Luke Piwalker

“E’s Already Got One – It’s Vara Nice”

Perhaps nature is telling me I should take a break and watch a movie. For the third time in a week I’ve somehow encountered occasion to get out a line from Monty Python and the Holy Grail.

Not the least of these is because Peter Harris (as in Harris-Stockdale Megalithic Foot) inquired about datasets for Mount Pleasant Henge, and I recently set about seeing what I could think of to try to sort out potentially conflicting data sets.

Once again, John Neal’s works prove valuable as metrological sourcebooks and as sources of inspiration, even if I cannot seem to bring myself to agree with Neal’s conclusions. In the case of Mount Pleasant, one of the authors cited is Wainwright, and because of Neal’s presentation, one possibility for assessing the credibility of Wainwright’s measures is trying to develop a track record for his data, potentially based on his also having reported measures for other structures, and particularly others with an intricate concentric design.

In the case of The Sanctuary, Wainwright’s data has an interesting character.

Even though his outer diameter is generally about 9/10 of a foot under what attempts to appear as a consensus value, it manages to give the sense of a design where the three outermost circles are all measured out in Thom’s Mid Clyth Quantum. Such a remarkable gesture may not be entirely without precedent, and certainly not for the outermost circle, and certainly not withou a great deal of potential purpose.

As the diameter of the outermost circle (Circle A) of the Sanctuary, Neal gives 129.761 ft and Thom gives 129.7, whereas Wainwright gives 129, which bring is very close to being yet another classic case of 129.0994449, i.e., the inverse form of the proposed (sqrt 60) ft Mid Clyth Quantum which is synonymous with a proposed Egyptian Sacred Cubit, that seems to appear in ancient American structures and in Beijing’s “Altar of Heaven” that is described by Neal in the very same chapter.

This might be purely wishful thinking, were it not that Wainwright’s data also suggests (1 / sqrt 240) x 10^n ft as the possible diameter of Circle B, and sqrt 2160 ft as the possible diameter of Circle C. Rather than representing overkill, it may represent a particularly potent combination of building blocks, and it is in fact precisely how the “Alter of Heaven” seems to have attained unprecedented mathematical potency, by the “overkill” of combining 60 with sqrt 60, even when of course 60 is the square of sqrt 60.

If nothing else, it’s a powerful demonstration of how and why the number system relates to sexigesimal values that we use even today to measure circles or time, but it’s also the current record holder for showing the exponential value of sqrt 60. What the “Altar of Heaven” seems to have done is take the most powerful mathematical probe yet discovered, and showcase its profound usefulness.

Neal’s values for the Sanctuary are ultimately (the numbers in parenthesis are the number of Iberian Feet proposed by Neal). (The values implied by Wainwright’s data about be A 129.0994449 ft, B 64.54972244 ft, C 46.47580015 ft, or 1000 Inverse Mid Clyth Quanta, 500 Inverse Mid Clyth Quanta, and 60 Mid Clyth Quanta respectively).

A 129.761 (140)
B 64.88 (70)
C 46.343 (50)
D 32.44 (35)
E 19.464 (21)
F 13.903 (15)
G 12.976 (14)

One of the remarkable things about this data to me is that Neal has opted to identify values D and E in “Iberian Feet”; 20 Assyrian Cubits = 26.6666666 Remens = 32.44622940 ft; 12 Assyrian Cubits = 16 Remens = 19.46773764 ft).

Things also seem curious to me in that Neal identifies the fundamental unit as 1/3 of the Vara – as in the silly title of this post, because I already have a candidate for the Vara and it’s “Vara” nice, differing from Neal’s value by approximately .008 feet. I’ve never actually declared this value to be a Vara, but almost 20 years ago I nominated the value after reading about the Vara while getting to know the work of Thom, wherein it was mentioned that the Vara and the Megalithic Yard were vey roughly similar (i.e., page 34 of Thom, MSIB, where five different values for the Vara are given, ranging from 2.7425 – 2.778 ft are given, thus being very vaguely similar to a Megalithic Yard of about 2.72 feet.

On this basis I proposed a useful value near the upper end of this spectrum, of 2.788548009, distinguishable from a figure in LSR Units of 16.76727943 / 6 = 2.794546572.

The fact is that there hasn’t been any need for me to “lobby” for this suggestion for the ideal value of the Vara – it would be like lobbying for the Greek Foot while already in possession of the Remen (which is why I rarely even think about Greek Feet or Assyrian Cubits, and less and less about the Squared Munck Megalithic Yard, because they’re all expressions of the Remen, and I already work in Remens).

Just because the ancients were geniuses at math doesn’t mean we need to over-complicate things – in fact, quite the contrary – we want to simplify as much as possible, because whatever system we work with, there may be some fiendishly complex math ahead, such as that describing the temporal mechanics of the Solar System in considerable detail.

Just the simple fact that sqrt (7776/1000) = 2.788548009 should clue us in immediately that such a Vara would belong to the Thom Mid Clyth Quantum / Egyptian Sacred Cubit family, which can be proven simply by checking their ratio

We can in fact go after the data with 1/3 of a Vara of 2.788548009 ft — 2.788548009 / 3 = .9295160031 ft — it’s a nice enough number in the context of metrological units derived from astronomical cycles because in reciprocal form it’s 10^n to the most obvious, useful, and likely approximation of the 10,759.22 day Saturn Orbital Period: 1 / .9295160031 = 10758.287078 / 10^n — although the results of applying it as Neal suggests don’t necessarily give sensible results in the context of Inductive Metrology, which Neal is apparently attempting to use.

Sacred Cubit 2.091411997 ft / 2.788548009 ft = 1.333333333 = 4/3, the very same relationship as between Remen and Assyrian Foot: 1.622311470 / 1.216733603 = 1.333333333.

We also saw here that the proposed Vara value is linked through the geometry of circles to other units, and some other very important ones. The particular unit cluster also includes the Indus Foot, the Petrie Stonehenge Unit (PSU = Inverse Hashimi Cubits), and the value that I use for the HSMF. Hence there may indeed be some point to using the 2.788548009 value specifically even though it is synonymous with the Thom Mid Clyth Quantum and and Sacred Cubit, just as there is great purpose in using the Hashimi Cubit value of 1.067438159 even though it is for intents and purposes synonymous with the Egyptian Royal Foot

As usual, not that many metrologists necessarily seem to be aware of the possibility of the use of multiple units in a construct, even though we don’t have to look far to find almost undeniable evidence of this, such as Egyptian pyramidia (pyramid capstones plural). It’s just not that uncommon to find things like pyramidia measuring at the base 1 Royal Cubit by 1 Remen, meaning of course that their length is one unit while their width is is another, while their base diagonal is in a third (in this case the Palestinian Cubit).

The Great Pyramid’s King Chamber also shows such metrologically diverse attributes in spite of how simple it may seem metrologically at first glance.

Beyond that, I’ve also largely declined “lobbying” for the Vara just because its story is somewhat weird. Like the meter, academicians may unabashedly suggest that somehow ancient Americans were already using it when the Spanish arrived, even without batting an eyelash in the direction of this being potentially a serious challenge to “isolationist” views of history, which some of us welcome.

Personally, if one is going to speak like a heretic, they might as well dress for the occasion in a “Proud to Be a Heretic” t-shirt. With any luck, they might help to spare future generations a lot of painful nonsense when they try to embrace their own roots.

The Vara is also difficult to work with, however, because it’s more or less the unit that “put the word Vara into var(i)ability.” As seen in Thom’s book, the values for the Vara are literally all over the place. Even if one can nail down an ideal value, the matter is still probably destined for considerable controversy, should one be that lucky for that much attention to be paid to it in the first place.

Anyone wishing to can read here what Maud Cunnington wrote about The Sanctuary. There may be some valuable clues therein, and some of us may benefit from them after as much as authors on the subject have tried to turn elaborate concentric ring structures into very simple arrays of a single metrological units.

It’s why I haven’t gone near structures like this – Woodhenge, The Sanctuary, Durrington Walls, or Mount Pleasant Henge – much in general. It’s a great way to end up indoctrinated with someone else’s notions of inexplicable simplicity, even when common sense probably dictates there may be something very wrong with idea of Thom’s “Stone Age Einsteins” counting off a very simple minded 10, 20, 30, 40, 50 of the very same unit as the diameters or perimeters of concentric rings.

Of course, the utility of this may also depend on the unit itself – i.e., it may be much less sensible to do a thing like that with the Megalithic Yard, for example, than with the Thom Mid Clyth Quantum, just as Geoffrey Wainwright’s data may imply was done.

It of course continues to trouble me that few metrological researchers if any seem to really “get it” why the math I use sacrifices certain whole numbers in order to maintain maximum compatibility with ancient sexigesimal systems, but then again I can point to Neal’s “21 Varas” and “35 Varas” as additional evidence of exactly why certain whole numbers like these would have been discarded to better accommodate sexigesimal systems – because of the confusion they’re capable of creating.

Optionally, we can go back to the example of the “Altar of Heaven” which combines the root of sexigesimal systems with the square root of sexigesimal systems to create a profoundly powerful display of mathematical resonance, to get a sense of why such concessions might have been made to sexigesimal. As always, the very name of the game is data storage and retrieval.

Unless it is surpassed by Wainwright’s data for The Sanctuary, the “Altar of Heaven” will remain the most dramatic single example of data storage and retrieval using metrological unit values.

By the way, yes, that would be an putative example of the use of the Egyptian Sacred Cubit in China, and rather late in its history at that. It’s ironic and then some that it features on the same pages of Neal’s book where we also find confirmation, or at least corroboration, of the use of the 1.067438159 ft Hashimi Cubit / Egyptian Royal Foot, in spite of Neal ostensibly conflating it with the Persepolitan foot.

“When it was rebuilt in 1749, by instruction of the Emperor Quainlong, the ‘customary’ module of the Qing was the exact length of the Persepolitan foot of 1.0666 ft”

Once again when it comes to ancient metrology, the fruit needn’t fall that far from the tree. Among other things, metrological equivalences were supposed to be facilitating the almighty, all-important practice of commerce, whereas NASA has reported disasters arising from simply mixing up Imperial and Metric units.

–Luke Piwalker

Mathematical Relationships Between Ancient Units of Measure

I’ve written before about the geometric relationships between proposed values for ancient units of measure. Ancient metrological units can be linked thought the geometry of both squares and rectangles, and the geometry of circles.

In terms of squares and rectangles, this is a concept I was introduced to by the work of John Michell, where we see direct relationship between the Egyptian Remen and other ancient established units of length measure through the classic square roots associated with and generated by the Vesica Piscis. Examples of this also occur in the work of Berriman, where we can observe the given Chinese “Ch’ih” value of 14.14 inches as 10 sqrt 2, making it (outrageously) the diagonal to the Imperial foot and a probable cognate of the Harris-Stockdale Megalithic Foot, and also in the works of WMF Petrie, where very simply, the diagonal of a square of 1 Royal Cubit per side, is the Remen (or the diagonal of a square of 1 Remen is 1/2 of a Royal Cubit).

