engbren at GHMB has been working on what I think is most likely a very worthwhile academic paper regarding the possibility of Saturn exerting an influence on the design of Egyptian pyramids though various possible mathematical fingerprints.
It reminds me that mathematical fingerprints for the planets seem somewhat difficult to come by. While I wouldn’t put anything past the ancients personally, the ancients knowing the distance of the planets from the earth or the sun or the diameter of Mars may be rather far-fetched to academicians and I don’t exactly blame them.
Even the question of whether the ancients were able to calculate the sizes of the planets has proven difficult. Our Founder Carl Munck had determined an interesting figure for the size of Mars at the equator. It escapes me now because there seems little point in remembering it, because only a few years after I’d been working with it, NASA completely revised their figure courtesy of improvements in areography (Mars geography) which was brought about by new data from further successive Mars missions. In a very real way, asking the ancients to know this data is asking them to know what even NASA may not know.
Angular values can also be difficult; I think in one place an angular value for some of Saturn’s retrograde functions was posited, a value of about 3.5 degrees with a stipulation of +/- 1 degree from an academicians. In our terms, roughly 1/4 of all possible valid numbers occur between 2.5 and 4.5, probably making it absolutely impossible to get a mathematical lock on Saturn using this parameter.
Still, retrograde values may be a great place to look because it’s a rather dramatic phenomenon from “the view on the ground”, and one that is accessible to even the “primitive” astronomer.
I’ve been hesitant to move forward with working retrograde (and prograde) values because astonishingly, the only data I have on the subject is still what we get from Wikipedia, which looks very much rounded to the nearest whole number.
However, even if we can’t rationalize the ancients sticking to whole numbers, one premise of my work is that they did use use some rather sophisticated decimal numbers to emulate simple whole numbers from calendars – 353.9334578 for the 354 day Lunar Year of technically about 354.36 days, and so forth.
For that we can hope that even rounded numbers for figures related to planetary retrograde cycles can at least tell us part of the story.
In a way it’s not really fair for me to just plop this data down on the table; there are some important comments that could be made and there are a number of interesting observations that could be made. On the other hand it’s not probably not fair for me to just sit on the data sets until I can figure out what to do with them, if they might be of assistance to others in the meantime.
One important thing that I think may be apparent in general is a number of additional “serendipitous coincidences” of the sort that likely lent themselves to the original development of such a calendar system that could meet such a grueling demand as being all-encompassing. One of them is that Mars Orbital Period / Mars Days in Retrograde looks very much like yet another instance of the Royal Cubit about literally falling from the sky.
Out of curiosity, while I was at it I included Uranus and even Neptune to see if we can learn more about whether the ancients devised a calendar system that could incorporate them too if they had somehow known about them. Some of what I’ve read about the possibility of the ancients having observed Uranus doesn’t it make it sound all that unlikely that they may have managed to spot it.
I don’t think these numbers can really tell us whether they knew Uranus or Neptune were there or not, but I am interested to know if ancient calendars or calendar-related metrology could have accommodated them or not, and little devoted study has been given to the subject so far on my part. I’m intrigued that Neptune may be capable of generating the same Mayan Wonder Number that seems to be something we may be able to generate with the formula of Venus Synodic Period / Venus Days in Retrograde.
I think that even for being offered with so little in the way of commentary, there may be some very rewarding things in store for anyone who wishes to linger over it for just a little bit. Anyone who’s been able to follow this blog very closely for very long may see a surprising number of familiar-looking figures, but perhaps even more importantly, there are some hints that we may still not quite a complete vocabulary of ancient astronomical constants.
There probably even some important clues here to the absolute identities of a few planetary cycle values that may remain somewhat uncertain such as Mercury’s Synodic Period. It’s always almost too easy to take that one as being equal to 80 Indus Feet in Imperial, but somewhere in previous posts I’ve been able to make some alternative suggestions that may have considerable merit themselves.
We may even see some equations here that ancient people tried to communicate thought mythology; there may be more than once instance of that related to Saturn visible in the tables.
Without further ado then, I will hope that I was successful in weeding out the last of the errors, and put this at this disposal of anyone who might be interested or assisted by it. I should have done this much sooner since of course that really is by far the best way to gain an overview of extensive data sets is to try to lay them out in some kind of orderly fashion.
Frequently in my writing, I use the term “resonance”, referring to “mathematical resonance” or “resonant” numbers, but I’m not still not sure if I’ve ever given any adequate explanation for the term.
In acoustics,
noun
the quality in a sound of being deep, full, and reverberating.”the resonance of his voice”.
(physics) the reinforcement or prolongation of sound by reflection from a surface or by the synchronous vibration of a neighboring object
I’m not certain who first applied the term to our work, but I suspect that if I could find the documentation, it would be Michael L Morton who deserves credit for first having used it, and not someone like me who tends to come up with rather unimaginative names for things.
It really is a very apt term for what it attempts to describe in mathematics, particularly the part about resonating with neighboring objects (or in our case, neighboring numbers).
In practice, I use to “resonance” to mean when a number has strong connections to notable number of other important numbers, or when a number has a notable quantity of aliases – aliases simply meaning different names or formulas that we can use to refer to a particular numerical value.
((sqrt 3.75) x Pi) / 5 = 1.216733603 is one aliase for the 1.216733603 ft Remen, sqrt (.15 x (Pi^2)) = 1.216733603 is another, and so on – and yes, the Remen value has so many meaningful aliases that we can consider it one of the more resonant numbers in our vocabulary.
Although each ancient monument represents a data repository unto itself, Munck and his students tend to see each monument as also belonging to a larger network, or web – or as Munck called it, a “matrix” – of numbers larger than itself that also incorporates and “speaks to” a larger number of ancient monuments.
One analogy is plucking a single strand of a spider’s web – the whole web will shake, or reverberate, or resonate.
The most resonant numbers being those with the strongest or largest number of meaningful connection to other important numbers, or those that have the largest numbers of impressive aliases, should also mean that these numbers will appear in many places because they can be constructed so many different ways, so that they can be found to be fairly widespread in their occurrence even though radical difference in proportion or measure or shape may be seen between any two constructs that can generate the same number.
The Great Pyramid and Stonehenge, which have little in common in terms of shape or structural design, manage to both incorporate a surprising collection of the same important numbers.
When I talk about using a “strength of the numbers” approach to architectural interpretation, or starting out our interpretive experiences with “good numbers”, I’m referring to “resonant” numbers. It’s this resonance that makes them “strong” numbers, or “good” numbers, or numbers with “strong connections”.
From there you can probably guess that I’d say that the two most “resonant” numbers I know are the two that I keep referring to as “the two most powerful mathematical probes ever discovered” – these numbers “reverberate” or “resonate” throughout the network to sometimes remarkably high exponential values (current record is still sqrt 60 or (1 / sqrt 60)) resonating throughout the system to as high as the 33rd power or possibly even higher.
Resonance is, at least for me, what helps define Munck’s system of numbers as having outstanding qualifications as an ancient system for data storage and retrieval, and that is again what so many of us mathematical and metrological researchers are attempting to do, is to mine ancient architecture for data that we suspected to be stored within them.
This phenomenon of mathematical resonance also creates resonance between seemingly unrelated monuments. I often mention cases of this; I will be talking about Egyptian pyramids and suddenly bring up data from a Mexican pyramid or stone circles, which some may accidentally read a little too much into.
I have no way of really knowing if the designers of Stonehenge were actually thinking of the Great Pyramid, or vice versa, but I do know that because of the mathematics resonance between the two, either one is a great place to go to learn more about how the other works.
I could add a reinforcement that resonance isn’t always measured in terms of exponential utility. The Hashimi Cubit value of 1.067438159 (ft) has rather poor exponential value to be honest, yet what makes this number particularly resonant is the number of places it can actually fit in with exact precision. Many equations point to it, and it is not only predicted that it will be found in a great many places because of this, but such predictions have largely already come true.
We can also say that because of some serendipitous coincidences which helped to make the impossible task of working out an ancient all-inclusive calendar system into an actual reality (the Mayan Calendar is an actual if somewhat crude example of exactly this), that the planets themselves express some surprising “resonance” with one another – they are “singing” in deep “resonant” voices that reverberate throughout their system.
Is “Stecchini’s Cubit” The Draconic Megalithic Yard?
Even though it hasn’t yet helped to give a historical identity to the “Outer Sarsen Circle Unit”, I may get to say that looking at various units (where applicable) as things that can be constructed from whole numbers combined with exponential expressions of Pi has already proved helpful in several ways. It does seem to have helped us to identify the nature of the Draconic Megalithic Yard, a viable form of the Megalithic Yard that provides the most accurate expression of the number of days in the Draconic Month when its value is given in Imperial measures.
It also already allows us to make some very valuable inferences about the nature of Megalithic mesurements and what we might expect from certain Stone circle designs. At far right at Pi to the 4th and 5th powers we can where the “Outer Sarcen Diameter Unit” first came from, because with the “AE” Megalithic Yard of 2.72017497 – the primary unit of the outer sarsen circumference – constructed from Pi^5, we know its diameter will be constructed from (Pi^5)/Pi = (Pi^4) because of course circumference / Pi = diameter.
By the very same token we can see that if a circumference is in “Outer Sarsen Diameter Units” made from Pi^4, its diameter will then be the unit that utilizes Pi^3 (Pi^4/Pi) in combination with a whole number.
Thus we have more reason to expect that we should find some number of stone circles that have the “Outer Sarcen Diameter Unit” as the unit of their circumference, because this is categorically a way to write the remarkable approximation that is the Draconic Month / Draconic Megalithic Yard value into the diameter of the same circle.
For reference, the Draconic Megalithic Yard is suggested to measure a calculated value of 27.21223280 / 10 feet, while the “textbook” value of the Draconic Month is 27.212220815 days.
For years now, I have been entering this neglected number into my calculator in the very same way I first discovered it, as (Generic Area of Circle 10313.24031 sq arc degrees OR Royal Cubit in Inches / 2) / ((Assyrian Cubit in Inches 19.46773764)^2) = 2.721223218).
In recent years, I’ve been finding out that some of the important equations formed by Stonehenge’s measurements are amenable to substitutions of Megalithic Yard values that are conducive to the discovery of this number in a manner that suggests its measures were very carefully and deliberately chosen in order to achieve this.
Almost twenty years ago, based on the dsecriptions from Stecchini’s text of three cubit rods described by Karl Lepsius, I tabled the possibility of a “Stecchini” or “Lepsius” Cubit. Michael Morton even in spite of the great pride he is entitled to as the first person to ever describe the Royal Cubit as it really seems to be, graciously entertained the possibility that I had discovered a valid secondary Royal Cubit value, which for a while I was referring to as the “B Version” of the Royal Cubit.
At the time, we did not know just how devoted the ancient Egyptians and others may have been to his geometrically standardized Royal Cubit value at the expense of all others. I have never needed a “Stecchini” Cubit to describe the exterior of a pyramid, and the only instance of it inside a pyramid that I’m certain of is one where “this is a very special case” is written all over it courtesy of other remarkably trend-breaking mathematics that goes with it.
