Le Menec Revisited

Some time ago back in July of 2020 I wrote a post on Newgrange and Le Menec. Not much else has been done with the subject since then, but Peter Harris recently mentioned Carnac in the context of the possible use of the Megalithic Foot there, which started me thinking about the subject of French Megalithic sites and whether I had done any work that supported the use of the Megalithic Foot in the region.

I’m generally not keen on working with the French Megalithic sites because I don’t know that we have ever had adequate data on the curious massive stone rows that we typically find in some of these places. There is an erratic nature to them that makes them seem if they could constitute entire libraries of significant mathematical values, but that seems like a very risky thing to just guess at.

However, at least the stone circles (cromlechs) that we find at some of these sites should be more easily fathomable, and better still, several of them were described by Professor Thom.

One reason the two subjects of Irish sites and French sites were lumped together in a single post is because there may be some important parallels in spite of the geography.

This rather rought sketch of Newgrange after Euan MacKie in The Megalith Builders shows that MacKie gave for the width of the outer ellipse a measure of 110 Megalithic Yards which is approximately 110 x 2.72 = 299.2 ft

I’ve annotated Thom’s diagram of the West cromlech of Le Menec so we can better see one of the parallels

The total length of the cromlech is 17 + 20 + 7 = 44 Megalithic Rods (the “Megalithic Rod” is something of a superfluous unit of measure since 1 Megalithic Rod = 2.5 Megalithic Yards)’ 44 x 2.5 = 110 Megalithic Yards = ~299.2 ft – so we can see that this unusual length was used in both Ireland and France.

In spite of David Kenworthy’s interpretation of this as simply 110 Megalithic Yards, as readers may be well aware, the number 11 is not in my vocabulary, for good reason, which I’ve recently gone into more detail about in my post about Exclusion of many whole numbers. One reason for this exclusion is because ratios like 22/7 = (11 x 2)/7 interfere with the use of the true Pi ratio, but perhaps even more importantly, the use of multiple units that the ancients demonstrably practiced is enough for metrological unit values to interfere with each other even with only a small number of units and with the pool of whole numbers already greatly diminished as in done in my work.

Add a couple more ancient units of measure or a few more whole numbers to the pool and good luck trying to figure out what units they were really using. Even Petrie himself was quite confounded by this, causing his Royal Cubit values to fluctuate almost beyond recognition, but of course that is one of the basic classic problems in ancient Egyptian metrology, the ease of conflation of Royal Cubits and Remens, which is probably one of the first things that has led Egyptologists to make an absolute mess of ancient Egyptian metrology.

I sometimes remark that Egyptologists with the Royal Cubit remind me of children who have just discovered ketchup or mustard who put it on everything – ice cream, birthday cake – which in a way is endearing, but for a parent who has to wash out all the resulting ketchup or mustard stains, sometimes it’s also downright exhausting.

At any rate, we will want to observe that “110” Megalithic Yards = 299.2 feet might likely be an attempt to reference the Jupiter Synodic Period fractionally; that is, 299.2 x (4/3) = 398.93333333, which puts us quite close to the “textbook” figure for Jupiter’s Synodic Period of 398.88 days, so that even in Imperial units (“modern” feet) we are still readily recovering important astronomical data from this measurement shared by both of these ancient monuments according to the data.

The East cromlech at Le Menec, from Thom, “Megalithic Remains in Britain and Brittany”. Here, the total length of the West cromlech, 44 Megalithic Rods, is repeated as a width of 44 Megalithic Rods = 110 Megalithic Yards = 299.2 ft.

I’m not at all certain yet what either of these two Le Menec cromlechs are really made of metrologically. In my experience I think it really is unheard of for any design to be made of a single unit, be it Royal Cubit, Megalithic Yard/Rod, or any other. That’s really why these multiple units of length measure exist in the first place. If we looked up “Ancient Egyptian Units of Measure” on Wikipedia and all that were these was the Royal Cubit, that would be one thing – but that’s not the way it works.

Even while I really know so little about what went into these designs, I wanted to bring up the subject again because I’m not at all certain that some important things got pointed out that may be somewhat more self-evident on examination.

For instance, note that according to Thom’s data, the perimeter of the West cromlech is 304.4 Megalithic Yards, and the perimeter of the East cromlech is 370 Megalithic Yards. What is the relationships between the two? Remarkably, it is

370 / 304.4 = 1.215505913

Is there anyone who isn’t going to immediately equate this first and foremost with the Egyptian Remen value in Imperial Units, of 1.216733603 feet?

Let’s attempt to examine this proposal just a little more closely.

Previously, it was proposed that the calculated 1006.4 ft perimeter for the East cromlech is a rather obvious “echo” of the 100.6036766 ft calculated mean perimeter of the Stonehenge sarsen circle.

If this is the case, it would make the perimeter of the Western cromlech, estimated at 827.968 ft to be, using the primary Remen value,

1006.036766 / 1.216733603 = 826.8340445 ft

Would this be a sensible proposal?

Well, it may be – starting with 826.8340445 / 2 = 413.4170223

Readers might recall that the amazing book by Harris and Stockdale identifies a “Unifying Value” of (Astronomy and Measurement in Megalithic Architecture, page 28) 413.42 days (15 Anomalistic Months and 14 Lunar Months, according to the text), which I believe to most likely be synonymous with 413.4170223 = (13.333333333 x (Pi^3).

Notice what happens if we use the value I use for the Megalithic Foot with this number

413.4170223 / 1.177245771 = 351.1739791, which is a number that we have been finding more often since beginning to examine the architecture of Egypt’s Faiyum Oasis, although it had long ago been found in the Great Pyramid model, where twice the perimeter of the now-missing apex section of the Great Pyramid would be 351.1730791 ft.

More importantly, 413.4170223 x 1.177245771 = 4866.934411, which is 100 times the 40 Remen inner diameter of the Stonehenge sarsen circle (40 x 1.216733603 = 48.66934411 ft).

In turn, 413.4170223 x (1.177245771^2) = 5729.577951, which is 100 times the Radian value of 57.29577951 (360 / (2 Pi) = 57.29577951), so we see that the Unifying Value of Harris and Stockdale appears to intimately related to and linked to both the Megalithic Foot and to the Stonehenge sarsen circle.

There is at least one other very noteworthy ratio that may be trying to leap out at us from the data – the length of the East cromlech is 52 Megalithic Rods, while while the width of the West cromlech is 34 Megalithic Rods

52 / 34 = 1.529411765 = 3.058823529 / 2

We may wish to recall at this point that the inner circumference of the sarsen circle calculates as 360 / (1 Megalithic Foot in feet 1.177245771) = 305.7985077, and that this very important number in fact appears to be incorporated into the basic proportioning of Thom’s Flattened Megalithic Rings.

Thom’s data for the Type A Flattened Megalthic Ring. The exact value for the perimeter / major diameter ratio Perimeter / MN = 3.0591 is thought to be 3.057985077. In a manner of speaking, what else would it be?

We hopefully know by now of course that circumference 305.7985077 / (2 Pi) = radius 48.66934411

This would be the same thing that we find often, that many Megalithic sites contain some of the same basic data that Stonehenge contains, although Stonehenge represents such an usually well-packaged bundled of this data that it’s very easy to see why it merited such a special architectural design to accompany such a special collection of numbers. There may really be nothing else quite like Stonehenge, architecturally or mathematically.

One way of looking at what is proposed here then, is that for

1006.036766 / 1.216733603 = 826.8340445 ft

Something it showing us is that a value in the ancient Egyptian Mystery Unit of 1.676727943 / the Remen = a value in another metrological unit.

I’m having a bit of trouble placing it, but this equation may well demarcate 826.8340445 as a metrological value, and it may be something of a Unifying Value metrologically speaking

For instance, a Megalithic Yard of 2.720174976 ft / a putative ancient Meter of 3.289868134 = 8.268340445 / 10. Likewise, the greater unifier of ancient metrological units, 2 Pi, turns 8.268340445 into

8.268340445 x 2 Pi = 51.95151515 = outer radius Stonehenge sarscen circle

Likewise, a value in Egyptian Sacred Cubits (see Thom Mid Clyth Quantum aka TMCQ) times Hashimi Cubits, will be in the same unidentified unit as is a vlue of 8.268340445

TMCQ / 2 = 3.872983357 ft; 3.872983357 x Hashimi Cubit = 1.067438159 = 8.268340445 / 2 = 4.134170224

So again, 4.134170224 may be a Unifying Value metrologically as well as a Unifying Value astronomically.

We may also wish to be observant that 4.134170224 / Remen 1.216733603 = 3.397761198, which is 4 Inverse Megalithic Feet

4 / 1.177245771 = 3.397761198, and this value (as 339.7761198 ft may have been built into Avebury partly for this reason, whereas Thom would have tended to identify it as 125 Megalithic Yards, again driving home the point about the risk of confusion of units even with a radically limited palette of ancient units and whole numbers. Once again, try and imagine what chaos this would be without such dramatic restrictive measures being in place.

While we are at it, why don’t we try and see if we can find out anything else about these subjects since it seems like on this occasion we may have spotted several things that were previously overlooked.

By the way, I am still experimenting with a value of 18.60376601 (6 x (Pi^3) / 100) as the number of years in the Lunar Nodal Cycle. This quantify may not have had a fixed value in ancient calendars- and it may not help that there is a variety of ways we can express the number of days in a year, but we might note that

18.60376601 / 4.134170224 = 45/10, so there is another possible facet of the Unifying Value form Harris and Stockdale’s book.

For the Le Menec West cromlech, with a perimeter (P) of an estimated 304.4 MY = 829.768 ft, and estimated length (L) and width (W) of 44 MR = 299.2 ft and 17 x 2 = 34 MR = 231.2 ft respectively, (without yet converting the perimeter into Megalithic Rods to match the length and width values which are in Megalithic Rods because this way we may be able to generate a meaningful “second opinion”) we obtain ratios of

Pmy/Lmr = 304.4 / 44 = 6.918181818
Pmy/Wmr = 304.4 / 34 = 8.952941176

As well as a length / width ratio of

L/W = 44 / 34 = 1.294117647

All of these three ratios look challenging to identify at first glance; however a reciprocal check shows that 1 / 8.952941176 = 1.116951380 / 10. Normally things like this will turn out to be one of three things: 1.115419203, 1.116225960, or 1.117818629; 1.116225960 = (360 x (Pi^3) / (10^5) has been surprisingly rare, so the correct fintended figure may well be one of the other two.

(There is also 1.117441171 – this is connected to the standard value for polar circumference in miles: (1 / 360) / 1.117441171 = 24858.28047 / 10^n – textbook polar circumference value about 24860 miles – but although we can find 1.117441171 in the Great Pyramid, generally speaking it has also seemingly been rather rare).

The other two ratios from the West cromlech of Le Menec I am going to try to refrain from attempting to qualify for now, but I do want to go back to the figure and make what may be a new observation, which is that while 34 MR = 231.2 ft, the standard Megalithic Yard / Megalithic Foot ratio is 2.720174976 / 1.177245771 = 231.0626246 / 100 = 346.5939368 / (15/10).