In terms of relating units through circles, it would be one of the more important breakthroughs of 2020 that having followed Geoff Bath‘s lead on this, I was treated to not only resounding but absolutely unprecedented success after experimentally applying this concept. The understanding of ancient units seems to have grown by leaps and bounds thanks to Bath’s guidance.

In short, the geometry of circles allows us to link and organize ideal ancient unit values with absolute precision, whereas linking them through squares and rectangles requires slight approximation, giving us slightly different and somewhat flexible values for sqrt 2, sqrt 3, or sqrt 5. The approximations of these values that we would use are significant in themselves – for example, sqrt 2 is for intents and purposes the Harris-Stockdale Megalithic Yard in “Imperial” – but this set of relationships between units does not otherwise accommodate the possibility of exactitude, and may in fact try to automatically exclude it.

It occurred to me to write this post that some of what appear to be the primary ideal values for ancient units of measure can often also be connected to one another via themselves, and this phenomenon in general affords us with a set of rules regarding the interaction of unit values in combination.

Rather than trying to gather up and organize my notes so as to be able to compile some of the observations regarding this, I’m just going to try to conjure up a few random examples for the sake of illustration.

I should perhaps note that part of this might include the idea of metrological units as mathematical constants, meaning that beyond simply converting from one to another, we can apply some of these values in equations exponentially. Important data may be lost and design logic may be lost therefore if we limit ourselves to thinking of unit values as only that.

The other extreme end of the stick might be that if we become overly accustomed to thinking of unit values as mathematical constants with exponential data recovery value, it may become easier to overlook where import unit values may be authentic ancient units. Such has been the case not only with the Pied du Poi / Hashimi Cubit, which I was quite proficient at working with long before I ever dreamed it was a metrological unit, but also with the Harris-Stockdale Megalithic Foot.

Originally, making a metrological unit of the HSMF seemed like taking three steps backwards because as a mathematical constant, it’s particularly important. When I finally realized that like the Megalithic Yard, the HSMF is directly linked to ancient Egyptian units through geometry, from which (in my opinion) it draws considerable legitimacy as an ancient unit of measure.

These remarks about the power of the HSMF (or at least the number I use to represent it) also apply to mathematical relations between units.

For example, let’s take the standard primary Palestinian Cubit value of 2.107038473 ft, and divide it half: 2.107038473 / 2 = 1.053519237.

Now let’s multiply this by the value I use to represent the HSMF, which is synonymous with Carl Munck’s “Alternate Pi”: 1.177245771

1.053519237 x (1.177245771^1) = 1.240251066 = (4 x (Pi^3) / 10^n)
1.053519237 x (1.177245771^2) = 1.460080323 = 1.2 Remens of 1.216733603
1.053519237 x (1.177245771^3) = 1.718873385 = 1 Egyptian Royal Cubit
1.053519237 x (1.177245771^4) = 2.023536423 = 216/10^n inverse Hashimi Cubits

Thus some unit values can not only be linked in series through the ratio (2 Pi), but they can also sometimes be linked in series via one another, at least for those units we can expect to have good exponential value, such as the Megalithic Foot, the Squared Munck Megalithic Yard, the Remen, or the Thom Mid Clyth Quantum (~sqrt 60 ft).

Readers may recall a recent report here that the Thom Mid Clyth Quantum was discovered to be none other than the proposed Egyptian Sacred Cubit, whose proposed value is conjugated from what are probably the two most prominent ancient Egyptian Units, the Remen and the Royal Cubit. Very simply, Royal Cubit x Remen = Sacred Cubit, or to use my usual numbers, 1.718873385 x 1.216733603 = 2.091411007.

Some, such as Newton, may have conflated this with the Palestinian Cubit, which it is distinct from. (I in fact discovered this value while perusing Newton’s metrological writings and observing that the ratio between his Sacred Cubit and his Royal Cubit was about 1.213, highly suggestive of the Remen. I then went on to discover that this Sacred Cubit value of 2.091411007 ft has geodetic properties, and finally that it is synonymous with Thom’s Mid Clyth Quantum, which we have seen appearing persistently in architecture as diverse as that of ancient America and China.

The confirmation of this comes through squaring the Sacred Cubit and observing the square is a whole number, which is a property of unit values in the Mid Clyth Quantum series. 2.091411007^2 = 4374/1000.

That’s another example – and a very important one – of precise mathematical relationships between units.

We can also see therein another rule that belongs to any collection of such rules regarding unit relationships

Remen x Royal Cubit = Sacred Cubit / Mid Clyth Quantum

Here’s another example – this one may require more caution because we’ll use the Mid Clyth Quantum represented as sqrt 60 ft. In exponential use, of course every second application represents 60, so of course Royal Cubit x (Mid Clyth Quantum)^n where n is an odd number will also be in Royal Cubits. By the same virtue, all odd numbered powers will also give the same unit as output.

Nonetheless, even though it adds little to our collection of rules, the Mid Clyth Quantum can be used to align significant metrological series such as (correct decimal placement aside a moment)

1.718873385 / ((sqrt 60)^1) = 2.219055998 = 27/10 inverse Remens
1.718873385 / ((sqrt 60)^2) = Radian^2 / 10^n = 10^n x (Royal Cubit) / 6
1.718873385 / ((sqrt 60)^3) = 3.698426664 = 1/2 Squared Munck Meg Yard
1.718873385 / ((sqrt 60)^5) = 6.164044439 / 10 = 1 / Assyrian Cubit
1.718873385 / ((sqrt 60)^7) = 1.027340740 = 1 / (Roman-Egyptian Foot)

Thus it also serves to illustrate mathematical relationships between units and how they can lend themselves to aligning units in series, facilitating maximum data output in return for relatively minimal inquiry.

This is ideal because ideally every monument should acknowledge the possibility that is the first, and perhaps the only, monument we have analyzed. The architects presumably wish for important keys such as demonstrated here to leap out at us immediately so that we can understand them, rather than understanding requiring us to spend 20 years crunching the numbers from 40 different monuments.

In fact, the ancient habit of dropping clues is seen so often that there probably should be a name for it, although a good one continues to escape me (and the possibly synonymous “KISS” principle – “Keep It Simple, Stupid” is already well known). You can see it in the recent work I posted on Giza, where we find the Royal Cubit as ratio as many as three times – not 3 Royal Cubits, not 17 Royal Cubits, ONE Royal Cubit – it doesn’t get much simpler, or more obvious, than that). They clearly seem to be trying to bring us into things at “ground floor level” as much as possible.

Returning to the subject of unit relationship rules, let’s try to generate a few more examples of that for the sake of the discussion.

1 Royal Cubit 1.718873385 x 1 “LSR” Unit 1.676727943 = 2.882083035 ft = 27/10 Hashimi Cubits – so now we know the rule that Royal Cubits x LSRs = Hashimi Cubits.

1 Remen 1.216733603 x 1 “LSR” Unit 1.676727943 = 2.040131231 = 2.720174976 / 1.33333333 – so now we know the rule that Remens x LSRs = “AE” type Megalithic Yards (which we could also have inferred from 2.720174976 / 1.676727943 = 1.622311471, knowing as we do that an Assyrian Cubit of 1.622311471 ft = 4/3 of 1 Remen of 1.216733603).

Many other such observations are possible, such as 10.67438159 / 2.107038476 = 5.066059170 = 10.13211834 / 2, proving that “short” Greek Feet, and “short” Remens can be constructed from Hashimi Cubits and Palestinian Cubits.

It of course isn’t necessary to catalog such relationships or memorize them, yet clear a compilation of such relations would have the potential to remove a lot of guesswork, particularly from more metrologically-oriented analyses — and that is after all what I continue to experiment with, with much success, the premise that by applying Petrie’s Inductive Metrology, and by being aware of a small cluster of vital ancients units of measure, that no matter how complex the things that we can build from these various pieces, that the pieces themselves can generally be reduced to very simple metrological values.

Here are a couple of surprises though, from a page of notes I managed to locate, where the following formulas are listed as being true

AEMY Megalithic Yards / Sacred Cubits = Palestinian Cubits

IMY Megalithic Yard x Remens = AEMY Megalithic Yards.

There might be an important statement about the nature of the Incidental Megalithic Yard (IMY) in there that is going over my head?

It was also noted that values in LSR Units x Sacred Cubits = the diametral unit of the Stonehenge sarsen circle. This unit is poorly understood, but is the consequence of essentially accepting Thom’s proposal that the outer circumference of the sarsen circle should be 120 Megalithic Yards.

Thus, LSR Unit 1.676727943 x Sacred Cubit 2.091411007 = 3.506727276.

Fellow fans of Munck’s work may recognize that 360 x (Pi^4) = 35067.27277, but also, 350.6727276 / 216 = 1.623484851, which may be an oversized form of the Assyrian Cubit of 1.622311470 ft, but can definitely be seen (optionally) as a fundamental unit of the outer sarsen circle radius or diameter

1.623484851 x 32 = 51.9515151 = outer radius sarsen circle = outer circumference (120 x 2.720174976) / 2 Pi.

What 1.623484851 definitely isn’t, however, is 4/3 of the “short” Remen analogous to 1.622311470 / (4/3) = 1.216733603 – hence the lingering shroud of mystery.

–Luke Piwalker

“Progress Report” on the Giza Layout

Earlier today, I posted this the following to the Megalithic Portal. I am not certain what to make of it but I find it very intriguing, and thought I’d share it with readers:

Well, let’s see – I was aspiring to have a look at data for the Mount Pleasant henge and see if I could develop an opinion on which available dataset might behave most like I’m accustomed to, and have ended up sidetracked back to the previous project I was working on.

It’s one I’ve never solved even after years – it’s the proportioning of the spatial distribution of the three main Giza pyramids.

Petrie’s basic descriptions of the NS and EW distances and diagonal distances between the apex of each pyramid expand into a considerably larger group of measurements when the proposed sizes of the pyramids themselves are considered

on pg 68 of the HSMF thread, I wrote

I found this link posted by Andy B about 7 years ago
http://digitaldigging.net/models/mount-pleasant-timber-circle.html

“Wainwright demonstrated that the surviving earthworks were the eroded remnant of an enormous earthen enclosure comprising a massive bank with internal berm and ditch, the total dimensions being 370m east-west by 340m north-south. Geophysical survey suggested 4 entrances in total, located to the west, north, east and south-east. The geophysical survey also turned up an unexpected bonus – a massive and continuous palisade running within and concentric to the henge ditch. Enclosing an area measuring circa 270m east-west by 245m north-south, the palisade trench would have supported a wall of timbers perhaps 6 metres high, yet only two extremely narrow entrances could be found through this enormous timber barrier.“

And pointed out that “their raw ratios are 370 / 340 = 1.088235294, which is very close to 4/10 Megalithic Yard in feet (2.72 x 4 = 10.88), and 270 / 245 = 1.102040816, very close to the Northern / Indus foot. (Some similar aspects seem to be present at Stonehenge, in terms of ratios).”