(For those wanting to know more about that very unusual construct, it is the passage chamber of the Lahun pyramid recently discussed here.
Still, we wonder why such a Cubit Rod even exists if it isn’t an actual Cubit. I have been happy to follow along with some archaeological trends and refer to it as a “ceremonial” or “votive” Cubit value in that this may roughly correspond to how I think of it. The “Stecchini” Cubit may never have been considered a Royal Cubit at all and yet it would be advantageous to commemorate it with a measuring rod because there are times when we can use this value AS IF it were a Royal Cubit.
I would have to dig for it now, but I know that one scholarly article made reference to Petrie’s description of another probably so-called and equally “ceremonial” Cubit Rod that appears to measure 1.6 Hashimi Cubits. The distinction has caused me a bit of confusion before, and may even have caused a bit for researcher Geoff Bath as well.
Munck’s model of the Great Pyramid (“from the proposed pavement level”) is a prime example of this, which is something that initially caused confusion over which would have been the primary Egyptian Cubit and which of them was used in the Great Pyramid’s design, back when we were almost sure that this “Stecchini” Cubit could have been a genuine Royal Cubit; it also caused a bit of confusion over the proportions of the Great Pyramid’s missing apex section this way.
Essentially, even though we want to measure Munck’s model of the Great Pyramid in Morton’s Royal Cubit of 1.718873385 ft, once we have done this to recover its original proportions in that unit and hence in Imperial as well, then we can go back and remeasure it based on these original proportions, in a Cubit of 1.722570927 ft, in order to to recover additional important data (essentially, if we follow this lead, one thing it illuminates is the Egyptian Sacred Cubit), regardless of whether 1.722570927 was ever a genuine Royal Cubit or merely a mathematical constant similar to Morton’s Royal Cubit value.
This “Stecchini” Cubit drew some plausibility from being able to boast an origin and a standard in basic components of circular geometry, just as Morton’s Cubit could.
Morton’s Cubit can be constructed by squaring 360 and dividing by 2 Pi to obtain the Royal Cubit value in inches x 10^n; “Stecchini’s” Cubit can be constructed by halfing and inverting 360 then multiplying by Pi^3.
360 / 2 = 180; 1 / 180 = 5.5555555555 / 10^n; 5.5555555555 x (Pi^3) = 1.722570927 ft x 10^n.
For reference, the value was given by Stecchini as 525 millimeters
525.0000 mm = 1.722440945 ft
We may also wish to note that in addition to this Cubit Rod, Stecchini also called attention to a cubit rod of 524.1483 mm x .0003281 = 1.719735072 ft, which may either be Morton’s Cubit of 523.912607 mm or the incidental “Long Cubit” of 524.2915417 mm; another rod was cited of 528.3231 mm x .0003281 = 1.733428091 ft, which may be another “ceremonial” or honorary Cubit value, and may well be the honorary Cubit-like value that was included in Teti’s pyramidion as the true link between two very important unit values, even though it may never have been considered a Royal Cubit at all. (We now know that this cubit-like value is actually a measure in Stonehenge’s “Outer Sarsen Diameter Unit”).
I have stated in the past that we may occasionally see values of 1.722570927 or its simple multiples or fractions as ratios or products generated by architectural designs because it has some relatively minor astronomical significance, but I didn’t realize at the time that it was also intimately related to the Draconic Month. It appears there is still more to be learned about these things.
Now that we are learning more about metrological units that can be built on Pi exponents, the (Pi^3) in (Pi^3) = 1.722570927 tells us that this “Stecchini Cubit” should be a member of the Draconic Megalithic Yard family. I might never have guessed without taking a more analytical approach.
I’m curious to see what future discoveries might be made concerning a family of metrological units based on whole numbers and (Pi^3).
When Grasp Equaled Reach
In commenting on the palette of units on display at Stonehenge that seem most integral to its design, several times I have used the expression that “sometimes their reach exceeded their grasp” – that they aspired to show us a few things they didn’t actually have at their disposal, but almost did, because it comes with the territory.
One of them is that a side effect of combining the Remen and the Hashimi Cubit as major units of the same structure eventually results in a disappointment, and the Megalithic Foot, another major metrological unit of Stonehenge, is also involved in this.
Remen = 1.216733603 ft Hashimi Cubit = 1.067438159 ft Megalithic Foot = 1.177245771 ft
Thus in “Imperial”, our Great Universal Ancient Reference Unit, Remen / Megalithic Foot is almost the square root of the Remen, and the Hashimi Cubit is almost the cube root of the Remen – almost.
I presume the designers were aware of this, just as I’m aware of it, but decided to go ahead with this collection of units anyway, in spite of it being able to potentially cause some confusion or disappointment in just this way because of the great many benefits of such a combination.
Naturally we want to start tweaking these numbers, adjusting them slightly to make these “almost” equations come out true, but speaking from experience, it’s probably not just worth it. It may not even be mathematically possible. This may be about as close as we’re ever going to get. If anyone can demonstrate otherwise, please be my guest.
In fact, the real cube root and square root of the Remen don’t even belong to this system of numbers. The very system itself considers them invalid.
On the other hand, we should be careful and not be too hasty to think of the Remen-like numbers generated by Stonehenge’s “almost” equations as useless mathematical miscreants. Technically, because they are the product of multiplication operations involving two or more valid constants, they are valid numbers, and I have seen them rise to the surface just often enough in the course of experiments to know that however obscure they may be, they may have some important functions that have yet to be appreciated.
It’s also really not worth second guessing the particular primary unit values listed above for the Remen, Hashimi Cubit, and putatively the Megalithic Foot – they are time honored and true. I have been putting them to the test for the better part of 20 years now.
I recently had the good fortune of stumbling over several equally ambitious metrological equations that actually do worth, and then an hour ago while searching for something totally different (I was checking to see what’s been written on the Hipparchic Cycle – not nearly enough apparently), I stumbled across another older formula that I’ve been trying for several weeks to recover.
Here are the two more recent equations – the first one may not really be that recent a revelation, but I cannot yet locate any other history of its discovery besides the note I put to paper two days ago.
Talk about something that makes me feel better about suggesting the crazy idea that a squared Megalithic Yard at Stonehenge is represented in root form as a pair of two different values, an “Alternate e’ Megalithic Yard (AEMY) and an “Incidental Megalithic Yard” (IMY) that together multiply to form the Squared Munck Megalithic Yard. It’s a bit bizarre and complicated both, but it is what the mathematics mandates, and more important, it actually works.
That is the nature of the squared Megalithic Yard, though – it is such that Munck’s value is what has the desired exponential value, even while its natural square root isn’t even valid according to the system it belongs to. “Squared Munck Megalithic Yard” is actually a misnomer because no true Megalithic Yard is involved in creating it as a square in the literal sense, but even though know we know it is 9 / Remen, no better descriptive has come along to date than “Squared Munck Megalithic Yard”.
Assyrian Cubit Cubed = 4 Hashimi Cubits
1.622311470^4 = 4 x 1.067438159
Here is the equation I have been several weeks trying to figure out where to look for. I posted about it at least as early as May 19 of 2020.
(2.719715671^2)^3 = 404.707285; 404.707285 x 4 = 1618.829140 = “Not-Phi” x 10^n
This is yet another fact that increases my confidence in the value of 1.618829140 as legitimate and intended. In spite of being able to say that this is a close adapation of a real Phi-like value common to every true pyramid with perimeter / height ratio of 2 Pi like some of those we find in Egypt, I realize that some people must find this number (and some of the places it has been found) bizarre enough that I still feel a bit apologetic about it. Suffice it to say it has proven itself time and again, and numerous architectural designs have been found that justify taking it to be second only to mighty 1.622311470 (the ordinary Assyrian Cubit) as Phi approximations go.
Recently we learned because of Stonehenge’s promoting, that as a metrological unit, “Not-Phi”, or the short Assyrian Cubit, is actually a form of the Hashimi Cubit, itself a form of the Egyptian Royal Foot. Some sources will make the Egyptian Royal Foot out to be a somewhat pointless 2/3 of a Royal Cubit, but the Great Pyramid’s chamber is one source that gives us the real story.
The often-seen diagram of the Pyramid of Niches at El Tajin (in Veracruz, Mexico), showing how the two most important ancient approximations of Phi appear as its diagonals.The missing figure at the right of the stairs at the bottom of the diagram is thought to be 11.1519204 ft, an expression in Egyptian Sacred Cubits, while the suspected figure at the left of the stairs is in Hashimi Cubits or Egyptian Royal Feet. The width at center is the most straightforward sort of expression possible of the Megalithic Foot.
1.618829140 even seems to have built in to an entire class of Thom’s Flattened Megalithic Rings, even though a subclass seems to be indicated; working with data from Ronald Curtis in Records in Stone: Papers in Memory of Alexander Thom it appears to me that at least several and probably a few flattened stone circles from this class may have successfully substituted 1.622311470 for 1.618829140.
Note the minor diameter / major diameter ratio of the Thom type B flattened ring (AB/BN contains a typographical error; it should read AB/MN like figures i and iii). We can take 0.8091 as half of Phi: 0.8091 x 2 = 1.6182 (1/2 Phi = 1.618033989 / 2 = 0.8090416994) but for years I have using 1.618829140 successfully with this design.With much respect for Phi proper, there is more to life and astronomical mathematics than Phi, such that it can be greatly advantageous to merely approximate it.
So there we have three different metrological formulas that are just as ambitious as some of those implied by the collection of units of measure prominently used at Stonehenge, but unlike the Remen root formulas, these metrological formulas work, and they work precisely.
An Emphatic Summary of Metrological Unit Relationships
Here is a more concise graphic summary of metrological unit relationships.
Sadly, three of these have only come to light for me in recent years. For years, I had only the concept of relating units though the geometry of squares or rectangles, as proposed by John Michell and others. It was only because of Geoff Bath’s timely emphasis on the concept of diametral vs circumferential relationships (the unit of the diameter of a circle vs the unit of the unit of its circumference).
It was timely because it’s the same consideration applied to circles and their measurements that we have with squares when we are talking about square based pyramids – if the Remen is sqrt 2 to the Royal Cubit and the base of a pyramid is measured in Royal Cubits, why shouldn’t its diagonal (sqrt 2 to the side of the base) be measured in Remens and not Royal Cubits?
I’ve been looking at pyramids that way for ages, but the same may not hold true for Egyptology.
With circles, this also holds true that the unit of the diameter shouldn’t be the same unit as the circumference. It’s a wonder I didn’t trip over this sooner because of Thom’s inclination to want both the diameter and the circumference of stone circles to be in Megalithic Yards.