To harmonize the perimeter / (lenght or widht) ratios then, we next convert the perimeter into Megalithic Rods, and discover that 304.4 MY / 2.5 = 121.76 Megalithic Rods, which greatly resembles 100 Remens = 121.6733603 feet, but we will have to be careful with that. Instead then let’s first use the values in feet to detimeter the true perimeter / (length or width) ratios

With a projected perimeter of 1006.036766 / 1.216733603 = 826.8340445 ft, a possible width of 231.0626246 ft, and an estimated length of 299.2 feet

826.8340445 / 231.0626246 = 3.578398047 = 1.789199023 x 2
826.8340445 / 299.2 = 2.763482869

I wrote a bit about 1.789199023 on my Megalithic Portal thread just yesterday. So far so good, if we are lucky.

I’m still at a loss for 299.2 means exactly. In accordance with some researchers, we might take it as 110 Megalithic Yards, with an adjustment of 110 to one of the more likely Indus / Northern Foot values (currently, Long Indus foot = 1.100874628 ft, Short Indus foot 1.100078967 ft, and that’s the way they’ve been ever since way back when I didn’t know what an Indus Foot was, but was trying to relate these values to the Royal Cubit at ratio of 1:64 — i.e., 110.0078967 / 64 – standard Royal Cubit 1.718873385 ft).

Starting with the most likely figures 1.100874628 x 2.720174976 = 299.4571616. However, 299.4571616 x 1.333333333 = 399.2762155, which is thought to be the C value for the Jupiter Orbital Period. They may have wanted to put a better foot forward than that here.

What they might have done at Newgrange and at Le Menec might be something they seem to have done at Callanish and elsewhere, which is switch the emphatic on the Megalithic Yard from the “AEMY” of 2.720174976 ft to the “IMY” of 2.719256444 for various practical purposes.

Hopefully they have not done something even more elaborate, but even Stonehenge does seem to represent a small variety of Megalithic Yard values.

A few paragraphs ago, I wrote, “304.4 MY / 2.5 = 121.76 Megalithic Rods, which greatly resembles 100 Remens = 121.6733603 feet, but we will have to be careful with that.” and I wrote that because we have a slight problem.

If we believe the perimeter value to be 826.8340445 ft, 826.8340445 / 2.5 = 330.7336178, and 330.7336178 = 2.720174976 x 1.215854203 x 100 ft, or 2.718208957 x 1.216733603 x 100.

If we conserve the Remen value at the Long Remen of 1.216733603 ft rather than the short Remen of 1.215854203, we get a Megalitihic Yard value of 2.718208957, which some readers may already know as a Megalithic Yard that is made out of Megalithic Feet (3.2 / 1.177245771 = 2.718208957) and which is the radial/diameteral unit to a circumferential unit of 1.067438159, as was discovered while following up on the remarkably insightful work of Geoffrey Bath.

Specifically, if for example the radius of a regular circle is 27.18208957 ft, the circumference will be radius 27.18208957 x 2 Pi = circumference = 170.7901058 ft = 160 x Hashimi Cubit 1.067438159.

Technically (in the “Inductive Metrology” sense where we should see whole numbers of units in “forward” rather than Inverse form), a Megalithic Yard is of 2.718208957 ft is the Megalithic Yard of the inner sarsen circle circumference of Stonehenge

305.7985077 / 2.718208957 = 1125 / 10

So we can see that by conserving the ideal Remen value, we haven’t gotten into any trouble we haven’t been in before at Stonehenge, so to speak.

Let’s go back to this now for a moment, “Starting with the most likely figures 1.100874628 x 2.720174976 = 299.4571616. However, 299.4571616 x 1.333333333 = 399.2762155, which is thought to be the C value for the Jupiter Orbital Period. They may have wanted to put a better foot forward than that here.”

If we take this equation and make one or more Megalithic Yard substitutions,

1.100874628 x 2.719256444 = 299.3560426 = 399.1413902 / 1.333333333

1.100874628 x 2.721223220 = 299.5725600 = 399.4300799 / 1.333333333

1.100874628 x 2.718208975 = 299.2407275 = 398.9876367 / 1.3333333331

IF we have the mechanics right and the measure in question is indeed linked to the Jupiter Synodic Period by 1.333333333 rather than by some other number such as 1.331433599 or etc, then conserving the more ideal Indus Foot value, either the IMY of 2.719256444 ft has been used as at Callanish (see also “Callanish Number,” etc), or the DMY (Draconic Megalithic Yard of 2.721223220 ft, representing Draconic Month of 27.212220815 days the most accurately of any of the known Megalithic Yard values).

Since the IMY is generally more useful all-round than the DMY, it may be more likely that the IMY has been used as at Callanish, with the result that the premiere A value for the Jupiter Synodic Period, 399.1413901, has been spelled out.

That’s still a long way from solving the puzzle of Le Menec’s West cromlech in total, but hopefully on revisiting the subject we have been able to chip away more of the mystery this time.

As I look at the diagram of the West cromlech that 17 MR = 115.6 feet stands out – it might tempt one somewhat to make the Egyptian Royal foot something like this rather than the more established ~115.3 / 100 value, but again, we have experimentally qualified the cromlech’s 17 x 2 = 34 MR = 231.2 ft width as probably (2.720174976 / 1.177245771) x 100 = 231.0626246 ft, thus “17 MR” = 231.0626246 / 2 = 115.5313123 ft, and the significance of this number has been discussed in some detail before (115.5313123 does not translate well into Megalithic Feet, but instead it becomes rather talkative when divided by Inverse Megalithic Feet exponentially).

115.5313123 / 10^n serves as a viable approximation of a rather important astronomical figure — in crude numbers, it is no less than “Tzolkin” 260 days / Venus Orbital Period 225 days = 1.155555555, hence it is important astronomical data, even while still in in “modern” “Imperial” feet.

Let’s go back here for a moment now

IF 304.4 MR = 826.8340445 ft and 44 MR = 299.3560426 ft and 34 MR = 299.3560426, then

826.8340445 / 299.3560426 = 2.7620422735 = 1 / 0.3620509394

IF 304.4 MR = 826.8340445 ft and 44 MR = 299.3560426 ft and 34 MR = 299.3560426, then

826.8340445 / 299.5725600 = 2.7600459952 = 1 / 0.3623128026

Either I have made some terrible mistake, or as often happens, the ancients are still five steps ahead of us with their remarkable mathematical prowess. The only one of these four numbers that I actually recognize is 3.620509394, and offhand I have no explanation for what it’s doing here.

We may not be seeing the Harris-Stockdale Unifying Value doubled to form the perimeter length pf the Western cromlech, although it would eminently sensible to find it here for the reasons described, and more.

If we’re forced to look for alternatives, perhaps it’s worth noting that in crude terms, Anomalistic Month 27.55 days x 3/4 of Jupiter’s Synodic Period (399 x (3/4) = 299.25 days) = 27.55 x 299.25 = 8244.3375 = 20.61084375 x 400, and the standard Royal Cubit in inches is 1.718873385 ft x 12 = 20.62648062 inches.

Thus, while once again it may be easy enough to see roughly what the ancients were up to, it can be a different thing to determine exactly what they were up to. Hence, while our understanding of he cromlechs may have grown by leaps and bounds, the jury may still be out on this one as a whole.

Before I “sign off” on this post, let’s have another look at the East cromlech again, shall we?

We think the perimeter may be 1006.036766 feet, or exactly ten times the mean diameter of the Stonehenge sarsen circle, with a width of ~44 MR = ~299.2 feet, and a length of ~52 MR = ~353.6 ft

This gives us internal ratios of about (using the raw value 1006.4)

P/L = 1006.4 / 353.6 = 2.846153846
P/W = 1006.4 / 299.2 = 3.363636363
L/W = 353.6 / 299.2 = 1.181818181

These too are potentially rather confusing. Perhaps the 4 most likely things for 1.181818181 to represent are the Megalithic Foot of 1.177245771, the Alternate Megalithic Foot of 1.1797781993, (57.29577951^2 x 36) / 1^n = 1.181810286, and (12 x (Pi^2)) / 100 = 1.184352528.

Before putting this back on the back burner, lets’ try looking at one more aspect for possible guidance.

For the West cromlech, l/w = ~44 / 34 = 1.294117647.

If 44 MR = 299.3560426 and 34 MR = 231.0626246, “44 / 34” = 299.3560426 / 231.0626246 = 1.295562374 and 1 / 1.295562374 = 0.7718655771; 299.5725600 / 231.0626246 = 1.296499425 and 1 / 1.296499425 = 0.77130770806

None of the four numbers thus generated are familiar and I can find no prior history on them.

Talk about a prime example that not just “any old thing” works!

One possibility here might be that they really did resort to using the C value for the Jupiter Synodic Period — note that 231.0626246 / 299.4571616 = 1.296, and that is something that we do recognize and which has considerable important.

Another possibility might be that because the Meg Yard of 2.720174976 may not be at a premium here, perhaps its a “faux paus” to use 231.0626246 which again has been conjugated from 2.720174976 / 1.177245771

Yet another possibility might be that a mistake has been in seeing Jupiter’s Synodic Period where perhaps we are actually seeing Venus’ Orbital Period.

I’m going to use for this demonstration a rarer form of the Venus Orbital Period that was discovered at Monte Alban in the Americas

(15 / 1.000723277)^2 x 10 = 224.6748781

224.6748781 / .75 = 299.5665042, while 299.5665042 / .75 = 399.4220056, just missing the current B value for the Jupiter Synodic Period of 399.4300799 days.

399.4300799 x .75 = 299.5725599 and 399.4300799 x .75 = 224.6748783, the F value for the Venus Orbital Period. The architects of this design may have navigated some of the most treacherous waters in the calendar system with this because the .75 ratio will try again and again to make us think it is working here, and exponentially — but is it, and did the ancients really go so far out on a limb to catch this one as an F value from the calendar tables? So far we really shouldn’t have to get any further out than the C group.

This may be in essence where the last attempt got fouled up as well.

There may even be other possibilities as to what’s gone seemingly wrong here. A particularly interesting one can be illustrated thus

399 x .75 = 299.25; 299.25 x .75 = 224.4375

Is anyone thinking what I’m thinking here? Do you happen to recall, dear reader, the Aubrey Number from experiments with the Aubrey Circle?

Remember that we can conjugate the Aubrey Number 224.4305459 as (1.315947254^2) x (360^2) = 224.4305459 x 10^n, although the .75^2 trick is apparently not going to serve this possibility either, at least not if one of the primacy values (values A-C) for the Jupiter Synodic Period is going to be honored. It’s a novel, very challenging problem, and exactly how the ancients got around it may require more understand that we currently have.

As always, ancient “rocket science” is by rights not necessarily any simpler than modern “rocket science” – but at least readers have been afforded the opportunity to see some of what can go into an effort to interpret ancient monuments.

In the long run, the usual rule should apply that since the “name of the game” is to record and communicate astronomical data, that whatever combination of possibilities best serves that purpose is going to quite likely be the right answer.