Peter Harris posted data to the same page of the same thread from
Alex Gibson and John Neal

(for Gibson’s data, Circle B 30m diameter / Circle C 24.6m diameter = 1.219512195 which is the range of the Remen or a similar number that seemingly relates to the Lunar Month and the Meter (3.280839895 x 9 = 29.52755906); 360 / 12.19512195 = 29.52755906).

I was inspired to hunt down my copy of Neal’s book, which also includes figures for average spacing between posts: Circle a 2.10 m, Circle b 1.93 m, and

2.10 m / 1.93 m = 1.088082902, which is the ostentibly same number I retrieved from Wainright’s data as “370 / 340 = 1.088235294”

Because I had interpreted the data attributed to Wainright as a possible pair ofmetrological units in Imperial expressed as ratios, for the benefit of those who might be skeptical of this, I attempted to reflect back to some of the instances where I’ve seen this very same thing working with Egyptian or Megalithic Architecure.

I was going to say that this idea of metrological units as ratios was generally suggestive of the same thing I’d just run into at Giza returning to the question of the layout of its main pyramids, but it’s in fact this reminiscent of Giza, where we also see what could easily be the very same figure, as the ratio between the NS distances from the apices of Mycerinus to Chephren and Chephren to Cheops.

The center group of equations in green attempt to define in the intended figure, and demonstrate some possible relationships between this ratio and some of the surrounding proportions.

I had been developing that as sort of a progress report on how far I think the interpretive metrology may have gotten, and ended up marking it out as an example of seeing metrological unit values in Imperial as ratios. Those are of course numbers that are built into the design regardless of what metrological units are applied and not at the mercy of the wrong choice of unit.

The equations in green at top and bottom of the diagram show what appear to be the Royal Cubit in both Imperial feet and inches built into the blueprint based on Petrie’s data (and eventually Petrie’s data combined with my models. Doing so gives 250 Royal Cubits as the distance from S edge of Cheops’ pyramid to the N edge of Chephren’s.

I wasn’t certain I could fit it onto that diagram, but it happens again

So in total so far for finding the Royal Cubit in Imperial as a ratio at Giza, we have

The equations in green in the middle of the third diagram also attempt to explain what is so special about a value of 4 Megalithic Yards that we might seem to see it repeated thusly from Giza to Mount Pleasant, in relation to some lower square roots.

I could have also added to the diagram that 1.088069890 (4/10 Megalithic Yards) ft / sqrt 5 = almost exactly 1/10 of 40 Remens, which is what I see as the inner sarsen circle circumference, and which is fundamentally the same as Petrie’s assessment, and those of some others.

Here is a slightly more extended view of possible internal relationships in the design

In order to maintain harmony and accuracy in the equations therein that involve sqrt 2, sqrt 2 is substituted for by the particular value I use for the Megalithic Foot, as given in earlier posts in the thread (1.177245771). That’s what seems to be mandated by the interpretive values assigned the other parts involved, and it’s also what see in some of the more direct relationships between ideal units (i.e., Royal Cubit / Remen = 1.718873385 / 1.216733603 = (1.177245771 x 12) / 10).

The data from Neal for the average spacing of the posts provides another interesting ratio: Circle d 2.21 m / Circle d 2.11 m = 1.047393365. The indispensable Pi / 3 = 1.047197551.

I’m not sure I’ve heard of anyone approximating Pi as 663/211 before. I thought for a moment I might have missed it by seeing it previously in reduced form, but 211 is apparently without factors?

So I have no idea what any of that is worth (except that it still doesn’t tell us much about Mount Pleasant henge), but it’s what landed on my plate.

Postscript: I’ll also include this here, which of course contains some of the data that is combined with Petrie’s data (first illustration in this post) to project a more complete model of the Giza Layout

–Luke Piwalker

Paving The Way

At present, I seem to be doing too much juggling for my own good. With one hand, I am working on the ultra-Herculean task of attempting to make sense of the larger layout of the Giza pyramids, while at the same time I have been tempted to start looking the engineering of Egypt’s ancient pyramid with the other.

The idea that the Great Pyramid was built using ramps, however prevalent the idea may be, manages to strike me as unlikely for a number of reasons and worse, to me represents the same lazy-minded, “lackadaisical” thinking on the part of many Egyptologists that has made such a demonstrable mess of ancient Egyptian metrology and mathematics.

Were I wiser person, I would probably avoid the subject of ramps and keep my eye on the task of understanding Petrie’s data for the placement of Giza’s major pyramids – indeed, the questions of how the pyramids were built, or even why they were built, really aren’t my area, as opposed to the question of what the pyramids are, metrologically, mathematically, geometrically or astronomically speaking.

Yet the minute we get beyond the question of how Cheop’s pyramid was built and into comparative studies of the anatomy of Old Kingdom Egyptian pyramids, the question of how inevitably tries to cross paths with the question of what.

The Bent Pyramid at Dahshur is one of the first things to come to mind here. The orthodox perspective often tries to attribute its novel “bent” design to a late course correction related to aversion of structural failure, even while a number of us “alt” researchers are able to come up with mathematical models that may suggest otherwise – that the “bent” shape may have been planned from the very beginning.

There is even a surprising amount of consensus among some of us, so that even while we may not agree on the precise specifics, we may be in general agreement about what some of the Bent Pyramid’s proportions represent – i.e., that the base perimeter of the Bent Pyramid in “modern” “Imperial” feet seems to represent the Earth’s circumference in “modern” miles.

This idea might be absolutely ludicrous even to my own sensibilities were geodetic data not peeking out of seemingly every corner at Giza.

Indeed, I often wish I didn’t have the excess baggage of trying to defend the prospect of the ancients possessing advanced and specialized geodetic knowledge about the size and shape of the Earth, when it seems like enough a battle already just trying to get some to consider the possibility that the ancients could actually do arithmetic better than is evidenced by a scant few generalized mathematical papyrii – but the appearance of this data in ancient architecture happens with a remarkable frequency that I cannot believe is mere coincidence.

In my writing over the course of the last few years, I have given many examples of how this geodetic data is expressed at Giza – yet we can almost guarantee that Egyptologists are having none of this whatsoever, and that it certainly isn’t likely to enter into any orthodox considerations of whether or not any “bent” pyramids show metrological or mathematical signs of planning and forethought, as opposed to their final proportions being desperate last-minute course corrections due to structural failures.

I have generally tried to avoid the subject of 3rd Dynasty Egyptian pyramids even though sooner or later addressing them is probably vital to an understanding of the history and evolution of ancient mathematical and architectural science – especially in the case of Sneferu’s pyramid at Meidum, yet now I find myself tempted to return to the subject of this pyramid, which to my reckoning remains shrouded in too mysteries, not matter how we try to look it at.

There may be something of a paradox in that good data projections might help to have an understanding of exactly what happened there, while an understanding of events surrounding the Meidum pyramid’s state might be of great assistance in making reasonable data projections. In short, I have never really had an idea of what the architects were trying to accomplish there that I feel very comfortable with.

In the course of trying to articulate some of my misgivings about Jean-Pierre Houdin’s ideas about the Great Pyramid having been constructed with the aid of internalized ramps, I find myself drawn back to the troublesome 3rd Dynasty pyramid because it’s been proposed with some frequency that the Giza pyramid may have been built around stepped cores that may be similar to the stepped components of the 3rd Dynasty pyramids of Djoser or of Sneferu’s Meidum pyramid, and trying to determine if what was typical for them might be compatible with Houdin’s ideas about the Great Pyramid as they are continually being depicted.

Offhand, I remain skeptical of Houdin’s proposal, but the exercise involves trying to have some idea of what was intended at Meidum. In the course of this I’ve begun examining photographs of the Meidum pyramid and making traces and overlays. I’ve become more convinced that for the stepped core the intent may indeed have similar to that for Djoser’s pyramid, of steps of equal height, which is a minor breakthrough that may help permit better projections of the original proportions of Sneferu’s Meidum pyramid.

In general, though, I have a lot of catching up to do with the 3rd Dynasty pyramids for having so long ignored them as potentially insurmountable challenges analytically.

It’s here that the subject begins to ostensibly overlap with the subject of the often-mentioned “hypothetical missing pavement layer at Giza”, which is why the preceding remarks, which mostly amount to non-news, serve as preface here.

It’s only been some 3 and a half years ago that I drafted the proposal of the missing pavement layer, which reconciles Carl Munck’s utterly brilliant Great Pyramid model with the current physical reality at Giza. There is a great deal that still isn’t known about it, including how it affects the Giza Layout, which may first require having some understanding of what is being affected.

I can say that recently the first experiment into this matter has been attempted only a few days ago, and successfully.

I don’t what lies ahead in this area, but I certainly encouraged by this.

Specifically, the distance due East-West from the W edge of Cheop’s pyramid to the E edge of Chephren’s that is obtained from combing Petrie’s data with my models of the Giza pyramids is a projected 365.9187465 ft. This looks a great deal like some form of the Solar Year, or possibly the Leap Year.

The Leap Year (~(366 days) is still largely a gray area, and moreover has again been superseded by other concerns, this time by the very recent attempts at identifying how an abbreviated Solar Year of 364 days was accommodated by the same mathematical system already expressing multiple forms of the 365 day year.

One possible identification of this number of approximately 365.9187465 is the “Usk River Number” 365.8052902, which was discovered in Prof. Thom’s data for the Usk River stone circles site. Its importance has been demonstrated on a number of occasions now, although it seems to remain unknown for now how to classify this number – is it an oversized figure for the Solar Year? Is it an undersized figure for the Leap Year? What is its nature exactly?

This may be a rhetorical question at Giza if this nomination doesn’t survive the assessment of the immediate mathematical environment in which it occurs, but to get to the point, if we add the hypothetical missing pavement layer, the number draws compellingly close to the standard, ubiquitous 365 calendar year, the “civil calendar” of the ancient Egyptians.

The history and evolution of the missing pavement proposal is that Munck’s remarkable figures the Great Pyramid, as far as I am able to tell, came about almost by chance. Understandably put off by the discrepancies between some of the different data sources for the pyramids, Munck apparently seized on a passing reference from Ahmed Fakhry to a baseleng of about 754 feet, considerably shorter than the mean of about 755.7 obtained from some of the most harmonious data sources.

It’s rather surprising that Munck didn’t seize upon the general consensus between Petrie, Coles, and Lehner/Goodman for the Great Pyramid’s measures, and yet it’s a very good thing that he didn’t.

Beyond that, other than endless matheamtical evidence for the validity of Munck’s figures, I have little to say about Munck’s departure per se except that it wouldn’t be the first time that the reckless presentation of pyramid data by I.E.S. Edwards or others has thrown a monkey wrench (or spanner, for our friends “across the pond”) into the works. Because of Munck’s acceptance of Edwards as a data source for Mycerinus’ pyramid, I had to revise the Myceronus model from the ground up myself several years ago.

Regarding evidence for the missing pavement, it unfortunately seems to remain circumstantial, even if compelling through its cumulative nature. Overhead views of the current pavement around all three of the major Giza pyramids shows an erratic array of material that is almost begging for an additional “cosmetic” layer to tidy it up.

