Irregularity in Ancient Architecture: Upping the Ante on Data Capacity?

Structural Irregularity: Accidental or Deliberate?

(And a “New” 2 Pi Root ratio from the Valley of the Kings)

These are some notes on data from Kent Weeks from the Valley of the Kings. One reason these were of interest is because there’s some debate (or should be) concerning the meaning of apparent errors in ancient architecture. It’s in evidence the ancients were quite capable of remarkable precision, and yet there’s often irregularity in their work nonetheless.

One school of thought – the one I currently subscribe to and have for a long time – is that a great many of these irregularities may be intentional. If the idea of ancient architectural design were to express numbers or mathematical concepts, it might be redundant to have the exact same length expressed on opposite sides of a temple or a room, and better to have the lengths be slightly unequal in order to store and express more data.

The data from Miroslav Verner’s The Pyramids on some Giza subsidary pyramids suggests to me that this principle holds true, and my impression of Kent Week’s data is similar (some of the numbers themselves are also rather similar between these two examples)

From Kent Week’s data for KV5 – ratios between corresponding North and South walls:

W Side Chambers 7b-7p (35)
N wall / S wall 
7b 3.12 3.12
7c n/a n/a
7d 2.57 2.60 —– 2.60 / 2.57 = 1.01167315175
7e 3.13 3.07 —– 3.13 / 3.07 = 1.0195439739 = 1.019328359?
7f 3.13 3.07 —– 3.13 / 3.07 = 1.0195439739 = 1.019328359?
7g 2.62 2.61 —— 2.62 / 2.61 = 1.0038314176 = 1.003877283 ???
7h 2.60 2.55 —– 2.60 / 2.55 = 1.0196078431 = 1.019328359?
7l 2.05 1.88 —– 2.05 / 1.88 = 1.090425531 = 1 / (2.920160646 x Pi)?
7j 2.63 2.60 —– 2.63 / 2.60 = 1.01153846153
7k 2.53 2.61 —– 2.61 / 2.54 = 1.02755905511 = 1.027340740?
7m 2.59 n/a
7n 2.95 n/a
7p 3.10 3.11 —– 3.11 / 3.10 = 1.0032258064 = 1.003877283 or? 2 Pi root of mean earth circumference 24883.2 / ((2 Pi)^3) = 1.003151727???

What I find most remarkable here in this sample is that we seem to see several ratios that are formed repeatedly by different sets of measurements. I’m not entirely surprised to see 1.019328359, this has been the ratio of the height of the Great Pyramid / the height of Chephren’s pyramid according to Carl Munck for quite some time now. 1.027340740 is another one we seem to see quite a bit of, such that it turning up here is hardly a surprise.

What are 1.0116731517 and 1.011538461?

They may both be 1.011768211, which is 1/4 of a member of the Equatorial Circumference 2 Pi root series. In order words, equatorial circumference of the Earth in miles (24901.19742 / 4) / ((2 Pi)^6)

1.019328359 works with 1.011768212 to provide several classic and important numbers. Hopefully 1.031324031 is recognizable as, among other important things, the number of inches in 1/2 of a Royal Cubit that uses the equation (360 / 2 Pi) = Radian 57.29577951 as its standard of reference.

1.011768212 x 1.019328359 = 1.031324031 
1.011768212 x (1.019328359^2) = 1.051257832

There’s another possible pair here that can accomplish one of these feats – fancy that

1.003877283 x 1.027340740 = 1.031324031 

Try it at home! Don’t know about anyone else but that looks rather deliberate to me, and I didn’t do it, lol.

Hence while, for example, the King or Queen’s chambers of the Great Pyramid may look for all the world like very simple exercises in Royal Cubits, don’t be too surprised if data as respectably detailed as Petrie’s is able to show a different and more remarkable story upon more careful inspection, and don’t be too surprised if that story imay be similar to that that the subsidary pyramids of Giza or the tombs of the Valley of the Kings might be attempting to tell.

–Luke Piwalker

Some Recent Observations on the Mycerinus Pyramid at Giza

Recently Jim Wakefield on GHMB brought up Petrie’s data for distances between major pyramids at Giza and proposed there is an order to them that Petrie might not have dug deeply enough into the data to spot.

In the course of looking at Petrie’s data again, I decided maybe it was time to take the bull by the horns concerning more of Petrie’s Giza data. Jim quoted a passage from Petrie that mentioned “peribolus” walls, and I’ve generally had some encouraging results looking at the data from enclosure walls with pyramids outside of Giza, as if indeed they are proportioned so as to “speak to” the proportions of the pyramids they enclose.

There’s also the matter of pyramid platforms. With Giza, looking at the relationship between the proportions of the Great Pyramid and those of its platform have been very rewarding, even if slightly vague (I may not have enough data to tell if the measures I obtain refer to the width of the platform on top or the bottom, it’s edges are tapered so that there are two similar figures for this rather than just one).

I don’t seem to have a clarification of whether these are interior or exterior measurements for the pyramid enclosure, but for the Mycerinus pyramid, Petrie has several values relating to the enclosure wall of 4450 inches and 8897 inches. The larger is just 3 inches away from being twice the other.

These don’t seem to relate to rational numbers of Royal Cubits or Remens, the apparent premiere metrological units of the ancient Egyptian.

4450 / 12 = 370.8333333 = 741.66666666
8897 / 12 = 741.4166666

And I’m curious whether either one of these is wanting to talk about the 7.412765010 value that I reposted a “dossier” about elsewhere that is now embarrassingly dated.

In the event that this repetition were redundant, a possible second number worthy of discussion is 7.4275533026, if it’s not too high for these estimates. It might be, I think Petrie has been accurate even in the measuring of things as large as enclosures, but the state of preservation of Giza’s enclosure walls could affect the accuracy in unusual ways.

There’s also 7.402203303, which is 5 x (1.216733603^2), and if we got away with going that low we perhaps might as well go all the way to 7.396853329, the Squared Munck Megalithic Yard.

So the jury’s still a bit out on this one, and then there is 3401 (the distance from the base of the Mycerinus to the interior of the enclosure wall on the W side.

The outer/inner distance ratio on this side is 3401/3309 = 1.027802962, which is probably the rather ubiquitous 1.027340740, the reciprocal form of the Egyptian foot (or the reciprocal of 80 Remens if one prefers) according to me.

On the south side, the enclosure length for Mycerinus is a remarkable 14049 inches

14049 / 12 = 1170.75 ft. That looks a lot like one of the 1170- figures I hesitate to talk about for fear that someone “just tuning in” might confuse them with the all-important 1.177245771 (Munck’s “Alternate Pi”).

From some of my notes, here is one of them – “For what it’s worth we sometimes get things like (1 / 1.025135529) x 12 = 1.170576930 = 360 / (61508.13172 / 2). 1.170576903 is important to place in the presence of both 1.177245771 and (Pi/3) for both the series that are formed at higher powers.”

I’ve run into things like this both working with Andrews’ data for ancient Mayan sites, and Angelopoulos’ data for ancient Greek sites and a fair amount has already been learned about their possible place in the scheme of mathematically-based ancient metrology..

The ratio between the estimated 1170.75 value and the adjacent section estimated at 8897 in = 741.4166666 may be 16 Pi, that’s certainly what it looks like from the Petrie’s raw data.

1170.75 / 741.4166666 = 1.579071596 = 1.599934033 x Pi) / 10

The wall length on the West is given as 7698 in = 640.75, for which I’m included to deduce is 640.4628973, which is a repetition of the perimeter / height ratio I gave to the Mycerinus during it’s revision (made necessary because Munck’s Mycerinus model was apparently based on bad data from IES Edwards).

The ratio between adjacent sections here would be 14049 / 12 = 1170.75 / 7698 in = 640.75 = 1.827155677, which might either be Radian / Pi, or the square of the reciprocal of the Squared Munck Megalithic Yard (SRSMMY, lol)

1 / 7.396853331 = 1.351926225; 1.351926225^2 = 1.827704518, which is also 15 “Thoth Remens” in feet, highlighting another way in which the probably highly speculative “Thoth Remen” is integrated into other previously established metrology.

Thoth Remen x 3 = 365.5409035, which is that value that I keep wondering about being present in the Great Pyramid every time I look at E. Raymond Capt’s book, and of course it has plenty of business being there because it’s 2 Pi / Morton’s Cubit in feet.

There’s another, smaller enclosure wall on the south side of the larger one containing Mycerinus’ pyramid

On the West side, the measure is given of 6275 in, for which 2 Pi (6283.185307 / 1000) would always be a great guess, although for the figure in inches, 6275 / 12 = 522.9166666 = 1 / 1912350598, which does resemble the 1.911240674 (500 x 1.216733603 x Pi) figure which is the square root of 1.911240674^2 = 365.2840194 / 10, which may be about as close to the Solar Year as the math I use will ever need to get, although we can do better for what it’s worth.

Equatorial Circumference of Earth in Miles (ECEM) 24901.19742 (the figure I use) x the 5280.37936 mile built on 2^n to my shorter experimental Indus foot will do (1.100078968 x 480 = 5280.379036; 24901.19742 x 5280.379036 = 365.2437801 x 10^n),

If however this more precise value of 365.2437801 for the solar year it were such a good number, I’d probably run into it more often than just that once, lol. This math seems to be far more concerned with the calendar year of ~365 days than the literal year of~365.25 (why we have “leap years”), and likewise with the cycles of other planets as incorporated into calendars as well.

On the East side, the enclosure length is given as 6196 inches / 12 = 516.3333333 ft, looking like perhaps 300 of someone’s Royal Cubit? (300 Morton Cubits of 1.718873885 ft = 515.6620156 ft).