Once again, at least we can say for that for this brain-twisting exercise, we have advanced our understanding of the Le Menec cromlechs considerably. If the next effort manages to advance things any further, we should have the answer at last.

–Luke Piwalker

Miscellaneous Metrological Musings

Happy New Year everyone!

I’ve ended up with a few mathematical thoughts the past several days that I thought I might share.

First, a bit of follow up – several posts ago I implied that as far as ancient meters go, 3.289868134 might take precedence even over the well-pedigreed figure 3.282806350, because it’s been observed on several occasions that 3.289868134 shows exponential value, and apparently often enough it shows it in excess of the exponential value of even some of the most respectable metrological values like the Hashimi Cubit, Palestinian Cubit, and others.

I went back and tried combining 3.289868134 with some other metrological units and found it working usually about the fourth power with various units, and as high as to the eight power with the Remen.

The series generated by combining 3.289868134 with 1.216736603 in operations of division and multiplication by 3.289868134^n links 360 / Preferred Eclipse Year to 2 x Preferred Saros Cycle and also includes the reciprocal of the “Real Mayan Annoyance”, 1/2 of Venus Orbital Period A, 1/2 of the Squared Munck Megalithic Yard (9 / Remen 1.216733603), and the present “B” value for the Jupiter Orbital Period 4332.424205. Powerful stuff!

We may wish to emphasize that the root unit of 3.289868134 ft can be seen as the Inverse Short Remen, as was demonstrated in the same very recent post about Ancient Meters. 4 / 1.215854204 = 3.289868134. Perhaps the ancients didn’t keep a Short Remen around for occasional use for nothing.

Intriguingly 2 Pi / 3.289868134 = 6 / Pi and I’d been talking again recently how (6 / Pi) can be used to link the other candidate Meter to the Royal Cubit.

I might have also mentioned this about the “Thoth Remen” — 2 Pi / Royal Cubit 1.718873385 ft = 365.5409038, which appears one of the lesser values for “symbolizing” the Solar Year. It we treat it like primary Solar Year values and divide it by 300 to generate a Remen, the Remen in question is 365.5409038 / 300 = 1.2184699679 ft, the Thoth Remen. The Thoth Remen thus has that additional and rather powerful pedigree lending to the sense of its legitimacy as an actual albeit less prominent ancient unit.

Also, I recently posted a “Tale of Two Numbers”, involving 1.424280286 and a very similar number, and probably focused more on the other number since it is newer, rather than wanting to try to round up as complete a presentation on 1.424280286, and older discovery for which equations are by now scattered all sorts of places.

However I hope I managed to make it clear – and if not I will try to do so now – more about how 1.424280286 got chosen in the first place. At the time I had the sense that 1.676727943 was very important to the Maya, because of its relationship to the Calendar Round values I work with and it was probably also observed that 1.676727943 can be seen as the integral or basic unit of what appears to be the height of the El Castillo Pyramid at Chichen Itza (that number has also been in Egypt, where it appears to be included in the Bend Pyramid according to Petrie’s data).

As the ratio of two very important numbers then, 1.676717943 / 1.1772435771 = 1.424280286 it has substantial pedigree, but “extra points” are awarded as well because it is the subject of multiple metrological convergences. It is not only 1.676717943 / 1.1772435771, but also Palestinian Cubit / 2 Squared Munck Megalithic Yards and possibly other things.

This same consideration has successfully gone into the interpretation of Stonehenge.

I think the conclusion was reached that both 1.424280286 and the other similar number 1.423799344 were both featured at Stonehenge.

I had also noticed the other day that a possible Meter of 3.289868134 ft has an important square

3.289868134^2 = 1.082323234 / 10. I may forget half the reasons by now, but I’ve long been saying that 1.082323234 is an important number to astronomy that may therefore have also been important to the Maya because of it.

One reason that this was proposed is because 1.082323234 x 4 x 1000 = 4329.292936, which is thus far though to be the primary representation of the Jupiter Orbital Period’s “textbook” value of 4332.59 days.

I think I might have discussed some of the metrological convergence on the same number in relation to Callanish in Scotland, but I got to experimenting a little more in the context of (1.082323234 x 10) having the valid and useful square root 3.289868134.

4329.292936 / 100 also has a valid and useful square root, which is sqrt 43.29292936 = 6.579736268, which is 1000th of one of the two primary values for the Saros Cycle. (The “natural” value is sqrt 4332.59 = 6582.24126 / 100, and the “textbook” figure for the Saros Cycle is 6585.3211 days).

To the best of my knowledge, this is a new discovery: 43.29292936^2 x 12 = 224.9138281 / 10^n, Half Venus Cycle (Calendar Round) C value, something we may not be able to do successfully with the “natural” Jupiter Orbital Period value.

224.9138281 is another figure reluctantly accepted on my part originally because of being the subject of multiple metrological convergences.

If memory serves, 4329.292936 was presented as being a possible rationale for some departure at Callanish from the primary “AEMY” Megalithic Yard of 2.7270174976 to the secondary “IMY” Megalithic Yard of 2.719256444 ft. The standard Megalithic Foot value is strong at Callanish, and

1 Megalithic Foot 1.177245771 / 1 “IMY” Megalithic Yard 2.719256444 = 4329.292936 / 10^n, whereas “AEMY” / (2 Pi) = 4329.292936 / 10^n. Thus there is metrological convergence upon 4329.292936 just from the Megalithic Yard alone, but 2.719256444 better fits as the correct unit value into the particular circumstances at Callanish.

I should probably point out that 4329.292936 has metrological value of some kind. It’s a simple division of the Stonehenge outer sarsen circle radius of 51.95151515, which is believed to be based on an unidentified unit of measure (possibly an oversized Assyrian Cubit of the standard value 1.622311470 x 1.000723277 ft).

51.95151515 / 4.329292936 = 12.00000000

Note that 51.95151515 x 4.329292936 = 43.29292936^2 x 12/10 = 224.9133281, Venus Orbital Period C

Another thing we can do with this is recycle the number that turns 51.95151515 into the Megalithic Yard

51.95151515 / (6 / Pi) = 27.20174972, and 27.20174972 / (6 / Pi) = 14.24280285

(Since the square of (6 / Pi) = 364.7562611/10^n, there’s a reference to the Solar Year in there too).

There are also some metrological convergences taking place through squared unit values. If we didn’t know not to square our units for basic metrological analysis, we might be tempted to think that the (Inverse) Squared Megalithic Foot was the base unit value of 51.95151515, but curiously, we could say the very same thing about the (Inverse) Squared Short Remen!

This may not help any to identify the unit of the outer sarsen circle diameter and radius, but it does seem to help to corroborate some good choices of numbers, and most likely should count as metrological convergence.

I probably shouldn’t avoid the geodetic significance of 1.424280286, that seemed to be part of why it was so important to the Maya. For example

1.424280286 / 57.29577951 = 24858.38047 / 10^n, standard representation of the Earth’s Polar Circumference. (At Tikal, it was also discovered that (1 / Lunar Year) / (Venus Orbital Period A)^3 = 24858.38047 / 10^n).

Regarding 4329.292936, it’s also worth noting that

4329.292936 x 1.216733603 = (1 / 18983.99126) x 10^n

I should probably emphasize that since 3.289868134 shows unusual exponential value, and since 3.289868134 x 4/10 = 1.315947254, we should also expect unusual exponential value to 1.315927254 as well.

1.315947254 is also a valid square root, of a false (or alternate) square root of 3 (1.315947254^2 = 1.731717175; sqrt 3 = 1.732050808). 1.315927254 x Pi = 4.134170226 which is very much like a number identified in the work of Peter Harris and Norman Stockdale as an important “Unifying Value”.

Regarding the Jupiter Orbital Period of primarily the value 4329.292936 days, 1.731717175 / 4 = 4329.292936 / 10000.

–Luke Piwalker

A Druidic Cubit?

I’n a little hesitant to broach this subject for fear of ruffling a few feathers again, but I think it might be appropriate because it may touch on one of the more controversial features of Stonehenge.

William Stukeley proposed a unit of measurement in use at Stonehenge of purportedly of about “530 mm” or “52.83 cm” = ~20.8 inch. I am speaking in vague terms instead of historical ones because I may not have an appropriate reference work immediately at hand adn thus am relying on Internet quotes for the purposes of discussion, although Wikipedia’s discussion of “Cubits” makes reference to a discussion of the “Druid’s Cubit” by Burl.

Burl, Aubrey (2004). “A. D. Passmore and the Stone Circles of North Wiltshire”. Wiltshire Archaeological and Natural History Magazine. 97: 197 Retrieved 9 April 2020.

Often enough, someone may come to the realization that 25 / 12 = 2.083333333 and assign this value in feet to a “Druid’s Cubit”.

I have a few struggles with this, not the least of which is not having presently at hand Stukeley’s measurements which inspired this observation. Offhand I would trust Stukeley even less than I trust Burl to provide accurate measures of a Megalithic Monument.

It’s true that if we keep throwing the major metrological key 12 at it, 2.083333333 will produce some important values in Imperial Feet, but if one is going to embrace ancient use of Imperial Units, it may not be a good place to start.

The next thing that may happen is that someone may try to divide it by 12 and declare that yet another Egyptian Royal Cubit has come forth this way.

Thus although to the best of my knowledge, researcher David Kenworthy has championed a Druidic Cubit of 2.083333333 ft, for more than one reason this would have to be another point of departure between David’s work and mine.

There’s rather a deep divide here – I hope none will be insulted by this since there are few if any metrological researchers whose work I don’t admire, and I’m sure all hard work in the field must pay off with some findings of genuine merit, but there are a lot of people whose work I look at just sort of scratch my head, or roll my eyes, but my general complaint is that it seems to be taking them umpteen version of the Pi ratio AND of the Royal Cubit for them to patch together their mathematical systems, and yet what results still seems to be just as tangled and complex as my own system of numbers for anyone who isn’t the author of it.

If there is any island of calm in the storm of numbers that I’m aware of, it’s picking ONE value for Pi, and doing your best to pick ONE – or at least one primary – version of the Royal Cubit.

From my perspective, these two are the closest thing to “sacred” possible, and they are not to be trifled with lightly – not just for the sake of keeping things from getting unnecessarily complicated, but because of some of the things these numbers represent.

In terms of astronomy, when it comes to the Royal Cubit, it is first and foremost the mathematical link between Solar Year and Lunar Year – quite possibly the first two calendars ever discovered by ancient humans, and that, along with both the geometry-oriented nature of both Pi and Morton’s Royal Cubit of 1.718873385 (54 / Pi / 10) ft, may be something else that is not to be trifled with lightly.

Another complain of mine about a putative Druidic Cubit with a value of 2.083333333 ft, is that this is none other than the reciprocal of 1 / 2.083333333 = 480 / 1000, and since 48 / 12 = 4, this is really the same thing as 12. There simply is no need to make a metrological unit commemorating something that is already a firmly established part of metrology.

For what it’s worth, this should establish once and for all that 2.083333333 ft is a value in Inverse Imperial Units, and there is virtually nothing in all the data I’ve collected to suggest that literal use of Imperial Units was any kind of frequent event at Stonehenge.