What I didn’t realize is that so many of the unit values that I’d worked so hard on to figure what was their ideal value, were all interconnected, precisely, by the Pi ratio. You can imagine my shock last year to discover what had been right under my nose the whole time, because certain things had gotten in my way on the road to this conclusion, including that I was still thinking of some very important unit value as mathemaical constants, rather than as mathematical constants embodied by metrological units.
If one doesn’t believe that 1.177245771 or 1.067438159 is a metrological unit, it breaks the exquisite chain that is formed from ancient units through the geometry of circles, joining them together very much like beads on a strong. I would say that our bundle of ancient units must most definitely have a great deal of circular geometry in its lineage.
The “Swiss Army Knife” concept of ancient metrological unit relationships again. Our core group of ancient units and unit families are all “joined at the hip” like the tools in this pocket knife, in as many as FOUR different ways – some of them approximate, some of them exact.
Some readers may have already been following along last year as I began to take more and more notice of the way that the expanding collection of units were related to one another through multiplications and division because of a rapidly growing number of cases of architectural ratios between adjacent proportions in diverse units, whichwould spell out of the value of different units in “modern” “Imperial” feet as a ratio, which I was referring to as the “unit a times unit b” or “unit x times unit y” phenomenon (division is equally involved).
That started way back when I first realized that the missing section of the Great Pyramid related to the whole pyramid at a ratio of 10 Royal Cubits in feet, but it wasn’t until more recently when I began to avail myself more heavily of Petrie’s publications and data that were becoming more widely available, that this line of inquiry really began to build up steam.
I’m going to try to find a way around the apparent error in the data for the cistern of Hadrian’s library in the diagrams by Athanasios Angelopoulos.
The specific issue is that if all of the proportions shown describe its measures inside the rim, 57.82 – (5.706 x 2) = 46.408 m, whereas it is labelled 46.16 m.
I presume that these are all interior measures of the cistern because taking pixel measures of both diagrams shows the ratio between inner width of the cistern and edge around the cistern to be quite close to equal especially considering the method. By projection then the cistern edge would have a width of very roughly 2.5 feet, which is about 4 times the error of 46.61 – 46.408 = 0.202 m = ~.6627 feet, so that this later difference does not seem to reflect the cistern’s edge width, but rather seems to present a smaller error of some kind.
Hopefully we can rest some extra confidence in the figure of 57.82 m since it is given twice as the length of the cistern and also as the width of the courtyard, and Angelopoulos specifically states that these two measures are equal.
If we can rest some confidence in the measures for the semicircles and their remarkable interpretive value, then the actual inner length figure for the cistern may be more likely 46.408 rather than 46.61.
46.408 m = 152.2572178 ft which is quite reminiscent of the Greek Cubit of 1.5 Greek Feet. There is some question about accuracy of approximation here which implies that perhaps they were using a different “360 / Half Venus Cycle” figure for the total length than the “A” version. 1.5 of the Greek Foot value I use customarily would be 1.013944669 x 1.5 = 1.520917004 ft.
If we re-work the equations, it rather looks as if out of necessity they may have meant the “C” version of the Half Venus Cycle that was first taught to us by Stonehenge, but I’m not certain of this and it might interfere with the parallels between the semicircular ends and the calculations for the Aztec Sun Stone unless perhaps the Sun Stonehenge somehow also incorporates the C value for the “HVC” as well as the A and B values.
Interestingly, the mean of the A and B values, 18983.99126 and 18997.72194 respectively, is (18983.99126 + 18997.72194) / 2 = 18990.85660; the HVC C value is 18990.40328 (accuracy of approximation for addition/subtraction = 18990.40328 / 18990.85660 = a very acceptable .9999761582), so it may indeed be possible that the Aztec Sun Stone and its apparent Mayan counterpart at Yaxchilan are even more clever than previously thought, and that the architect referenced the HVA C value of necessity without departing from the “Aztec Sun Stone” theme.
At this point, however, the use of the Greek Cubit is only speculative, and I don’t have the highest confidence level that “1.5” is the true ratio between Greek Foot and Greek Cubit either. I believe this is about the point where we may encounter some turbulence in the metrological schemes of Michell. We have seen the relationship between Egyptian Royal Foot and Egyptian Royal Cubit given as an equally questionable “2/3” even when it is demonstrably NOT the relationship between primary values of these two units.
The inner dimensions of the courtyard outside the columns are given as 73.468 m = 241.0367454 ft and 95.352 m = 312.8346457. The first expression, the width, does not appear to be in Greek Feet although perhaps it’s possible that the second expression, the length, could be 1/324th a Greek Foot x 10^n ; 1.013944669 / 324 = 312.9458855 / 10^n. This is somehow an unfamiliar number so I don’t know yet know what to make of it. The ratio between the two raw data figures is 95.352 / 73.468 = 1.297871182, which may be 51.95151515 / 4 / 10 = 1.298787879. I went though several phases where I was specifically looking out for this number and raw values that looked like this often seemed to turn out to be exactly that.
Assuming symmetry in design, we can obtain the width outside the columns as 73.486 – (7.47 x 2) = 58.546 m = 192.0800525 ft; the length outside the columns is given as 84.454 m which appears to be adequately consistent with the determination (95.352 – 5.45 x 2) = 84.452. 84.454 m = 277.0734908 ft. I am not sure offhand what either of these values is supposed to represent. Neither figure appears to be in Greek Feet in the Inductive Metrology sense.
277.0734908 could represent the Sidereal Month and/or reciprocal of the Solar Leap Year; intriguingly 192.0800525 in Greek Feet is approximately the HVC “B” Builder Figure. The ratio between the two raw data figures is 84.454 / 58.546 = 1.442523827. One of the more attractive figures in this range might be 1.441041519 = (1 / Metonic Cycle 6939.425799) x 10^n, although a fraction of the Eclipse Year (ideally 346.5939351 / 240 = 1.444141396) might also be possible.
It appears as if the inner courtyard length is supposed to be 80.664 meters with 80.479 m as a speculative alternative, as it appears may also be the case with the figure of 57.82 m where 57.76 is seemingly offered as a speculative alternative. The figure of 80.479 as an actual label would seem to be be misplaced as it appears over a longer rather than a shorter part than is labelled 80.664 m. 80.664 m = 264.6456693 ft, which is roughly ten times the value of the width of the Propylon (8.05 m = 26.41076115 ft).
These do not appear to be in Greek Feet unless perhaps 1/384 of a Greek Foot times 10^n, but rather may be measures in Indus Feet. The ratio between the two raw data figures for length and width is 80.664 / 57.82 = 1.395088205, which might be either 1.3941274004 (a simple fraction of Munck’s “Giza Vector Grid Point”) or 1.397273286 (a simple fraction of the proposed primary meaning of “56” at the Aubrey Circle).
For the two figures for the span outside the columns to the inner perimeter of the outer courtyard, 5.45 m = 17.88057743 and 7.47 m = 24.50787402. One thing this second figure in feet might be is (1/48th of a Megalithic Foot) x 10^2, or perhaps possibly 24.46388062, a simple multiple of the inner circumference of the Stonehenge sarsen circle (46.61 m taken literally probably would have meant 1/2 the inner sarsen circle circumference). 17.88057743 does not appear to be in Greek Feet. It may be the reciprocal of the suggested primary approximation of “56” at Stonehenge (1.789199025 x 10^n), or perhaps alternately, the product of Egyptian Mystery Unit (“LSR”) x Hashimi Cubit (1.789803389 x 10^n).
The ratio betwen these two figures is 7.47 / 5.45 = 1.370642202, a range that sometimes requires some thought as to what we are seeing. For example (Pi / 365.0200808) / 2 Pi = 1.369787654 = 1 / 73.00401616) x 10^n, while Michael Morton had a notable fondness for 1.370778390 = (1 / (72 x (Pi^2))) x 10.
That is most of the readily available data on the courtyard given a cursory treatment then; the ratio 97.24 / 95.352 = 1.019800319, which surprisingly might also be valid data and could represent numbers like 1.019328359 or perhaps even 1.021521078, which may be the two best candidates in this particular range.
Two of the next steps we might take are to consider any possible good alternatives to some of these suggestions, and then to begin to see how well any of the suggested component values actually fit together into a cohesive whole.
The relationships between successive courtyard lengths are 95.352 / 84.454 = 1.129040661 = ~(1 / Lunar Year) x 4 x 10^n, and 84.454 / 80.664 = 1.046985024, quite likely to prove to be (Pi / 3) = 1.047197551 (probably a nice Tikal-style style complement to some of these numbers that may be effective at mining additional data from them).
The corresponding relationships between the courtyard widths are 73.468 / 58.528 = 1.255262438 (4 Pi = 1.256637061) and 58.528 / 57.82 = 1.012244898, which may be the Greek Foot in Imperial as ratio, and presumably the intended value is a bit closer to the traditional long Greek Foot value 1.013944669 although the short value of (10 / (Pi^2)) ft may also be possible. 1/16th of at least one form of Phi is probably also in this range (1.618829140 / (16/10) = 1.011768213)
The projected diagonals are: for 73.468 and 95.352 m, d = 120.3725506 m = 394.9230661 ft; for 58.528 and 84.454 m, d= 102.7521528 m = 337.1133621 ft; and for 80.664 and 57.82 m, d = 99.24632636 m = 325.6113070 ft. These do not appear to be in Greek Feet with the possible exception of 337.1133621 (Greek Foot / 3 = 337.9815564 / 10^n).
The diagonals across the corner of the cistern after correction would be about sqrt ((12.49^2 m) + (46.408^2 m)) = 48.05936500 m = 157.6750820 ft, which may be 1296/10 Remens = 157.6886749, or in Greek Feet, 15552 / 10, with the value as expressed in Remens probably being the more economical whole number. 157.6886749 x 2 x 10^n = 31537734.98, the number of seconds in a year of 365.0200808 days.
The length between two circles projected from the semicircles at the ends would be about 57.82 – (5.706 x 2) = 34.996 m = 114.8162730. This may be 114.83806170, which was found at Tikal as its reciprocal, Squared Munck Megalithic Yard x 1.177245771.
The width across the squarish alcoves of 9.72 m = 31.89976378 ft may not be written in Greek Feet per se, but may represent the Apsidal Precession Cycle converted into Greek Feet: 31.89976378 x 1.0139 = 3233.303150.
Thus we conclude for now the first phase of the analysis of the courtyard of Hadrian’s Library.
Once again, the mathematics I am using uses a deliberately restricted pool of whole numbers.
Over time, the rationale for this has evolved somewhat. Fifteen or twenty years ago, if you had asked Michael Morton or I to explain this, we probably would have told you the reason for this concerned a desire for compatibility with sexigesimal math (the math that gives us 360 degrees to a circle, 60 minutes to an hour, sixty seconds to a minute, and so forth.
Since the discovery that Petrie’s “Inductive Metrology” apparently can be successfully applied after all to ancient architecture as a ground level analytical tool with only a slight expansion of the list of recognized metrological units, it also comes to light that even with this restricted set of whole numbers and this small number of ancient units of length measure, once the basic simple multiples and fractions are projected, there is already such a complex set of numbers generated that to not impose such stringent restrictions on the pool of acceptable whole numbers would generally result in unfathomable mathematical chaos.
