I think I might have found a few items worthy of mention, although none would merit a post by themselves.
Selective Omission of Numbers
I’m still somehow feeling rather apologetic about this, which was part of my post about Exclusion, referring to all of the numbers in red (and their multiples and dividends) being excluded from the proceedings.
It’s a terrible thing to hear and a terrible thing to tell someone that for intents and purposes, the numbers 7, 11, 13, 17 and etc “don’t exist” in this system of math, but that is how this math works. To the best of my knowledge, we will never see Carl Munck use numbers like this in his works unless perhaps there are occasionally truly novel or extenuating circumstances.
There are apparently, exceptional monuments – the occasional 7-tiered step pyramid or the odd stone circle of purportedly 11 or 13 or 17 stones, for example, but the intended protocols still aren’t absolutely clear. It’s long been hypothesized that these gestures represent approximations of more complex numbers, as with “56” at Stonehenge’s Aubrey circle, but some of this view was based on the El Castillo pyramid at Chichen Itza.
It was assumed not only that we are entitled to round up the raw figure of El Castillo’s “91 steps” per side (364 / 4) to something more like a true 1/4 of the Solar Year, but also that one of the steps would prove to be slightly larger than the rest to bear this out, but in recent weeks it seems to come to light that 91 steps apparently differs from Maler’s data and may be a fantasy invented by modern archaeologists.
The hypothesis that “illegal” numbers of structural details like steps or columns or posts or stones represent approximations may be perfectly sound, but there may be less recognized support for this idea than was previously thought.
I want to re-emphasize that I think the reason for this – that is, the justification for the numerous exclusions of numbers – isn’t that that is what we have to do to get the equations to work out, so much as that trying to accommodate different metrological units can generate such a mess that we’re more or less obligated to try to simplify matters through such exclusions of numbers.
Even working with the Remen and the Royal Cubit only, we see problems of this nature. “7 Remens (8.517135221 ft) or 5 Royal Cubits?” (8.594366925 ft) is a classic problem in ancient Egyptian metrology and one which may have helped to lead Petrie astray from looking further into the common use of the Remen for lengths independently of the Remen being the diagonal of the half Royal Cubit.
It may also be such concerns that could have led expert Marshall Clagett astray. Dividing 5 Royal Cubits (as specified) by 7 produces an oversized Remen of 8.594366925 / 7 = 1.2277660704 ft rather than 1.216733603 or less; Wikipedia claims that a Remen is 37.5 cm = 1.230314961 ft, and attributes this seemingly rather inflated figure to Clagett’s Ancient Egyptian Science.
What certainly seems as if it didn’t happen is trying to ward off mathematical chaos by restricting the number of units of measure in use instead of restricting the quantity of whole numbered values in use; quite the contrary, the metrology of numerous ancient nations is almost inexplicably – and very confusingly – diverse, rather than minimalist.
Khentkawes II’s Pyramid
This comes to mind again because the last several days I’ve been looking at data for the pyramid of Khentkawes II and its satellite pyramid, and trying to see if I trust the data, but I’ll still struggling to make any fully coherent models out of any of it, and as of this hour another classic Egyptian metrology problem hands overhead: given the data, is the satellite pyramid’s baseline 10 Royal Cubits (17.18873385 ft) or 16 Hashimi Cubits (17.07901054 ft)?
Very strangely, my first model is almost perfect. We can create a lovely pyramid of baselength 16 Hashimi Cubits, height 2 Squared Munck Megalithic Yards, base diagonal 72/10 Egyptian Mystery Units of 1.676727043 ft, and vertical edge of 60 / Pi ft.
The problem with this model is that the resulting perimeter / height ratio, or why they would have tolerated it, seem almost completely incomprehensible.
The subject is confusing, but appealing, particularly given the data I have.
Note that the figures quoted by Lehner (Verner seems to have an error in height data in The Pyramids, pg 486, giving a very implausible 72 meters for the height) make it look like a pyramid with base in Inverse Remens, with height of “The Giza Vector” number and Perimeter / Height ratio of 1/2 of Megalithic Feet: 11.77245771 / 2 = 5.886228855.
I’m skeptical of the figures from Fr.Wikipeida in this case – why write Phi in meters then pit it up against 100 of same? We don’t need pyramid designs to do our reciprocal checks for us and pyramids design this way may suffer heavy losses in data storage and transfer capacity, so what would be the point?
(I’ve traced the figures from Fr.Wikipedia for the satellite pyramid to a publication by Miroslav Verner, Abusir III. The Pyramid Complex of Khentkaus and the measures stated by Fr.Wikipedia in their list of pyramids are as stated by Verner).