On the South side, the enclosure length is given as 9059 / 12 = 799.9166666. That will look for all the world like 800.0000000, more proof that the ancient Egyptians knew and used the modern foot, and can be taken as such if one chooses, but in my experience, that’s not something that went around trying to prove to us.

Occasionally we get round numbers of feet because they serve an important mathematical purpose in relation to their immediate mathematical environment (their relationships with numbers included nearby), but 8 or 8 x 10^n is hardly something anyone needs to post that way, we already can figure out to halve or double a number multiple times, so that 8 is essentially built into our own analytics and doesn’t need to be part of the mathematical message being interpreted per se.

The more purposeful thing here would probably be the usual: 8 x 1.000723277, not as 1/8 of the proposed 6.404628973 x 100 adjacent to it, but because they are both important and very useful constants in their own right, although there could be hints in there of a common metrological root unit, even if neither Remen nor Royal Cubit in the common sense of something being a fundamental metrological unit (i.e., not a whole number of either)

9059 / 6196 = 1.462072305, which might be 1.462163615, but perhaps the bidding should start at 1.460080323 = 2.9201960646 / 2 (i.e., 1.460080323 = 1.2 Remens in feet [i]as a ratio[/i], not as a linear measure, something I seem to see quite a bit of in ancient architecture).

9059 / 6275 = 1.443665339, several of the stronger possibilities being 2.882083038 / 2 = 1.441041519 and perhaps 2 / (1.177245771^2) = 1.443097644.

6275 / 6196 = 1.012750161, for which my first suggestion would be the Greek foot as a ratio, i.e., 1.216733603 / 1.2 = 1.013944669

Notice the possible symmetry of

1.216733603 x 1.2 = 1.460080323 and
1.216733603 / 1.2 = 1.013944669 / 10

I’m honestly not sure what I think is intended for the distance from the pyramid base to the outside of the wall on the West being 3309 / 12 = 275.75, but perhaps it’s interesting that 275.75 / 1.2167 = 226.6376284, while the same measure to the outside of the same wall is 3401 / 12 = 283.4166666 = 2267.333333 / 8?

The wall length / outer distance is ~640.75 / 275.75 = 2.323662738 – a good starting guess here might be sqrt 540 = 2.323790008; the wall length / inner distance = ~640.75 / 283.4166666 = 2.260805645

I should say concerning what amounts to about 8897 inches on the angled East side of the larger enclosure that 8897 / 2 = 4448.5 = 1 / 224.7948747, looking a lot like the Venus Orbital Period that’s best written in this math as 224.8373808 (and may match a metrological unit proposed by Petrie at Stonehenge).

If so, that would give the wall a value of ((1 / 224.8373808) x 2) / 12 = 741.2764998, as suspected at the opening here.

741.2764998 ft x 12 = 8895.317998 to Petrie’s 8897. A nice thing about this is that 8895.317998 x 12 = the highly prized and remarkable 1.067438159, which tends to recur emphatically at Giza – perimeter Cheops pyramid / perimeter Chephren pyramid, Munck) and 1.067438159 x (360^2) = 1383.399855, my perimeter value for the Mycerinus, revised from Munck’s values for the Mycerinus which were based on bad data from I.E.S. Edwards (not a good author to go to for working pyramid data).

Perhaps the wisdom, logic, and harmony of it are beginning to emerge here?

I’m a bit excited about this next one. Petrie remarks that the courses of red granite at the base of the Mycinerus pyramid cover 1/4 of the pyramid.

It may be harder to achieve consensus on the height of the Mycerinus pyramid because pyramid heights are often extrapolated from base measures and slope angles, and some of the red granite courses are not in a finished state.

I gave it a height of probably 216.000000 originally, one of those times when there may be considerable mathematical rationale for a round number of modern feet (the ancient universal standard of measure, although its usually painfully boring and uninformative to write whole numbers with it).

Interestingly, although Petrie estimated a height about 2 1/2 feet shorter for the height of the Mycerinus pyramid, possibly at least in part because of the unfinished state of the red granite courses casting uncertainty about the intended slope angle, he says (Pyramids and Temples of Giza, “updated” edition with section by Zahi Hawass, pg 37)

“The granite probably ceased at the level of 645.2, i.e., including the lower sixteen courses”, and then proceeds to enumerate the reasons for this. “This being settled, it is worth notice that the granite just covered on quarter of the height of the Pyramid, the total height being 4 x 641 +/- 4”.

645.2 inches / 12 = 53.7666666 ft. 53.7666666 x 4 = 215.0666667, now bringing Petries’ estimate to just under one foot less than my proposed value of 216.0000000.

Perhaps then it is also worth notice that 216 – 53.7666666 = 162.2333333, looking incredibly like another number just as sacred as 1.177245771, namely 1.622311470, to an accuracy of .9999865235, high above the usual “Giza Standard” of .9995 for when approximations are forced upon us.

This is of course even more remarkable because addition and subtraction often do strange and unwanted things to a system of numbers otherwise geared almost exclusively to multiplication and division. Obtaining that kind of result is easily the sort of thing that should have required premeditation, trial, and error.

Another possible strange result of the necessary use of addition and subtraction in this case, could be a small amount of discretionary liberty afforded to interpreters of this design.

Of course in the end there is an overall best interpretation, but for now I wanted to point out that that the estimated figure of 53.7666666 could be taken as 216 / (1 / 24901.19742) = 53.78658643 (accuracy .9996296518) or (1 / (6 x (Pi^3)) = 5.375255739 (decimal placement not critical here) to an accuracy of .9997375832.

While expression of geodetic figures like the earth’s equatorial circumference seems to have a very high priority (my Great Pyramid model is about bursting with them), keeping numbers in harmony with their mathematical environments was also a high priority that is in evidence, hence the relationship 16223.11470 / 3018.11029 (again, 3018.110298 ft is Munck’s Great Pyramid perimeter value) = 5.375255739 is therefore difficult to be dismissive of also.

The more demands we place on these numbers to offer us evidence that they were deliberately chosen by ancient architects, the harder we may have to think about how well they’re living up to those demands sometimes.

The important thing to remember here is that even being able to make either proposal concerning this in the first place is virtually remarkable, especially contrasted against the backdrop of the picture of ignorant ancients our educators have dogmatically painted for us.

From my point of view, this would be an immensely gratifying thing to see, because it’s of paramount important to try to work numbers like Pi, 1.177245771 and 1.62231147 into monumental architecture because they’re hugely important numbers.

With the Great Pyramid, having already specified that the height without the pyramidion was probably 452.389140 feet, I was thrilled to learn than in Morton’s Cubits of 1.718873385 ft, this is

452.389140 / 1.718873385 = 263.1894508 Royal Cubit = 16.22311471^2

Thus is the same thing built into the Great Pyramid’s height, in a model that was based on anything but this particular consideration.

I should hope I get to the bottom of this soon enough, but for now I wanted to share with others a view from the top of it.

Postscript: A Christmas bonus – a little more on the subject of 7.412764998, from an alternate version of this post, if I’m not repeating myself.

WMF Petrie’s data on the Mycerinus pyramid in Pyramids and Temples of Giza gives the distance from the pyramid’s base to the enclosure wall on the west as 4450 inches. 4450 / 12 = 370.33333 ft = 740.66666 / 2; he gives the length for the angled enclosure wall on the southwest side as 8897 inches. 4450 x 2 = 8900, just three inches over being exactly twice the first figure. 4450 / 12 = 741.166667

Candidate figures for 370.33333 ft might be 1/2 Squared Munck Megalithic Yard x 100, or possibly figures that can express the Uranus Synodic Period 369.66 days if we want to stick out our necks and speculate that the ancients were able to see Uranus with the aid of some clever optical apparatus, although some proofs of that would certainly be owed.

3.75 x (Pi^2) = 370.1101650  or 365.5409035 / Pi^2 = 370.370370 might be worth a little consideration?

However, is one of these really supposed to be exactly twice the other?

There is a number that resembles 741.166667. It’s 741.2765010 / 10^n. In March of 2006, I posted an alias list for this number that filled up remarkably fast. It’s an amazingly conversant number for how seldom it seems to be seen, apparently for lack of ancient architects cueing it into their designs, although some fairly obvious permutations like 741.2765010 x Pi might have been used in the vicinity of Tikal.

In standard Remens of 1.216733603 feet, it’s the concave apothem length of the Great Pyramid without pavement, or 500 “Thoth Remens”, in my present model of the Great Pyramid; in standard Royal Cubits it may be something of a nonsensical figure. What is the point of using a metrologically awkward figure like that?

Well, 7.412765010 belongs to a striking enough 360 series

(Bear in mind here I’m ignoring where the decimal point goes, it doesn’t matter at the moment)

7.412765010 x (360^1) = 2.668595404 = 1.067438159 / 4

7.412765010 x (360^2) = 9606.943453 = Great Pyramid Perimeter paved 3018.110298 ft x Pi = 9606.943453

7.412765010 x (360^3) = 345.8499643, revised sidelength Mycerinus Pyramid

7.412765010 x (360^4) = 12450.59872 = 24901.19742 / 2

Giza has found yet another way to tell us what the circumference of the Earth is? How many is that now? We’re well into the dozens, at least

7.412765010 is also the so-called “Real Mayan Annoyance” (see preceding posts) / ((2 Pi)^2). 2 Pi will also draw more data out of 7.412765010 at various powers, as will the Radian 57.29577951.

It also answers to one of the Great Pyramid’s resident forms of “Phi”, if that’s what we can call them, as 12 / 1.618829140 = 7.412765010

Also, I think it’s interesting that (57.29577951 x 2) x 1.177245771 = 1 / 7.412765010, when 1.177245771 / (57.29577951 x 2) = the ever-popular 1.027340741, which if we like we can think of as the reciprocal Roman foot (1 / 1.027340741 = .973386881 ft = 4/5 of 1 Remen of 1.216733603.

