At present we are having a bit of discussion at GHMB again about Chephren’s pyramid. We seem to have at least a small amount of consensus on its measurements, and the ability of those measurements to communicate the Lunar Year at a 1:1 ratio of Imperial Feet to days.
It reminds me that I am probably overdue in offering further explanation of the specifics on Chephren’s pyramid to the extent that they may be known.
I’m probably also overdue to have another look at its metrology to see if enough is known now about ancient units to better understand its underlying units of measure.
I still work with Carl Munck’s model of the Chephren Pyramid. It may seem a bit quirky at times, but it’s still solid enough that in 20 years of working with these numbers, I have never found sufficient cause for a revision, in contrast to Munck’s work on the Mycerinus pyramid which was rather precariously based on data from I.E.S. Edwards’ book on the pyramids, which eventually resulted in my attempting a complete revision of the Mycerinus several years ago.
One of the very first things I should point out about the Chephren Pyramid, is that in Munck’s model, the ratio between base lengths of Cheop’s Pyramid and Chephren’s is
Perimeter Great Pyramid (Cheops) 3018.110298 / Perimeter Chephren 2827.433388 = 1.067438159
For those who may be beginning to understand the importance of 1.067438159, the Hashimi Cubit value in feet, I will repeat some of the remarkable history of this remarkable number.
To the best of my knowledge, I am the first one to report on 1.067438159, but I cannot take credit for its discovery. There is absolutely no way that this number was not known to Carl Munck well before I discovered it.
To this day, I am still not sure why he didn’t publish it – it’s as if he were saving it as his secret weapon in a debate with a worthy detractor who never came along.
It was right there under everyone’s nose the whole time, as the ratio between the bases of Giza pyramids “G1” (Cheops) and “G2” (Chephren). I even took great pains to try not to overlook things like that. I made a list of every ratio and product that could be formed from every number that Munck has associated with Giza, but I had used one of the Microsoft word processor programs so that my list of numbers would automatically sort its self out into numerical order so we could just go down the list to check if a particular number was known to have been found at Giza.
The massive list went on some 10 pages, double columns, single spaced, both sides of the page, and I hadn’t had the chance to familiarize myself with the voluminous contents before some 24 rows of numbers of numbers that had just been added were apparently accidentally highlighted and deleted upon temporarily closing the file, and as bad luck would have it, those 24 deleted rows just happened to include multiple incidents of 1.067438159, which would have immediately stood out for recurring throughout some of the most important Giza numbers.
Rather than leave a gap where the deletion took place, the word processor automatically filled in the gap so that when I opened the file again, I was none the wiser that a terrible accident had taken place.
As a result, I was completely unaware of 1.067438159 and its recurring presence at Giza. It wasn’t until I had not only attempted to fill in missing pieces of the Stonehenge model, but had actually mastered some of the bizarre mathematical properties of the Megalithic Yard, that I first found 1.067438159 as the ratio between the outer and inner aspects of the sarsen circle.
When at last I asked myself, “Does Giza know about this?” that was when the word processor accident was finally discovered. I nearly fell out of my chair when I saw the significance of 1.067438159 to Giza, and in the case of its presence as the Cheops / Chepren base ratio, it withstands any cartographic concerns.
So that is one of the first things I should say about why 1.06743159 is very much sacred stuff – it registers as both the ratio between inner and outer sarcen circle at Stonehenge, and as the ratio between the bases of Cheops’ and Chephren’s pyramids. We want to do everything in our power to respect and conserve this number.
Regarding Chepren’s pyramid itself, we can describe it as a graphic representation of Pythagoras’ theorem, in spite of Pythagoras himself being dated to the last millenium BC. Where we can describe Cheops’ pyramid, as Munck did, as “a three dimensional model of the number 2 Pi” because 2 Pi is its perimeter / height ratio, we can describe Chephren’s pyramid as “a three dimensional model of the number 6” (perimeter / height ratio Chephren = 6).
Munck proposed that the height of Chephren’s pyramid was intended to be 150 Pi feet: 150 x Pi = 471.238898; height x 6 = perimeter 2827.433388 ft (900 Pi), side length 2827.433388 / 4 = 706.8583471 ft (225 Pi), and those are the same figures I use for it to this very day.