One consequence of this restriction, is that it simultaneously loosens restrictions on the interpretation of excluded numbers; the rejected number 7 might be represented as 360/10^n Assyrian Cubits, or as ((1 / 45) x Pi) x 10^n and probably others.
In previous work on Stonehenge we see how this affects the “56” post holes of the Aubrey Circle. Since 56 is an excluded number, multiple approximations of 56 come into play there. Thus “56” is being used at Stonehenge as a “wildcard” with multiple correct interpretations, which hails back to the question of how we represent such decimal numbers as 55.89093145, which is probably the primary meaning of 56 of Stonehenge, through a number of architectural features.
This flexibility seems to correspond to the flexibility that may be required to use numbers near to 56 in astronomical equations.
Because of these, I have never tried to map out a list of likely primary approximations or definitive interpretations of excluded numbers, as there may not be any “definitive” interpretations; rather the interpretive value of indicated excluded numbers may depend on situation, context, and even some artistic licenses on the part of the architect.
As such, I really have no stock answer to the question of how we are to represent the excluded number 19, yet it is ultimately a question that needs to be dealt with to gain a better understanding of the representation of the Metonic Cycle. It’s probably fair to say that there is considerable mathematical evidence that the Metonic Cycle was known and used well before the time of Meton, its namesake who is generally credited with its discovery, yet the exact mathematics that is applicable has thus far remained somewhat elusive.
The great likelihood is that the ancient Maya were also quite aware of the Metonic Cycle and its utility, and yet they how they would have dealt with it and applied it exactly even in orthodox terms seems to also remain rather mysterious.
I’ve mentioned on a number of occasions that I don’t seem to have received much guidance from the ancients as to how to use approximations of 19, because every time they get up into that range, they seems to have some mandate to talk to about the Half Venus Cycle instead, and even that the Half Venus Cycle may often be what we are really seeing if we think we are seeing “19”.
So soon after the possible finding of some extremely noteworthy Aztec and Mayan calendar math in the cistern of Hadrian’s library complex, we can observe that while approximation of the Metonic Cycle as “19” Solar Years generally rests on the inference of the role of the canonical Solar Calendar Year, introduction of the Solar Year proper into this “Greek” formula tends to generate numbers resembling some of what would probably have to be considered the more controversial approximations of the Mayan Calendar Round.
So you can better see what I mean, let’s look at some equations
Textbook value” of Metonic Cycle (Wikipedia) 6939.688 days 6939.688 / 365 = 19.01284384 years 6939.688 / 365.25 = 18.99983025 years 6939 / 365 = 19.01095890 years 6939 / 365.25 = 18.99794661 years “B” value for Half Venus Cycle = 18997.72194 days
The Metonic Cycle being a relatively large number, it is somewhat insensitive to small changes in input parameters in operations of division – it almost seems at first glance that there is no real distinction between “6939.688 / 365 = 19.01284384 years and 6939.688 / 365.25 = 18.99983025 years”, yet the same equation can also be very sensitive to changes in input parameters when seen as operation of multiplication. It may withstand that since we may be given some tolerance; representing 18980 as 18997.72194 adds some 17 days per approximately 19000 days, the maximum acceptable error or approximations thus far; proportionately, the figure of ~6939 would be allowed approximately +/-0 6.2 days, making the valid range for Metonic Cycle approximations to be about 6932.8 days-6945.2 days.
However, adding to the confusion is the fact we can generally find ways to approximate the Metonic Cycle with reasonably good accuracy so that we do not require such a wide tolerance. The cube of the reciprocal of 2 Remens, for example, has proven to be fairly pleasing and useful – and rather accurate – approximation of the Metonic Cycle.
From an ealier discussion,
“Step 1: Obtain the reciprocal of the Double Remen, as previously shown.
(1 / (1.216733603 x 2)) x 10 = 4.1093622960
Step 2: Multiply by 10
4.1093622960 x 10 = 41.093622960
Step 3: Cube this figure
((41.093622960 / 10)^3) = 6939.421817 x 10″
Mathematical probes have also returned some rather peculiar things regarding this formula of 19 Years = Metonic Cycle. I’m seeing at least one almost valid equation with an error ratio so low that I can’t recall seeing it ever before (making it that much more important that we work with tend digits).
This may prove to be related to the possibility of approximating the Metonic Cycle through a different formula:
((1 / Remen) x 10^n) / (12 x (Pi^2)) = 6939.425318
Where (12 x (Pi^2)) also represents the selected link between Venus Orbital Period and Half Venus Cycle, and which could therefore be conveniently recycled to facilitate a Metonic Cycle formula.
We might also recycle another number thought to be involved with calendar operations, 1.541011111 x 2 = 3.082022222, and multiply by the canonical Venus Orbital Period (“my” VOP B): 3.082022222 x 225 = 6934.55, which also falls within the projected tolerance range of 6932.8 days-6945.2 days. Currently, one candidate formula for the Eclipse Year is Venus Orbital Period x 1.541011111.
3.082022222 is thought to be encoded in an entire class of Thom’s flattened Megalithic rings; however it may be incompatable with Venus Orbital Figures for this particular purpose.
Likewise, we have another possible formula that may fail to deliver on promises of exactitude.
(365.0200808 / 192) x 365.0200808 = 6939.565593 / 10.
The cube of the reciprocal of 2 Remens nonetheless remains a promising possibility, including that recently we seem to have seen numerous examples from actual architecture showing the strong possibility that the ancients frequently referred to the Saros Cycle through a cube root figure as well.
One tempting possibility for a solution or alternative solution is that some branches of ancient astronomical mathematics may have opted to represent the Metonic Cycle simply by doubling the Eclipse Year: Eclipse Year 346.62 days x 2 = 6932.4, which depending on specifics might just pass for the lower end of the tolerance suggested for Metonic Cycle (min ~6932.8 days) that is proportional to the tolerance afforded to the Half Venus Cycle of necessity.
To do this with our best representation of the Eclipse Year, that is 346.5939350 x 2 x 10 = 6931.878703. Divided by the shorter calendar year of 3600 / Pi^2 = 300 Short Remens = 364.7562611 days, 6931.878703 / 364.7562611 = 19.00413904. However, 6931.878703 may prove to have little true affinity for more accurate representations of the Metonic Cycle.
For the record, 6931.878703 has already made an appearance as a projection in a set of experimental Metonic Cycle values based on the experimental formula, Saros x (1 / 1.053519238), which is some ways quite an attractive potential link between Saros and Metonic Cycle – yet the dynamics of this have managed to exclude what are otherwise probably the best candidates for the for Metonic Cycle.
Another experimental set places 6939.425316 in the “A” position and 6944.444444 ((1 / 144) x 10^n) in the “B” position, which also falls just within the proportional tolerance range of 6932.8 days-6945.2 days. This too may be worthy of some consideration.
Perhaps one other possible solution to at least some of this ((such as the larger errors) is that the Metonic Cycle was important enough to have been allowed two sets of values, which has seemed to prove the case for some of its possible components as well i.e., our mathematics hasn’t been forced to choose between different versions of calendar figures but instead has proven to be versatile enough to represent multiple versions of the Solar Year, Lunar Year and other figures that may merit such accommodation. It doesn’t seem that implausible that there were multiple forms of the Metonic Cycle, possibly depending on which version of the Solar Year we select as input (i.e., ~365 vs ~365.25).
If nothing else, this may be a good opportunity to review several points related to this.
In the early days of this work, it actually discouraged the idea that this math was about calendars in that the true Solar Year figure of about 365.25 days seems to extremely difficult for this mathematics to represent so accurately, believe it or not. Within the past year or so I discovered that it could be done – by dividing the Megalithic Foot by 12^n – but to do this requires such high exponential use of the number 12 that it will be rather difficult to justify its use in retrieval of the number, so essentially by the time we generate this more accurate Solar Year figure, it is already lost to us. For that, I’m not even going to mention what the number actually is, but anyone sufficiently curious can easily find out the answer to that for themselves given the clues.
What we seem to have been given by our numbers in the way of representation of 365.25 thus far tends to look more like 365.2767076 (133333.3333 / 365.0200808) and 365.2840914 (365.0200808 x 1.000723277).
(Note that Wikipdia informs us “Meton of Athens judged the cycle to be a whole number of days, 6,940” and that used in the canonical formula, 6940 / 19 = ~365.263).
Generally, on occasions when our equations give us something more like 365.25, it will be 365.2767076 or 365.2840914. At least one of these may be found in the Megalithic landscape as well as in Egypt; Alexander Thom’s Type A Flattened Ring possesses a “theta angle” of 19.10660535* which may have been intended as a square root of such a figure: 19.10660535^2 = 365.0623680. In terms of valid ways of representing this, sqrt 3.75 x (Pi^2) = 5 Egyptian Remens x Pi = 19.11240674 = sqrt 365.2840914.
19.11240674 is 1/16th of the calculated inner sarsen circle circumference of Stonehenge, and 1/32nd of the calculated slope length of the Great Pyramid from the base (and yes, that’s one the more obvious Stonehenge-Giza parallels of which there are a great many, that the Great Pyramid’s slope length works out to precisely twice the Sarcen Circle’s inner perimeter).
We might even be tempted for a moment to think that 19.10660535 could pass for “19” but again the multiplication operation is sensitive to variance, so that (sqrt 365) x 365 = 6973.315289, adding an unacceptable 34+ days to the Metonic Cycle – roughly twice the acceptable error even for the much larger Calendar Round.
To interpret these results so far as optimistically as possible, we may already have the very pieces we need to complete the picture; it may merely be a questions of finding an ordered way of organizing them.
Since I’m still not certain if the subject quite merits its own post, I will add here that our best value for the Eclipse Year also poses something of a metrological problem. Although, like many important numbers, it can be fashioned from established metrological units, in this case even the convergence of multiple metrological unit pairs on this particular Eclipse Year value does NOT seem to produce a quantity that can be readily identified as any of the 12 metrological unit families thus far.
Thus either something highly unusual is afoot here with the Eclipse Year value, or there may be a 13th family of unit values that has been overlooked.
I know that once I’ve made comment to GHMB about finding a metrological synonymy that was “masked”, which is so highly unusual as to be virtually unique so far, but I’m unable to find the information now. (I was certain I had specifically used the term “masked” but it has managed to evade search engines so far).
Could it be a case like that, or is it case of a still unidentified family of ancient units at work?
At any rate, even while it’s challenging to determine exactly what the ancients did about the “Metonic Problem”, we hopefully at least have what we need to see why it may not be unreasonable that the Metonic Cycle may have enjoyed universal use, well before the time of Meton of Athens.
If it takes 18 pages of calculations to do it, one of these days we will hopefully see the answer to the “Metonic Problem”.