Petrie gives the distance from the base of the Mycerinus pyramid to the outer side of the west wall as 3309 inches, and the distance to the inner side of the west wall as 3401 inches. 3401/3309 = 1.027802962 — 1.027340741, anyone?

Season’s Greetings,

–Luke Piwalker

The Tower of Babel Story: (Another) Ancient Number Puzzle?

Member Ginkgo on the GMBH recently inquired whether the Gobekli Tepi site could be the legendary site of Babel and after a few days of looking at the thread sitting there, it dredged up a memory of hearing a version of the Tower of Babel story that had mankind being divided into 12 tongues and 12 tribes.

Since my very best advice to anyone wanting to explore possible relationships between ancient metrological units is to divide (or multiply) them by the number 12 (as in the 12 inches in a foot) in order to help sort through the Babel (read: confusion) of ancient metrology, the story of Babel began to remind me of metrology, which naturally caught my attention.

According to Wikipedia, the number 72 is seemingly a lot more common as to the number of different languages involved, but same difference, 72 is a simple multiple of 12 (12 x 6 = 72), a number of which seem perfectly effective for examining metrological relationships, probably for being made out of the number 12.

That includes 36 or 360 (12 x 3 = 36), and 24/25 = .96 (12 x 8 = 96), and so forth.

So my hypothesis hasn’t sunk, but while visiting Wikipedia, I might have found something even more interesting.

I ran into a lot of interesting numbers in the Wikipedia article with interesting relationships to each other, that make me think the story of the Tower of Babel could be largely a vehicle for ancient number puzzles much more than having any aspirations toward being accurate history keeping.

I’ve attempted to solve what looks like a similar one in a terribly gruesome but probably rather ludicrous “historical” anecdote featured in Robert Shoch’s book, Voyages of the Pyramid Builders (pgs 106-107), but in the case of the Tower of Babel story I think some later sources might perhaps have been making their own interesting contributions to the number puzzles?

Some interesting stuff in The Book of Jubilees it looks like, and in later works as well. The question may remain for now just how late various authors might have still been making interesting contributions to the story’s math puzzles? Even as late as Giovanni Villani’s account dated 1300 AD?

I don’t know if I want to jump right into A.R. George’s discussion of metrological concerns, but apparently attempts have been made to identify the Tower of Babel with the Etemenanki

https://en.wikipedia.org/wiki/Etemenanki

And a suspiciously round figure of “91” meters appears concerning it in both Wikipedia and George’s text. In spite of the blunted looking figure in meters, perhaps this value may be recognizably linked to well known ancient metrologies via squares or rectangles.

Something that looks very much like it has already appeared in my tables of metrological experiments

91 m = 298.556 ft
in cubits, remens, and Meg Yards, and Palestine cubits approximately:
298.556 / 1.72 = 1.735790698 (sqrt 3 = 1.732050808)
298.556 / 1.2167 = 245.3817704 (not 2.452595355 = x / 2^n, is it??? keep looking?)
298.556 / 2.72 = 109.7632353 = ~109.6622711 = ~4 / Earth Year
298.556 / 2.107 = 141.6971998 ((sqrt 2) x 10^n = 141.4213562) 

I’m not certain of the specifics, but 298.556, whatever it is, resembles a potential metrological unit that is related to the main Egyptian units. We can spot that back where the mystery figure is divided by 1.72 to get ~sqrt 3. That may not be implausible in spite of the implausibly round meter value?

Anyway, if I take some of the numbers that come from different sources and examine their relationships, however randomly (and I’m sure others can see even more in here than I do),

5433 / 72 = about sidelength Cheops pyramid (Munck)
2354 / 2 = 1177 see 1177.245771?
2354 / 700 = 168.1325571 x 2 = ~1 / 2.98 (= 1/2.973661852)
700 / 107 = ~2 / inner circumference Stonehenge sarcen circle (Munck)
2354 / 107 = 22 = ?? (1 / 4523.893421) = 22.1048532 / (10^n)
2354 / 1656 = 1.421497585 = ~ 452.3893421 x Pi
30/13 = ~base Great Pyramid in meters???
13 stades of (1.216733603 x 500) = 7908.76842 ft (7912.668258 mi = “polar radius”)
30 stades of (1.216733603 x 500) = 18251.00405 ft = (365.0200808 / 2) x 10^n
70 / 72 = .9722222222 see inner diameter Sarcen circle (Munck) .9733868822 x 100 ft
1656 = ~5 / 3018.110298 (3018.110298 = perimeter Great Pyramid, (Munck)

1656 x 365 = 604440 (18980 / 604440 = 3.140096618 = ~Pi); perimeter Great Pyramid unpaved 3022.41664 ft — 3022.41664 x 2 = 6044.83328
1645 x 365.25 = 6048054 = 168.015 x 360
5433 / 1656 = 3280.797101 / 10^n; my usual meter – feet converter uses 3.28084 ft for the meter.
107 / 43 = 24883.72099 / 10^n

Notes: 

Venus Orbital Period 224.701 / Saturn Synodic Period 378.09 = 1.682636036
Some candidates:
377.8020786 / 224.8373803 = 1.680334818
378.4528198 / 224.8373803 = 1.683229093
377.8020786 = 1/2 Base Great Pyramid, unpaved

(The length of a stade can be variable so I am just using my usual one to see what happens here).

I’m rather surprised at the amount of lip service that might be paid to the Great Pyramid’s proportions therein, although I probably shouldn’t be.

What we may have in such number puzzles from ancient myth and folklore may be similar to what we may have in cases like ancient America’s “pecked crosses” – blunt numbers that are selected and grouped very carefully so that their ratios can closely approximate import numbers with surprising precision.

https://en.wikipedia.org/wiki/Teotihuacan

The same may also be true of certain ratios between between numbers of parts of ancient monuments. Munck may have been the first two use approximations with his own style of math in the way he interpreted the physical parts of the El Castillo pyramid at Chichen Itza (this formula was still found on the Internet when last I checked a few months ago):

https://old.world-mysteries.com/chichen_kukulcan.htm

You’ll note that he uses the formula

“9 terraces x 365 steps x 4 sides x 4 stairways = 52,560”

To approximate the valid value 52562.8916

I like to take similar liberties with El Castillo’s 91 steps per side, for 91.18906528 x 4 = 364.7562611, an approximation of the 365-day calendar year that we can also make from 3600 / (Pi^2)…

Even though we can also use 91 as-as for 9 terraces x 91 steps per side = 819, a significant if sometimes strange Mayan calendar number

In the past I’ve worked extensively with the idea of ancient myth as a vehicle for both folk wisdom and the sciences; perhaps in the further I will be working more with the idea of myth as a vehicle for mathematics.

BTW, yes you read that right back there. I’m still not done ruling out that part of the multifunctionality of the Great Pyramid’s base length could have been as a tribute to the length of Saturns’s Synodic Period. Something to keep an eye on.

–Luke Piwalker

Lessons from Asian Pyramids?

I’m still missing some old posts by Michael L. Morton, so I was digging in some old files and stumbled over a folder on Asian Pyramids. Several of them have little bits of data from unknown sources and a couple of architectural diagrams, although there may be some confusion as to which diagrams and measurements describe which.

Some of that material (see Zangkunchong step pyramid) is here https://www.crystalinks.com/pyramidchina.html

Some good material on others is here

http://mesosyn.com/Koguryo.html

Previously, I’ve backed down from working with the data I have for being somewhat intimidated by it. I wasn’t sure what to expect from it for one thing, and for another there’s so little that it makes it much harder to feel very certain of any interpretive efforts.

Having a little more experience and hopefully a little more vision that last time around, I decided to stick it out and try to make the most of the data I have for now.

What do I think I see in the data?

For one thing, what looks like a surprising level of devotion to the Megalithic Yard, perhaps in myriad forms, in spite of the fact that these pyramids are in Asia.

Another thing that emerges from the data looks like a particular focus on the number 104, or nearby numbers that can be used to represent it, since the number 104 per se does not belong to the system of numbers I work with (“Munck’s”). The number 104 and its simple divisions are very important to ancient calendar systems, and still important to contemporary calendar systems.

52 weeks in a year x 2 = 104

520 / 2 = 260 days, the Mayan “Tzolkin”

and etc.

Attempting to work with this Asian pyramid data has more or less rubbed my nose in the fact that although I work with two primary calendar systems separated by the common 1.000723277 ratio, I have a still unsolved problem in that my two main figures for representing 104 and it’s simple fractions, are not separated by the 1.000723277 ratio, which is probably a long-ignored red flag.

Presumably, there’s a number in this range that I need to learn more about, to help me figure out which version of “104” goes with which group of calendar numbers, but it’s very surprising that after this much work with calendars, I haven’t been forced to face this fact until now.

Surprisingly, when I look at the relationship between these two current candidates primary representives for “104”, which are the 103.9030303 foot calculated outer sarcen circle diameter of Stonehenge, and the 104.0913798 (Michael Morton’s “Great Pyramid Apex Displacement Ratio” which belongs to the long form of the story of how the value 24901.19742 was obtained to represent the earth’s approximately 24901.5 mile equatorial circumference)…

104.0913798 / 103.9030303 = 1.0018127430

And that is the “New” “2 Pi Root” I just wrote about in the last post.

Let’s look a little more closely at the available data on both counts

For one pyramid (this is from a page on the Zangkuchong pyramid), I have a quoted base size of 31.58 meters per side and a height of 12.4 meters.

31.58 m = 103.6089239 ft

So already we’re brushing up close to 103.9030303 here – in modern feet, as always.

12.4 m = 40.68241 ft

This pyramid would have a base diagonal of about 103.6089239 x (sqrt 2) = 146.5251454 ft

On the very same page is the diagram still seen at Crystallinks, giving one of these pyramids a base size of 29.34 meters per side and a height of 11.28 meters.