For many years I’ve been familiar with 2.083333333 = 480 / 1000, as well as another number that is similar. The reciprocal of Munck’s height for the Great Pyramid, 480.3471728 ft, is 1 / 480.3471728 = 2.081828596 / 10^n ft.

Thus the difference here is the customary difference between Imperial Units and the very similar (but not identical, and for good reason) Hashimi Cubit/Egyptian Royal Foot. We can equate the two if we like, but in doing so we may only serve to help obscure the possible origins of both of them.

A key question however when proposing any metrological unit value should probably be “How well does it work in situ?” – do we really get anything sensible if we measure Stonehenge with it? – but of course since the Hashimi Cubit has long since been an integral part of Stonehenge’s mathematics and metrology, that may be something of a rhetorical question – in theory, at least, a Druidic Foot 2.081828596 should work well at Stonehenge because a Hashimi Cubit of 1.067438159 ft works quite well at Stonehenge – but because of this, it may not tell us much that we don’t already know.

(By the way, an “Imperial Hashimi Cubit” of 1.066666666 ft is indeed embedded in my model Stonehenge, but we have to dig so deep for it that few could find it, which only goes to contraindicate the idea of this being a Hashimi Cubit of great importance). Most would have an easier time finding it dividing Thom’s outer 48 Megalithic Rods for the sarcen circle by the inner 45 Megalithic Rods, but that’s not necessarily where it’s hiding and there were demonstrably a great many things Thom did not know about ancient metrology, the same as with Petrie, including that it stands to reason for the Megalithic Yard (times 2.5 = Megalithic Rod) values used in the sarsen circle to be different for the exterior than for the interior, for very good reason.

The reason I being this up most of all is that I’ve noted that a number of researchers into ancient mathematics seem to honor John Neal’s interpretation of the Stonehenge sarsen circle having a mean value in Greek Feet. Neal’s figure of 101.376 ft here for 100 Greek Feet is very near to my own primary value for same, 101.3944669 ft Imperial.

The mean of the indicated figures is (104.27245 + 97.32096) / 2 = 100.796705 ft, rather than 101.376, but technically ~97.32096 should be the inner diameter of the Sarsen Circle rather than the Lintel Circle. Few would be happier than myself to see the Greek Foot of ~101.376 here since it is the Remen at the ratio 10/12, but that’s not what it appears to be.

DavidK and rodz seem to want to take this mean figure in the direction of ~100.8, while I take it in the direction of 100.6036766 based on the other established parameters of my model. My model also indicates that the outer Lintel Circle diameter should be 104.0913798 ft rather than 104.27245 ft.

I’m not suggesting that’s what it is (it’s actually an unusual approach to the Hashimi Cubit of Egyptian Royal Foot like that of the Megalithic Yard as used in the Great Pyramid’s base length), but for the record, 104.0913798 ft would be 50 Druidic Feet of 2.081828596 ft each.

What the Lintel Circle’s own width is may still be up for debate; perhaps too few have noticed how similar Neal’s figure of 347.57485 / 100 actually is to the 346.62 day Eclipse Year but 345.8499644 is also a significant figure in Hashimi Cubits / Egyptian Royal Feet – it’s in fact the base length of the Mycerinus Pyramid in the revised model.

I’d like to emphasize this in light of my recent post on “Exclusion” that not even throwing out as many whole numbers as has been done in this work has been enough to keep metrological confusion like the Sarsen/Lintel circle mean from trying to happen — too many whole number of unit figures in one unit still closely resemble whole number of units values in other units of measures.

Thus, from my perspective, the whole numbers you see omitted here…

Weren’t omitted simply to accommodate the true Pi ratio – perhaps above all else, they were omitted to accommodate the use of multiple metrologies.

–Luke Piwalker

Musings on an Ancient Meter

Many references have been made to possible values for an ancient meter. An ancient Meter very much like the modern meter is an idea that seems to occur to numerous researchers into ancient mathematics and metrology, myself included. I’ve been experimenting with the idea for more years than I’d like to admit, but it still isn’t clear to me what would be the primary intended value for an ancient Meter the way it is with other units.

1.216733603 and 1.718873385 are very obviously the primary values for the Egyptian Remen and Egyptian Royal Cubit respectively, 1.177245771 and 1.067438159 are very obviously the primary values for the Megalithic Foot and Hashimi Cubit respectively, and so on – but what the primary value for an ancient Meter would be seems to remain mysterious.

A number of candidates for an ancient form of the (1 / .0348 = 3.280839895) ft Meter have been known for a long time, including the “Radian Squared Meter” (Radian 57.29577951)^2 / 1000 = 3.282806350 ft, and the “Shaved Meter” 3.282806350 / 1.000723277 = 3.280433687 ft.

The “Radian Squared Meter” differs from the contemporary Meter value by 3.282806350 – 3.280839895 = 0.001966454645 ft or 0.001966454645 x 12 = 0.023597455 inches, about 2/10ths of 1/10th of an inch, and the “Shaved Meter” of course differs from the modern Meter by even less.

A possible formula has been noted that may link some ancient Meters to the Egyptian Remen, which is 4 / Remen = ~1 Meter.

4 / 1.216733603 = 3.287490368
4 / 1.215854204 = 3.289868134
4 / 1.218469679 = 3.282806351
4 / 1.219350970 = 3.280433687

It seems uncertain whether 1.219350970 was recognized as a form of the Remen or whether this would be “mixing metaphors” mathematically. Likewise, 3.287490368 hasn’t seemed to attract all that much support as an ancient Meter and may simply be another expression of standard Remens of 1.216733603 ft.

1.218469679 is the so-called “Thoth Remen”, so-called because Our Founder Carl Munck made a big deal out of the number 9 and its reciprocal (along with the number 240 and its square root) being symbols of “Thoth, Father of Number”, and demonstrated that if we square the reciprocal of 9, it produces all the numbers (except 8) in the correct order

1 / 9 = .1111111111; .1111111111^2 = 1.234567901 / 100, making “Thoth” the “Father of Numbers” literally.

Michael Morton and I also discovered that if we multiply this by the number missing from the sequence, 8, we get all the numbers in descending order

1.234567901 x 8 = 9.876543210

This wonderful phenomenon doesn’t seem to see all that much usage, largely because 2 / 1.234567901 = 162/100, which is far too easily overshadowed by more powerful and more useful numbers like 2 / 1.622311470 = 1.232808888 or 2 / 1.618829140 = 1.235460834, and yet it is part of our vocabulary.

Finally, 1.234567901 x (Pi^2) = 12.18469679, or 10 “Thoth Remens”, and that is why the name was used.

We see in the list one of 1.218469679’s claims to legitimacy, which is

4 / 1.218469679 = 3.282806351, which is equivalent to 4 / (360 / (1.718873385^2)). Because basic circular geometry, involving the 360* degree circle, 2 Pi, and 360 / (2 Pi) = Radian 57.29577951, it’s a fairly strong vote of confidence for the legitimacy of 1.218469679 as an actual Egyptian Remen Value, no matter how little use it may see for being greatly overshadowed by the two most important forms of the Remen, 1.216733603 and 1.215854204.

This may bring us to a struggle for “primacy” between 4 / 1.215854204 = 3.289868134 and 4 / 1.218469679 = 3.282806351.

Certainly it’s a huge vote of confidence for the possible legitimacy of 3.289868134 as an ancient Meter value that such a figure is technically the same as the “Short Remen” of 1.215854204 ft, but it’s a difficult battle between the important circular geometry origin of 3.282806351 and the relatedness of 3.289868134 to the “Short Remen” as to whether 3.282806351 or 3.289868134 is the first Meter value we should try when confronted with a possible ancient Meter.

3.289868134 may have something important going for it which may not, which is exponential value.

For example, while 3.282806351 ft is currently thought to be the intended width of the Stonehenge sarsen circle (outer diameter 103.9030303 ft – inner diameter 97.33868822 ft = 2 x 3.282171039 ft, which can represent 3.282806351 but cannot represent 3.289868134, as accurately as we would prefer to require, we nonetheless have only to drop 3.289868134 into this situation to see some of what it’s capable of

2.733689474 x 3.289868134 = 29.58741331 = 4 Squared Munck Meg Yards
29.58741331 x 3.289868134 = 97.33868822 = 80 Remens in ft
97.33868822 x 3.289868134 = 320.2314486 = 300 Hashimi Cubits in ft
320.2314486 x 3.289868134 = 1053.519238 = 500 Palestinian Cubits in ft
1053.519238 x 3.289868134 = 346.5939370 x 10^n, preferred Eclipse Year

So we not only see exponential value in excess of that of most metrological units from 3.289868134 here, but we see that it is directly linking other metrological units to each other, and to the preferred Eclipse Year value, which should put it on at least equal footing with a possible meter of 3.282806351, which should be hard pressed to show such exponential value even for being an exponent of the Radian.

3.289868134 has been considered as a possible meter for long enough that it’s tried to take on several names. It’s been called the “Guachimontones Meters” before because in his paper on one of the Guachmontones circular step pyramids, Dimitrios Dendrinos asks whether an ancient Meter could have been in use there, and the Meter value as dictated by the suggested geometry formula would be 3.289868134.

That’s when I began to take the question of an ancient meter more seriously again.

3.289868134 has been referred to before as the “Third Meter” before, not only because it was the third candidate meter to be taken more seriously, but the name is also appropriate because One Third x (Pi^2) = 3.289868134.

This is also (Pi / 3) x Pi, and given as fond of (Pi/3) as ancient American mathematician seems to have been of (Pi / 3), perhaps at Tikal especially, they may therefore have looked favorably on the idea of 3.289868134 ft as a meter.

I have probably mentioned previously that the archaeologist and explorer Auguste le Plongeon contended that ancient Americans used the Metric System, although personally I found him difficult to take seriously because of the seemingly fanciful nature of some his published works. As absurd as this may sound to some, later less fanciful scholars have made the same contention, only more convincingly and with reference to historical sources (see Tlalcuahuitl).

I would happily agree with any of these sources, except that a problem that remains here is that I am very skeptical that ancient Americans used any single unit of measurement – the data from sources like George F. Andrews or even Teobert Maler simply does not seem to support this idea, very similar to how the overall data available on ancient Egyptian architecture simply does not support the idea of the ancient Egyptians having only measured in Royal Cubits.

What the data actually shows us seems simply too diverse for that, even though it may be enough to support the frequent use of some form of ancient Meter by either culture.

A possible Meter of 3.289868134 ft not only works to at least the 5th power with the inner diameter of the Stonehenge sarsen circle, it likewise works to at least the 5th power with Munck’s values for the Great Pyramid, which are herein presumed to be the Great Pyramid’s final measurements following the placement of a hypothetical layer of now completely missing pavement.

Likewise, it should be mentioned that 2 Meters of 3.289868134 ft = 6579.736269 / 1000 fr likely represents a probably ancient approximation of the Saros Cycle of 6585.3211 days. It’s a bit of a departure, but is roughly proportional to the departure represented by representing 18980 as 18997.72194, a gesture that finds some justification from certain likely ancient formulas, as has been mentioned on a number of occasions here.