Because they have different perimeter / height ratios, the ratio between their perimeters or sides, 1.067438159, is of course not the same as the ratio between the two heights. In Munck’s models, their height ratio is going to be Height Cheops 480.3471728 ft / Height Chephren 471.2377980 ft = 1.019328354. This figure also manages to evade the radar a lot of the time, but it’s also an important number because whereas the ratio between the perimeters is in feet the Hashimi Cubit (a form of the Egyptian Royal Foot), this ratio as a physical measure would be in Megalithic Feet of 1.177245771 ft each: 12 / 11.77245771 = 1.019328354.
This number has a major Stonehenge connection also. It is the square root of 1.019328354^2 = 1.039030304, or 1/100th of the diameter in feet of the outer sarsen circle.
Any consensus in the direction of Chephren’s baseline referencing the Lunar Year is something I’m glad to see, because Chephren’s proportions put it a fairly unique position to do this.
Chephren base length 706.8583471 / 2 = 353.42917335, which we can optionally take as a reference to the 354 day Lunar Year. If the departure seems excessive, remember this is calendars we’re talking about and larger liberties than this were farily commonplace. The real criteria will be whether 353.42917335 can fit into any scheme that projects multiple versions of calendar values or whether some special allowance needs to be made before it is appropriate to refer to this figure as the Lunar Year.
(If we go so far as to accept a seventh group of calendar figures 353.42917335 will appear in the projection as the “G” value, but it is still not that clear at all whether there need to be that many sets of calendar figures. Almost all of the “heavy lifting” is already being done by groups A-C. We may well be much better off with only three sets of calendar numbers).
Harris and Stockdale’s works find figures as low as Megalithic Foot (10 x ((sqrt 2) / 12) x 300 = 353.5533906, and if I use my standard value for the Megalithic Foot of 1.177245771, it goes as low as 1.177245771 x 300 = 353.1737313. This later figure appears to be built into an entire class of flattened Megalithic Rings, although I do still have some reservations about whether this formula should be seen as giving us the Lunar Year or not.
What happens at Stonehenge is more or less that the measures of the lintel circle are so similar to the measures of the slightly narrower sarcen circle that metrologically about all we can do with them is render them the same way as the sarcen circle, resulting in slightly different unit values that may indicate alternative measures that were occasionally resorted to out of mathematical necessity.
In a way, it’s even somewhat arbitrary because the output of the lintel circle is intimately related to the output of the sarsen circle; the alternate Megalithic Foot value implied by this arrangement can be constructed from the standard Megalithic Foot, because the outer perimeter of the lintel circle is the reciprocal of the inner perimeter of the sarsen circle.
For a long time, I doubted that fact; I thought it was redundant to throw in the reciprocal, I even though it little more than showing off – okay, so they knew numbers forward and backward and it’s most impressive, but writing the same number both forward and backwards really doesn’t increase our knowledge or our data pool, it diminishes them, so what is the point?
It’s really the metrological exercises that more recently have shed light on the reasons for this, and whereas 360 / inner sarsen circle perimeter = 1.177245771, 360 / inner lintel circle perimeter = 1.179778193, and whether or not it was ever considered a Megalithic Foot, it is the number we can use in the Lunar Year formula to generate an improved Lunar Year value of 1.179778193 x 300 = 353.9334578 = ~canonical calendar value 354 days.
Thus, this particular model of Stonehenge ultimately gives us a very clear picture of the intent of the mismatch between the sarcen circle and lintel circle, and shows that ancient people were well aware of how to use numerical inversion to reduce or offset mismatch errors or drift.
This same consideration very much applies to the proportions chosen for Chephren’s pyramid as well, which is part of the reason I’ve brought it up here, in addition to what is illustrated by Stonehenge in this fashion concerning the Lunar Year per se.
Before I go any further, I should finish explaining why Chephen’s pyramid is in a unique position to offer commentary on the Lunar Year. We saw that Chephren base length 706.8583471 ft / 2 = 353.42917335, but we can also observe that 250000 / 706.8583471 = 353.6776513, which is an improvement on 353.42917335, and which actually does appear among the primary projected values for the Lunar Calendar Year. 353.6776514 is the projected “A” value for the Lunar Calendar Year (the “B” value is 353.9334578).