29.34 m = 96.259843 ft; 11.28 m = 37.007874 ft

1 / 96.259843 = 103.885480054 / 10^n

So there we are – already brushing up again against number in the neighborhood of 103.9030303 or 104.0913798.

This pyramid would have a diagonal of about 96.259843 x (sqrt 2) = 136.1319755

As I’ve previously mentioned, I work primarily with several forms of the Megalithic Yard – the Squared Munck Megalithic Yard (“SMMY”) of 2.719715671^2 = 7.396853331 ft, and also a Megalithic Yard (“mine”) of 2.720174976 ft.

7.396853331 / 2 = 3.698426666 – compare to this same pyramid’s height of 11.28 m = 37.007874 ft

2.720174976 / 2 = 1.360087488 – compare to this same pyramid’s estimated diagonal of 136.1319755 ft.

Of course, more could be said here – including that in terms of planetary cycles Lunar Year 354 days / 260 (“Mayan” Tzolkin) = 1.361538462, or that 260 x 4 = 104 x 10, or that estimated width 96.259843 = ~Lunar Year x 2.720174976.

There may be additional Lunar data at work here (such as the Draconic month 27.212220 days), but one possible way of looking at it is of course that the Megalithic Yard, like the Remen and Royal Cubit, may be another major ancient metrological unit that “descended from heaven” – that is, came to us through our studies of planetary, solar, and lunar cycles and our attempts to organize them into calendar systems.

Also note that the estimated height of the pyramid in the first dataset, 12.4 m = 40.68241 ft is suspiciously near to 2.720174976 x 15 = 40.80262464 (unless perhaps we have some Egytologists intrepid enough to prefer to try to make 23 and 2/3 Royal Cubits out of it?)

When we get to a bit of data on the Tomb of the General https://en.wikipedia.org/wiki/Tomb_of_the_General

We are told that at the base it measures almost 75 m.

75 m = 246.063 ft; (1 / 406.3999869952) / 100

Compare this to the height given in the first pyramid dataset, 12.4 m = 40.68241 ft

I can’t guarantee that all of these observations will hold up for very long, but for me this is pretty much customary, to be apparently dealing with ancients who knew their mathematics forward and backwards – literally, even if history as we know it isn’t able to account for that very well.

The perimeter of the Tomb of the General based on this data (and assuming it is square at the base as more of these Asian pyramids seem to be)

75 m = 246.063 ft x 4 = 984.252 ft

That’s (1 / 1.015999999) x 10^n

The most important thing I know of in this range is 1.017140346. As near as I can tell, the ancient Maya were quite fond of this number, and I think it turns out that one reason is because 360 / 1.017140346 = 353.9334582, which from my perspective (and presumably theirs) is a more useful way of representing the “354” day Lunar Year.

It doesn’t take long to find the number 353.9334582 at Giza, either.

A pyramid with a base length of about 246.063 ft would have an estimated base diagonal of

246.063 x (sqrt 2) = 347.985613

That one, I’m going to leave alone for now, but I will note in passing that Hugh Harleston managed to obtain a “Standard Teotihuacan Unit” of ~3.475721758 ft (1.0594 meters), and while I’ve never had much use for it, I did make recently make a proposal what such a unit might actually be if there really is such a thing.

I also have a couple of crumbs of data for the Andong Pyramid

https://www.megalithic.co.uk/article.php?sid=50335

Base 13.2 by 12.7 meters, height about 4.5 meters

4.5 meters = 14.7638 ft

Compare the doubled Squared Munck Megalithic Yard (SMMY)

2.719715671^2 = 7.396853331; 7.396853331 x 2 = 14.79370666 ft

Also note the base length/width ratio

13.2 / 12.7 = 1.03937007874

And there we are once again creeping up on numbers in the 103.9030303 range

Also, the width/height ratio is about 12.7 / 4.5 = 2.8222222222, and that’s rather suggestive of the reciprocal of the Lunar Year

1 / 353.9334582 = 2.8253898489 / 10^n

I think perhaps that is all I want to say about that one for now?

So I haven’t really figured out much of the why of some of these numbers seemingly recurring even in such limited data, although it does come of light that two of these recurring figures form a third important figure

103.9030303 / 1.017140346 = 1.021521079

1.021521079 is a “wonder number” I “found” first at Tikal, then at Stonehenge and Giza. It’s more of a wonder number for what other numbers can do it than for what it can do to other numbers, it’s part of at least powerful and important series and I’ve written about it in some detail here.

http://grahamhancock.com/phorum/read.php?1,1198957,1203926#msg-1203926

There’s one more interesting piece of data I stumbled across on Asian pyramids that I should mention

I’m also having more trouble accessing some of the pages for whatever reason – there is or was recently a page on Chinese pyramids by Stijn van den Hoven. This is in the version still in the Google cache

“German engineer and architect Helmut Fuernreder has discovered that some of pyramids in China were built according to the golden ratio. For instance, if we divide the height of the (Great) White Pyramid (300 m) by the length of its base (485 m), we get 0.618.”

Google does not seem to know much about anyone named Helmut Fuernreder as far as I can tell, but I’m quite intrigued with this. That’s not quite the ratio we actually get, but it may be close enough.

I’ve no idea where the data comes from, but for the record 485 m = 1591.21 ft; the reciprocal of 2 Pi is 1 / (2 Pi) = 1591.549431 / 10^n.

That’s on top of someone pointing out the similarity between the layout of three of these large Chinese pyramids and the three main Giza pyramids.

300 m = 984.252 ft. 

Rather than speculate on what this is, let’s extrapolate based on the suggestions

1591.549431 / 1.622311470 = 981.0381424 ft

981.0381424 belongs to the 2 Pi series that features “New” “2 Pi Root” 1.001812742 at the bottom.

I am not looking for 1.001812742 ever since I posted it to this blog. I am much more keen to avoid it than fool around trying to justify it further. It is now looking for me apparently.

Again, I have no idea about data sources or data quality here. Generally, it’s probably as smart or safe to accept data without provenance as it is to accept candy from strangers, but there are amazing coincidences adding up into patterns here, which does imply some valid data even with the ambiguity about sources.

I just wanted to see if I could learn something from the experience, which is something else we can hope for from valid data, and indeed I have.

If you’ve made it this far reading this, here’s a little bonus for you. My now-standard figure for the height of the Great Pyramid without pavement and without capstone is 453.2094072, 452.3893421 ft with pavement.

It’s sort of a long story, but 453.2094072 in spite of being a strange number, is an impressive responder to application of the 2 Pi ratio.

453.2094072 / 452.3893421 = 1.0018127418

Hence the “New” “2 Pi Root” has been lurking in my own model of the Great Pyramid for some time now.

–Luke Piwalker

A “New” “2 Pi Root”, the Pyramid of Meidum, and Ancient Geodesy Part 1

Ancient Geodesy

I almost hesitate to leap from metrology to geodesy (Wikipedia: geodesy, also named geodetics, is the scientific discipline that deals with the measurement and representation of the Earth) – I have enough unbelievable things to tell people about the mathematical prowess of ancient people as it is.

Even some of the most open-minded people seem to struggle to get past the idea that the ancients could only work in fractions. Perhaps in some backwards way it’s good fortune that I’ve seen what it looks like when a pyramidologist takes that premise to such extremes that the intrinsic folly of it starts to become obvious.

I must be about as open-minded as they get but I still cannot get my head around the idea of ancient Egyptians doing actual rocket science in fractions. That has to be going too far with what must have started as a relatively plausible premise.

Sooner or later, both Egyptology and pyramidology may have to face it that the surviving papyrii showing math using fractions may be simplified, generalist works, not mathematical treatises reflecting the state of the art. Some of these papyrii cover so many subjects that they very much seem to be generalist works, in fact.

It’s inevitable, though, that the next incredible thing I have to share is that the ancients seem to have been keenly aware of the size of the earth, and seemed to delight in finding ways to incorporate this knowledge into the proportions of ancient monuments and architecture with an insistence that seems to imply that once upon a time, this knowledge was lost with potentially disastrous consequences, and they were determined that it should never be lost again.

I should note that it’s remarkable that no matter how different the mathematics of individual proposals, this basic idea comes up time and time and time again in interpretive works on the pyramids, as if we are all intuiting the basic truth. Again, even Sir Isaac Newton is among those who seemed to have such a “gut feeling” that the study of Egyptian pyramids would reveal knowledge about the size and measurement of the Earth that exceeded that of his day.

How did they know this? The ancient Greeks were able to work it out fairly closely, depending somewhat on exactly how one chooses to interpret the historical record. It doesn’t seem that implausible for someone to have beaten them to it, and gone after the problem more diligently or creatively until they eventually achieved accuracy to rival modern standards.

https://en.wikipedia.org/wiki/Eratosthenes

“Geodetic 2 Pi Roots”

When working with Tikal intensively for the first time several years ago, it wasn’t long before I encountered a “geodetic 2 Pi root”. There are at least three of these, and I call them “2 Pi roots” because when they’re multiplied by 2 Pi to an unspecified power (in this case, the 3rd power), they generate significant figures, and the “geodetic 2 Pi roots” generate significant geodetic figures.

The system of mathematics I use allows representation of the Earth’s equatorial circumference (Wikipedia, Earth: 24901.461 mi) with great fluidity and resonance as accurately as 24901.19742 (miles), accurate to about 1/4 mile, notably better than a number of modern mapping datums of the 20th century.

24901.19742 belongs to a whole series of numbers linked together by the ratio 2 Pi. The Great Pyramid, which probably commands more attention than any other, has been reckoned by numerous researchers ever since the science of modern pyramidology began, to be an embodiment of 2 Pi, the ratio between the circumference of a circle and its radius, because of its apparent 2 Pi perimeter/height ratio.