In fact, before closing, I will check one more archival source for information about the Metonic Cycle that might have come from previous work, and I should also want to remind people of Jim Wakefield’s remarkable paper on the Rollright Circle, From the Rollrights to Stonehenge, in which he deftly developed a model in which both the Lunar Month and the Metronic Cycle are incorporated though the mathematics of a circle.
I haven’t felt the need to adapt Jim’s model although I’m aware of some possibilities of doing so, but this may also be a place to look for general guidance if the Metonic Cycle insists on being problematic.
This is from some of my notes which are dated to Jan 11, 2021 and are associated with experiments with projecting hexagonal geometry onto the Stonehenge model. While speculative in terms of where such a thing can actually be found in ancient architecture, it nonetheless identifies a notably well-integrated astronomical series.
“One reason that 43.60169520 may be attractive in spite of possibly being a relatively large departure from the raw data, is because of the way it responds to the all-important circular constant 2 Pi
When exposed to 2 Pi, it seems to becomes at very least a minor masterpiece of astronomical mathematics, forming a series at least as long as being bookended by both Metonic and Saros Cycle candidates, with other figures significant to astronomy also in the series
43.60169520 x (2 Pi)^1 = (1 / 365.0200808) x 10^n 43.60169520 x (2 Pi)^3 = Jupiter Orbital Period B / (4 x (10^n)) 43.60169520 x (2 Pi)^4 = 6795.522395 (Lunar Nodal Precession 6793 days) 43.60169520 / (2 Pi)^1 = 6939.425318 / 10^n (best value to date for Metonic Cycle of 6939.688 days
This Metonic Cycle value is the cube of the reciprocal of 2 Remens.
(1 / (1.216733603 x 2))^3 = 6939.425318 / (10^5)
I want to emphasize that the “Lintel Circle Megalithic Yard”, 1/16th of 43.60169520 is simply a variant of the standard value I use for the Megalithic Foot
Therefore it seems perfectly accurate to suggest that the preceding display of astronomy and geometry magic in a series is brought to us courtesy the Harris-Stockdale Megalithic Foot.”
I also find a note reminding us that “Pi Jedi” have known 6939.425138 in its reciprocal form for quite some time now.
“144 x 1.000723277 = 144.10141520 might be preferable here because 1 / 144.10141520 = 6939.425313 / 10^n, most likely one of the better approximations of
the ~6393.688 day Metonic Cycle.”
There is also a note on how what is still ostensibly the best suggestion for the value of the Metonic Cycle has a geodetic connection.
“It also seems to be linked to geodesy: 6939.425322 / (12^3) = 1 / 24901.19742”.
I still don’t have any definitive answer, but that is hopefully most of the relevant observations concerning the Metonic Cycle to date, now all in one place to help offer guidance to any future attempts.
This is something of a difficult subject for me to get into for fear of ruffling more feathers among fellow researchers. I’ve no wish to do that, I very much wish to encourage the work of other researchers. I have no idea where their journeys may take them or what they may discover – perhaps even something that is capable of making me reconsider my own views – and I’m very much of the opinion that it’s unlikely that anyone can play around with a pocket calculator for long without making significant and important discoveries related to ancient mathematics, so more power to them.
Frequently, the work that others do, and the work that I do, or at last significant parts of it, can ideally be translated from one system to another very simply. If we map out a solid metrological infrastructure, hopefully its expressions are conserved even when the unit values vary somewhat from researcher to researcher. One example of this that is still fresh in my mind for having just mentioned it in a GHMB discussion, is that regardless of the suggested Remen value, hopefully the expression “300 Remens in Imperial feet = Solar Year in days” remains a valid one in anyone’s proposed system of numbers.
Beyond that, I’m generally so preoccupied with my own direction of inquiry that it tends to exclude the possibility of knowing the systems of mathematics that others propose intimately enough to be in any position to criticize. I always find it very uncomfortable when my work faces criticism from individuals whom I can tell have very little grasp of what they’re criticizing – typically half the critical remarks refer to what these critics only think is being said, and the other half of the criticism is a parroting of the dogmatic mainstream views of the ancients “not having this” and “not knowing that” when our history is in fact so tattered that it’s very difficult to feel certain of what they did or didn’t have or know.
I have seen others suffer this too. One researcher who comes to mind is Don Barone, who posted to various places his research on planetary cycles being expressed in the design of Giza. Something I’d like to do but still haven’t found the time to do, is to compare our work and see how compatible it may be. Don’s work uses a number of classic lower square roots, which are often one of the factors that are translatable from one system to another, requiring only very slight adjustment to the traditional exact values of sqrt 2, sqrt 3, and sqrt 5 to make them compatible with idealized or geometrically standard values for units of measure. I had to grimace (and still do) when a would-be detractor replied to Don, “The ancient Egyptians didn’t know what decimals were, I just looked it up on Wikipedia”.
I generally prefer then to do more constructive things that criticize others, including learning from them what I can, because no doubt that most persons working in areas like ancient metrology tend to have the encyclopedic knowledge that may sometimes be required of them, and indeed it’s difficult to think of a fellow researcher to whom I am not indebted for knowledge, inspiration, or valuable references, referrals, and links.
However, occasionally I come across an objection to someone else’s proposal that tries to stand on its own merits and own terms, rather than having to do with the fact that I tend to see and use mathematics in a slightly different way. I can think of several who, after the fashion of even Petrie and other scholars, have set about trying to interpret all of ancient Egypt in Royal Cubits. My objection to this does not stem from my own opinion that the ancient Egyptians knew and used a dozen metrological units simultaneously, but rather the observation that if it is at least accepted that the Remen existed as diagonal to the Royal Cubit, as Petrie taught, with as many Egyptian pyramids as there are whose bases seem to measure in nice round numbers of Royal Cubits, shouldn’t their base diagonals at least be in Remens and not Royal Cubits?
There is a fair amount of alternative metrological research, however, that continues to have me moaning under my breath about the “increasingly absurd collection of fractions required to keep it from falling apart”, whether or not that that really is a fair or accurate description.
We have thankfully managed to move the goal posts in a more positive direct on this matter in a cooperative fashion – that is, rather than asserting the premise that ancients used simple whole numbers because they were too primitive and ignorant to do anything else, that they may have done so simply because they found them particularly effective.
Nonetheless, this may still only go so far toward painting a sensible picture.
Curiously, I am not aware of many researchers who acknowledge the academically accepted idea of multiple metrologies who doesn’t nonetheless ultimately end up trafficking in fractions so mind-numbingly complex that most of their own expressions end up written in the decimal notation that they usually end up insisting themselves that the ancients did not possess, because that often seems to be what we modern intellects have to do in order have comprehension of what these complex fractions represent.
I know of NO researchers advocating the use of these “increasingly absurd fractions” who finds them suitable to work in exclusively, and I find myself questioning the logic of using such fractions or of the ancients having used them, if the proceedings inevitably lapse back into decimal. Unfortunately, even such eminent names in metrology as John Neal can be cited as prime examples of this.
Back at the beginning of my re-acquaintance of ancient mathematics – at the time I have no intention of getting involved again but I happened to stop in on a thread on the Great Pyramid and it set me reminiscing about Munck’s work and Morton’s and mine, and eventually one thing led to another – I crossed paths (and light sabers) with one researcher who had taken the idea that “the ancient Egyptians only knew how to work in fractions” to such an unfortunate extreme that his readers were seeing proposals of complex expressions from modern physics like Planck’s Constant or what-have-you, written in fractions approaching ten digit numbers over eight digit numbers – fractions so complex that if we really stuck to what we know about Egyptian math from the surviving shreds of evidence, they’d have likely needed a whole roll of papyrus just to write a single one of these absurd numbers in authentic ancient Egyptian unit fractions.
Even for as little as I care to disparage the work of others, about all I can think of that is, “You GOTTA be kidding me – you think the ancient Egyptians were smart enough to figure out Planck’s Constant but you still think they were too stupid to invent the decimal system?!?”. Methinxt someone got hold of a rudimentary education on Egyptian mathematical papyri and an on-line decimal-to-fraction converter, and the intoxicating combination must have gone straight to their heads.
In a way, I can say I know what lies further down that road of using “increasingly absurd fractions” to hold systems together because I’ve seen it for myself and in my opinion, it isn’t very pretty – or sensible.
Sadly, because I don’t necessarily have to the time to go through other people’s work and convert these fractions into decimal (you should see how much time I already spend just going through architectural data and converting meters to feet) so that they are compatible with the rest of the same work, I don’t have quite the understanding that I should of just how much trouble some may have gotten themselves into this way or not.
If the work advocating fractions were presented in fractions entirely, we might already see whether or not some may have gotten into trouble with these elaborate fractions possibly lacking common denominators that would permit their reduction into simpler and more useful expressions, or whether some of these elaborate fractions and statements formed from them cannot effectively be reduced.
This is in no way meant to suggest that the ancient mathematicians weren’t smart enough to not be stumped by such matters, but it’s more of a question of how to get from point a to point b mathematically. If it requires conversion into decimal just to have a common denominator to solve a fractional expression, then why not just give up the fractions and use any ultimately inevitable decimal notation exclusively? There is no more surviving historical evidence of elaborate fractions like many of the ones in question, than there is of the ancient having used decimal.
In my work, one of my personal objections is not so much to other people’s systems of numbers, but to many of the numbers that are in them. Some of these numbers belong to “my” system of numbers as well, but the model I am building emphasizes how exponential use of certain numbers, and the use of certain numbers that are amenable to exponential use, can expand the data storage and retrieval capacity of ancient artifacts and architecture by as much as 30 times, or even more. The same is not likely to be able to be said about many whole numbers, including many of the ones that find their way into fractions commonly seen in the work of many other researchers, and even some of the ones I (reluctantly) sometimes work with as well.
Many of these numbers find greater service to us, or to ancient architects, if they’re “souped up” by the ratio of 1.000723277. Particularly mindful readers may note that for all that has been said by others about Egyptian “Eye of Horus Fractions” and a “hekat” of 64, my perimeter/height ratio for the revised model of the Mycerinus pyramid isn’t 64/10, it’s (64 x 1.000723277) / 10 = 6.404628973, and the reason for that is that it’s a much more useful number, including that if we’re fortunate, we may see exponential utility from the number and a corresponding increase in data storage and retrieval functions and in connection to other important numbers.
The very same goes for 64 / 4 = 16 or 6.404628973 / 4 = 16.01157243. While the number 16 may occasionally be a fact of life (it does have some important roles to play), not only can we expect much more out of the number 16.01157243 than we can the number 16, but we can even find 16.01157243 built right into the standard model of the Great Pyramid, making the suggested perimeter/height ratio I’ve given for Mycerinus’ pyramid its close kindred.
Perhaps an even better example is the number 81. It belongs to “Munck’s” system of numbers and has some remarkable mathematical properties, including 1 / 81 = 1.234567901 / 10^n, but we may not be able to expect much from it exponentially, whereas 81.11557390 is half of 162.2311470.