Remnants of the Great Pyramid’s casing stones in places show signs suggestive of having been protected from weathering by an additional now absent layer. There are also places along the base where unevenness of execution is implied, something else that may call for a cosmetic cover-up with additional, now missing, pavement.

Egyptologist’s view of the Red Pyramid indicate a pavement layer encroaching upon the lowest casing stones exactly as the Giza “missing pavement” model projects having been present there. (This is purported to be according to diagrams by Josef Dorner which I currently do not have in my possession).

Thus far I am of the opinion that this missing pavement would have consisted of prized material perhaps similar if not identical to the material found underneath the Pyramid Temple at the East foot of Cheop’s pyramid, and may have been fitted into the pyramid’s enclosure wall (rather than under it), so that this prized and relatively thin missing layer would have been easily accessible, easily carted off, and thus may not only have been the very first thing to be plundered for material to recycle, but because of this, was also thoroughly picked clean, leaving little to attest to its former presence but the circumstantial evidence cited, unless I’ve underappreciated more possible evidence.

Remnants of black basalt paving from beneath Cheops’ pyramid temple. Photo: Jon Bodsworth.

In the case of Mycerinus’ pyramid with unfinished dressing at the base, it may be very possible that the stage of applying this pavement there was never achieved, since this missing pavement layer is presumed to to have been placed against the finished casing.

The Great Pyramid’s missing pavement would have been a bit over a half a foot thick, and that of Cheop’s pyramid somewhat less, the exact figures being the difference between the “paved” (Munck) and “unpaved” (mine) heights. For the Great Pyramid, that is 481.0325483 ft – 480.3471728 ft = .685375457 ft.

An early breakthrough in this area was the realization that the Great Pyramid (and others) would have metrological and mathematically significant values with or without the hypothetical pavement layer.

Please note that these same considerations may apply to any and all ancient Egyptian pyramids. On this account, there may ultimately be no such thing as an ancient Egyptian pyramid that has singular height and base length measurements.

At any rate, why this is of relevance here is that although I am not necessarily in possession of data for Sneferu’s Meidum pyramid nor Djoser’s pyramid at Saqqara, and although accepting pyramid data from authors like Edwards, Lehner or Rossi is even riskier than accepting candy from strangers (one reason that I do not have finished models of these pyramids), if we look at the data that Lehner presents in The Complete Pyramids, we have

Djoser: base 121 x 109 m, height 60 m

Sneferu: base 144 m, height 92 m

Recalling that Munck’s proportions of the Great Pyramid are height 480.3471728 ft x 2 Pi = perimeter 3018.110298 ft, for Djoser’s pyramid we have

121 + 121 + 109 + 109 = 460 m = 1509.186352 ft = 3018.372703 / 2

Perimeter / height 460 m / 60 m = 7.666666666

For Sneferu’s, 92 m = 3018.372703 / 10 ft

Perimeter height (144 x 4) / 92 = 6.260869565

To my sensibilities, that is about one random coincidence too many, but there is more.

2 x Pi = 6.283185307, which is thought to be the Great Pyramid’s perimeter / height ratio, and which the final perimeter / height ratio of Sneferu’s Meidum pyramid may easily have been also.

Munck’s height for the Great Pyramid divided by 2 Pi = 480.3471728 / (2 Pi = 7.644962696 (compare 7.666666666).

Between the two of them, then, these very speculative heights for these these two 3rd Dynasty pyramids would frame the ratio

Djoser 7.644962696 / Sneferu 6.283185307 = 1.216733603, the value of the Egyptian Remen in “modern” feet.

Thus some of the best circumstantial evidence for Munck’s unorthodox Great Pyramid values, and the missing pavement proposal that supports them, may come from the proportions of other Egyptian pyramids, including even older ones than those at Giza.

This data accuracy reflecting the original intent here would also indicate that not even Egyptologists can spoil all of the data, all of the time.

By the way, if anyone spotted it in those figures, you are correct – that is one of many important things that the Great Pyramid’s height according to Munck means:

480.3471728 / 2 Pi / 2 Pi = 12.16733603, the Remen in feet

I’m probably not up to task of trying to track down this missing layer of pavement, nor do I need another thing to juggle at the moment, but obviously the logical place to begin a search might be some of the the oldest Egyptian structures to be built from pillaged material taken from the Giza plateau.

For the moment, I thought I would attempt a fresh explanation of why Munck’s Great Pyramid seems to be a foot shorter than everyone else’s, and why it may matter to both the study of, and the story of, the 4th Dyansty pyramids at Giza, and the 3rd Dynasty pyramids at Meidum and Saqqara.

These numbers may still hold the key to understanding whether what we see at Meidum and Dahshur is the result of accident, or the result of careful and organized planning.

–Luke Piwalker

A Number Never Mentioned

Well, perhaps it’s more like a number very rarely mentioned.

At present I am continuing to try to find ways to fathom the data resulting from combining WMF Petrie’s data for the distances between centers of the three main Giza pyramids with my established models of these pyramids.

In the course of this, I seem to have encountered a number I have known about for a very long time, from back in the early days working with Michael Morton. I’ve always felt it would be a likely number to find at Giza because of some of its mathematical properties, but it never seems to have appeared in a particular place with any certainty.

It may be that it’s simply easily overshadowed by other numbers, like 1.06748159^2 or another very similar number, but in spite of the number of times it’s been looked at, it seems to struggle to “make a name for itself”.

The number is 113.7800777 (decimal placement optional).

sqrt 1.137800777 = 1.066677448, which probably isn’t a valid number, to the best of my knowledge, but the resemblance to the standard 1.067438159 ft Hashimi Cubit (equals Egyptian Royal Foot / 1.08) is difficult to overlook.

The metrology would actually be 113.7800777 ft = 54 Palestinian Cubits, where we already have 360 Palestinian Cubits as the width of the Great Pyramid’s platform, and the proposed distance North-South from the center of Chephren’s pyramid to Mycerinus’ from Petrie’s data is 15170.4 inches / 12 = 1264.200000 ft, proposed to mean 15170.67702 in / 12 = 1264.223085 ft (7200 Palestinian Cubits), which may also be a number that recurs within the Giza layout in some different metrologies.

1264.223085 x 9 = 113.7800777

It may have also been found in the King’s Chamber of the Great Pyramid, in an elaborate mathematical scheme that recognizes some of the chamber’s actual reported measurements, rather than simply its mean measurements.

Nor may it be all that difficult to find at Stonehenge – if we take the 316.0557713 mean circumference of the sarsen circle and multiply by the 360 degrees of a circle,

316.0557713 x 360 = 113.7800777

113.7800777 is actually part of a fairly nice 2 Pi series, although it may help a bit to appreciate that if we double it before we begin dividing it consecutively by 2 Pi. We also get an important series by dividing it consecutively by Pi that includes the Egyptian Royal Foot and the Inverse Lintel (Circle) Megalithic Yard, along with what is thought to be the number of Royal Cubits in the base of the Bent Pyramid at Dahshur, if the equatorial circumference it seems to be trying to express is the common equatorial one.

That looks like this

113.7800777 / Pi^2 = 11.52833214 = 10 Egyptian Royal Feet in modern feet

113.7800777 / (Pi^3) = 3.669582092 = 1 / (Lintel Meg Yard 2.725105952 ft / 10)

113.7800777 / Pi = 36.21732356; 36.21732356 x 10 Royal Cubit 1.718873385 = 622.4299355, possible sidelength of Bent Pyramid in feet = 24901.19742 / 4.

(36.21732356 = Palestinian Cubit 2.107038476 x Royal Cubit 1.718873385)

Surprisingly, 113.7800777 links to the proposed Great Pyramid slope length of 575.1793153 without the apex section but with proposed missing pavement layer, via the square of the primary Venus Orbital Period

5751793.153 / (224.8373808^2) = 113.7800777

It can also be seen as Radian^2 x Eclipse Year

(57.29577951^2) x 346.5939372 = 113.7800777 x 10^n

So yes, 113.7800777 is “something”, although it may be more likely to appear as a ratio or product of other measurements than an actual measurement itself, and the two calendar related equations seen here were actually discovered while writing this, so there may still be a lot we don’t know about this number.

Let me play around with it for another minute here and see what else we might find… Let’s try the number 12 as a probe and see if we find any known metrological units, or other important data

113.7800777 / (12^1) = 948.1673148

That’s Munck’s Great Pyramid perimeter (3018.119298 x Pi) / 10

113.7800777 / (12^3) = Saros Cycle 6584.495241 days / 10^n

I’m glad I decided to share it with everyone.

–Luke Piwalker

Giza’s Dirty Little Secret?

I have a lot of pieces of paper flying about with calculations relating to the Giza layout, so I sat down to put some of them into HTML tables.

One of the first things that happened is that I set up a ratio matrix (a list of the dividends of all the numbers divided by one another). Ratios are one of the best places to store data because ratios are independent of the actual measurement units that are used in design or applied to interpretation.

The highlights of this show us some of what was “hiding” in Petrie’s data.

36857.7 / 17873.2 = 2.062176890 = 20.62648062 (Royal Cubit in Inches)?
29102.0 / 19168.4 = 1.518227916 = ~1 / Saros? (1 / 6585.3211 = 1.5185288380)
22616.0 / 19168.4 = 1.179858517 = 1.179778193?
22616.0 / 13931.6 = 1.623359843 = 1.622311470?
22616.0 / 13165.8 = 1.717783955 = 1.718873385?
19168.4 / 13931.6 = 1.375893651 = 27.52186571 / 2 = 137.6093286? Anom Month / 2 (~27.55 / 2 = 13.775)
19168.4 / 9450.20 = 2.028359188 = 20 / (Pi^2) = 2.026423673, or 2.027889338 (16.22311470 / 8.0)?
17873.2 / 15170.4 = 1.178162737 = 1.177245771?
17873.2 / 13931.6 = 1.282925148 = 1.290994449 (1 / sqrt 60)?
13931.6 / 13165.8 = 1.058165853 = 9 x 1.177245771 = 10.59521194?
15170.4 / 13931.6 = 1.088920152 = 4 MY / 10 (4 x 2.720174976 = 10.8806999)?
15170.4 / 13165.8 = 1.152258123 = Egyptian Royal Foot 1.152833215?

Two of the things that I mention often are how we seem to find metrological unit values in “modern” “Imperial” feet as ratios in ancient designs (something I first noticed with ancient Egyptian architecture), and how two of THE most important numbers to incorporate into every piece of ancient architecture if at all possible are 1.622311470 and 1.177245771

If we follow in this direction, using 22616.0 = 7200 Pi = 22619.46711 and 15170.4 = 15170.67702 = 600 Palestinian Cubits of 2.107038476 as fixed values, we get

22619.46711 / 1.622311470 = 1394.274005 inches = 11618.95004 ft
22619.46711 / 1.718873385 = 13159.47254 inches = 1096.622712 ft
15170.67702 x 1.177245771 = 17859.61537 inches = 1488.301280 ft
15170.67702 / (4/10 x 2.720174976 = 1394.274004 inches = 11618.95004 ft
15170.67702 / 1.152833215 = 13159.47254 inches = 1096.622712 ft
22619.46711 / 1.179778193 = 19172.64384 inches = 1597.720321 ft

Which seems harmonious enough, and able to provide significant justification to the values proposed here

There seems to be a possible surprise here (in addition to the appearance of 1488.301280 (480 x (Pi^3)) – Munck was rather fond of this number it seemed, no doubt for some good reasons).