In this “geodetic 2 Pi series” at 24901.19742 / 2 Pi to the third power, we find

24901.19742 / ((2 Pi)^3)) = 1.003877283

I seem to have found this ratio in the Great Pyramid, repeatedly. The ratio between its adapted apothem were it not concave on the sides, and its actual apothem which gives truth to ancient authors describing its apothem length as “one stadium” (alt. stadia or stadion). Taken as 500 Remens of 1.2166733603 ft, 1.216733603 x 500 = 608.3668015 ft, which requires slight indentation of the sides to achieve (this concavity of the Great Pyramid sides has been both photographed and measured; I refer to Flinders Petrie’s data on the matter).

Without this indentation, the apothem (again, the length of a line from the center of a side of the base to the very peak at the top) would be 610.7875012 ft. Because the use of addition, subtraction, or trigonometry often produce figures that require slight adjustment to be valid, this was adjusted to the valid figure of 610.7256118 ft ((1944 x Pi) / 10).

610.7256118 / 608.3668015 = 1.003877283

This is only one of MANY ways of referencing the size of the earth that has been found in my model of the Great Pyramid, which is an extension of Carl Munck’s basic model. The apothem values are part of my own contribution to it, since to the best of my knowledge he never published valid figures for them, nor do I know if he ever acknowledged the concavity of the sides as Flinders Petrie did.

I also seem to have “discovered” 1.003877283 as the ratio between the size of the Great Pyramid’s base and the size of the platform it sits on, although even with the data from Lehner and Goodman via the impeccable Glen Dash, it’s hard to be certain if this involves the size of the platform at the top or the bottom of it.

The three main “geodetic 2 Pi roots” I recognize are 1.003877283, 1.002151142, and 1.003151727

1.003877283 x ((2 Pi)^3) = 24901.19742 (miles) / 100, 1/100 of the Earth’s equatorial circumference (Wikipedia: 24901.461)

1.002151142 x ((2 Pi)^3) = 24858.38047 (miles) / 100, 1/100 of the Earth’s polar circumference as best represented by the system of numbers I use (Wikipedia: 24859.73)

1.003151727 x ((2 Pi)^3) = 24883.20000 (miles) / 100, 1/100 of the Earth’s average (mean) circumference as best represented by the system of numbers I use. Some other pyramid researchers use this mean figure also.

We may find another member of the equatorial 2 Pi root series at Teotihuacan, where the complementary data retrieval tool, 2 Pi, may be posted repeatedly and rather obviously. This is 24901.19742 / ((2 Pi)^2)) = 6.307546992 x 10^n, which may be the ratio between the size of the Quetzalcoatl pyramid and its surrounding enclosure. 6.307546992 may also be the perimeter/height ratio of an unspecified major Egyptian pyramid, perhaps one that is presently being taken as 2 Pi.

Thus far, the pyramids that seem to have been solved at Giza seem to have their own unique perimeter/height ratios, which is a trend that may continue so that we may yet find there (or elsewhere in Egypt), important figures like 6.307546992 and 6.115970155 as the perimeter/height ratios of various pyramids.

Geodesy at Tikal

Both Carl Munck and Michael Morton share in the credit for “my” “discovery” (re-discovery) of the figure 24901.19742. Both got within a hair’s breadth of it but stopped, so to the best of my knowledge I am the first to ever publish it. It was first discovered working with the Great Pyramid, originally according to Munck’s “geomathematical” (map-based) data on the Giza pyramids.

Munck also observed that according to Teobert Maler’s data, the platform width of Tikal Temple I was, when converted from meters to feet, 24.901, which he astutely recognized as 1/100 of the earth’s equatorial circumference at a modeling ratio of modern feet to modern miles. You can see this drawn and labelled in Munck’s own hand just before halfway down the page here

http://www.viewzone.com/bigpicture/bp112311.html

Lest anyone think this is mere coincidence, Maler’s data for the El Castillo pyramid at Chichen Itza shows that they seem to have done the very same thing there too, only they wrote it “backwards” as its reciprocal. Several years ago I managed to extract a second figure for the earth’s equatorial circumference from the platform length of Tikal Temple I. It’s not as good as 24901.19742 but it is valid and it is sometimes a mathematical fact of life, and since it rivals the accuracy of some 20th century mapping and geodesy, it’s hard not to accept it.

So by all appearances they were doing at Tikal exactly what Munck thought they were doing at Giza, posting geodetic data at the summits of important pyramids.

My first geodetic discovery at Tikal happened many years ago. I’d just come in the door with photocopies from a college library of Maler’s study of Tikal, and in the first five minutes I discovered that the data gave us a diagonal for a pyramid base that looked like 111.5419203 ft (an important number, for a number of reasons).

As it turns out, 111.5419203 is the square root of ten times 1/2 of the earth’s mean circumference as expressed here

111.5419203^2 = (24883.2 x 10) / 2

As with Giza, it seems rather insistent (and rather creative) the way these geodetic figures are expressed at Tikal.

The Pyramid of Meidum and A “New” “2 Pi Root”?

I’m filing this under “2 Pi Roots” for now since I don’t know quite know what to do with it.

Several years ago working with Flinders Petrie’s data for various Egyptian pyramids, in which the sides of these pyramids often appear to be somewhat unequal, I noticed that some of the ratios between uneven sides resembled some of the “geodetic 2 Pi Roots”. I began to wonder if these pyramids were deliberately made irregular in order to express these “geodetic 2 Pi Root” as ratios between sides, and designed a hypothetical pyramid that could express the three main geodetic roots as well as the main equatorial / polar circumference ratio.

I’ve mostly left it alone since then because I probably want to retract the idea that this was achieved by using different values for the Royal Cubit for each side, even though this idea was heavily influenced by some of Petrie’s comments during his struggles to make simple numbers of Royal Cubits out of everything.

It’s been so long since I looked at it then, I was curious what pyramids might have originally inspired this hypothetical pyramid model, and working afresh with the step pyramid of Meidum, I again encountered several ratios between sides resembling the “geodetic 2 Pi Roots”. I re-examined the ratios between adjacent sides for additional Egyptian pyramids examined by Flinders Petrie, hoping to see if there was one that actually matched my hypothetical pyramid model.

Sneferu’s step pyramid at Medium is strange and intriguing – today its remains stand as a step pyramid, even though it’s supposed to have been finished into the same sort of pyramid we see in the major pyramids of Giza.

The step pyramid of Meidum represents a number of research opportunities, including the chance to examine the mathematics that might have been concealed beneath the surface of an Egyptian pyramid in the proportions of the individual steps and the layers used to build them.

It also represents the possibility opportunity to examine relationships between interior and exterior mathematics and proportions of an ancient pyramid.

Since I began working with this subject, I’ve held the view that ideally, any data concealed in a pyramid should also be presented on the exterior – it seems rather senseless to have to tear a pyramid apart to retrieve its data – but even now this is still only a hypothesis that this is what we will eventually find.

The step pyramid of Medium is a little strange in its exterior however. I’m not absolutely certain if these remarks are still valid today after over 100 years of additional discoveries, but Petrie stated that it was built just before the Great Pyramid, and probably unbeknownst to Petrie his data for the pyramid of Meidum makes it look like exactly that mathematically.

Petrie and other sources suggest that like the Great Pyramid, the finished pyramid of Meidum was a 2 Pi pyramid (perimeter/height ratio = 2 Pi), but the resemblance between the two may go deeper than that. Petrie’s measurements and calculations give 3619 inches as an approximation of its original height.

3619 inches / 12 = 301.5833333 ft, which is strikingly close to 1/10 of the Great Pyramid’s 3018.110298 ft perimeter as determined by Munck.

In other words, at least at first glance, Sneferu’s Medium pyramid looks a lot like the Great Pyramid with its mathematics and proportions re-arranged. Thus far, it literally tends to look like the “father” of the Great Pyramid mathematically.

In re-examining some of Petrie’s data for various pyramids and the ratios between unequal values for their sides, a pattern may emerge (data from Petrie in inches)

Sesostris II (El-Lahun) 

At pavement N 4161.4 E 4174.5 S 4168.9 W 4169.3 

4169.3 / 4161.4 = 1.0018983995

4168.9 / 4161.4 = 1.0018022780

Sesostris II subsidiary (Queen’s) pyramid (El-Lahun)

N 1071.2 E 1069.7 S 1072.3 W 1073.2

1073.2 / 1071.2 = 1.0018670649

Out of three additional pyramids, figures starting “1.0018…” appear at least twice already.

So is there a figure in this range that is significant?

It turns out that there actually is – it’s 1.001812743, and it’s significant as a “2 Pi” root

This is about where it starts to become recognizable

1.001812743 x ((2 Pi)^6) = (1 / 1.622311470) x 10^n

1.001812743 x ((2 Pi)^7) = sqrt 15 x (10^n)

1.001812743 x ((2 Pi)^8) = (1.216733693 x 2) x 10^n

1.001812743 x ((2 Pi)^9) = 152.8992541 x 10^n (half the inner circumference of the Stonehenge sarcen circle in feet

1.001812743 x ((2 Pi)^10) = 9606.943469 x 10^n – that’s 3018.110298 or 301.8110298 / Pi, which were just discussed up the page

The series continues to be useful to at least as high as (2 Pi)^13, 2 Pi to the 13th power. I’m rather impressed.

You can also see in Petrie’s pyramid data from El-Lahun that there may be a second ratio that is very similar to 1.001812743. I don’t have a guess yet what it might be.

It seems as if the ancients still have a lot to teach us. They probably should, they had a very long time to get that good at math, believe it or not.

–Luke Piwalker

More About Stonehenge

The Mean Circumference of the Sarcen Circle

With the model we have so far, which thus far only describes the sarcen circle, we have circumference values of 326.4209971 ft (outer circumference) or 120 Meg Yards of 2.720174976 feet, and 305.7985077 ft (inner circumference).