We can then of course also invert 706.8583471 — 1 / 706.8483471 = 1414.710605 / 10^n — then divide by 4 to achieve the same result as 250000 / “x”.
1414.710605 / 4 = 353.6776513.
The bottom line then is that forward or backwards, Chephren’s pyramid “wants” to talk about the Lunar Year, although it seems to do a slightly better job of it if we permit it to talk backwards.
One of the most neglected subjects concerning Chephren’s pyramid may be the effect that an extension of the model of Cheops’ pyramid has on it.
Those who’ve been reading awhile are probably quite familiar by now with the idea of there being “paved” and “unpaved” models of the Great Pyramid. Several years ago I was called upon to try to reconcile the difference between Munck’s model of the Great Pyramid, and the current reality described by data from multiple sources.
If you’ve seen any of Munck’s displays of pyramid data from differing sources, you can probably deduce what seems to have happened. Amid the massive confusion caused by different values from different sources, Munck opted for a “strength of the numbers” approach to the data, drawing support from a chance reference from Ahmed Fakhry as to the base length of the Great Pyramid, rather than investigate Petrie’s work more carefully so as to allow Petrie to accrue the reputation as a trustworthy data source that he so richly deserves.
All I can say it that it’s a very good thing that Munck made this mistake, because it isn’t a mistake. The “strength of the numbers” approach has worked perfectly well here, but the missing piece seems to be that the disparity between Munck’s figures and the relative consensus between the most respectable sources on the Great Pyramid’s proportions most likely owes to a gesture we have seen in other 4th Dynasty pyramids like Sneferu’s Red Pyramid at Dahshur and others, which is a layer of pavement that encroaches on the bottom of the casing, rather than the top of the pavement sitting at the same level as the bottom of the casing.
Munck’s model then describes the Great Pyramid’s proportions from the level of the pavement face, while my extended model also describes the proportions of the Great Pyramid from its base. It was a remarkable thing to first discover that both figures could be valid and significant.
Before that, I’d always imagined the figures from the base would be insignificant and would have been concealed beneath the pavement where they weren’t accessible but as it turns out they would have accessible though mathematics, and I’ve ended up with the same basic model for the Great Pyramid from the base as John Michell and Hugh Franklin, namely perimeter = 1111.111111 Megalithic Yards, and once again, there seems to be little case to be made for the ancient Egyptians having been unaware of the Megalithic Yard given its relationships to their most important units.
So, we have important ratios between perimeters and heights of Cheops and Chephren’s pyramids to preserve, but dual specifications for Cheops’ pyramid.
Perimeter of Cheops’ pyramid at pavement level: 3018.110298 ft
Perimeter of Cheops’ pyramid at base level: 3022.416640 ft = 1111.111111 x Megalithic Yard 2.720174976
Many times I thought that the ideal secondary value for the base length of Chephren’s pyramid would be 706.3474625 ft but if we conserve the Cheops / Chephren ratio of 1.067438159,
3018.110298 / 1.067438159 = 2827.433388 = 706.8583471 x 4 = perimeter of Chephren Pyramid at pavement level
3022.416640 / 1.067438159 = 2831.467679 = 707.8669182 x 4 = perimeter of Chephren Pyramid at base level
707.8669182 = 5000000 / 706.3474625
Thus we obtain 706.3474625 anyway, but the inversion in the figure brings the more accurate calculation into the forward position at the base, so that we simply have to divide the base in half to obtain the more ideal Lunar Year figure
707.8669182 / 2 = 353.9334591 standard figure for ideal Lunar Calendar Figure = ~354.
What also happens because of the inversion, is that the Venus reference slips from the forward position into the background as the backward (inverse) form.
706.8583471 = 225 Pi, and this canonical version of the Venus Orbital Period, 225 (days), made its way into the projected planet cycle values as the “B” value for the VOP.
The “A” value of the VOP times Pi is 224.8373808 x Pi = 706.3474625, and again 707.8669182 = 5000000 / 706.3474625.