Yet this very ideal looking set of possibilities inspired by the data from Lehner, as much as we should have been expecting pyramids with the hugely important number 1.177245771 in the base or the height or perimeter / height ratio do not seem fit together properly, once again disproving what some onlookers seem to think about working with numbers, that was can “just slap together any old thing and make magic”.
More like the relationships between parts of structures we are talking about rely on careful selection and deliberate design – but that still leaves us with seemingly little to show at this hour for Khentkawes II’s pyramid.
About the only real clue I might have had so far is that the ratios between Khentkawes’ pyramid and its satellite resemble possible tributes to fractions of the Great Pyramid’s baselength.
Senusret (Sesostris) II’s Pyramid at Lahun Again
I went back to the Lahun pyramid to sketch out the measures at the East of the chamber
Something I am wondering, although I haven’t begun to try to confirm it yet, is if the width values seen here were intended to express a certain number.
123.3 / 123.1 = 1.001624695. This could be another vote of confidence for the idea that we see meaningful and deliberately designed irregularities at work. I find this number rather reminiscent of a notable valid number, 1.001812743.
This remarkable number seems to have only appeared in 8 previous blog posts although its discovery was headline material.
It has been found at Stonehenge at least once, it has been found in the Great Pyramid, it has been found in Kent Week’s data for the Valley of the Kings. In accordance with referring to it as a “2 Pi Root” (like certain geodetic “root” figures), if we multiply it by (2 Pi) repeatedly, we get a remarkable, high-quality series of data, including the reciprocal of 1.622311470, 11.77245771 / 12, sqrt 15, the Double Remen, and others.
The Power of The Megalithic Foot as Mathematical Constant
I have a communication from Peter Harris that I need to get back to that I managed to overlook until about an hour ago, touching on a subject I’d meant to ask him about, which is the nomenclature of the Megalithic Foot. I’m calling it the Harris-Stockdale Megalithic Foot (HSMF) but I’m using Carl Munck’s value for it of “Alternate Pi” = 1.177245771 vs the standard HSMF value of (10 x sqrt 2) / 12 = 1.178511302 ft.
It’s a good question how these are most accurately named, but as far as I’m concerned, if I’m going to say “Megalithic Foot”, as much as Munck or Morton or I know about 1.177245771, “Megalithic Foot” is a metrological function, and as far as any values in this range being metrological units, it was Harris and Stockdale and no one else who put them on the map – and firmly – as a metrological unit.
On the other hand, I’m quick to want to get away from the idea that the Megalithic Foot value is merely a metrological unit, because it is also a mathematical constant. In metrology, we may never see the point of multiplying or dividing by the Megalithic Foot value more than once to find the amazing mathematical properties of some values in this range.
I might have found and shared this before, but I stumbled on an equation series that I think serves as a good example of this.
Let’s take the Petrie Stonehenge Unit in inches / Venus ORbital Period A value of 224.8373808, and square it: 224.8373808^2 = 50551.84781. Now let’s divide it by the putative Megalithic Foot value of 1.177245771, probably the single most important number at Stonehenge, repeatedly:
50551.84781 / (1.177245771^1) = 42940.77673
50551.84781 / (1.177245771^2) = 36475.62623 (100 x Solar Calendar Year A value = 100 x (3600 / Pi^2)
50551.84781 / (1.177245771^3) = 30983.86686 (1000 x sqrt 960)
50551.84781 / (1.177245771^4) = 263.1894514 = 162.2311470^2 = (5 / Half Venus Cycle B 18997.72188) x 10^n
50551.84781 / (1.177245771^5) = 22356.378263 = “false sqrt 5” x 10^n; this is what we get for sqrt 5 because of a Megalithic Yard of 2.270174976 and a Remen of 1.216733603: 2.720174976 / 1.216733603 = 2.2356378263. This number divided by 4 gives the most useful approximation of 56 for the Aubrey Circle and is tied into the metrology of the missing apex section of the Great Pyramid
50551.84781 / (1.177245771^6) = 18990.40386 Half Venus Cycle C
So all of that is built into Stonehenge because 224.8373808 and 1.177245771 have been combined there, and we see that this value of 1.177245771 continues to do at higher powers what the HSMF is known for, recovering astronomical data from Megalithic sites. The series is actually able to find two different forms of the Half Venus Cycle as well as the Solar Calendar Year, and other numbers with astronomical significance.
–Luke Piwalker
Luke Piwalker 