36857.7 / 17873.2 = 2.062176890 = 20.62648062?

I’m not entirely certain, but this number may not be what it looks it like — as with Teti’s pyramidion, they may have set aside the “sacredness” of the standard Royal Cubit to perform a departure. With their of course having worked extensively with the Royal Cubit, no doubt they were aware of many similar figures when their equations didn’t quite work out with the standard value, and they seem to have occasionally made use of some figures similar to the standard Royal Cubit to resolve a design scheme.

(As far as I can tell, Teti’s pyramid uses 1.7317171717 instead of 1.718873385, even at considerable risk of confusion, because in relation to its other aspects, 1.067438159 x 1.622311740 = 1.7317171717. The standard cubit of 1.718873385 simply doesn’t fit there.)

So — if 20.62648062 doesn’t mean 1.718873385 x 12 = 20.62648062, Royal Cubit in inches, what else could it possibly mean?

Well, let’s look at Petrie’s data some more for a minute, there’s something else I’d like to point out that’s in there

36857.7 / 29102.0 = 1.266500584
22616.0 / 17873.2 = 1.265358189

36857.7 / 22616.0 = 1.629717898
29102.0 / 17873.2 = 1.628247879

29102.0 / 22616.0 = 1.286788114
17873.2 / 13931.6 = 1.282925148

In terms of proposed candidates values

17859.61537 / 1394.274005 = 1.280925794
22619.46711 / 17859.61537 = 1.266514796

If we continue to project the figures in this direction we may obtain

36904.87899 = (29.52390320 / 8 = 3.690487899) x 10^n
36904.87899 / 22619.46711 = 1.631553866 = 1.177245771^3
17859.61537 x 1.631553866 = 29138.92450 = 36904.87899 / 1.266514796

29138.92450 / 22619.46711 = 1.288223297 = 3 / 3.881314680

This isn’t all certain, and involves a remarkable 37 inch error on Petrie’s part relative to his figure of 29012.0 inches N-S from the apex of Mycerinus’ pyramid to the apex of Chephren’s pyramid even while some of Petrie’s other figures seem far more accurate, but 29138.92450 does keep coming back from numerous mathematical probes as the possible original intended value, and the fact that, remarkably, 1.177245771^3 serves both of these equations may be nothing to take lightly.

(Of note here is that

29138.92450 / 19172.64384 = 1.519817754 = 1 / 6579.736270

6579.736270 is the A value for the Saros Cycle if 6584.495235 is the B value, which is most likely the most sensible proposal of what values for the Saros Cycle should look like and how they should be).

This would give us then

36904.87899 / 17859.61537 = 2.066387110 = 1.033193555 x 2

That, is a very curious thing.

I searched this blog to prior references to 1.033193555 and couldn’t find any, so even if I’ve taken a wrong turn there, this number is probably worth talking about.

What it is, something of a relic, an artifact that commemorates that sometimes even the most otherwise perfect mathematical scheme can have a minute flaw in it – or at least that’s what I thought for a very long time.

It was a “side effect” of Munck’s absolutely amazing scheme for location the Giza “Hall of Records”, and very little had been discovered about its usefulness in the years that followed.

It went, Grid Latitude Sphinx 63571.27163 / Grid Latitude Hall of Records 61528.90839 = 1.033193555

Even when it should have been the mighty 1.033542556 = ((1 / 3) x Pi^3) / 10 that Munck was so fond of, often seemingly having ouright reverence for sites he associated with this number.

61528.90839 was the square of the Great Pyramid’s Grid Point: 8 x (Pi^3) = Grid Point Great Pyramid 248.0502134; 248.0502134^2 = 61528.9083

These equations were up to their necks in 1.067438159, which I finally realized after finding 1.067438159 first at Stonehenge as described to the classic word processor accident story, where I accidentally deleted entries for 1.067438159 while making a complete list of Giza ratios from Munck’s figures years ago.

If I hadn’t, I would have spotted it right away in the Giza data because is shows up repeatedly, and Munck – who absolutely had to be aware of the magic number 1.067438159 – never mentioned it in print that I’m aware of even though it’s almost essential for anyone attempting to explore his numbers.

A very bizarre thing is that 1.033193555^2 = 1.067488923 = ALMOST 1.067438159. There IS another number very similar to 1.067438159 that I keep ignoring as junk – I don’t even know what it is offhand and I’m not so sure it’s 1.067488923 – and then there is a third one that’s an illegal square root.

Even more curiously, 1.033193555 apparently makes a poor false square root of 1.067438159 given that it’s complement would be
1.067438159 / 1.033193555 = 1.033144422, which is all likelihood is an even more useless number.

This is very reminiscent of Stonehenge, where at some point its metrological reach exceeds its grasp and we if we hope to keep worthing with the premise of ancient precision have to accept that a little of what it wants to show us isn’t quite mathematically possible. Stonehenge dramatically combines metrologically units like Remen 1.216733603 ft, Megalithic Foot 1.177245771, and Hashimi Cubit 1.067438159.

That 1.067438159^3 = 1.216264895 – ALMOST 1.216733603 but not quite, and (1.177245771 / 1.067438159^2) = 1.216322751 – also ALMOST 1.216733603 but not quite aren’t accurate enough to accept as useful, yet come far too close the target figure to not merit honorable mention.

A similar consideration then, may have been carried out at Giza, although Stonehenge would distinguish itself from Giza by expressing a viable false square root pair for 1.06748159 in the slightly skewed (of necessity) inner and outer sarsen circle proportions relative to the sarsen circles mean proportions, where Munck’s number 1.033542556 x 1.032795556 = 1.067438159, a true working false square root pair for 1.067438159.

Let’s try to put away the notion though that 1.033193555 is merely an incidental junk number arising from Munck’s breathtaking “geomathematical” scheme, even though that is what Michael Morton and I ourselves generally thought about it for a very long time.

What 1.033193555 really is is sort of a “Secular Wonder Number” – it’s difficult enough to work with that we could take it for useless, yet if we apply the right “key”, it functions just as some of the Tikal “Wonder Numbers” do – in a very similar fashion, actually.

1.033193555 belongs to a (2 Pi) series which also includes 54 x 1.177245771, 399.43300794 (Jupiter Synodic Period B), the Sacred Cubit in inches, and 1/4 of the Equatorial Circumference…

1.033193555 x (2 Pi)^11 = 24901.19742 / 4

Just the sort of thing the Maya seemed to like to do, starting a series higher or lower than we might normally expect.

For what it’s worth then, this possible ancient Egyptian series is merely 1/4 of the classic Tikal series – and for whatever else it’s worth, the Mayan series in question can be started from 1.177245771^3 = 1.631553864, which this particular set of proposals suggests is found in the Giza layout ratios, twice “for good measure”.

Regarding these projections,

19168.4 / 13931.6 = 1.375893651 = 27.52186571 / 2 = 137.6093286? Anom Month / 2 (~27.55 / 2 = 13.775)
19168.4 / 9450.20 = 2.028359188 = 20 / (Pi^2) = 2.026423673, or 2.027889338 (16.22311470 / 8.0)?

19172.64384 / 1394.274005 = 13.7509870880 = 1.718873385 x 8 –

We DON’T get 1.718873385 x 12 but we DO get 1.718873385 x 1 and 1.718873385 x 8? (8 Royal Cubits = Mars Orbital Period (A or B?) x 10^n x 2).

However, this seems to miss the opportunity to express half the Anomalistic Month, in favor of Mars.

Regarding 9450.20, the two proposed ratio values would give either one of these

19172.64384 / 2.026423673 = 9461.32050047 = 365.0200808 x 25920
19172.64384 / 2.027889338 = 9454.48229384 = 364.7562611 x 25920

In terms of accuracy,

13159.47254 inches – 22619.46711 inches = 9459.994566 inches

9454.482293 would fail the “Giza Standard” test, coming in at only 9454.482293 / 9459.994566 = 0.99941730695, just short of the Giza Standard of .9995 or greater, whereas

13159.47254 inches – 7200 Pi inches = 9459.994566 inches = 9461.32050047 to .999859857

9461.32050047 inches / 12 = 788.443375 ft = 3153.773500 / 4 = 63075.4700 / 8 = 1576.886750 / 2

3153.773500 = seconds in a year of 365.0200808 days

1576.886750 is part of the 2 Pi chain arising from 1.033193555.

63075.4700 = 24901.19743 / (2 Pi)^2

22619.46711 / 9461.32050047 = 2.39073045975
13159.47254 / 9461.32050047 = 1.39087060197
1394.274005 / 9461.32050047 = 0.14736568800 = 1 / 6.78584013367
17859.61537 / 9461.32050047 = 1.88764510927
19172.64384 / 9461.32050047 = 2.02642367300

1.887645109 = 2.222222222 / 1.177245771 = (1.179778293 x 16) / 10 = (2 / 1.067438159) x 10
2.390730461 = (1 / 41828.22014) x 10^n
1.390870601 = unknown; reciprocal = 718.9741441 <—

Notice the third figure especially – it’s the reciprocal of the projected 718.9741438 ft Great Pyramid vertical edge that has already become part of the discussion of the Giza Layout.

By the way, metrologically speaking, 9461.32050047 inches = 788.443375 ft = 648 Egyptian Remens of the standard 1.216733603 ft

Finally (for now), regarding

13931.6 / 13165.8 = 1.058165853 = 9 x 1.177245771 = 10.59521194?

13942.74005 / 13159.47254 = 1.0595211933 — as projected, (9 x 1.177245771) / 10

This of course is only one possible interpretation, but it seems like a fairly coherent one so far.

One is almost tempted to wonder if Munck worked with Petrie’s data but never mentioned it, It’s still a mystery to me what inspired him to accept 1.033193555 in his “Hall of Records” calculations, but if it were something he’d seen before at Giza, perhaps it’s more understandable?

–Luke Piwalker

The Great Pyramid’s Vertical Edge

As I continue to beat my head against the problem of the layout of Giza’s main pyramid – the “Giza Blueprint” as DUNE calls it – I find myself going back to the model of the Great Pyramid and some of its less referenced attributes.

Overall, the reasons for this include that combining my models with Petrie’s data for the distances from apex to apex of the pyramids of Cheops, Cheprhen, and Mycerinus (pyramids G1, G2 and G3 respectively), the distances directly East to West from the apex of Chephren’s pyramid to the West face of Cheops’ pyramid calculates from Petrie’s raw data as approximately 719.3479200 ft, while the vertical edge length of Cheops’ pyramid has “traditionally” been given by me as 718.9741438 ft.

That is, although this figure is relatively recent, being only about 4 years old, it has yet to require revision in the light of any newer considerations. It continues to withstand the great many new developments and discoveries.