We can also determine a mean circumference value — 326.4209971 + 305.7985077 = 632.2195048, and 632.2195048 / 2 = 316.1097524.

The system of numbers I use (i.e., “Munck’s”) is all about multiplication and division – addition and subtraction usually aren’t kind to it, and normally force us to accept approximations. A standard of accuracy of .9995 or higher appears to be a universal standard that we can use to help evaluate approximations. Although this standard may have been truly universal in the ancient world, I often refer to it as “the Giza Standard”, because it was developed in the present by working with the architecture of Giza, primarily the Great Pyramid.

Thus we can accept the raw mean circumference value 316.1097524 as meaning the valid and important figure 316.0557713, to an accuracy of 316.0557713 / 316.1097524 = 0.9998292330, better than the proposed universal .9995 minimum.

This mean circumference value relative to the inner and outer circumference values, gives us for ratios

326.4209971 / 316.0557713 = 1.03279555933

316.0557713 / 305.7985077 = 1.03354255610

Both of these are important numbers, particularly the second one, as some people who’ve followed Munck’s work may already know. It’s easily enough to make us think we’ve done something right here, although there is further exploration of the meaning of these values we can undertake later.

The ratio between these two figures, by the way, is

1.03354255610 / 1.03279555933 = 1.000723277

Anyone “just tuning in” might want to make a note of that figure, 1.000723277, which more than makes up for the slightly lopsided averaging here.

1.000723277 first appeared in print, to the best of my knowledge, when a reader named Doug wrote into Munck’s newsletter to inquire about it. Munck, who was of course already aware of it, called it a “gremlin” in the works, and Michael Morton and I followed suit.

What it really is, is THE most important fine ratio in the entire system of numbers I work with. In the past several years, it’s been revealed as the ratio between two primary sets of numbers that express ancient planetary cycles, and it’s also very important to ancient metrology in this mathematical system.

The Outer Bluestone Circle

For the outer bluestone ring, Prof. Thom (Megalithic Remains in Britain and Brittany, pg 145) determines a likely diameter of 28.65 Megalithic Yards. This is remarkably close to the half-Radian, 57.29577952 / 2 = 28.64788976. 28.64788976 x 2.720174976 = 77.92727284, the diameter in feet, and 77.92727284 x Pi = 244.8157478 gives us the circumference in feet.

Since we already seem to have discovered the use of two “Egyptian” units of measure, the Remen and the Royal Cubit at Stonehenge, 77.92727284 / 1.216733693 = 64.04628970 Remens and 77.92727284 / 1.718873385 = 45.33624962 Royal Cubits. Both of these are also significant. 64.04628970 is (for example) exactly ten times the perimeter/height ratio of Mycerinus’ pyramid at Giza in my recent revision; 45.33624962 is one-half of 90.67249924.For a better perspective, let’s backtrack here a little to a very important fact. For the circular monument in question, the 360 degrees of a circle divided by the inner circumference in feet gives us360 / 305.7985077 = 1.177245771, thus Stonehenge gives a dramatic display of Munck’s “Alternate Pi”, 1.177245771. I’ve also shown in the previous post about Stonehenge where Stonehenge dramatically displays the number 1.067438159 as the ratio between outer and inner circumference of the sarcen circle.

1.067438159 / 1.177245771 = 90.67249924 / 100

Relative to the circumferences values for the sarcen circle

326.4209971 / 244.8157478 = 1.3333333333

316.0557713 / 244.8157478 = 1.2909944484

305.7985077 / 244.8157478 = 1.2490965570

All of these ratios are significant, although I’m a little disappointed to see the 1.333333333 ratio, it’s a little bit “pedestrian”, it’s simply 4/3. What might be more interesting is if it turned out to be something like 1.331433599, which is half of an important timekeeping number, 2.662867199, that seems to be involved in the calendar formulas imbedded in the proportions of Mesoamerican calendar-related artifacts like the Sun Stone and Tizoc Stone.

As always, this is when ancient architecture and artifacts are measured first and foremost in modern feet, which my work (and Munck) indicates was an ancient universal standard of measurement that we’ve inherited without inheriting its history. It’s what gives profound meaning to other ancient measurements.

I do have some misgivings about this still, even after over a decade. 77.92727284 x 10 is very near to the Mars Synodic Period of 779.96 days, although I have yet to actually be able to make 779.2727284 stick as a representation of this ~780 cycle. It may not be entirely out of the question that what was intended for the outer bluestone circle was slightly higher than 77.92727284 and closer to 779.96 / 10.

1.2909944484 is reassuring to see, it’s the reciprocal of the square root of 60,

1 / (1.2909944484 / 10) = 7.745966692 (sqrt 60)

which is not only a particularly powerful data retrieval tool as previously mentioned, it’s an intrinsic part of Munck’s interpretation of Stonehenge. To Munck’s reckoning, the 60 stones of the sarcen circle and the 15 stones of the sarcen “horseshoe” indicate that we should apply the square roots of 60 and 15 to our Stonehenge calculations. (I’ll have more to say about this later).

Suffice it for now that sqrt 60 seems to have been quite popular in ancient American architectural math, and for good reason.

Perhaps if we see how the tentative 77.92727284 ft diameter of the outer bluestone circle relates to the sarcen circle circumference values we’ve already been looking at, it might lend more clarity?

326.4209971 / 77.92727284 = 4.188790204

316.0557713 / 77.92727284 = 4.055778673 = (1.622311470 / 4) x 10

305.7985077 / 77.92727284 = 3.924152566

These are significant numbers too, so hopefully the nomination of values for the measures of the outer bluestone circle look like less of a mistake.

77.92727284 also responds well to two of the numbers that Stonehenge seems most enthusiastic about expressing thus far, 1.177245771 and 1.067438159.

77.92727284 x 1.17724571 = 9.173955211 = 2.920160646 x Pi which is very significant (1.177245771 also works at higher powers to retrieve additional data from 77.92727284)

77.92727284 / 1.067438159 = 73.00401615 — this is the best valid representation to date for the number 73, which is important to ancient calendars, and 73.00301615 / 2 = 365.0200808 / 10, so there is the ~365-day calendar year as well.

If 77.92727284 is the correct diameter for the outer bluestone circle, you saw it here. There’s a lot more that can said about this, but for now let’s call that a start.

–Luke Piwalker

Loitering In Doorways, Mathematically…

…So to speak. This has to be one of my favorite activities the past few years. When I returned to my studies of ancient architecture and decided that Mesoamerica remained a neglected subject (it’s amazing the amount of attention that Egypt commands for its few pyramids compared to the astonishing number of pyramids in the Yucatan), I decided to go back and see what else I could find in Teobert Maler’s data.

One of the few things I really knew about ancient Mesoamerican math was that Munck proudly declared that the width of the exterior doorway of Tikal Temple II was 1 Squared Munck Megalithic Yard.

Revisting Maler’s data, I “discovered” that the ratio of height to width for this temple doorway appeared to be another of Munck’s favorite numbers, “Alternate Pi”, 1.177245771. I also discovered that not only the ratio of height / width seemed to important, but also the product of height x width. That’s a criteria I now use in evaluation any proposals for height and width of ancient doorways.

Obviously this makes ancient doorways seem very interesting subjects if they’re going to be “billboards” for some of our more important or more useful numbers, or if their proportions are going to be chosen with particular care as if they are a mathematical welcome mat.

To date then I’ve worked with data for the doorways of all of Tikal Temples I through VI. Temple VI is rather a challenge, having three exterior doorways and one of them trying to appear inexplicably redundant (in terms of mathematics expressed architecturally, that’s a waste of “paper”), that I still haven’t solved the original specifications for any of its outer doors, but I like to think I’ve made progress with the exterior doors of Temples I-V, which were full of surprises.

One thing I seemed to learn about ancient Mayan math is they generally seem to have been skilled at starting important series of numbers higher or lower than we might normally consider valid or useful. I also learned some new novel numbers that lately I’m getting better at finding in the data for Giza where they’d been overlooked.

I should say about working with Tikal, that I wasn’t getting very far until I acknowledged the Mayan affection for calendars, and until I noticed after measuring a photograph by Edwin Shook of one of the main pyramid temples, that the tiers of the pyramid seemed to be progressing outward as they progressed downward, at a rate of (Pi/3). Pi/3 turned out to be like a magical key to understanding what I was seeing “written” in the temple doorways.

I’ve also worked with, relatively successfully, the idea that both the height of Mayan doorways with and without taking their lintels into consideration may be valid as well as the ratios between their heights with and without their door lintels. Nothing goes to waste, it seems.

If I could find data, I’d like to explore the proposition that ancient Egyptian false doorways may be just as meaningful mathematically.

Sadly, even with the wealth of data provided by Maler and Andrews, the complete data for Mayan doorways is still rare to have. Mayan doorways often flared at the bottom so that the width at the bottom of the door was greater than the width at the top. In the few cases where I have data for width at both top and bottom, meaningful ratios between the two widths seem to consistently appear.

I could go into much more detail, and I hope I get the opportunity soon to do just that, but those are the general remarks I can make on the subject based on my experiences working with the data. I continue to expect to find great things studying how ancient doorways were proportioned, wherever they may be.

–Luke Piwalker

The Sharpest Tools In My Kit

When confronted with an unfamiliar number, I use various strategies to try to identify it. If someone wants to try their hand at Munck’s style of math and have their own adventures with it, they may often encounter numbers that they’re not sure belong to the system of numbers or not.

When we can take an unfamiliar number and turn it back into something familiar by means of an obvious tool, we know we’re still within the system, and we know we’re generally helping to keep the system fine-tuned and on-track.