1.622311470 is an ancient number so important that again and again and again we find the evidence that it was a number that ancient architects strove to incorporate somewhere, somehow into every one of their designs, which is really not that surprising considering its literally profound properties of facilitating ancient ancient data storage and retrieval.
At present, 2 / 1.622311470 still constitutes the second most powerful mathematical probe ever discovered, and over the course of my blog posts so far we have seen where this both this figure and its direct progenitor 1.622311470 seem to be represented in architecture over and over, over a span of thousands of years and at least 4 different continents.
Some may prefer to think that I’m simply conflating 1.622311470 with prevalent ancient use of the Phi ratio or “golden ratio” 1.618033989, but readers have seen me refer to the utility of 1.622311470 – frequently as high as the sixth power – and of 2 / 1.622311470 in powers into the twenties. I must insist that the ancients must have been just as capable of knowing the difference between the two as even I am.
Honestly, when was the last time you heard a researcher claiming to have the mathematics of the ancients all worked out refer to Phi having exponential value past the third power? Probably never, and I can’t recall such a thing either.
However that is ideally what we are after with any grand unifying scheme of ancient mathematics and metrology, is exactly that kind of infrastructure and internal connectivity. Whereas we could sit around scratching our heads until the cows come how about whether an ancient architect was working in digits or barleycorns, a more advanced level of metrological and mathematicalinfrastructure lends great truth to Richard Hoagland’s remarks that Munck’s math represents “the holographic memory of a race” – ultimately, it permits reconstruction of a vast mathematical network along paths of least resistance, from amazingly few fragments.
Unless I am tragically mistaken, we could if we chose to write the entire system I use and everything in it simply by writing the numbers 2, 3, 5 and sqrt 60, and store it, retrieve it, or reconstruct it from scratch the same way.
Everything that follows, a vast ancient vocabulary of numbers as “words”, can proceed from their interactions. If we conditionally allowed the use of invalid square roots as a temporary starting point, we could probably just draw a rectangle measuring 1 by 2 of the same unit, like the floor plan of the King’s Chamber in the Great Pyramid, although I somewhat doubt the King’s Chamber has ever been mined for data with the requisite zeal to extract the entire system from it methodically unless it was secretly Munck’s own starting point, although he may have been missing major parts of the system for glossing both over astronomy, and glossing over some high quality architectural data sources well.
To get back to the particular subject, intriguingly enough if our example 81.11557390 were a “souped up” number, it’s not one that’s made from a whole number “souped up” with 1.000723277. The ratio between 81.11557390 and 81 is 81.11557390 / 81 = 1.001426830, which also happens to be the curious ratio between the size of Great Pyramid measured from the base, and the size of the Great Pyramid measured from the projected pavement level (or in other words, the ratio between my model of the Great Pyramid and Munck’s model of the Great Pyramid). Even to that ancient madness there may have been precise and meaningful method.
I haven’t talked that much about the ratio between size from pavement and size from base for the Great Pyramid (also known as the “unpaved/paved” ratio), I’ve probably been happy to think any readers were making use of the opportunity to get to know 2 Pi better before I fling any of its dopplegangers at anyone, but if the “Pi Jedi Academy” can boast yet of any advanced students out there, they’re probably asking what business I have proposing that such an ungainly monstrosity as that has any place in the Great Pyramid when all things that live in the shadow of the Great Pyramid’s unparalleled physical modelling of 2 Pi must answer to 2 Pi.
Good point – if you run the numbers forward, it will look like I’ve committed the ultimate sacrilege and mutilated 2 Pi to appease some half thought out hypothesis that generates nothing but nonsense in actual practice. Pay no heed to any resemblances to 2 Pi in that part of the proceedings, 1.001426830 is an unusual number that has an unusual response to 2 Pi. Here’s how it looks “backhand”
2 Pi / 1.001426830 = 6.274233021 (that’s not 2 Pieither, don’t worry); 6.274233021 x 4 = 25.09693208, which in inches is not only a putative Egyptian Sacred Cubit, it’s the putative Egyptian Sacred Cubit that is forged from the union of what appear to be ancient Egypt’s two most sacred units: standard (Morton) Royal Cubit of 1.718873385 ft x standard Remen of 1.216733603 ft = 2.091411007 (ft) x 12 = 25.09693208 inches = 6.274233030 x 4.
I think that was almost literally biting Isaac Newton the nose at one point; it was from Newton’s own calculations that I made that discovery of the nature of the Sacred Cubit. He would have had 1.213-something where I have 1.216733603 and may not have been in a position to distinguish the Sacred Cubit from the Palestinian Cubit unless he’d dug a little deeper into his figures, but it’s nearly ironic that if you look at his figures and ask one of the most natural questions for a metrologist to ask, “How much bigger was Newton’s Sacred Cubit than his Royal Cubit” you have a good chance of it trying to bit you on the nose as well. Maybe it was just my lucky day that that happened?
Hence, even if this ratio 1.001426830 (I wonder if it’s available as a fraction?) that occupies a very prominent place in the Great Pyramid gives a very limited response to 2 Pi, it seems to give a particularly poignant one – and even more so if we recall that we have recently identified the unit family of the Sacred Cubit as none other than where the proposed value of no less than almighty sqrt 60 (the most powerful mathematical probe / data retrieval tool)as the identity of Thom’s Mid-Clyth Quantum.
That’s about a hair’s breath away from Munck pulling out sqrt 240 (sqrt 60 x 2 = sqrt 240!) and telling us that “Thoth Himself” endorsed the utter madness of 1.001426830 with his own (mathematical) “signature”, which is probably not anything I’m in any position to disagree with.
As far as I can tell, nothing, not even a little detail like that, goes to chance in these ancient designs. Everything is accounted for.
Well, I’m glad if I’ve hopefully found my way down off the soapbox about fractions now. There wasn’t much else – there are some numbers you cannot write with fractions, some decimal numbers you cannot convert into fractions, and I suppose ultimately that I suspect that if there is any true common denominator to ancient mathematical systems, decimal is exactly what it may be.
If I’m going to rant and/or ramble, I do still try to make it worthwhile somehow, so there there are some more hopefully interesting things about the Great Pyramid and speculative ancient Egyptian mathematics that so far have been very rarely talked about.
Still, it always seem to return to haunt me that deductive reasoning is supposed to be if it has to be, a matter of making a list of all of the possibilities and trying to find reasons to cross them off the list one by one which in theory should leave the correct answer left standing. I’m well aware aware of the work of numerous others and would be in a position to include their work on my list of possibilities, but this is not necessarily a two-way street. I don’t imagine Michell or Neal had Munck’s work or mine in mind when they made their own list of possibilities to start eliminating. While we pay endless homage to Petrie or Michell, would they themselves be paying endless homage to Munck were they present and aware?
What I’m fairly confident that I may actually know, is that far too often I have seen people who are mainly completely unsuspecting of the true possible palette of useful numbers, latch on to loose resemblances between their calculations and whole numbers (whether they announce it or not this is Petrie’s “Inductive Metrology” in practice) – in fact, in my opinion, those has to be exactly what has allowed contemporary Egyptology to keep Petrie’s terrible metrological mistakes afloat instead of helping to clear them up
As long as one can pass 22.17 or 22.21 Cubits off as “22 Cubits” there’s never any need to go back and check if the figure in might be actually represent the use one of numerous established units instead. Thus while even Wikipedia knows there we all sorts of Egyptian units like the Remen and the Short Cubit and the Span and yet all Egyptology ever seems to be able to find in use is Royal Cubits!
At any rate, in my view these are even more definitely cases of trying to apply deductive reasoning without having a full list of possibilities in the first place, and at least in this case its resulted in sheer metrological disaster, not to mention leaving in its wake a modest hoard of archaeology enthusiasts who often want to argue with you when to try to help put things to right. Sorry to put it bluntly, but Corinna Rossi can vacation in Hades if she pleases as far as I’m concerned – what I resent is the thought of how many unsuspecting she may be dragging along with her, and the same goes for other authoritative sources who may be propagating folly – even if Petrie himself was the first of their number when it comes to the Royal Cubit.
Curiously I think most of the objections I’ve heard from people about my “digit strings” and “decimal strings” sound like they come from people who simply don’t want to memorize ten digit decimal numbers. Here’s a little tip: I don’t want to either, so I don’t.
What I do is learn to recognize numbers based on the first four or five digits. If one comes up I the calculator and I’m in doubt whether its the same exact number that it reminds me of, I use any simple mnemonic formula that will stick, to generate the right number to compare it with. Pi numbers (numbers based on Pi and a whole number) are of course absurdly simple to memorize or generate from scratch for comparison or enter into a calculator as a formula rather than a series of digits. “144 Pi” is name and a formula all in one.
The total number of numbers I have ever memorized to ten digits requires approximately all the memory capacity it took to know the telephone numbers of your two best friends, your mother, your fiance, your favorite restaurant that delivers, your doctor, and the police in case of emergency, back when phones weren’t quite so smart. I’m amazed if I still know what Pi is to ten places, I never practice it, I always enter it into the calculator just by pressing the Pi key.
So there you have some confessions of a Pi Jedi with too much time on his hands – I still can’t quite decide on a study for today and maybe one will have to wait until tomorrow. I though about getting into the Mazghuna pyramids – there really isn’t much left to them but they really are a tragically overlooked pair of Egyptian pyramids, specially since Petrie and/or Mackie were at least able to provide us with some data about their interiors. As usual, Keith Hamilton has some excellent “Layman’s Guides” for anyone wanting to know more about them and the relevant work by Petrie and Mackie is available from archive.org as are most of Petrie’s writings, if anyone wishes to have a look at the data.
The thing that really needs more investigation is the Greco-Roman architecture, but I don’t want to overburden myself with it or burn myself out on it. It’s still a bit of a bumpy ride for me as well. One day you discover something that looks characteristically Greek, and the next you find something that makes it looks likely that Aztecs or Mayans must have paid a visit to Athens. Working with any major branch of ancient architecture can be much more difficult in the beginning if a sense of what can be called typical is hard to come by.
I think every once in awhile, one too many people comes along who make me think that John Neal has as if knowingly coaxed them up a tree that now they can’t really climb down from – I’m sure it all started innocently enough with 175/174 – that I even get ticked off at John Michell just for having gotten Neal started, and if I get ticked off at Michell – well, don’t get me started on the gematria. Yes, it’s a real thing, and it’s someone’s custom and I salute that, but if there’s anything that must make it easier for skeptics to slur honest hard-working alternative metrology researchers as “numerologists” …
I must feel like the schoolteacher who finds themselves called on to make a public statement about the abysmal condition of the public school system in desperate hopes of inspiring corrective measures. It must be about as much fun as being buried in an anthill, but it’s very easy to feel like it might have been the right thing to do.
A few posts back, we took a first look at Hadrian’s Library based on the plans and data from Athanasios Angelopoulos, focusing on the Library itself and the adjacent reading rooms and auditoriums. The general assessment is that rather than consistent use of Greek Feet in the design, what we actually see is a variety of different ancient units being deliberately used, as we often see in so many examples of ancient architecture.