Such things may also be facts of mathematical life given the structure of the numbers system itself, where the most common and most important fine ratio 1.000723277 is the ratio between the primary (A and B groups) sets of calendar numbers, for example 365.0200808 / 1.000723277 = 364.7562611 and etc. 6579.736269 x 1.000723277 = 6584.495241, which may be about the best representation of the 6585.3211 day Saros humanly possible given the particular building blocks available.

Awhile back I attempted to project a specrtum of unit values in a manner very similar to that of metrologist John Neal, which gave very pleasing results for the Meter and other units. For the Meter, at least four of the five strongest candidates are included

Projected values for an ancient Meter based on geodetic (measurement of the earth) schemes. This particular scheme manages to include both forms of the Indus Foot as a possible slightly larger meter consisting of 3 Indus Feet 3.300236898 ft and 3.302623883 ft (see also work by David Kenworthy).

However, the set of formulas seems to derange the possible 4 / 1.216733603 value, and a set of values this diverse is still something we should be hoping to avoid for most units if at all possible – not because ancient people were simple-minded when it comes to mathematics, but because we assume they weren’t gluttons for punishment either.

For good measure, I might add this from a post I posted elsewhere back in April of 2020

“The Aztecs, inheriting the vigesimal [base 20] counting system of the earlier Mesoamerican cultures, created a cadaster on which they recorded surveys of each family’s tract of land. The cadaster listed the distances measured by the surveyor for each tract. Their recording system consisted of maps that marked these distances by making a dot to record every 20 measuring units, called the “tlalcuahuitl,” which translates as “land rod” and which we will call “T”. For measures less than 20 T they marked the map with a vertical line, marking up to 19 of them, in groups of five. For measures less than one T they used a series of glyphs to represent fractions of T such as a heart [1/3 T], arrow [1/2 T], or hand [2/3 T]. But, what was the size of T, the basic Aztec measuring unit? 

The native historian Fernando de Alva Ixtlilxochitl, writing about 1600 AD, described a vast Aztec communal dwelling built by the Tezcucan chieftain Nezahualcoyotl. This building was said to be capable of accommodating over two thousand people. Ixtlilxochitl writes: “These houses were in length from east to west four hundred and eleven and a half [native] measures, which reduced to our [Spanish] measures make twelve hundred and thirty-four and a half yards (varas), and in breadth, from north to south three hundred and twenty-six measures, which are nine hundred and seventy-eight yards.” 5 In his description, he spells out the basic “native” measuring unit for land, T, as exactly equal to 3 Spanish varas. So the question then becomes, how many feet are there in a vara? This is not an easy question to answer.

1234.5 varas = 411.5 “native measures”

https://www.academia.edu/37786698/The_Aztecs_and_the_Vara

I attempted some work with a vara of 2.788548009 = sqrt (7776 / 1000) ft, but I go to apply this and the magically suspicious number 1234.5 / 411.5 = another suspicious number 3, so whatever the Vara is, apparently the Aztecs were using the Spanish Vara before the Spaniards got there, or they had already adapted the Vara by the time this historical account took notice and “native measures” means simple multiples of the Vara?

Something that still may never have been considered is how close “411.5” is to the “textbook” value of the Full Moon Cycle, “411.78443029 days”, or that “1234.5 varas” = (2 / 1.620089105) x 10^n, very similar to 2 / 16.2 = 1.234567901, might herein with these considerations be overshadowed by the usual suspects, 2 / 1.622311470 and 2 / 1.618829140, had the quoted authors been aware of them.

I suppose on that note we might observe this for future reference as well – the Megalithic Foot value of 1.177245771 ft links a particular series. Correct decimal placement aside

2.012073535 = mean diameter sarsen circle / 50 = 60 Mystery Units of 1.676727943 ft
2.012073535 x 1.177245771 = 2.368705060 (possible Ubaid Cubit)
2.368705060 x 1.177245771 = 2.788548009
2.788548009 x 1.177245771 = 3.282806357
3.282806357 x 1.177245771 = 3.864669901
3.864669901 / 2 = 1.932334951 = “Cuicuilco Number”
Based on Emil Haury’s data for the Cuicuilco Pyramid, the metrological values of Royal Cubits and Hashimi Cubits may be present in the design;
206.2648062 (120 Royal Cubits) / 1.067438159 (Hashimi Cubit) = 1.93233495 x 10^n

1.93233495 is rather a strange number, but it does appear to be a metrological fact of life that we presume the ancients harnessed toward good use. Perhaps it’s only foggy memory, but I’m a little bit surprised to see here how it can be generated from both 1.177245771 and 1.067438159. Perhaps it it’s more important than its getting credit for.

So that is the state of an ancient Meter as it remains – it’s something that’s very possible, yet the specifics of it remain seemingly difficult to sort out entirely. The ancient Meter has great potential for being actual historical truth, yet it seems to remain a perennial metrological problem child.

Maybe someday a definite link to other units will be found and recognized which will settle the matter once and for all. Maybe 2021 will be the year.

Cheers and Happy New Year!

–Luke Piwalker

Exclusion

Especially if anyone is “just tuning in”, they may wonder what on Earth I’m doing with whole numbers. I still struggle to find the correct way to explain it, but for purposes of illustration, for the whole numbers 0 through 99, the numbers in this chart in green are the ones I work with, and the ones in red are ignored.

To some degree, it can be explained that I work with whole numbers that are factors of the number 360 – whole numbers that can create the number 360 through multiplication or division, but at best that might be confusing, and at worst perhaps not exactly true.

If we looked at 360 / 64 = 5.625, we might not guess that this is keeping with a such a rule.

From https://www.calculatorsoup.com/calculators/math/factors.php

Alternately we might say we use the whole numbers that can be created from 2 through 9 excepting 7 and hope that that remains true in the bigger picture.

Adding more zeroes to 360 finds more valid whole numbers used in this work, yet it still can’t pick up the valid whole number series 27 x 2 = 54; 54 x 2 = 108; 108 x 2 = 216; 216 x 2 = 432, etc. (216 x 1000 = 60^3)

At any rate, I’ve elected to put up the chart so that people can have a quick reference to the lower whole numbers that I do and do not work with.

I’ve also put it up to help make a point.

Notice what is happening here – 100 possible whole numbers have been thinned down to only 30.

I should admit that sometimes I struggle to embrace the work of others who work with ancient mathematics. These rules have become so ingrained that watching other use numbers like 7 or 11 or 13 is like watching them set fire to their cat – such things simply aren’t done and I feel a bit as if as being a responsible citizen, I should perhaps summon the authorities.

The point to be made is that even by reducing the pool of available whole numbers from 0-99 by a whopping 70%, we have still not prevented the often confusing metrological overlap that happens just because ancient cultures for some reason demonstrably used multiple units of measure.

If you think what I’ve shown in the last few posts or more looks like a mathematical traffic jam, imagine what a mess it might be if 70% of the available whole numbers had not been excluded from the proceedings.

In the last post especially we tried to capture some of that metrological overlap and harness it to good use, but we still ended up with 4 different Palestinian Cubits to go with 4 different Megalithic Yards in order to do it.

Another way of illustrating metrological overlap might be that in spite of the case that has been made for the two numbers from “Another Tale of Two Numbers” and what we have discovered about the relationships to the Palestinian Cubit, we still get figures that run contrary to this such as

2 Hashimi Cubits / 1.5 = (2 x 1.067438159) / 1.5 = 1.423250879, which is neither of the two number from the “Tale”, 1.423799344 or 1.424280286.

Even with such a simplified pool of whole numbers, it’s still that easy to generate from a whole number and a metrological unit a number that so closely resembles others that also arise from very simple numbers of metrological units – and we’re not using that many metrological units either, especially after having reduced the relatively few that we have looked at to a smaller number of unit families.

Please don’t get me wrong, people can and do make functional schemes using the numbers that are banished here, and someone probably did the same in ancient times. On the other hand, probably everyone I have seen go that route may end up adjusting their Pi ratio and/or Royal Cubit value until the proverbial cows come home – and by way of equivalency, some other unit values related to the Royal Cubit along with it. If a Royal Cubit x 64 = 100 Indus Feet, changing the Royal Cubit value may mean the same as changing the Indus Foot value, creating an equal number of different speculative Indus Foot values, or else specialized rules like only changing the Indus Foot along with the Royal Cubit sometimes, and so on.

So while I often make a case in point out of having excluded 7 and 22 preventing clashes with the Pi ratio (22/7 = 3.142857143, Pi = 3.141592653), there is more to it than simply that.

The bottom line is that all sorts of great expectations from numbers that numbers may or may not be capable of living up to generate something of such complexity that we are best advised to take our simplicity where we can get it, and that is exactly what this exclusion of many whole numbers is providing, believe it or not, is simplification.

That’s not to say that with such a gesture we are necessarily accepting limitations. Quite the contrary, the invalid status assigned many of these numbers means that when we do encounter them in architecture, say a temple with 22 columns or a stone circle with 37 stones, that they may function is “wildcards” so that they are substituted for not by single values, but by a variety of different approximations that may make a wider variety of mathematical expressions true.

I’ve never really gone down the list and suggested substitutions; rather I rest somewhat assured that the correct substitution should be what a particular situation requires or benefits most from – sometimes even multiple substitutions.

For example the inverse of 1.423799344 that we have been talking about the last few posts, 1 / 1.423799344 = 7.023461587 might be considered a substitution for 7; other times 7 might be substituted for by 19.46773746 x 360 = 7.008385550 x 10^n, or even 6.981317008 (1 / (Radian 57.29577951 / 4)), and etc.

This is really nothing new because of course even though we divide a 365 day year into 52 weeks of 7 days, 365 does not really divide by 52 or 7 evenly

365 / 52 = 7.019230769 — see where I find justification for suggesting that of all things, 7.023461587 might substitute for 7? I don’t have to look far — and 365 / 7 = 52.14285714.

Of course, we can easily see that 52 x 7 isn’t 365, it’s 364!

I’m not the only one with some strange math, it seems.

At Stonehenge 260 / 10 seems to be substituted for by 259.7575758 / 10 (to name one); the number 56 suggested by the 56 holes of the Aubrey Circle may see multiple substitutions depending on both the cycles of the planets that are being expressed (56 or thereabouts seems to be important to a number of astronomy equations) and the parts we have been given at Stonehenge to construct things out of.

(By the way, I’m not sure if this is a new discovery, but decimal point placement aside, the Stonehenge inner sarcen circle circumference squared times 24 = the Aubrey Number that I’ve long thought may belong to scheme of the Stonehenge Aubrey Circle’s measurements)

305.7985077^2 x 24 = 224.4305456 x 10 — what a remarkable “coincidence”, yet another for Stonehenge.

11 and its multiples are of course well covered by the Indus Foot values I work with (1.1008744628 ft and 1.100078966 ft substituting for the “1.100000000” ft Indus Foot); 73 is substituted for by 6 Remens = 6 x 1.216733603 = 73.00401618, and etc.

At any rate, there is more again of why I work with numbers the particular way that I do – it’s the way that it makes sense to me for ancient peoples to have used them in order to have made their very grand and complex calendar schemes work best, even if has managed to elude the historical record.