The figure calculates from the model as sqrt (1068.585650 / 2)^2 + 481.0325481^2 = 718.9305496 ft, which is 718.9741438 to an accuracy of 718.9305496 / 718.9741438 = .9999393661, well above the experimental “Giza Standard” allowance of .9995 or greater when additional, subtraction, or trigonometry are involved. These operations are often a fact of life, but take their toll on absolute accuracy.

We also seem to be pointed back in the direction of the Great Pyramid by considering the layout around it, because the distance North to South from the South Edge of Mycerinus’ pyramid to the center (apex) of Chephren’s pyramid calculates at 1437.124985 feet.

I have also “traditionally” given the Great Pyramid’s slope length without the missing apex section as 575.1793153 ft (this is for the “Munck model” which involves a hypothetical missing layer of pavement that decreases its base length and height from their present state to the remarkable figures that Munck proposed for them; the vertical edge length figure of 718.9741438 is for the Great Pyramid in its present state and is not dependent on this proposed missing pavement layer).

Please note that 718.9741438 x 4 = 575.1793153 x 10, and 718.9741438 x 2 = 1437.948288.

These resemblances between calculated values around the Great Pyramid, and calculated values within the pyramid itself also give the inpression of a common metrology at work.

Thus I thought that perhaps before I attempt to proceed any further with the project, I would stop and take the time to over the figure 718.9741438 and what it is, and how it became part of the Great Pyramid model.

(Before I forget, this value could also be the base length of the Red Pyramid, which remains unsolved largely because the lack of consensus data, with the value being given as anything from about 717 or 718 feet, to 722 feet. As pointed out previously, this value of 718.9741438 ft possibly being the intended base length of the Red Pyramid would be part of a larger pattern of specific measures from one pyramid being “recycled” into another).

Metrologically speaking, 718.9741438 isn’t exactly a revelation – it’s 972/10 Squared Munck Megalithic Yards (SMMY = 2.719715671^2), which are equal to 9 / 1 Remen. (We could say therefore that the value is in inverse Remens, but the custom according to Petrie’s “Inductive Metrology” would be more to use “forward” unit values rather than inverse ones).

972/10 x (2.719715671^2 = 7.396853331 ft) = 718.9741438

This is one of a great many ancient architectural proportions that is amenable to exponential application of a metrological unit value.

718.9741438 / (7.396853331^1) = 9.72
718.9741438 / (7.396853331^2) = 13.14072291
718.9741438 / (7.396853331^3) = 1.776528791
718.9741438 / (7.396853331^4) = 2.401735862 / 10
718.9741438 / (7.396853331^5) = 3.24686969 / 100
718.9741438 / (7.396853331^6) = 43.89663485

All of which is significant data that increases our understanding of how the ancients used numbers, and may further illustrate how the relationships between numbers can allow us to get the same data “on the ground” that we could obtain from physically measuring or invading the pyramids — Munck’s height for the Great Pyramid 480.3471728 ft / 100 = 2.401735862; my projected value for the base length of a side of the missing Great Pyramid apex section is 43.89663485 ft.

Along the very same lines, 1.776528791 x 2 x 10 = 35.53057584 (360/100 x (Pi^2)), which in feet is the projected slope length of the missing apex section of the Great Pyramid. It calculates raw from the projected height of the missing section as sqrt (height 27.94546573^2 + half base 21.94831745^2) = 35.53417641.

A very interesting and similar thing happens we we go to test if the estimated 718.9741438 ft is a measure in Royal Cubits of 1.718873385 ft

718.9741438 / (1.718873385^1) = 418.2822016
718.9741438 / (1.718873385^2) = 243.3467208 (200 Remens = 243.3467208 ft)
718.9741438 / (1.718873385^3) = 141.5733834
718.9741438 / (1.718873385^4) = 82.36405580 = 1 / “Not A Remen”

All of this too is significant data that helps give a “jump start” to our vocabulary of ancient numbers, and contributes to our understanding of the relationships between the Squared Munck Megalithic Yard, the Remen, and the Royal Cubit, and others – thus further underscoring the legitimacy of the Squared Munck Megalithic Yard as an authentic ancient metrological unit.

It’s very interesting to see “Not A Remen” in there. That took me a very long time to find for overlooking its presence in these Great Pyramid proportions. It is (1 / 1067438159) x (360^2) / 10^n = 1.214121852 and is thought to be another number that is involved in certain calendar calculations.

The SMMY can indeed be seen as synonymous with Remen as well, although it is a specialized form of the Remen, just as the Hashimi Cubit is a specialized form of the Egyptian Royal Foot.

718.9741438 is also amenable to the exponential use of the Radian value (57.29577951) as a mathematical probe and data retrival key

718.9741438 / (57.29577951^1) = 12.54846605
718.9741438 / (57.29577951^2) = 21.90120488 / 10^n
718.9741438 / (57.29577951^3) = 38.22481350 (2 x sqrt 365.2840918)
718.9741438 / (57.29577951^4) = 66.71488515 = (625 x 1.067438159) / 10
718.9741438 / (57.29577951^5) = 1.164394406 / 10^n
718.9741438 / (57.29577951^6) = 2.032251618 / 10^n; 2.032251618 = 2 x (360 / proposed Lunar Year approximation 354.2868381 days)

Likewise, dividing 718.9741438 by 360 repeatedly yields another significant data series, including that at 718.9741438 / (360^3), we obtain 1.541011111, and important piece of astronomical data. Venus Orbital Period x 1.541011111 = Eclipse Year is one of two known experimental formulas for generating values for approximation of the Eclipse Year. Either one makes an optimal value of the remarkable approximation 346.593936 days, representing the “textbook” figure 346.62.

It may also be of some interest that the calculated value for the Great Pyramid’s vertical edge length without the missing apex section but with the hypothetical pavement in place is approximately

717.9062172 – 41.77436507 = 676.1318521 ft, which has been tentatively identified as 675.9631125 ft (accuracy .9997504339). This is 1/2 of 1000 inverse Squared Munck Megalithic Yards, or to use the preferred syntax, it is 10000/18 of the Remen

(1.216733603 / 18) x 10000 = 675.9631128

To write it another way, 675.9631128 x 360 = 1.216733603 x 2 x 10^n.

This combination of numbers also forms an important series which includes the number of hours in a year of 365.0200808 days, the number of seconds in a year of 365.0200808 days, and at 360^4, a number that may be intrinsic to the mathematics (the volume calculations, specifically) of both the King and Queen’s chambers of the Great Pyramid.

675.9631125 x (360^4) = 11353.58459 / 10^n

Volume King’s Chamber: 17.18873385 x 34.37746770 x 19.21388686 = 11353.58456 cubic feet

There is of course more to the story of 718.9741438, but let’s consider that a start for now.

–Luke Piwalker

The Valleys of Neptune

Don’t worry, dear reader – I’m sure I have enough controversial things to say already without seriously suggesting that the surprisingly modern-looking observatories of the Maya were once equipped with telescopes, but even if we can’t look to it as proof that they did, I do find it interesting to think about whether they selected calendar systems that could have encompassed the cycles of Neptune, had the ancients been aware of its presence.

I already have it to say that that I think we’ve gotten a great deal wrong when it comes to where astronomy fit into ancient cultures. Many orthodox historians would have it that, even when acknowledging the personification of planets in myth by ancient cultures, that the ancient storytellers who kept these tales somehow believed in their own hype and actually thought the solar system worked in the ludicrous manner described by myth.

To my reckoning, these tales likely serve the opposite function – bringing to the average person an understanding that the planets are so regular in their behavior as to exhibit all the sentience of a refrigerator light – you open then door, the light goes on, you close the door, the light goes off, and it’s not exactly a sign of “godly” intelligence.

In fact, to me that seems like what’s in the fine print of sentiments like those of Bill Saturno, noted for his efforts to calm those concerned that the end of the Mayan calendar meant the end of the world not all of 10 years ago.

“‘The ancient Maya predicted the world would continue, that 7,000 years from now, things would be exactly like this,’ he said. ‘We keep looking for endings. The Maya were looking for a guarantee that nothing would change. It’s an entirely different mindset.'”

Many of us may also have gotten it wrong as to why astronomical themes saturate the ancient cultures in question, thinking it’s because ancient lives revolved around ancient “gods”.

With calendar cycles so large, all the “fuss” may have involved to a large degree, simply the rarity of such celestial events as alignments or the completion of cycles. It may have been something of a case of “Hey, kids, come here and look at this, here’s something you’ll probably only see once in your lifetime. We’ve been waiting 52 years for this”.

At any rate, the “textbook values” for Neptune’s cycles are

Orbital period 60,182 days
Synodic period 367.49 days

Note that 1 / 367.49 = 2.721162481, very close to the Megalithic Yard, which we already know is integrated into ancient calendar systems, and quite likely represented the Draconic Month (particularly the “Draconic” Megalithic Yard).

18980 / 60182 = 3.153766907, which is almost identical to 3.153773498, which is not only a valid number, it’s the number of seconds in a calendar year of 365.0200808 days.

60,182 days / Venus Synodic Period 584 days = 103.0513699, yet another astronomical occurrence of a number likely to be commemorated as 6 Royal Cubits (1.718873385 x 6 = 10.31324031 ft), our first encounter with a number like this, if not as a geometric extrapolation of the Remen, quite likely being as the ratio between Solar and Lunar Years, 365 / 354 = 1.031073446 or 365.25 / 354.36 = 1.030731460, and probably neatly dovetailing with the prospect of representing Mars’ Orbital Period of 686.971 days as 1.718873385 x 4 x 100 = 687.5493542, which makes the ratio between idealized Mars Orbital Period and Venus Synodic Period values to be 687.5493542 / 584.032128 = 1.177245771.

If the set of Mars Orbital Period values which provides this relationship is finally accepted, it means that a suggested Megalithic Foot value of 1.177245771 ft is not only an astonishing key for unlocking astronomical data from ancient sites, it is itself a piece of astronomical data (not to mention what happens with we combine it with Pi, and so on).

All of which requires that the ancients were aware and used the “modern” foot along with other units, but that may be precisely where the “Imperial” foot got its start, is as 1/10 of the side of a square with a diagonal of, for all intents and purposes, 12 Megalithic Feet.

The plausible role of the modern foot as a reference unit may help to explain why it seems so rare to find it in ancient use. It seems 50 times easier to find ancient Egyptian units used in ancient Mexico than it is to find orthodox evidence of the Imperial Foot used there, just as it is in ancient Egypt – but to use the Imperial Foot in a way that would be detected by “Inductive Metrology” method would have given values so simple as to be meaningless.

We don’t need the ancients to tell us what 2 or 3 or 4 or 5 are, we can think of that ourselves. It’s reserving the “modern” foot as a reference unit that gives all kinds of splendid meaning to other ancient units of length measure.

At any rate, it may be a safe guess that indeed the ancients did work with multiplanetary calendar systems and cycles that could indeed have embraced the cycles of Neptune within their astoundingly ambitious grasp had they been aware of its presence.

For all we might know, they may have actually named Neptune in the few surviving records, if we only knew to put the name to the “face”. There do seem to be names in these works that we have yet to understand, even after having successfully placed a number of them with the identities of various planets already.