Likewise, when working with raw data, we can use the same analytical probes to refine the raw data by recognizing which numbers were likely intended, provided this ancient system of numbers was as universally distributed as it very much seems. (I’ve found the same style of math and the same design logic seemingly at work in data for ancient Greek architecture and in ancient Mesoamerican architecture as in ancient Egyptian architecture).

Here then are some of the most important analytical probes I use

2^n. I don’t know how many times simply doubling or halving an unfamiliar number by 2 or 2 x 2 or 2 x 2 x 2 etc turns it into something recognizable so we can see we haven’t strayed too far from the path. Other simple numbers may also function well for this such as 3^n, 4^n, 5^n, or 6^n but I’m not much in that habit of that even if perhaps I should be.

Select Pi Fractions: Pi/3 and Pi/6 can be powerful analytical probes, even when other simple fractions of Pi like Pi/2 or Pi/4 may not be. Pi/3 can coordinate some remarkable series of important numbers.

The Circular Numbers, based on the elementary mathematics of a circle: 360, 2 Pi, and 360 / 2 Pi = the Radian 57.29577951. 2 Pi is often an extremely powerful analytic probe and data extraction tool, but the last three years especially I’ve come to realize how powerful 360 and 57.29577591 can be as well.

Alternate Pi 1.177245771. Munck introduced this number and the name “Alternate Pi” and at one point expressed a preference for it even over the Pi ratio itself. I sympathize. It’s a very powerful tool for converting unfamiliar numbers into familiar ones, and linking up important numbers into series that are revealed by applying 1.177245771^n

Alternate Phi 1.622311470. Munck introduced this number in one of his newsletters and wondered if it could be an ancient form of Phi. He is probably exactly right. I therefore took the liberty of christening it “Alternate Phi”.

Another very powerful tool for identifying unfamiliar constants and linking up important numbers into series by applying 1.622311470^n. This seems to be even more heavily involved in ancient calendar systems than Alternate Pi 1.177245771 is.

I’ve been finding out lately how powerful some obvious variations on 1.622311470 can be. 2 / 1.622311470 = 1.232808888 may be even more powerful than 1.622311470 itself. Since I started working seriously with Mayan architecture, I’ve seen series where we can find 2 / 1.622311470 operating effectively as high as the 17th to 20th power.

1.622311470 may also turn out to be a metrological unit (in feet). 1.622311470 ft = 4/3 of 1 Remen of 1.216733603 feet

The Squared Munck Megalithic Yard (SMMY) 7.396853331 (ft) – used as a constant or probe, this number is often enough surprisingly good at serving analytical and data retrieval functions. This number can also be effective in its reciprocal form, 1 / 7.396853331 = 1.351926225 / 10

Not-Phi 1.618829140. I called it that because it is not Phi (the Golden Ratio 1.618033989) – see what I mean about my unimaginative names for things. I “discovered” this in an expanded model of the Great Pyramid. The natural ratio between apothem and half base length of a perfect pyramid with perimeter/height ratio of 2 Pi is 1.61899318661, which is an invalid number within the system in use.

1.618829140 is what I found in the search for a valid form, and makes a true statement of the apothem / half base ratio using the adapted valid version for the “ideal” apothem length of 194.4 Pi.

Not-Phi not going to outshine “Alternate Phi” by any means, and unlike “Alternate Phi”, Not-Phi performs poorly at higher powers, but sometimes it’s just the right tool to interpret a challenging number. Thus far it seems to gravitate toward the summits of important pyramids, such as the Great Pyramid and the pyramid temples of Tikal, Guatemala. Some surprisingly creative things may have done with this number at Tikal.

Square roots. I may not resort to them that often, but numbers like sqrt 15, sqrt 60, and sqrt 240 can also be very powerful interpretive tools and data retrieval keys, perhaps sqrt 60 especially. Again, I’ve recently found the reciprocal of sqrt 60 working at as high as the 27th power. It’s probably another tool I should use more myself.

1 / x = y. The reciprocal check. It happens quite often that a number that isn’t familiar turns out to be the reciprocal of one that is familiar, all we have to do is divide 1 by the unfamiliar number to convert it to its reciprocal to see what it looks like “backwards” – or “forwards” as the case may be when we are already looking at reciprocals and don’t realize it.

These days I’m getting better at using variations on this theme: 2 / x = y, 3 / x = y, 4 / x = y and so forth.

1.067438159. Similar to Not-Phi in that it gives poor performance at higher powers, 1.067438159 is nonetheless often enough effective as an analytical tool so that it deserves honorable mention at very least. It has a way with turning the unknown into the known.

–Luke Piwalker

“Pi Numbers” and “Square Root” Numbers

I have a confession to make – I’m not really the last of the Pi Jedi in that technically, I’m the first and last of the Square Root Jedi, trying my best to cover for the absence of the two venerable Pi Jedi, Carl P. Munck and Michael L. Morton while doing my own thing. What does this mean in plain English?

In working with Munck’s system of numbers, I work with two kinds of numbers, which I call “Pi Numbers” and “Square Root” numbers, since someone went and left the naming of these things to the unimaginative likes of me. I prefer to work with Square Root numbers and select them for proposed measurements of ancient architecture because they can be extremely powerful tools for unlocking whole series of important numbers.

I think just recently I found the reciprocal of sqrt 60 working to reveal important numbers at all the way to the 27th power. By pairing sqrt 60 with the right number, that’s writing 27 important numbers by writing just two!

To the best of my knowledge, “Pi numbers” will always resolve into whole numbers when multiplied or divided by Pi x number of times, which implies their ultimate derivation is from whole numbers and the Pi ratio. Some very large whole numbers can be tricky because they go on so many places they can look like they aren’t whole numbers, when the actually are.

“Square root” numbers to the best of my knowledge, will never resolve into whole numbers via the Pi ratio, but tend to resolve into whole numbers via multiplication or division by some of the most commonly seen square roots in “Munck’s” math like sqrt 15, sqrt 60, sqrt 240, sqrt 960, sqrt 3840 and etc.

( 15, 60, 240, 540, 2160, 4860, 8640, 31104, 77760, and 174690 are some of these numbers that have valid and important square roots within the system in question).

That may help explain why numbers like 6480 or 25920 don’t seem to have valid square roots, and of course we can see the pattern of multiplication of one whole number by 4 in there (15 x 4 = 60, 60 x 4 = 240, 240 x 4 = 960, and etc), because 4 has a valid square root (which is 2) in contrast to 2 because the square root of 2 doesn’t belong to the system. 

A Pi number multiplied or divided by another Pi number will always give another Pi number. 

A Pi number multiplied or divided by a Square Root number will always give another Square Root number. 

A Square Root number multiplied or divided by another Square Root number will always give a Pi number. 

For background here, a reader writing into Munck’s newsletter had noted seeing a lot of 1.000723277 in his own experiments with Munck’s math, which Munck referred to as a “gremlin” in the works. Sometimes things don’t work out the way we expect and we find they’re off by a ratio of 1.000723277.

Over the years, we’ve realized (or at least I have) that 1.000723277 is less of an annoyance that can foul up our best schemes for interpreting ancient monuments, and more of an essential link between important numbers – a link between what may be two separate sets of calendar numbers, and a link between what may be two separate sets of metrological numbers (give or take my hypothesis that ancient metrological numbers are directly descended from ancient metrological numbers).

Back to the subject at hand for now, I called them Square Root numbers because that’s where they seem to get a foot in the door, as soon as we take a system based on circular mathematics and throw in the Almighty Sacred sqrt 240 that is at the heart of the proportions of the Great Pyramid and Stonehenge according to the reckoning espoused by Munck. 

We could theorize that the height of the Great Pyramid is actually 5 x (Pi^6) = 480.6945969 ft if we chose, as opposed to (sqrt 240) x (Pi^3) = 480.3471728, but it’s the Square Root numbers that make life interesting and provide some of the most powerful tools for unlocking data. 

5 x Pi^3 = 15.50313824, but the more interesting and useful option is 15.49193338, sqrt 240, and that’s also where the most infamous “gremlin” gets a foot in the door, namely the fine ratio 1.000723277: 15.50313824 / 15.49193338 = 1.000723277. 

We can tell right there that 1.000723277 is a Square Root number because it’s formed here from a Pi number .5 x (Pi^3) and a very obvious Square Root number, the square root of 240. 

So when it comes to the concerns of calendars, we know that 364.7562611 is a Pi number because it can be formed from a whole number and Pi (3600 / (Pi^2)) = 364.7562611, and… 

364.7562611 x 1.000723277 = 365.0200808 is a Square Root number, because it is formed here from a Pi number and a Square Root number. Likewise, Venus Orbital Period (A) 224.8373808 is a Square Root number because it can be formed from 225 (Pi Number) / 1.000723277 (Square Root number) = 224.8373808 (Square Root number). 

Rather than leap off the deep end into calendar numbers, I’ll leave it at that for now, but suffice it to say understanding that the fine ratio 1.000723277 is at work has been vital to understanding that there are seemingly two separate sets of ancient calendar numbers, and why there are seemingly two separate sets of ancient calendar numbers, that are expressed by the proportions of ancient architecture.

For the record then, my preference for 365.0200808 over 364.7562611 and my preference for 224.8373808 over 225.0000000 probably make me a Square Root Jedi rather than a Pi Jedi proper, but pardon the pun, who’s counting? 🙂

–Luke Piwalker

The Great Pyramid and Stonehenge: Kindred Spirits?

This may be more fuel for discussion than a single post can accommodate. Over the years, I’ve developed a lot to say on the subjects of the Great Pyramid and Stonehenge. I’ve worked to expand Carl Munck’s remarkable but limited models of both monuments, and both continue to teach us some astonishing things.