–Luke Piwalker

Addendum to “Two Numbers”

While the subject of 1.423799334 and 1.424280286 is fresh in my mind…

At Stonehenge this past year or so, I discovered a third set of planetary, solar and and lunar calendar values. Before that there were only two sets, but as I began to see more about the metrological units that Stonehenge uses and how they were used there, gaining more respect for the “authoritity” of these numbers, it became clear that that I should take what I was seeing seriously and that therefore there was probably a third major set of calendar values.

Here is one of the ways we can generate the third or “C” value for the Venus Orbital Period from major metrological units associated with Stonehenge:

Palestinian Cubit 2.107038476 x Hashimi Cubit 1.067438159 = 224.9133272 / 100. 224.9133272 = Venus Orbital Period C

From there, we following the 1 / (Pi^2 x 12) link to the Half Venus Cycle (HVC) to find the HVC C value of 224.9133272 x (1 / (Pi^2 x 12)) = 18990.40377.

I soon found out that Giza had known about this all along, but I had never thought to ask in the right context. All one has to do is multiply the Great Pyramid’s perimeter length in feet by (2 Pi) – and what could be more obvious since from my perspective the Great Pyramid is first and foremost a model 2 Pi since that is its perimeter/height ratio – with the understanding that yes, there really and truly were probably more than two sets of calendar values observed and used by ancient people, and that that is what we are seeing.

Finding the three major Half Venus Cycle (Calendar Round) values in the Great Pyramid.

In retrospect, I probably missed a good chance back at Tikal, where my Mayan studies started, to recognize that there were at least 3 valid sets of calendar numbers, and this is because

1.424280286 x (4/3) = 1.424280286 x 1.333333333) = Half Venus Cycle C value 18990.40381 / 10000

Thus the two numbers that the latest tale is about have a simple and direct connection to the almighty Half Venus Cycle aka Calendar Round that coordinates ancient calendar systems. While the idea of ancient Egyptians or ancient British observing the Mayan Calendar may inspire controversy or dissent among some, the Mayan calendar is undeniably built on just such considerations.

All that any ancient culture really has to be on the same page with the Maya is to take a calendar just like the one we currently use where 365 days are divided into roughly 52 weeks of 7 days and also multiply the 365 days in a year by 52 by get the Half Venus Cycle of roughly 365 x 52 = 18980.

Accordingly, our Faiyum number 1.423799334 x (4/3) = 1.423799334 x 1.333333333 = 18983.99112 / 10000, and 18983.99112 is Half Venus Cycle A.

This may tell us that we might except a third such number related in the same way to Half Venus Cycle B.

Half Venus Cycle B 18997.72194 / (4.3) / 10^n = 1.424829146.

For now we will ignore this new discovery while we learn more about 1.424280286 and 1.423799334.

In the previous post, we looked at some of the series formed from our two numbers under discussion, 1.424280286 and 1.423799334, when combined with some of the more powerful and better known data retrieval tools like 2 Pi, Pi / 3, 1.177245771, 1.622311470 and square root 60 (or 1 / sqrt 60).

As often mentioned, the two most powerful data retrieval keys known thus far are square root 60 (or 1 / sqrt 60), followed by 2 / 1.622311470.

If we apply 2 / 1.622311470 = 1.232808888 to our two numbers under discussion, we find they are both linked to one another via exponential application of 2 / 1.622311470 = 1.232808888.

In addition to any important numbers appearing in the (2 / 1.622311470)^2 series formed, we can observe for ourselves that 1.423799344 / (2 / 1.622311470)^11 = 1.424280286 / 10^n, so we can say this pair of numbers has that extremely important value built into their relationship.

Another way of looking at this is simply as 1.424280286 / 1.423799344 = 1.000337788

So now we know that the annoying ratio 1.000337788 that fouls up many an otherwise perfect mathematical scheme, is actually the prestigious (2 / 1.622311470)^11 / (10^n).

1.000337788 also happens to be the ratio between the “Alternate e'” Megalithic Yard of 2.720174976 ft, and the “Incidental” Megalithic Yard of 2.719256444 ft.

As mentioned in the preceding post, the relationship between these premiere linear Megalithic Yard values is the ratio (Pi / 6), so we can develop the formula

(Megalithic Yard / (Pi / 6)) x 1.3333333333 = Half Venus Cycle / 10^n

Interestingly, to use this formula to generate the important Half Venus Cycle B value, requires as input the Draconic Megalithic Yard of 2.721223218 ft, a near perfect representation of 1/10th of the Draconic Month in days (textbook value Draconic Month = 27.212220815 days).

So with further study, we begin to get an even better idea of why both 1.424280286 and 1.423799344 were so important that we should indeed find them prominently featured in ancient architectural proportions, and we even begin to learn that there may be additional important numbers that are similar, and why they might be important.

Since discussion of the Megalithic Yard has been added here to a discussion that is largely about Palestinian Cubits that has been something of a thread running through the last few posts concerning the Faiyum Oasis and etc, let’s go back and take a look at the relationship between the Megalithic Yard and Palestinian Cubit.

Using the primary values,

AEMY 2.720174976 / Palestinian Cubit 2.107038476 = 1.290994449 = 1 / (sqrt 60). Simply by coexisting, these two ideal unit values generate as the ratio the most powerful mathematical probe known (which can also be seen metrologically as a value in Egyptian Sacred Cubits: ESC 2.091411007 / 1.290994449 = 1.620000000)

1.290994449 is equal to (sqrt 5) / (sqrt 3) precisely.

Thus we might expect that this value should link a number of Palestian Cubit values to a numer of Megalithic Yard values

AEMY 2.720174976 / 1.290994449 = 2.107038476 = 57.29577951 / 2.719256444 / 10
IMY 2.719256444 / 1.290994449 = 2.106326984 = 57.29577951 / 2.720174976 / 10
DMY 2.721223218 / 1.290994449 = 2.107850441 = 57.29577951 / 2.718208958 / 10
MFMY 2.718208958 / 1.290994449 = 2.015515605 = 57.29577951 / 2.721223218 / 10

This seems to reaffirm the two most recent additions to the linear Megalithic Yard family, the Draconic Megalithic Yard (DMY) and the “Megalithic Foot Megalithic Yard” (MFMY), which is generated from 32 / Megalithic Foot 1.177245771 / 10, so it is actually a value in Megalithic Feet rather than in Megalithic Yards, yet as a Megalithic Yard, it is the unit of a circle’s diameter when the circumference of the circles is in Hashimi Cubits = Egyptian Royal Feet.

It covers the three primary Palestianian Cubit values 2.107038476, 2.106326984, and 2.015515605 while casting a vote that there is a fourth important Palestinian Cubit Value we should know, 2.107850441. I wasn’t so sure of that until I looked at it this way.

This in turn may also cast a vote for the legitimacy of a fourth set of calendar values, because the formula 4 / Palestinian Cubit = Half Venus Cycle / 10^n was discovered some time after the present inquiry into the association between astronomy and metrology began.

One way of framing all this seems to be to say that the Egyptian Sacred Cubit links the Megalithic Yard to the Half Venus Cycle.

There is another potential formula that comes to mind (there are many formulas that should help simplify all of this) that is not known well enough to tell if it is truly useful, but it is

Palestinian Cubit x 24 = Venus Orbital Period^2 / 10^n

The formula is somewhat unstable because it generates some figures for the squared VOP that do not have true valid square roots and instead these squared VOP figures may have to be generated using assorted VOP figures in false square root pairs to match what is generated by using the Palestinian Cubit as the starting point. The first formula actually works without having to use a false square root pair. Simply,

VOP A^2 = 224.8373808^2 = 2.016326992 x 10^n

From there it looks like (VOP A x VOP C) / 24 = 2.107038475 x 10^n; (VOP A x VOP B) / 24 = 2.107850445.

The forth formula relating to a Palestinian Cubit value of 2.015515605 ft is probably not yet understood in this context.

At any rate, if these Venus Orbital Period formulas (which relate also to the proportions of the Great Pyramid’s King and Queen’s chambers) are valid, they may lend additional legitimacy to 2.107850441 as a fourth Palestinian Cubit value, which has long been suspected but never quite expected, if that makes sense.

We will hopefully hear more about 2.107850441 and hopefully I will be able to review the history of why and how I’ve encountered it before on a number of occasions.

For now, the discussion of 1.290994449 seems to have stimulated a new discovery, that of a reversing metrological 2 Pi chain. This comes from noticing that 1.290994449 (in Egyptian Sacred Cubits = Thom Mid Clyth Quantum) times 2 Pi is a figure in Remens, and 1.290994449 divided by 2 Pi is a figure in (inverse) Remens, and the progression continues onward in either direction of the known series of metrological units linked to the Remen through 2 Pi and or circles (circumference / radius = 2 Pi), i.e., Megalithic Foot, Egyptian Royal Foot (Hashimi Cubit) and etc. as described here.

Would all of this perhaps be part of why the Palestinian Cubit might have come into greater prominence, in at least the Faiyum Oasis but perhaps elsewhere as well in Egypt, in later Egyptian Dynasties?

Cheers and Happy New Year!

–Luke Piwalker

Another Tale of Two Numbers

Since I’ve been working with data from archaeological sites in Egypt’s Faiyum Basin lately I seem to be seeing a lot of the number 1.423799334. In accordance with what may be trend in seeing particular focus on the so-called Palestinian Cubit, 3 / 1 Palestinian Cubit in feet 2.107038476 = 1.423799334, which helps serve to reinforce the idea that it is 1.423799334 we are seeing and not some similar but distinctive number such as 1.424280286 especially.

I’m a little less confident now that 1.424280286 was what I discovered at Tikal, although it does belong to some vital series of numbers unlocked with the usual most important data retrieval tools. Much of the success working with Tikal comes from the realization that (Pi / 3) was a very important number there, and accordingly it links together numbers from Tikal’s temple pyramids and pulls forth much important data from many numbers found there.

(I’ve written about 1.424280286 in at least 15 previous blog posts).

Ignoring proper decimal placement for the sake of illustration,

1.424280286 / (Pi/3)^1 = 1.360087487 – 1/2 Megalithic Yard of 2.720174976 ft
1.424280286 / (Pi/3)^2 = 1.298787879 – 1/4 of the Outer Sarcen Circle Radius of Stonehenge
1.424280286 / (Pi/3)^3 = 1.240251066 – 4 x (Pi^3)
1.424280286 / (Pi/3)^4 = 1.184352527 – 1 / (Half Venus Cycle / Venus Orbital Period)
1.424280286 / (Pi/3)^5 = 1.130973354 – found in Egyptian Pyramids including Khufu’s (Great Pyramid)
1.424280286 / (Pi/3)^6 = 1.080000000
1.424280286 / (Pi/3)^7 = 1.031324031 – generic area of circle, 1/2 Royal Cubit in inches and Solar Year / Lunar Year
1.424280286 / (Pi/3)^8 = 9.848419048
1.424280286 / (Pi/3)^9 = (Radian 57.29577951^3) / 2

Thus simply by having situated 1.42428086 in a place where great importance is placed on (Pi/3), writing the number 1.42428086 is actually writing ten important pieces of data, including at least two essential expressions from astronomy and calendars.

