a daily basis, “Am I sure I have this right?”
Maybe someday we will learn how to know more of the story.
In the meantime, we can continue to throw more power probes like sqrt 60 or reciprocal of sqrt 60, or 1.177245771, or 1.622311470 or 2 / 1.622311470 at the proportions of the missing section to secure even more important data, further adding to our knowledge of the Great Pyramid’s encyclopedic collection of important data.
–Luke Piwalker
Luke Piwalker 