One always gets the sense that the architects expected and welcomed interpretive efforts, and frequently tried to put as many of the interpretive tools that we would need into our hands as expeditiously as possible. Part of that is proving to be the nature of the units themselves and their relationships to one another, and part of that is what we seem to see with Hadrian’s Library complex, where a deliberately diverse collection of units appear to be on display.
A good way of conceptualizing our basic metrological toolkit. (I’ve deliberately made the corkscrew the mile to reflect the fact that there may be some tools that are best suited only for certain tasks).
I will be honest and admit that I’m having a few interpretive troubles with Angelopoulos’ diagram, as to what measures belong to which parts exactly, so I’m not certain how much data I can recover from the plan. I may also be having a bit of trouble getting some of the parts to add up correctly as they are described.
I’m particularly intrigued with the pool (cistern), but this is one of the areas I find somewhat confusing. In this diagram by Angelopoulos, we can see that the cistern has a substantial rim that is perhaps as wide as a foot or even more. This would easily afford it with both inner and outer length and width values, whereas I only have single figures for these attributes.
However, we generally have reason to believe (based on experience) that the data is nonetheless valid and accurate until proven otherwise, and in that spirit I have attempted to look into the matter further.
Here I’ve adapted Angelopoulos’ diagram of the cistern and added several details along with conversions to “Imperial” Feet, our primary interpretive unit.
The semicircular sections at the ends are proving quite remarkable. Their radius value in feet would be 18.72047244, which hopefully readers are learning to recognize as 1/12 of the Venus Orbital Period in days, or as the value of the Petrie Stonehenge Unit in feet: 224.8373808 / 12 = 18.73644840, a value which can be reduced to Hashimi Cubits or Egyptian Royal Feet.
The diameter then would be 18.73644840 x 2 = 37.47289680. I sometimes comment that it seems curious that we don’t see this figure more, nor this value divided by Pi, because these figures have very significant and very powerful pedigree.
Readers have probably already guessed that the projected perimeter then is 37.47289680 ft x Pi = 117.7245771, so that the perimeter of the circle is measured in Harris-Stockdale Megalithic Feet (I am using the figure for the HSMF which I prefer of 1.177245771 ft rather than Harris and Stockdale’s stated figure of 10 x (sqrt 2 / 12) = 1.178511302 ft).
Let’s set that thought aside for just a moment so we can look at several other aspects. Assuming the cistern is a symmetrical construct, we can calculate that the widths excluding the diameter of the semicircles is at either side of the semicircle 0.539 m = 1.768372703 ft.
It’s possible that this figure represents 1.765868657 ft, which would be another expression in Megalithic Feet: 1.177245771 ft x 1.5 = 1.765868657.
Using these figures, the total projected width – I am presuming of necessity here the interior width – of the cistern would be 37.47289680 ft + 1.765868657 ft + 1.765868657 ft = 41.00463441 ft.
I find this too somewhat remarkable, because I’m fairly certain I recognize the figure. I’ve been seeing it since as long ago as my exchanges and collaborations with Michael L. Morton, but never before can I recall having seen it in a situation that is so suggestive that an ancient architect was aware of this unusual number and chose to commemorate it architecturally as directly as actual incorporation of the number as a physical measurement.
The number in question is, I presume, 41.00542110. While I cannot recall ever having found this number is ancient American architecture with any certainty, it entered the proceedings at very least as a useful abstraction. It did so because it is 1/2 of 82.01084221, which was part of an inquiry into whether whether the Mayan had used a larger number analogous to their known calendar number 819. 819 represents a formula that uses 364 days rather than 365: 819 = Venus Orbital Period 225 days x Solar Year 364 days = 81900; 225 x 365 = 82.125, and the reciprocal of the standard Remen is 1 / 1.216733603 = 82.18725919 / 10^n.
Thus 820.1084221 entered the list of potential candidates for the refined meaning of “225 x 365”. It boasted a Mayan pedigree, and even prompted the initial discovery of the “Best Lunar Month” value (giving rise also to rumors about a possible “Lunar Remen”).
The frequently seen diagram of Tikal Temple I by Carl Munck, labelled with Teobert Maler’s data.
82.01084221 = (1 / 1.219350970) x 10^n; 1.219350970 = (360 / 29.52390320) x 10; 29.52390320 = “Best Lunar Month”.
The “Mayan Pedigree” part is that the interpreted value of the length of Tikal Temple I’s temple platform = 38.81314681 exactly, and 38.81314681 x Pi = 12.19350969
The next thing I would like to call attention to regarding the diagram of Hadrian’s courtyard cistern is the length value 57.82 meters, which according to Angelopouos is also the width of the courtyard. 57.82 m = 189.6981627
Hopefully there are at least a few who are now equipped to recognize this number., or numbers like it. They belong to a class of numbers that we can call “Half Venus Cycle Builders” or “Calendar Round Builders”. They are too low to be valid figures for the ~18980 day Half Venus Cycle aka Calendar Round, but but we can divide 360 x 10^n by these “Builder” numbers to “build” valid figures for the HVC/Calendar Round.
The “Builder figure” for HVC A for example is 189.6334628: (360 x 10^n) / 189.6334628 = HVC A 18983.99126, and that may well be be what we are seeing here.
We find “Builder figures” along with actual HVC/Calendar Round values in Great Pyramid equations with relative ease.
It was probably beyond the scope of my blog posts to share this piece of history with readers, but it took me a long time to warm to the idea of these “Builder figures”. Originally I was against the very idea and thought of them as being little more than generally useless inferior versions of actual calendar Round approximations.
The first time I was finally forced to accept them, I was provided at the same time with the opportunity to understand why we are sometimes forced to write the HVC/Calendar Round “backward” as the “Builder figures” and why it can be very much worth the trouble to do so. This was early on in my Mesoamerican studies when I encountered the indisputable mathematical brilliance of the Aztec Sun Stone, from which I received tremendous encouragement that I was finally on the right track to understanding what all this ancient architectural math was for).
The Aztec Sun Stone, an ancient calendar formula calculator that uses measurements in “modern” “Imperial” Feet.
If we don’t deliberately invert the thickness projections for the Sun Stone into their reciprocals before performing the calculations, we will generate the “Builder figures” A and B rather than their “forward” forms, HVC/Calendar Round A and B, from the projected dimensions. Of course, all that we require to “build” the desired figures are these “builder” numbers and a 360* circle, which we seem to have been provided by the very shape of the artifact.
For what it’s worth, I can think of at least one example where the Maya seem to have built one of these things with the very same radius, diameter and circumference. The site is Yaxchilan and the location is structure S39 A4 where the artifact (“altar”) has the distinction of having been located indoors. The particular data point comes from older sources via Carolyn Tate’s book on Yaxchilan, but the subject was also covered by George Andrews.
Two different independent academic data sources provide the data from which the interpretation of the Aztec Sun Stone was based on.
Allow me to go ahead and cut to the “show-stopper” here – the presence of a HVC/Calendar Round “Builder number” isn’t all that Hadrian’s cistern has in common with the Aztec Sun Stone – the projected proportions from the semi-circles at the ends of the cistern, and the proportions determined for radius, diameter and circumference of the Aztec Sun Stone (and Yaxchilan “altar”) are identical except for the location of the decimal point.
I am greatly devoted to the proposition that until proven otherwise, these mathematical constructions are part of an universally well-distributed ancient mathematical language, but even then I am still rather surprised to see such a remarkable set of parallels as this.
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
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.
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.
Attempting to continue onward with a study of Grecian temple architecture, we now come to the Temple of Hephaestos at Athens. This surprisingly intact ancient temple is another for which Athanasios Angelopoulos provided us with plans and data.
It’s going to take some time and a fair amount of work to be sure to try to get a better understanding of the design. However, I can at least report that one of those numbers that I maintain ancient architects were eager to incorporate into every structure somehow (in this case 1.177245771, the numerical value of the Megalithic Foot in “modern” feet), has been found yet again in one of Angelopoulos’ data sets.
Detail from Angelopoulos’ diagram. 1.177539224 = ~1.177245771
Now for a bit more of the history of the craft as the prelude to the next item.
Not that we genuinely had the tools to succeed in our search, but Michael Morton and I were keenly aware that validating his astronomical work would probably end up calling for further validation, such as context-specific references.
What I mean by this is that if a temple or pyramid is aligned with a certain star or planet on a significant date like perhaps a solstice or an equinox (or whatever archaeoastronomy might define as a significant date for an alignment), we should expect that ideally, the architects will clarify by incorporating a numerical symbol of that star or planet into the architecture of that site in a fairly obvious way so we can be sure of what they wanted to talk about.
One problem is that I don’t think we quite realized what attributes would give us the best “lock” on the planets. I even spent considerable time trying to work out where the planets belonged among the stars in his “Archaeo-Sky Matrix” grid, but although there were some tempting prospects for planetary “Grid Values”, they never seemed to quite fit together as a system, which I think was when I first began to suspect we had hold of either the wrong methodology, a bad data source, or both.
In recent years, when I returned to this work, knowing the ancients wanted to talk about something with great gusto, but still not being sure what it was exactly, after one too many references to “calendar pyramids” it finally became clear that the Orbital Periods and Synodic Periods of the planets not only gave an excellent “lock” on the planets, but that once someone realized this, they could see the sun, moon and the planets being referenced virtually everywhere in ancient architecture.
I’m sure I’ve told people half a dozen times since then that “you can’t swing a cat in a Mayan temple without hitting the Venus Orbital Period”, which is almost literally true.
However, one of the things that helped make it possible for the ancients to achieve this grand planetary system and its mathematical descriptors is that the planets themselves are somehow rather amenable to it – multiply the Eclipse Year by 2 to approximate the Metonic Cycle, or multiply the Eclipse Year by 1.25 to get the Jupiter Orbital Period, and so forth. There are enough such convenient and helpful coincidences amongst the lunar and planetary periods that it’s almost enough to give one ideas about the solar system being the product of “intelligent design”.
This makes it hard to take Jupiter out of the Eclipse Year and harder to know which one was meant if only one was.
(There is more to it than that, I suspect. I predict that eventually it will found that ancient architecture is also making reference to things like Retrograde Periods, but it still remains to be confirmed what numbers they were using for this and where they are to be found. I’ve attempted it before but I think I have yet to really succeed).
In general, then, what we seem to actually find in practice is ancient structures brimming with all sorts of astronomical references that exceed the bounds of context-specific references. Expect to find the Venus Orbital Period no matter who a temple is dedicated to, and so forth.
While I was dwelling on the questions of just how important Hephaestos was, and what planet he might be associated with, I was reminded mythographically, I see Hephaestos as sort of a “Poor man’s Ares” (in other words, as actually being another form of Ares). He’s associated with fire, volcanos, and the blacksmith’s forge (where weapons of war are forged). As such, I imagine that an association with the planet Mars would be quite fitting.