The number 1.42428086 is easily fashioned from several of the most important numbers to know: 1.676737943 / 1.177245771 = 1.424280286.

It can also be formed from 1.177245771 and more of them: (2.720174976 / 1.177245771) / 1.622311470 = 1.424280286

1.424280286 has an interesting response to the Hashimi Cubit value of 1.067438159

1.424280286 / 1.067438159 = 10.67438159 / 8

1.424280286 / (1.067438159^2) / 1.25000000

Subsequent to its discovery at Tikal, 1.424280286 was also found in the metrological and mathematical models of Stonehenge and Giza.

Beyond being able to construct 1.424280286 with 1.177245771, 1.424280286 responds reasonably well to 1.177245771

1.424280286 x (1.177245771^1) = 1.676727943
1.424280286 x (1.177245771^2) = 1.973920881 = 2 x (Pi^2) / 10
1.424280286 x (1.177245771^3) = (sqrt 540) / 10
1.424280286 x (1.177245771^4) = 2.735671962 = ((Royal Cubit 1.718873385) / 2 Pi) x 10^n
1.424280286 x (1.177245771^5) = 3.220558248
1.424280286 x (1.177245771^6) = ((360 / 18990.40379) x 10^n) x 2

If we divide 1.424280286 by 1.177245771, we begin to get unwanted numbers that might serve as clues that there may be another very similar to 1.424280286 out there. Correcting the familiar looking but mutilated numbers we get by dividing 1.424280286 by 1.177245771 gives us

1.424280286 / (1.177245771^2) = 1.027687763
1.424280286 / (1.177245771^3) = 8.729594010

(1 / (57.29577951 x 2 = 114.5915590) = 8.726646260

8.726646260 x (1.177245771^1) = 1.027340741 (inverse Egyptian/Roman ft)
8.726646260 x (1.177245771^2) = 1.209432542
8.726646260 x (1.177245771^3) = 1.423799346, the Wonder Number of the Faiyum

1.423799346 may belong to a false square root pair representing the double long Greek Foot, now confirmed here

1.423799346 x 1.424280286 = 20.27889339 = 1.013944670 x 2

This pairing of similar numbers also provides

1.423799346 x (1.424280286^2) = 28.88282808 /10, 1/120 of the Eclipse Year 28.88282808 x 12 = 346.5939365

At Stonehenge, we may be able to find both numbers in question if we remember that Munck’s Squared Megalithic Yard (2.719715671^2) = 7.396853331 = 9 / 1.216733603 is important to Stonehenge and that we should try it on Stonehenge numbers

Hence, 73.96853331 / outer sarcen radius 51.95151515 ft = 1.423799346, while

51.95151515 / 1.424280286 = 36.47562608, 1/10th of the shorter of the two primary representations (364.7562608 and 365.0200808) of the Solar Calendar year, a definite positive response.

1.424280286 from Tikal also works well enough with the essential probe 1.622311470, and consistently recovers astonomy data

Ignoring correct decimal placement once again,

1.424280286 x (1.622311470^1) = 2.310626244 = 2.720174976 / 1.177245771
1.424280286 / (1.622311470^1) = 8.779326981 – possible representation of Mercury Orbital Period, and 1/2 base perimeter of Great Pyramid missing apex section
1.424280286 / (1.622311470^2) = 5.411616168 important number championed by Michael Morton = Jupiter Orbital Period / 8
1.424280286 / (1.622311470^3) = 3.335744256 = 106.7438159 / 32
1.424280286 / (1.622311470^4) = 6579.736270 (one form of Saros Cycle?) / 32

There is more to it of course, but there is some of why 1.424280286 is important.

Regarding the “Wonder Number of the Faiyum”, 1.423799346, as we saw its series forming ability with 1.177245771 works downwards rather than upwards (by division rather than multiplication), also including

1.423799346 / (1.177245771^4) = (1 / 144) x 1.067438159
1.423799346 / (1.177245771^5) = 1 / 1.588133133 (1.588133133 is thought likely to have been used in the Stonehenge ovals)
1.423799346 / (1.177245771^6) = 1.069734371 / 2

1.069734371 is a wonder number in its own right, although it may otherwise lack usefulness for being easily overshadwed by 1.067438159

1.069734371 belongs to what is probably the most classic (Pi/3) series from Tikal, hence in the presence of (Pi/3) it can generate the entire series, which one of is of paramount importance.

1.423799346 also belongs to a coherent 1.622311470 series

1.423799346 x (1.622311470^1) = 2.309846007
1.423799346 x (1.622311470^2) = 3.747289671 = 11.77245771 / Pi
1.423799346 x (1.622311470^3) = 6.079271014 = “5 Thoth Remens”?
1.423799346 x (1.622311470^4) = 9.862471096 = 1 / 1.0139944670 Long Greek Foot
1.423799346 x (1.622311470^5) = 16.00000000
1.423799346 x (1.622311470^6) = 25.95698349
1.423799346 x (1.622311470^7) = 4.211031294 = 2 x 2.105515602, a slightly shorter (and rarer) form of the Palestinian Cubit of normally 2.107038476 ft; (1.315947254 x 16 = 21.05515602)

1.423799346 may not necessarily may not give the best responses to (Pi/3)^n, but it does give a series of important data in combination with 2 Pi^n

Correct decimal placement once again ignored,

1.423799346 x ((2 Pi)^1) = 8.945995119 = 1 / 1.117818629
1.423799346 x ((2 Pi)^2) = 5.620934509
1.423799346 x ((2 Pi)^3) = 3.531737312 = 1.177245771 x 3
1.423799346 x ((2 Pi)^4) = 2.219055999 = 887.6223994 / 40
1.423799346 x ((2 Pi)^5) = 1.394274005 1/8 of Munck’s “Giza Vector” = 5577.096019 / 8 = 139.4274005
1.423799346 x ((2 Pi)^6) = 8.760481940 = 72/10 Remens; 365.0200808 x 24 = 8760.481939 hours/year
1.423799346 x ((2 Pi)^7) = 5.504373141 base radius Silbury Hill 550.4373141 ft
1.423799346 x ((2 Pi)^8) = 3.458996450 side length Mycerinus Pyramid 345.8996450 ft
1.423799346 x ((2 Pi)^9) = 2.173039415
1.423799346 x ((2 Pi)^10) = 27.30721865 / 2; 27.30721865 = Tropical and/or Sidereal Month and/or reciprocal of ~366 day Leap Year?

Just this combination of two numbers with one used at high powers gives us powerful insights into the ancients used math, and provides us with some of the data commemorated by no less great monuments than Silbury Hill and the pyramid of Mycerinus at Giza.

As it does with (Pi/3), 1.423799346 also seems to give an almost inconsequential series when combined with (1/sqrt 60), one of the two most powerful data recovery probes known, although interestingly at

1.423799346 / (1 / sqrt 60)^6, it gives us 3.075406583, which may (or may not) be part of the design of at least one of the Faiyum pyramids.

There is at least another interesting metrological relationship concerning the Tikal number 1.424280286, as was actually touoched upon near the beginning of this post as

“1.424280286 / (Pi/3)^1 = 1.360087487 – 1/2 Megalithic Yard of 2.720174976 ft”

Megalithic Yard of 2.720174976 / 1.424280286 = 6/Pi

We also see 6/Pi as the relationship between a putative ancient Meter of (360 / (2 Pi))^2 / 1000 = 3.282806349 ft

Meter (?) 3.282806349 ft / Royal Cubit 1.718873385 = 1.909859317 = 6/Pi

So the world of ancient numbers is probably big enough for both 1.424280286 and 1.423799346.

Happy New Year!

–Luke Piwalker

Phi in the Faiyum

Yesterday I attempted to write a paper on ancient architecture of Egypt’s Faiyum Oasis. It seemed to serve as a good opportunity to go over some important points, but I’m a little discontented that it mostly asks questions rather than providing answers.

Currently under study are the pyramids of El Lahun, the pyramid of Sesostris II (Senwosret II, Senusret II) and the smaller accompanying “Queen’s pyramid”, both of which, very fortunately, Flinders Petrie left us data for. I am really still sifting through that data, looking for common threads and themes that might offer guidance with an interpretation.

These two pyramids, like some others, present something of an understated challenge (not to mention that an ancient pyramid is not something one should try to solve overnight in the first place. Even though still incomplete, it took years to construct the current model of Giza’s Great Pyramid).

Two complications that are or may be present are that we have data both for the pyramid proportions from the pavement, and from the pyramid footing, making them each two separate pyramids to interpret, and that the measurements may reflect still more originality on the part of the architects, resulting still more unexpected or unfamiliar numbers.

While unexpected or unfamiliar numbers can sometimes represent errors or inadequate data quality, typical for Petrie’s data we do seem to have some familiar points of reference in the data to reassure us of its quality.

One of these possible “plausible points of reference” may be the projected 60.91666666 ft height from pavement level of the Queen’s pyramid, which is quite ostensibly 50 Remens = 50 x 1.216733603 = 60.83668015 ft.

As I try to better organize Petrie’s data and the data we can extrapolate from it, I do notice that according to it, there seems be a bit of focus on the “Golden Ratio” Phi, or one of the usual numbers that are very similar to it and serve to represent it while also serving other functions.

We can see this where the projected height of Sesostris II’s pyramid from the footing calculates at a raw value of 161.757421 ft

We can also see it where the raw calculated diagonal of the Queen’s pyramid at the pavement level is 123.5786951 ft, or 200 / 1.61840190.

Both of these figures then, 161.757421 / 100 ft and 200 / 1.61840190, wax very close to Phi (1.618033989). In the present work, it is expected that the most likely intended values here are eight 1.622311470 or 1.618829140, for these are variations on the Phi ratio that better relate to astronomical (calendar) formulas.

Enthusiasts of theories that the ancients were well aware of both the size of the Earth, and the “modern” mile of 5280 ft may wish to note both the half-diagonal value for Sesostris II’s pyramid at the footing level 248.9251572 feet, and the projected height of the Queen’s pyramid from the footing level, 62.25281327 ft.

Both of the values bear great resemblance to geodetic figures, namely 1/10th of the earth’s circumference in miles, and 1/400th of the earth’s circumference in miles respectively.

It is noted here that this value for the height of the Queen’s pyramid from the footing appears to be very similar to 1/10 of one side of the Bent Pyramid at Dahshur, which has been recognized by multiple independent researchers as being likely to represent a geodetic figure, even if there is not absolute agreement as to whether the polar, equatorial or mean value is intended there.

I will aspire to refrain from sharing interpretive efforts until a more complete and more confident model can be developed, although I would like to note that the base length of Sesostris II’s pyramid at the pavement level could be a representation of the Eclipse Year (4168.5 in / 12 = 347.373 ft; Eclipse Year = 346.62 days).

It’s somewhat difficult to imagine what else could be meant here. To try to frame the figure as a simple number of Royal Cubits (200 Royal Cubits = 1.718873385 x 200 = 343.7746770 ft) would imply either an implausibly sizable error of about 3 and 2/3 feet, or an alarming departure from the traditional, time-honored Royal Cubit value.