Perhaps finding the name would prove more feasible than trying to figure out where their telescopes made off to, but if there happened to be any gold used in their construction, at least one possibility may come to mind, and two possibilities if these purely speculative optical accessories were advanced enough to be able to afford anyone with any militaristic advantage at the time.

So, I’m not going to try to tell anyone that the ancient discovered Neptune, but with fond remembrances of my collaborations with Michael Lawrence Morton (my “Obi-Wan” as it were), let’s try not to pre-judge that something that might even be plausible is “impossible”.

–Luke Piwalker

Do We Decimal?

First, perhaps a disclaimer is order – sometimes, I’ve had to do little more than try and explain the logic of my work with numbers, or explain how certain ideas are not tranferrable across diffeent proposed ancient number systems, to give other independent researchers the impression that I intend something other than mere constructive criticism at worst.

Nothing could be further from the truth – I cannot think of another mathematical researcher who isn’t making important discoveries and valuable contributions to our collective understanding of ancient mathematics, often on a frequent basis.

That said, all the same I try to be very careful what I accept as logic. I am certain I have witnessed both of my most helpful mentors go off the deep end with pouring vast effort into ideas that in the long run, cannot be successfully supported – Munck’s “geomathemics” or Morton’s “Archaeo-Sky Matrix” – in what amounts to a phenomenal and tragic squandering of keen human intellect.

Few things are more disheartening to me that to read discussions like some of the ones that involved researcher Don Barone, who posted work on connections between Giza pyramids and planets to a number of “fringe” forums where they’d presumably have been of considerable interest. Don was once told something to the effect of “You’re wrong, the ancient Egyptians didn’t know what decimals are, I just looked it up on Wikipedia” as if one minute’s worth of education somehow beats years worth of education and hands-on experimenting.

Obviously, when alternative researchers like myself suggest that the ancients were using decimal mathematics, we’re talking about something that has not only escaped Wikipedia, but all of history in general. There seems to be little telling what was in the hundreds of thousands of scrolls of ancient writings that went up in flame at libraries like that of Alexandria, to name only one.

Equally tragic and no doubt equally detrimental to a better understanding of human history and capability are the pre-Columbian codices torched en masse by Spanish conquistadors on account of their “superstitious” “heathen” content.

Still, I’m sure none of us are content to rest our arguments on written works that could have existed or even probably existed. When someone insists to me that the ancient Egyptians could only work in unit fractions, I’m well aware that the scant few surviving ancient Egyptian mathematical works have all the earmarks of being generalist works rather than specialist works – they tend to attempt to cover a very diverse range of applications of mathematics with a single style of mathematics that may not even be as simple as it looks at first glance.

I’m also aware of what the alternatives to the idea of ancient use of decimal math look like. One pyramid researcher I encountered was quite devoted to the idea that the ancients possessed all sorts of complicated knowledge from astrophysics to physics outright, yet equally devoted to the orthodox dogmatic proposition that ancients were nonetheless still so simple minded as to be restricted to working with fractional numbers, resulting in pyramid theories saturated with fractions so complicated as to be insoluble for all intents and purposes.

Makes one want to ask if fractions are so well suited for astrophysics, why NASA isn’t using them.

I am rather wary by now of any proposition that the ancients were doing their “rocket science” in fractions, particularly unit fractions, but there’s the rub – how many researchers out there insist on claiming the ancients used fractions because of the historical basis for this, but proceed immediately into elaborate theories using types of fractions for which there is no historical support for them using?

I’m sure I must have unintentionally ruffled someone’s feathers when I attempted to turn around a criticism that alternative researchers often face, on the individual who issued it, to say I wonder if we’d be so keen on the idea of the ancients using nothing but fractions if someone took away our pocket calculators, because that’s what most if not all of us do these days is enter these fractions into our pocket calculators where we don’t really get a good idea how hard they may be to work with in real life.

On a bad day, everyone from metrologist John Neal on down insists the ancients did their calculations in fractions but presents their work in decimal. I have truly never seen anyone actually do their projected “ancient fraction calculations” in the actual manner that the vestiges of historical sources attempt to dictate that ancient math was actually done.

Some of my other misgivings about the ancients having used fractional math to render their astronomical calculations include that while we can get around the reflexive objection that “the ancients didn’t know how to do long division” by asserting that they may have used inverted multipliers instead, which is the same thing without division – for example,

360 / 72 = 360 x (1/72) = 5

In many cases, the simple round numbers that people often like to attribute to the ancients aren’t so simple when they become inverted. Instead, they become complex decimal strings more readily expressed in decimal form. The simple divisor 360 inverts to the inverted multiplier 1/360 = 2.777777777 / 10^n, and so forth.

A great many of these are of course numbers that the “conventional wisdom” insists that the ancients weren’t able to manage.

Setting aside fractions for a moment, another pitfall of attributing simplistic math to the ancients is that it refuses to stay simple very long. Even for the amount of academic works that I’ve read that discuss Mayan calendar math, I’m still left with little idea what academia in general thinks was the ancient protocol for dealing with this fact.

Some of these academic works simply exclude calculations that don’t work out perfectly in nice round whole numbers, virtually gutting the broader applicability of these calendar systems, while others proceed as if the Maya simply knowingly rounded off these unruly decimal numbers that often result from the breakdown of accepted calendar numbers and “supernumbers”. It seems as if the question of the Maya would have dealt with numbers like 346.62 days, the Eclipse Year, may remain an open one that still begs for consideration of whether it would have required decimal math to overcome some of these obstacles.

Sure, we’d make that into 346 and 2/3 in two seconds flat – but have you ever heard anyone give the Maya credit for even knowing what a fraction is?

While I don’t have a particular example prepared, this is something that anyone can discover for themselves in a hour simply by picking any accepted Mayan calendar number and attempting to break it down by the obvious subordinate cycles of the moon, or the Orbital and Synodic Cycles of the planets, and observing how simple some of the resulting numbers aren’t.

I continue to have great concerns that what some researchers may be doing is conflating ancient marketplace math – the “common math” – with the mathematics required for many astronomical calculations. There is no doubt that historical examples do point to astronomy having been simplified for the “average person” – our own calendar of 365 days rather than the true ~365.25 day year is itself an example of this – yet even while we make and use such mathematically egregious approximations, our own astronomy requires knowing the difference between a Tropical Month of 27.321582 days and a Sidereal Month of 27.321661554 days!

Importantly, just because most of us including myself still use the 365 day year plus the Leap Year when required, doesn’t mean that none of us can tell the difference between the Tropical and Sidereal month.

In my own work, the working premise is that ancient man had means, motive and opportunity to develop complex math. “He” (or quite possibly she) was able to deal with math to develop calendars in the first place, and was motivated to do so beginning with knowing when winter was approaching and how much goods needed to be stored up to survive until the next harvest season.

As far as opportunity goes, we find examples of what we believe are calendars that are a minimum of three times as old as the oldest pyramids, and even older.

The opportunity part of this equation then seems to be that we had thousands or tens of thousands of years to develop the complex math required to keep complex calendar systems on track without a veritable encyclopedia of necessary corrections even if the scattered remains that survive now are thus far exclusively devoted to math for the “average person”.

More than one published researcher has asserted that developing base-ten mathematics is an eminently logical development for ten-fingered beings such as ourselves. Likewise, it seems a rather logical development if the earliest need for mathematics is to effectively divide a ~365-day year into smaller, more manageable units such as months, since the only real factors of 365 are 73 and (10/ 2) = 5.

Yet 365 is otherwise so indivisible by nice round numbers as to provide great impetus to find constructive ways to deal with remainders even as early as the first discovery that there were 365 days in a year.

I nearly fell out of my chair when I first started working with calendar math as the possible original intended application of the mathematics that Carl Munck was teaching. Again, we have only to try to divide the 365 day year into months just like our very own – of 30 or 31 days – to be confronted by uncanny approximations of numbers I already knew like the back of my own hand.

365 / 30 = 12.16666666; Remen = 1.216733603 ft

365 / 31 = 1.177419355; Munck “Alternate Pi” = 1.177245771; Harris-Stockdale Megalithic Foot (HSMF) = 10 x (sqrt 2) / 12 = 1.178511302 ft (?)

In spite of the immense respect as I have for Peter Harris, that difference of some 12/10000 of a foot between “Alternate Pi” and the HSMF isn’t likely to be recognizable through any field measurements. To me, for all intents and purposes they are virtually the same thing and I have 20 years experience with 1.177245771 and its properties to tell me that it is the superior number for practical purposes.

Once again, I will express my belief that what the ancient Egyptians were working with for metrological units, were probably believed by them to be the oldest heirloom numbers possible. Anything else would be just too much of a coincidence, and on top of the years squandered by Munck and Morton there are the years I squandered myself by looking time and again at the equation 365.0200808 / 300 = 1.216733603 thinking “What an interesting coincidence” and never thinking to ask “What if it isn’t a coincidence?” Been there, done that, OOPS…

Again, some 20 years now I have been working with Munck’s numbers, constantly testing their capability for expressing important data, and their value as a networked system of numbers as a whole. The entire time I’ve felt literally apologetic about the numbers that, per se, end up excluded from the system, constantly asking “Why can’t we have Phi in this system? Why can’t we have sqrt 2, sqrt 3, or sqrt 5 verbatim? Why can’t we use 7 or 11 or 17 or 22, etc etc.”

The past several years more than ever, I’ve seen why, and why we use approximations of things like this where necessary to create a system that is even more versatile than ordinary numbers, yet ultimately predictable enough to be understood. It may not immediately so (few if nay learn a new language overnight), but picking the best parameters immediately helps to take most of the endless guesswork out of things. Although the ancients continue to surprise us, 50-80% of the time my first guess is right because, well what else would it be? If it’s near to Pi, Pi is obviously the best first guess.

If there’s anything David Kenworthy’s work has shown me, it’s that numbers like 7 and 11 are mutually exclusive to the actual Pi ratio – if you use 22/7 as Pi for a moment, you’ve begun creating a system that is going to be hostile to Pi. Pi and Phi are also mutually exclusive, which is something I should have accepted long ago, especially since I seem to have discovered most of our alternatives to Phi myself.

Because Pi is the stuff of circles and Phi is the stuff of pentagons, and because of the prevalence of ancient circular architecture and the general absence of ancient pentagonal architecture, Pi obviously won out over Phi a long time ago.

There’s probably little point in feeling apologetic about this math having rejected prime numbers – it’s probably what anyone who worked with fractions would have done soon enough. Please correct me if I’m wrong, but operating fractions involves reducing numbers to common denominators, and of course prime numbers aren’t reducible because they lack factors by which they can be reduced thus.

The great irony that motivates this post, though, is that for all the talk about how “the ancients didn’t know what decimals are”, is that except for a few instances such as root functions (anyone remember the “root rules”?), the ancients didn’t need to care what decimals were.