Imagine my surprise then looking at the proportions of the innermost sanctum of Hephastos’ Athenian temple, and finding these diagonal values, facilitated by the curious flaring of the width of the walls near the ends.
To me, this looks remarkably not only like someone wanted to reference the 779.96 day Mars Synodic Period (in “modern” feet) but wanted to do so in a manner in which they might incorporate two of its primary values if they are using the same system I’m using, and multiple values are one of the very characteristic things about the system I’m using.
It remains to be seen if this proposal will fit with an overall interpretation of the proportions seen here, but I find this rather exciting because at least at first glance, it looks very much like it might be the sort of context specific archaeo-astronomy reference that Michael Morton and I always hoped we might might find some day, and the first thing it does is make me wonder how many others might have been overlooked for more or less giving up the quest some ways back.
Beyond that, there is still so much to be considered here. This is another case, and so very soon after the last one, where I almost get the impression that the architect was consciously working in Imperial and a variable ancient metric at the same time. (It may be that ancient metric can only have a highly variable nature given the unit values it co-exists with, which might well explain why its values have been so difficult to pin down that I’m still not all that sure of them after 20 years). Anyone attempting that is a far braver soul and a far better better mathematician than I am.
If we can ever prove that this happened, I will only be able to stand in awe of it.
Peter Harris recently posted his HSMF thread at the Megalithic Portal concerning the Waun Mawn site in Wales, where not only has evidence been reported of a stone circle, but it has been speculated that bluestones at Stonehenge may have originally stood in Waun Mawn circle before being brought to Stonehenge.
Being ever curious and happy if I can offer assistance, I jumped in with a bit of speculation.
Unfortunately, I think that the combination of staying up past bedtime trying to learn more, and juggling three of four different projects at once probably caught up with me.
I recently discovered that Athanasios Angelopoulos, a data source for a number of examples of Greek architecture that I’ve been posting about, also gave us a description of the Temple of Hesphaestos in Athens. This surprisingly intact example makes for quite an attractive subject, but the emerging picture is also one of what appears to be another challenging study, which makes it plenty to be dealing with at one time.
However I wanted to get in on the more recent discussions taking place at GHMB about the locations of the Giza pyramids in relation to data both from WMF Petrie and from Glen Dash, since I’ve done some work with Petrie’s Giza layout, but this is another subject that is probably also plenty to deal with at one time.
In the chaos then, I managed to conflate radius and circumference when it comes to Waun Mawn, and quite surprisingly even though the correction ratio for the error would simply be circumference / radius = Pi, it may make a difference to in the interpretation and to how Waun Mawn relates to Stonehenge.
As I mentioned to Peter, Stonehenge strikes me as a particularly good “repackaging” of mathematical ideas that had already been written at other probably older but generally less impressive sites. I used the analogy of a “Greatest Hits album”, which I think may be a fairly good one.
It’s very much as if the architects and builders of Stonehenge had said, “Let’s make this a really nice one because I think we’ve really outdone ourselves with the the mathematics this time” – hence the unusual features such as the lintels and the general tidiness of the sarsen ring from which we are able to extract minimum, mean and maximum diameter and circumference values, which is still unheard of for almost virtually any other stone circle. There is a great deal more we would already know about stone circles if only data of that detail were more often available.
It probably speaks of impetuousness on my own part from me to dive in an start wrestling with the available data; earlier today I was trying to relocate some poignant passages from Aubrey Burl concerning the accuracy (or lack therefore) in many British archaeological surveys. Last year I had to replace some of Burl’s books that went missing and might now have a different edition of the work. The comments in question were attached to display much as we see in the appendices of Burl’s The Stone Circles of the British Isles (1976-1977) showing proportions and numbers of stones.
In the passages in question, Burl wryly stated something very much to the effect that either many British archaeological surveys were rounding off their measures to the nearest 5 or 10 feet, or somehow ancient people had a keen awareness of the modern foot. Those passages seem rather ironic in a way since it’s a well tested premise of the current work that they were indeed quite aware of the “modern foot” (but generally reserved it as a universal reference unit, not a design unit).
The original text from Burl goes a long way to explain why I remain generally mistrustful of most data sources on Megalithic sites except for Thom, and indeed it’s unfortunately proved risky before to accept metrological from even from Burl in spite of his demonstrable awareness of the problems with archaeological data quality.
In the case of Mike Parker Pearson, I should like to hope that, like Thom, he is geared toward reasonable accuracy his data because he himself “has a pony in the race”; like Thom with his Megalithic Yard, Pearson and Chamberlain are champions of the use of the “Long Foot” at Megalithic sites, although regarding the unit itself I fear that it can only be folly to try to measure all of the Megalithic landscape in whole number of a Long Foot (or any single unit of measurements), and likely can only be little more than an unnecessary repetition of the tragic mistake made by many Egyptologists of trying to measure all of ancient Egypt in Royal Cubits in spite of the accepted existence of other ancient Egyptian units.
That said, for the unit itself, Pearson gives a value of 110 meters for the diameter of Waun Mawm which equals 360.8923885 (for which, oddly enough, neither this diameter value nor its projected circumference value, seem to be reducable to Long Feet of 1.056 “modern” feet in the sense of Petrie (the identification of whole numbers of units), in which Pearson seems to be working.
Were this the circumference value, it would be easy enough to suggest that it could be a case of a very “generic” circle with a circumference of 360 feet (generic because of 360 feet over 360 degrees), which would resonate easily with Stonehenge because it’s these generic characteristics of a circle (360 / 2 Pi = 57.29577951) that are what was used to make their original discoveries of Stonehenge mathematics, by comparing the proportions of the sarsen circle to just such a “generic” circle.
In this context, we could have seen Stonehenge as a refinement of Waun Mawn in the sense of Stonehenge having the generic qualities of a circle implied or self-contained so that there is no need to spell them about as blatantly as actually having a circumference of 360 feet).
Things may be rather different somehow after the correction that 110 m is Pearson’s figure for the diameter rather than the circumference of Waun Mawn, but aside from which part of the circle is being referred to specifically, a very promising candidate would have been 360.774411 ft (109.9640405 m).
I will have to go back to the drawing board on Waun Mawn, but for the sake of this number, I would like to continue a little further forward first.
Clearly not only does 360.7744112 register as a neglected number that deserves more attention and understanding, but given its qualities and its resonance with Stonehenge, we have reason to expect that this would have actually been used as the original circumference by design of more than one regular Megalithic Circle (and perhaps some ancient math prodigy even found a way to incorporate the figure into one of the flattened Megalithic Rings or “egg-shaped” stone circle designs.
360.7744112 has been part of the Megalithic landscape for some time now. In reciprocal form (1 / 360.7744112 = 2.771815211 / 10^n), it was Munck’s “geomathematical” “Grid Point” for one of the major British monuments. (Historically, I believe it was one of Munck’s own revisions in this area that first prompted me toward the realization that his British cartography would require further revision for being drawn on what we call the OSB36 mapping data, rather than the WGS84 datum).
We can conceptualize (and remember) 2.771815211 as simply 270 / (Pi^4).
I have been working on the concept more lately of Pi exponents as roots for a few of the familiar metrological units after still having been able to place any simple division of the outer sarcen circle diameter of Stonehenge as a historical unit of measure, although analysis shows it to be constructed from a whole number and (Pi^4).
I put a question mark after Draconic Megalithic Yard only because I’m still a little bit in disbelief that nature of the “DMY” was that easy to unravel, which I discovered (or was reminded of?) only after start starting to work on that modest diagram. I nearly posted it with only a question mark in that box.
Thus we really only have to see that (Pi^4) is involved to guess that this relates to other numbers with that can be seen as having (Pi^4) as a root.
We are not too terribly surprised then to discover that 144 / 51.95151515 = 2.771815211, or that 1333.3333333 / height Great Pyramid from base 481.0325483 ft = 2.771815211; the “Pi^4 factor” in each number will cancel each other out and give way to a whole number as can be originally used with (Pi^4) to create these numbers.
In inverse form, 1 / 2.771815211 = 360.7744112 / 10^n – so we can also guess then that 360.7744112 relates readily to 51.95151515, which is one reason we can think of 360.7744112 being a somewhat “Stonehengy” number.
Perhaps a more or interesting way of looking at is that if we square the Megalithic Foot, then invert it, when divide it by half, we get 360.7744112.
At Stonehenge, this plays out something like
360 / Megalithic Foot 1.177245771 = 305.7985077 ft, the inner sarcen circle circumference = 40 Remens x 2 Pi, and
So the Megalithic Foot links the inner circumference to the 360* of a circle, and the Megalithic Foot squared links the outer diameter to the 360 degrees of a circle.
The outer diameter value actually originates with taking Thom seriously that the outer circumference of the sarsen circle is 120 Megalithic Yards, and this is what happens if we use a Megalithic Yard of 2.720174976 feet ((120 x 2.720174976) / (2 Pi) = 51.95151515). I still find it remarkable how this gesture continues to relate Stonehenge to a Megalithic Foot value of 1.177245771 ft.
If we divide 360.7744112 by 305.7985077 we get 1.179778194 which is a number that Stonehenge went out of its way to teach us in a different way already (Lunar Month 353.9334582 / 300 = 1.179778194), and likewise if we multiply the two
360.7744112 x 305.7985077 = 110324.2766 (Best Eclipse Year 346.5939368 / Pi) x 10^n, which is thought to also be presented by the perimeter of the Stonehenge bluestone Oval With Corners as 4 x 346.5939368. (We can also form a simple fraction of the Best Eclipse Year simply by dividing the standard Megalithic Yard by the Megalithic Foot).
Does Giza know about this? Naturally. If we covert the length of one side at the base of the projected missing apex section (see preceding post) into Egyptian Remens, that will be the value:
perimeter 175.5865397 ft / 4 = side 43.89663492 ft; 43.89663492 ft / 1.216733603 = 36.07744112 Remens
This is a little bit of why 360.7744112 would be a very “Stonehengy” number to find, why we should probably expect to find it from time to time in the ancient Megalithic landscape.
To be honest, I really don’t have much business lobbying for any figures for Waun Mawn just yet, not am I certain how much faith I have in any of Pearson’s proposals, but Pearson’s figure of 100 m caught my attention because of the resemblance to 360.7744112 ft.
I would feel much more comfortable finding 360.7744112 as a circumference than as a diameter, but even then, as we can hopefully guess by looking at the Pi Factors Table above, if the diameter of a circle is 360.7744112, it is using a unit based on (Pi^4) so that multiplying diameter x Pi to generate circumference generates a number based on (Pi^5), which is the standard linear Megalithic Yard of 2.720174976 ft.
If 360.7744112 were the diameter of a circle then, its circumference would be 2400 Inverse Megalithic Yards of 2.720174976 ft each, and that is something else that I find most interesting about Pearson’s figure for the diameter of Waun Mawn.
Luke Piwalker 