Munck’s model of the Great Pyramid features a perimeter of 3018.110298 ft, a height of 480.3471728 ft, and a perimeter/height ratio of 6.283185307 = 2 Pi, the ratio between the circumference of a circle and its radius (radius = 1/2 diameter). While it’s contentious whether the Great Pyramid was intended to embody 2 Pi or the Golden Ratio Phi, other authors before Munck proposed the intention on the part of its designers to express the 2 Pi ratio.

Munck’s model for the Stonehenge sarcen circle involves a circumference of 305.7985078 ft, a radius 48.66934411 ft, and a diameter of 48.66934411 x 2 = 97.33868822 ft. The starting point here, a radius of 48.66934411 ft, is suitably close to Flinders Petrie’s data (Wm. M. F. Petrie: Stonehenge: Plans, Description and Theory, 1880).

Munck observed that for these values, they are linked by the Pi ratio, and can be derived from the square root of 240.

(sqrt 240) x Pi = 48.66933441

(sqrt 240) x Pi x Pi x Pi = 480.3471728

Stonehenge’s proposed 48.66933441 radius reduces to 40 Remens of 1.216733603 feet each. Certainly this raises questions to be able to model the inner circumference of Stonehenge in an even number of ancient Egyptian metrological units. The first thing anyone might want to ask (I did) is whether this suggests ancient Egyptians designed and constructed Stonehenge.

Personally I lean toward the opinion that the builders of Stonehenge and of the Great Pyramid had both already inherited these units of measurement from an older source, but if someone wanted to make the argument that at very least, whichever of these two monuments came second, it’s designers designed it with the other in mind, there is more that can be considered.

Last time I checked in with Petrie’s data on the Great Pyramid (Wm. M. F. Petrie, Pyramids and Temples of Giza), if memory serves we can also find 80 Remens, equal to the inner diameter of the Stonehenge sarcen circle, as the distance down the ceiling of the descending passage from the entrance until it branches off into the ascending passage, leading to the “Grand Gallery”.

“The Pyramids and Temples of Giza” Online

http://www.ronaldbirdsall.com/gizeh/

What is more, having worked on both “ideal” and “concave” apothem values for the Great Pyramid, the value I have for the “ideal” apothem of the Great Pyramid turns out to be precisely twice Munck’s circumference for Stonehenge.

Some explanation is due here: for a pyramid, the apothem is the line running from the center of the base to the point (apex) at the top. As Petrie’s measurements noted and as photographs have shown, the Great Pyramid is slightly indented or concave at the center of each side, giving a slightly different length for the apothem that it otherwise would.

Hence, just as we can project a height for a pyramid even if it’s missing its uppermost part by following its slope angle upwards to a point, we can also project a line upward to the point from the center of a line linking two corners to obtain an “ideal” value for the apothem length even when the reality appears to be a “concave” apothem that starts roughly 3 feet toward the center of the pyramid from a line connecting two corners.

What I call the “ideal” apothem, then, represents the apothem value of an idealized perfect pyramid, rather than the apothem of a pyramid like the Great Pyramid which appears to be altered slightly toward achieving a different objective. In my model that different objective seems to be making the concave apothem for Munck’s model to be 500 Remens of 1.216733603, which lends truth to ancient Greek authors who described the Great Pyramid’s apothem length as 1 “stade” or “stadium”.

Also, these remarks about a Great Pyramid apothem that is precisely twice the inner sarcen circle circumference of Stonehenge are essentially true for an experimental model of the Great Pyramid which is missing approximately eight inches of courtyard pavement from within its enclosure wall that would have altered the height of the Great Pyramid once in place – in other words for the Great Pyramid in its present condition and not its original condition.

My expanded Munck model of the Great Pyramid has grown to include this “unpaved/paved” explanation for why Munck’s model of the Great Pyramid is the better part of a foot shorter than most people’s models which are often based on the best measurement, data and reasoning that can be applied.

Circumstantial evidence for this hypothetical missing pavement aside from the mathematics itself may include both examples of surviving casing stones that give the suggestion of having been protected from erosion along the bottom (as if they had been protected by this proposed missing pavement) and the untidy appearance of the present pavement around the Great Pyramid (and others), suggesting the incentive for an additional cosmetic layer of pavement that might have been quick to be plundered for having been made of a desirable material, and for having been more accessible than any materials used for casing the pyramid itself.

At any rate, my expanded unpaved/paved model of the Great Pyramid provides different but meaningful values with or without the pavement, and simply means what what most people are studying in modern times may be the unpaved model and not the finished product. (Later on, I’m sure I will want to go into more detail about the unpaved model to demonstrate why its values are significant).

The better part of two decades ago, I made an interesting “discovery” about Stonehenge. From my perspective, all that “Pi Jedi” do is re-discover ancient things, but I was clearly not the first to make such a discovery in the present. Due to a word processor accident that took place while I making a list of all of the ratios that occur between numbers Munck associated with Giza’s architecture, I lost the data on a remarkably important number and didn’t have a fateful encounter with it until sometime later.

Had the word processor glitch not happened, I’d have seen immediately that the number recurs at Giza with an insistence that suggests it may have considerable importance.

Some time later, I started working with the materials from Professor Alexander Thom, who despite his meticulous work as surveyor and mathematician, is sometimes denigrated for the unorthodoxy of his ideas. This unfortunately sometimes goes as far as marginalization of a proposed metrological unit he obtained from the data from a great many surveys of Megalithic stone circle sites, a unit of about 2.72 feet which he called the Megalithic Yard.

https://en.wikipedia.org/wiki/Alexander_Thom

Carl Munck developed his own version of the Megalithic Yard, of 2.719725671 ft. This is a perennial problem child however, because while this number performs brilliantly when it is squared (2.719715671^2 = 7.396853331 ft, the Squared Munck Megalithic Yard or “SMMY”), before it’s squared it doesn’t even belong to Munck’s own system of numbers.

With this in mind, after much experimentation, I developed a version which does belong to his system of numbers before it’s squared, of 2.720174976 ft. I generally refer to this as the “Alternate e’ Megalithic Yard” or “AEMY”, because this value is remarkably close to the value e’ from tetrahedral geometry of 2.720699046.

If it helps any to remember these entities or keep them sorted out, you might think of them as your two new friends, Sammy (SMMY) and Amy (AEMY).

Being confident in having the inner values for the Stonehenge sarcen circle correct, it began to trouble me all the more not to have any values proposed for the exterior of the sarcen circle. Finally, after much reassurance through experiment and calculation, I decide to take Prof. Thom up on his “120 Megalithic Yard” value for the outer circumference of the sarcen circle, giving the Meg Yard for this application as “AEMY” or 2.720174976 ft.

120 x 2.720174974 = 326.4209968 ft

Later, when I became curious whether the outer/inner sarcen circle ratio was relevant, I obtained

outer circumference 326.4209968 ft / inner circumference 305.7985078 ft = 1.067438159

When I tried using this number 1.067438159 with Munck’s data for the Giza Pyramids, that was when I finally became aware of the word processor accident that had removed this number from my list of Giza ratios. It turned out that among other things, this number is the ratio between Munck’s perimeter length for Cheops’ (Khufu’s) pyramid and the perimeter of Chephren’s (Khafre’s) pyramid.

Thus Munck was obviously quite aware of 1.067438159, but seems to have kept it to himself.

Recently I attemped to revise Munck’s model of the Mycerinus pyramid which was apparently based on a bad data source, and arrived at a model whose edge length is 326.4209968 ft to a remarkable degree of accuracy.

Thus, the “Stonehenge – Giza Coincidences” continue to accumulate, but still it’s difficult to tell for certain what this means regarding what people built and designed Stonehenge. Even for finding what is essentially Thom’s Megalithic Yard repeatedly at Giza, this too may simply suggest common inheritance of more ancient mathematics and metrology.

As authors such as John Michell and Bonnie Gaunt have pointed out, a number of notable ancient metrological units are loosely linked by simple square root relationships. The double Remen is essentially the diagonal of a square with sides of 1 Royal Cubit in length, the Palestine Cubit is essentially the diagonal of a rectangle with side of 1 Remen and 1 Royal Cubit in length, and as it turns out, the Megalithic Yard is the diagonal of a rectangle with sides of 1 Remen and 2 Remens in length, making the Megalithic Yard seem to be rather Egyptian in origin if it cannot be traced back further in time.

Another remarkable find that wasn’t even even taken into consideration at the time of deciding to honor Thom’s assessment of the outer proportions of the sarcen circle is this:

outer circumference 326.4209968 ft – inner circumference 305.7985078 ft = 20.622489000

That figure in feet is strikingly close to the value of the Royal Cubit in feet when given in inches. Using the “.03 Radian” Royal Cubit of Michael L. Morton of 1.718873385 ft, that’s 1.718873385 ft x 12 inches = 20.62648062 ft, thus the difference between the inner and outer diameters of the Stonehenge sacren circle is as precisely as possible 12 Royal Cubits, another “Egyptian” flourish at an ancient British monument.

Thus even if I’m not very firmly convinced that ancient Egyptians had a hand in Stonehenge, I would find it hard to argue with anyone who was.

For what it’s worth, for this number

outer circumference 326.4209968 ft / inner circumference 305.7985078 ft = 1.067438159

I could write a whole chapter on the things I’ve learned about 1.067438159 since it came to my attention, including that Stonehenge seems to want us to understand that it is almost but not quite the cube root of the Remen in feet, seen as 1.216733603 ft.

1.067438159^3 = 1.067438159 x 1.067438159 x 1.067438159 = 1.216264895

In my revised model of Mycerinus’ (Menkaure’s) pyramid at Giza, the perimeter value turns out to be (1.067438159 x 360 x 360) / 100 feet.

Those are always some of the things I find the most compelling – the things we didn’t even think about when we chose the most likely numbers for the measurement of a monument. It’s harder to say we soured our own experiments with preconceived notions when that happens.

— Luke Piwalker

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