Thus, while the Chephren measurement values can look remarkably similar to (sqrt 2) / 2 = 707.1067812 / 10^n, these skillful inversion tricks and subtle variations on sqrt 2 may be required to really get the most out of these numbers, because due to the properties of sqrt 2, (sqrt 2) / 2 = 707.1067812 / 10^n = 500000 / 707.1067812. This would restrict the Venus Obrital Period x Pi equation to an oversized 225.0790790, which is enough to upset the correspondence of the Venus Orbital Period with the Half Venus Cycle, and would restrict the Lunar Year value to 707.1067812 / 2 = 353.5533906.
Ultimately, what we seem to end up with is that 1/2 side of Chephren’s pyramid at the base = 707.8669182 / 2 = 353.9334578, our best approximation of the 354 day Lunar Calendar Year, while 1/2 side of the Great Pyramid at the base = 755.6041600 / 2 = 377.8020800, our best approximation of the 378.09 day Saturn Synodic Period – not necessarily best in terms of accuracy, but best in terms of solvency, resonance, and utility.
As with Stonehenge, then, so too at Giza — while it may look at first glance that they are simply showing off that they know their numbers forward and backwards, there are particularly reasons for the skillful mathematical gymnastics we see, and there are reasons for the slight departures from the standard square root values of “sacred geometry” such as sqrt 2, sqrt 3, and sqrt 5.
In reality, Munck’s math is the rightful domain of the square roots of larger valid whole numbers like sqrt 15, sqrt 60, sqrt 240 or sqrt 960.
From there, we can project the height of Chephren’s pyramid from the base simply as (707.8669182 x 4) / 6 = 471.9112788 ft, and deduce the missing pavement around Chephren’s pyramid as having been
Height from base 471.9112788 ft – Height from pavement 471.2388980 ft = pavement thickness 0.67380761 ft = 8.06859138 inches, slightly thinner than the calculated pavement thickness around Cheop’s pyramid, Height from base 481.0325483 – Height from pavement 480.3471728 = 0.685275457 ft = 8.224505486 inches
These pavement thickness values likely have mathematical / astronomical significance (1/4 of a Long Meter of 3.28968134 ft = 0.822467033 ft) although being such relatively small numbers, extra caution is advised in attempting to interpret them.
Because 1.177245771 x 6 = 706.3474626 and because the perimeter height ratio of Chephren’s pyramid is thought to be 6, we can surmise that the base unit of both the perimeter and height of Chephren’s pyramid from the base will be in Megalithic Feet or one of the units immediately derived from it.
I have commented before that the Megalithic Foot may be the “most prodigious” of the ancient units of measure given the sizable number of units which reduce to Megalithic Feet. This may also say something about the vintage of the Megalithic Foot as well. I’ve hypothesized that the origins of the Megalithic foot and Remen may lie in the first attempts to divide the calendar year of 365 days into rather contemporary months of 30 days (Remen) and 31 days (Megalithic Foot) respectively.
365 / 30 – 12.16666666; Remen = 1.216733603; 365 / 31 = 11.77419355; Megalithic Foot 1.177245771 ft.
To me it seems quite possible that both of the units have been present since the very beginning of mathematics.
We can also say of the Megalithic Foot that it appears to be quite well integrated with other ancient Egyptian units though geometry.
Given the thickness projections for the pavement around Cheops’ and Chephren’s pyramids, we can attempt to deduce what became of the pavement.
A number of factors, including the relatively small size of the pavement blocks compared to the pyramid casing, their accessible location on the ground rather than 3/4 of the way up the slope to the pyramid summit, and quite possibly a somewhat more extravagant choice of paving material at Giza than may have been seen with other 4th Dynasty pyramids such as those at Dahshur, might easily have resulted in the pyramid courtyard paving at Giza have been picked thoroughly clean before the dismantling of the Giza pyramids themselves ever started.
In the case of Mycerinus’ unfinished pyramid, things may have never reached the stage of the courtyard pavement being installed because pavement imposing upon the bottom casing stones would have required dressing of the casing at the bottom prior to the pavement being laid into place.
At the moment, it still isn’t 100% clear to me whether the projected measures from the revised model of the Mycerinus pyramid represent the measures from the base, or from the pavement. With due diligence, perhaps it will ultimately be possible to complete the deductive process to know which it is, and what the thickness of its courtyard pavement was going to be.
–Luke Piwalker
Luke Piwalker 