If you’re working to ten digits, simply move the decimal points 20 places to the right, and now you don’t need to know what decimals are because now for intents and purposes, you’re working with whole numbers. Move the decimals 20 places back to the left when you’re finished if you prefer – or if you’re the ancient Maya breaking down the calendar number 819 (not 81900!) into 364 x 2.25 (not 364 x 225!) perhaps you needn’t even bother putting the decimals back where you found them.

I’m tempted to throw out decimals myself, although it’s a little bit late now. Imagine how it plays out searching my files for references to various numbers. How much easier life would be and how much faster progress might occur if all I had to search my files for was 1177245771, rather than 0.1177245771 and 1.177245771 and 11.77245771 and 117.7245771 and 1177.245771 and etc.

That’s what the argument about ancient decimals really seems to be – little more than semantics, and usually brandished by those who’ve apparently given little forethought to the matter.

Anyway, if someone is trying to convince me that before a Sumerian can sell a Babylonian a fish, the fish has to be weighted and the weight converted to volume, and the volume has to be converted from cubic Sumerian units to cubic Babylonian units by an elaborate series of maneuvers including a shift from base 10 math to base 77 that takes an hour to work out on papyrus in unit fractions or some such, no, I’m probably not buying, sorry.

If that’s the kind of thing we get for insisting that the ancients only knew how to use simple fractions, I hope readers can see my issue with the premise.

I even have misgivings about whether legal standardization of ancient measures results in more, or less, standardization. Ideally, yes, by Royal decree ancient unit “x” shall be equal to umpteen barleycorns so that all trade in the kingdom shall be fair and honest, but it’s also an incentive and an opportunity for any dishonest persons to tamper with the standards to earn a little extra.

To put it simply, in my opinion, the standards of the marketplace make poor standards indeed for ancient metrological values, and obviously there is little if any need to apply the kind of math it takes to distinguish the Tropical Month from the Sidereal Month for the village fishmonger to peddle a fish or the local baker to follow a pastry recipe. The mathematics of the marketplace or kitchen is simply NOT the mathematics of astronomy, and vice-versa.

This might help explain to readers why I gravitate toward ancient metrological values that find standardization though geometry, numbers that will always be exactly the same according to the equations that generate them, like 54 / 10 Pi = Royal Cubit 1.718873385 ft, or sqrt 15 x (Pi / 10) = Remen 1.216733603 ft – constants we call them, because that’s what they are, constant. Predictable. Fathomable. Recognizable. Reliable – or as the ancients may have thought of them, numbers that are as old as time itself, literally.

Mix them with numbers that they clash with, however, and they may not stay that way long.

Again, I’m really not meaning to knock anyone else’s work here, I am greatly indebted to a staggering number of researchers – probably every one of them that I can name – for insights and inspiration, but I thought that in case some of my logic seems hard to follow, it might help readers for me to gather up some of my thoughts on why I approach numbers the way that that I do, and why I’ve chosen to work with a proposed ancient system of numbers that rejects many numbers that other researchers would see nothing against working with.

I’m not sure how to find it now, but there was a post on the GHMB from forum member “loveritas” that described a similar preference for certain whole numbers that were described by this poster as factors of 360 – numbers by which 360 is evenly divisble. Naturally, 7, 11, 17, 19, 21 etc aren’t on that list – what is on the list, are factors of 360 (and for us many of their simple products and dividends as well — 9 is a factor of 360: 360 / 4 = 9, so 9 x 2 and 9 / 2 and 9 x 3 etc are also valid).

This was a good way for me to see these numbers described, because it highlights the compatibility of these numbers with sexigesimal (base 60) math the way we use it nowadays to measure latitude or longitude or azimuths or or time – 360* (60 x 60) degrees in a circle, 60 minutes per degree, 60 seconds per minute.

I doubt it’s any coincidence that the most powerful combination of numbers I know of is 60 and sqrt 60 – this combination gets sqrt 60 working at data retrieval at least as high as the 33rd power – an unprecedented thirty-three pieces of data for the “price” of two!

How many examples of ancient architecture can we find by the usual suspects where we can’t find sqrt 60 as part of the design? It’s one of those numbers they understandably seemed to want to build into virtually everything, as if they actually wanted and intended for us to be able for us to find it along with other critical data retrieval keys like 1.177245771, 1.622311470, Pi, or (Pi / 3) so that with the very first architecture we attempt analyze we can begin to see what’s going on.

Again, that is what any of us who are looking for anything in the proportions of ancient architecture are working with, is the inherent proposition of data storage and retrieval. The more the better, obviously.

It certainly seems to highlight the sense of heritage that seems to go with the idea of recording astronomy data in architectural proportions generation after generation, generally in particularly durable form, and the idea that this may represent a sort of guarantee that we are inheriting the very same world our ancestors lived in, with the very same cycles and the very same seasons.

I’ve probably strayed far from the point of this by now, the point being that whether or not the ancients knew what decimals were really isn’t the question, not at all.

–Luke Piwalker

Metrological Unit Families

The one-man debate continues whether or not a system of metrological units similar to John Neal’s should be applied to the units I use. Even then, one of the things that continues to make such a system of units questionable besides the inconvenience of eight or more values for each unit, is that even if we attempt to project such a system of units, there are certain values that stand out from the rest as the most useful, values we might then regard as the primary unit values.

As such, some primary units can be reduced to other primary units, and distinct families of units can be tentatively identified. Referring exclusively to primary unit values here,

EGYPTIAN REMEN FAMILY (see also Algernon Berriman Historical Metrology for Remen fraction relationships)
Remen 1.216733703 ft
Greek Foot 1.013944669 ft (1.216733603 x 10/12)
Squared Munck Megalithic Yard (9 / 1.216733603)
Assyrian Cubit 1.622311470 ft (1.216733603 / .75)
Egyptian Foot / Roman Foot 0.973386882 ft (1.216733603 x .80)
Roman Pace 2.433467205 ft = Double Remen = 1.216733603 x 2
“Short” Palestinian Cubit 2.102515665 ft (1.216733603 x (1728/1000)) (rare?)

MEGALITHIC FOOT FAMILY
Megalithic Foot 1.177245771 ft
Indus Foot 1.100874628 ((360^2 / 1000) / 1.177245771)
Pole / Rod 16.51311941 ft (1944 / 1.177245771 / 10)
Alternate Megalithic Foot 1.179778193 ft ((100/72) / 1.177245771)
Megalithic Yard 2.718208958 ft (32 / 1.177245771) (dimeteral unit of circumferential unit of 1.067438159)
Lintel (Circle) Megalithic Yard 2.725105951 ft (1.177245771 x (432 / 100))
Nippur Cubit (speculative) 1.695233910 ft (1.177245771 x (144 / 100)
Nippur Cubit (speculative) 1.698880598 ft (2 / 1.177245771)

EGYPTIAN ROYAL FOOT FAMILY
Egyptian Royal Foot 1.152833216 ft
Hashimi Cubit / Pied du Roi 1.067438159 ft (1.152833216 x (100/108))
Petrie Stonehenge Unit 224.8373808 inches = 18.73644840 ft (20 / 1.067438159)

EGYPTIAN SACRED CUBIT FAMILY
Egyptian Sacred Cubit 2.091411007 ft (1.718873385 x 1.216733603)
Thom Mid Clyth Quantum 7.745966692 ft (2.091411007 x (27/100))

MEGALITHIC YARD FAMILY
Megalithic Yard 2.720174086 ft
Karnak Cubit 1.700109360 ft (2.720174976 / (16/10))
Ubaid Cubit (primary value uncertain) 2.361263000 ft (2.720174976 / (1152/1000))

EGYPTIAN ROYAL CUBIT FAMILY
Royal Cubit 1.718873385 ft (diameteral unit to circumferential unit of 1.000000000 ft x (54/10))

UNIDENTIFIED
“Long” Palestianian Cubit 2.107038476 ft
“Le Serpent Rouge” 1.676727943 ft*
Stonehenge Outer Sarsen Circle Radial/Diameteral Unit (outer sarsen circle radius 51.95151515 ft)
Draconic Megalithic Yard 2.721223218 ft
Incidental Megalithic Yard 2.719256444 ft
Putative Ancient Meter Values (primary ancient meter value unknown)

*”Le Serpent Rouge” 1.676727943 ft – Approximate geometric verification
1 remen x (sqrt 2) = 1 Royal Cubit
1 remen x (sqrt 3) = 1 Palestine Cubit
1 remen x (sqrt 4) = 1 Roman Pace (Double Remen)
1 remen x (sqrt 5) = 1 Megalithic Yard (MY)

1 Palestinian Cubit of 2.107038476 ft x sqrt 2 = 2.979802389 ft
1 Royal Cubit of 1.718873385 ft x sqrt 3 = 2.977176035 ft
((6 Petrie Stonehenge Units of 18.73644840 ft) / 100) x sqrt 7 = 2.974318975 ft
1 Remen of 1.216733603 ft x sqrt 6 = 2.980376480 ft
((10 Pied du Rois of 1.067438159 ft) / 8.0) x sqrt 5 = 2.983580357 ft

5 / 1.676727943 = 2.981998374

*”Le Serpent Rouge” 1.676727943 ft – Exact geometric verification

Simple multiples of 1.676727943 are Radial/Diameteral units to circumferential values of simple multiples of the primary Palestinian Cubit, i.e., diameter 1.676727943 x Pi = 21.07038475 / 4.

Regarding Neal’s unit systems: While variations on unit values can be generated and rationalized geodetically, it should be considered whether the most practical approach is really to generate multiple versions of the same unit for all units, or to simply vary only the value of the mile which is used in their conversion.

Note on “Pi” Numbers and “Square Root” numbers

I’ve written before that all of the numbers I work with belong to one of two groups, “Pi Number” and “Square Root Numbers”. They can be distinguished because “Pi Numbers” should always resolve into valid whole numbers by applying Pi^n (Pi to an unspecified power).

(If they are a Pi Number and they do resolve into a valid whole number or the reciprocal of a valid whole number by about Pi^12 or so, they may be a whole number so large that it appears to be an irrational number, which can be tested by attempting to break them down into smaller numbers. I can only think of once or twice when I’ve encountered valid whole numbers this large).

Their properties are such that

Pi Number x or / Pi Number = Pi Number
Pi Number x or / Square Root Number = Square Root Number
Square Root Number x or / Square Root Number = Pi Number

Of the units discussed here, the Pi Numbers are

Egyptian Royal Cubit 1.718873385 = (54/10) / Pi — 54 is a valid whole number so 54/10 is also. Changing the decimal place does not affect the integrity of whole numbers.

Megalithic Yard 2.720174976 and family — 2.720174976 / (Pi^5) = (1 / 1125) / 10

Draconic Megalithic Yard — because it can be conjugated from 10313.24031 (Pi Number) and (19.46773764^2) (Pi Number), the Draconic Megalithic Yard of 2.721223218 ft is therefore a Pi Number. Knowing this, we apply Pi as a mathematical probe and discover that 2.721223218 x (Pi^3) = 84375 / 100

Stonehenge Outer Sarsen Circle Radial/Diameteral Unit 51.95151515 = (1 / 1875) x Pi^4 x 10^n

The rest of the metrological values appearing here are Square Root Numbers.

–Luke Piwalker