While the ratio between the proportions at footing level and the proportions at the pavement level for Sesostris II’s pyramid seems rather self-explanatory as being probably one of the most common values in “modern” feet for the Greek Foot, the same ratio for the Queen’s pyramid seems to remain rather mysterious.

It calculates at approximately 1.021934997, and while we might have expected the “Wonder Number” 1.021521078 (first found in the architecture of ancient America and then laster at Giza), some of my experiments are throwing other numbers back at me, including, rather surprisingly, 1.023280583.

I’ve known about this number for a long time now, almost 20 years, but it generally sees so little usage that there has probably never been occasion before to even mention it in anything I’ve written in the last 4 years.

Even more curiously, it would have been very much expected that these two ratios would bear a dynamic relationship to one another

In terms of a common thread that may run throught the architecture of the Faiyum region, what 352.033333 feet could represent in spite of the departure from the data is 351.1730792, which would seemingly tie the design of the pyramid of Sesostris II to other architecture of the Faiyum Oasis, and in turn back to Giza. (Other possible “echoes” of Giza architecture may also be present at El Lahun).

Depending on the final outcome of interpretive efforts the so-called “Real Mayan Annoyance” (RMA) might also be present at El Lahun. The raw projected vertical edge of Sesostris II’s pyramid from the pavement, 292.9371530 ft, bears considerable resemblance to the “RMA” 29.26442328.

The RMA is another “Mayan Wonder Number” that was found in ancient Merican architecture and then later at Giza in the Great Pyramid itself.

There may also be some emphasis on the “Palestinian” Cubit in the Faiyum architecture (see also the Pedestals of Biahmu from the preceding post), and an appearance of the RMA would be in accord with that since 29.26442328 x 72 = 2107.038476 = 1000 Palestinian Cubits in Imperial feet.

If nothing else, the fact that we can spot some plausible values for both of these pyramids at both the footing level and pavement level, should lend further credence to the idea that the Great Pyramid works the same way, having different intended meaningful values both with and without a hypothetical missing pavement layer.

The character of other Egyptian pyramids such as pyramids at Dahshur or elsewhere, may also lend credence to such a view of the Giza pyramids as havng such a “dual” nature as suggested.

Happy Holidays and thanks for reading!

–Luke Piwalker

A Small World When Measured

Apologies for things being slow around here again. I much enjoy writing this blog and am always on the lookout for interesting subject material, but lately I’ve gotten a bit into some experimental things that I don’t have quite so much certainty or optimism about as to be eager to post.

Most recently, I’ve been browsing through the database at the Megaltihic Portal. I’ve looked before there at stone ships (ship settings), and at stone circles that occur in less expected places, following leads that soon seemed to overturn the idea that most of the world’s stone circles are in the UK.

This time I’ve been looking again at pyramids in less expected places, as well as having another look at the possibilities concerning a “Long Foot” of 1.056 ft or thereabouts (quite close to the ratio between the Egyptian Remen and the Egyptian Royal Foot), and before that I spent most of a week revisiting the question of the Greek historian Herodotus’ lost Lake Moeris pyramids in Egypt. I had begun to wonder if there were any recent progress on the problem, and stumbled over Stijn van den Hoven’s paper that proposes that the pyramids mentioned by Herodotus are in fact the pedestals of Biahmu.

I cannot quite bring myself to accept this proposition yet although I am mindful of Petrie himself also, if perhaps somewhat inexplicably, having declared the same thing. Although Herodotus’ measures for the two pyramids in Lake Moeris must most likely be exaggerated, there are still a number of probably unanswered questions about whether they could be the Pedestals of Biahmu, or whether the Biahmu structures might simply be a pair of much more humble constructs with related adornments to the two pyramids that Herodotus describes.

I’ve done a fair amount of pouring over Petrie’s data for the East pedestal, a particular challenge because of what appear to be several curious inconsistencies in Petrie’s data.

No matter how many times I seem to be able to find a slender golden thread running through the last two or three things I worked on, I try not to expect it lest I help it to happen when it shouldn’t. I certainly wasn’t expecting it here, but the possible discovery of at least one such thread linking several recent inquires is why I think there’s an occasion to post after all.

Diagram of the East Pedestal of Biahmu after WMF Petrie, “Hawara, Biahmu and Arsinoe“. Thanks to Jim Alison for pointing out that the 125 Indus Feet labelled at top is no doubt more sensibly interpreted as 80 Royal Cubits (especially since non-standard Indus Feet would have been required).

The main reasons I prepared the diagram are because I wanted to show the possible appearance of metrological unit values in modern Imperial feet as ratios between parts of a structure, yet again, and that one of them, 1650 / 1400 = 1.178571429, would appear to be the Harris Stockdale Megalithic Foot.

My interest in the lost Lake Moeris Pyramids was re-kindled by rumblings about ground-penetrating imaging that supports Herodotus’ claims of the fabled and fantastic lost Egyptian Labyrinth being adjacent to the pyramid of Amenemhet III at Hawara, and sustained even through the difficulties with Petrie’s Biahmu data because eventually the Biahmu data seems to produce some unusual numbers that are related to those associated with my previous efforts at interpreting the Hawara pyramid.

Furthermore, these unusual numbers are also related to those seen in Petrie’s data for the Qasr el-Sagha temple, which according to Keith Hamilton’s paper is located not far north of the north shore of what’s left of Lake Moeris, in the form of the present day Birket Qarun.

What does all this look like mathematically? Well, here is Petrie’s reconstruction of one of the colossi atop the Biahmu pedestals with Petrie’s data (such as it, in too round of numbers of feet for anyone’s own good), labelled onto it.

If I take the equation 35 / 21 = 1.6666666666 and adapt it to the 2.107038746 ft Palestinian Cubit, 

21.07038746 x 1.666666666 = 35.11730793 as the height of the statue. This is equal to twice the perimeter of the missing Great Pyramid apex section / 10^n. 

Curiously, 5 / 35.11730793 = 1.423799344 = 1 / 7.023461587 

Because 1 / 60 = 1.666666666 / 100, if we look at the ratio between Petrie’s “60 feet” total height for pedestal and statue, projective though it is, we get the same thing from the opposite direction: 60 / (10 Palestinian Cubits = 21.07038476) = 2.847598689 = 1.423799344 x 2 .

Although not previously posted, based on Petrie’s data, 7.023461565 has already already suggested to be the perimeter / height ratio of the Hawara pyramid.

I’ve cited 7.023461565 and its reciprocal 1.423799344 previously as an ancient Egyptian Wonder Numbers, and currently I’m finding it very hard to ignore these unusual numbers turning in three of three different structures in the Faiyum Oasis, even though Petrie’s height measures for Biahmu are otherwise somewhat questionable.

There may be other unusual numbers as well that could be common to sites in the Faiyum Oasis.

To make a long story somewhat shorter, as part of a closer look at pyramids in less common places, I managed to end up having another look at one of the South Korean pyramids. We’ve taken a look at several of them before here.

I’d actually intended another look at the legendary Glastonbury Pyramids – Wikipedia: “Both Giraldus and Ralph say that the spot lay in between two pyramids in the abbey. Willian of Malmesbury does not refer to Arthur’s tomb but elaborates on the pyramids of varying height, upon which were statues with inscriptions…” but the Asian pyramids are quite good at catching my attention, and as I began to revisit that subject, realized that the Asian pyramids in question may be linked by the same slender golden thread that may link the sites of the Faiyum together as well.

The discovery in question is that a certain number, the often mentioned “Real Mayan Annoyance”, may be part of the proceedings with the Asian pyramids, along with an apparent modest wealth of Lunar references that they seem to contain.

RMA / (1 / 24) = Faiyum Wonder Number as 7.023461575 rather than 1 / 7.023451575 = 1.423799344 / 10
RMA x (1 / 24) = 1.219350968 = 360 / 29.52390325 / 10

The (1 /24) actually seems to be built into the Andong pyramid data that I have. I still don’t know the provenance of that data, but normally if someone goes to the trouble of giving measures to the centimeter, it seems to mean they actually bothered to try to measure so accurately.

Only several hours ago I discovered an additional connection, that may link the golden thread to the ancient Americas and then back to Egypt again, to Giza – which may help to explain what some of these numbers are doing in any of these places to begin with.

Still an unfinished project even after this year’s efforts, one of the more confident interpretive gestures that is possible here is taking 15170.4 in = 1264.2 ft to mean 15170.67702 inches = 1264.23985 ft = 600 Palestinian Cubits of 2.107038476 ft each.

We may wish to note here that 1264.23985 / 180 = 7.02361586, and in fact, we take the 360 degrees of the sacred circle, and thread a small series with together with it, starting with the reciprocal of 1264.23985.

Ready?

(1 / 1264.23985) = 7.90999635 / (10^n)

7.909996357 x 360 = 2.847598689 = 1.423799344 x 2

2.847598689 x 360 = 1.025135528 x 10^n, 1.025135528 being one of the Tikal Wonder Numbers

1.025135528 x 360 = 369.0487900 = (29.52390320 / 8) x 10^n, 29.52390320 being the “Best Value” for the Lunar Month

Suddenly, all of these numbers and the frequency with which we find them, seem to make a little more sense to me.

There is one more recent observation I’d like to share

Unless I’ve made some terrible mistake, all three of these Asian pyramids, in spite of their differences, seem to have a perimeter / diagonal ratio of approximately 2.825389852, the main reciprocal form of the Lunar Year of ~354 days (1 / 2.825389852) x 10^n = 353.9334578.

What are the odds of this being accidental or coincidental?

Happy Holidays!

–Luke Piwalker

Save Stonehenge

Peter Harris (as in Harris-Stockdale Megalithic Foot) posted this to his thread at the Megalithic Portal.

“It appears that a legal challenge may be made against the Stonehenge tunnel decision. 

Here is a link to a crowd funding project to possibly make it happen.”

https://www.crowdjustice.com/case/save-stonehenge-world-heritage-site/

See also: [stonehengealliance.org.uk] [www.theguardian.com] 

I’m not sure what I think they should do, but I suspect that what shouldn’t be done is planning a tunnel any number of miles in length and telling people that archaeological discoveries made in the process will be respected. 

I can’t see how it will be realistic to expect them to stop the machinery and call in the archaeologists every time an archaeological find is made, and actually living up to that seems like it would push any benefits of the tunnel further and further into the future, so I do have the sense that the plan may have been approved under false pretenses and may be very destructive to heritage, and that the matter should be subject to review…

Not to mention that 1.7 BILLION £ is still A LOT of money, and there may be much better uses for it in times like these.

I will leave it at that for this post, but suffice it to say I’ve posted a great deal to this blog about why Stonehenge is not only an archaeological treasure, but also a mathematical treasure of the highest order. I’m greatly saddened at the thought of this being disrespected in any way, and in spite of hardships am willing to risk a small donation if it helps to oppose any insensitive threats to Stonehenge or its archaeological landscape. 

As always, deepest thanks to readers for their time and consideration.

–Luke Piwalker

 

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