More About Stonehenge, Pt 8

Let me try to retrace my steps to get to what was supposed to have been the point of the previous post in this series, and go back to where I was just before the train seemingly took leave of its track.

I was observing that potential Eclipse Year candidate 346.4769015 / 225 = 1 / 6.49393934, which is 1/8 of Stonehenge’s 51.95151515 ft outer sarcen circle radius.

The next entry in my notes that makes any sense says (1 / (Pi/3)) / Eclipse Year = ~Anomalistic Month; indeed, (1 (Pi^3) / ~(Anomalistic Month^2) = ~5.030183830 / 4 as previously mentioned. I should also add that (1 / (Pi/3) x Anomalistic Month = ~(1.622311470^2), but I don’t really know if we can find numbers that can accomplish all of that.

For what it’s worth, (1.62231147^2) / (1 / (Pi/3)) = 27.56113481 again, so we may be seeing Anomalistic Month values that can function at at least the second power.

It must be pretty handy then that Pi/3 seems to have been provided for us in Stonehenge’s ellipses so we don’t have to just pluck Pi/3 out of the air to do that, and I found a series where (Pi/3) can take over as constant when it hits bottom. These notes elude me at the moment (I have lots of notes…) but 9.315155240 / 2 = 4.657577620 was taken to be the point where the former series probably bottomed out.

Not idea how I got there, but it was as I recall a long enough series, which should narrow down the possibilities, although surprisingly it doesn’t look to have been been Pi, 2 Pi, 1.177245771 or 1.62231147, the sort of things that string together long series. That may leave sqrt 60 as the one we want.

Next it is written that something times the Anomalistic Month = ~13238 / (2^n), I think that’s where the thread got lost.

The very next thing that is written suggests I somehow got a geodetic Remen value that must be close to Jim Alison’s in his paper here

http://home.hiwaay.net/~jalison/blu5.PDF

Next there is a note about a meter-like figure of 3.281541775 and that it associates with a polar circumference figure of ~24860. A Remen-like figure of 1.215385842 follows, with the notation that this figure divided by 2^n equals 18990.40378. “n” must be unusually high here, but 18990.40381 is the figure suspected of being an additional version of the Long Count. n=6, as a matter of fact. 18990.40381 x 64 = 1.215385842

Okay, since that was mentioned in the context of polar circumference, we’ll estimate with 24860 x 5280 = 131260800.0, and 131260800.0 / 1.215385842 = 107999283.4, near to the ideal of 108000000.0.

Okay, so now I know where I ended up, although I still have no idea how I got there, especially from ~13238 / (2^n).

So this is what I was talking about: 18990.40381 x 64 = 1.215385842 – a suspected variation on the half Venus cycle that might have unsuspected geodetic value.

At least now the idea can be investigated with the basic premise having been retraced.

There is also a note that some figure or other ~ 1.215385842 “works with a 5280.3 ft mile”. We may have to see about that soon, I’m still not sure 5280-anything works with this system of numbers.

Okay, while I try to let my brain cool down, here’s another factoid to nibble on

Did you know that the base perimeter for the Great Pyramid in my unpaved model in inches = (1 / Pi) x (1.067438159^2)? Neither did I, but sure enough, that comes to 3022.416624.

I’ve probably made something of a case by now of how important 1.067438159 is although I hardly suspected it the fateful day I tried to make the outer sarcen circle into 120 Meg Yards of 1.067438159, which is how it was finally captured. I had no idea Stonehenge might be veritably swimming in it, or just how many ways it might be written at Giza.

THE MEAN OF THE TRILITHON ELLIPSE, Take One

How about another look at the Trilithon ellipses next?

We have the stones of the Trilithon ellipse (“horseshoe”) framed by two ellipses, ovals that measure according to Prof. Thom 30 x 20 Megalithic Yards and 27 x 17 Megalithic Yards.

The mean values would be about (30 + 27) / 2 = 28.5 MY as the mean of the major diameters, and (20 + 17) / 2 = 18.5 MY as the mean of the minor diameters.

28.5 MY x 2.72 = ~77.52 ft, and 18.5 MY x 2.72 = ~50.32 ft

Remind anyone of anything? 50.32 reminds me of 50.30183830, which is already established as my suggestion for the mean radius of the sarcen circle.

Also, remember that Carl Munck declared that sqrt 15 and sqrt 60 were important to the mathematics of Stonehenge and were specifically indicated by the unusual displays made of 15 and 60 stones (the “horseshoe” and sarcen circle respectively).

28.5 MY x 2.72 = ~77.52; (sqrt 60) x 10 = 77.45966692

If we avail ourselves of an ellipse calculator such as

https://keisan.casio.com/exec/system/1223289167

and input major radius (28.5 / 2 = 14.25) (Meg Yards) and minor radius (18.5 / 2 = 9.25) (Meg Yards), we get circumference = 74.66534855219 MY; 74.66534855219 x 2.72 = ~203.0897481 ft.

That’s a bit on the mysterious side, but there is a figure 203.0782797 that we can construct thus: (1 / 3018.110298) / (1.177245771^3) = 203.0782797 / 10^n.

On the other hand, since 360 / 203.0782797 = 354.5430861 / 2, we might be forced to try to decide if 354.5430861 was approved by the ancients as a legitimate way of expressing the Lunar Year of ~354 days, or whether we are falling slightly afoul of a more conventional expression for the Lunar Year here.

If I go back to the ellipse calculator and input (77.45966692 / 2 = 38.72983346 (sqrt 15) and (50.30183830 / 2) = 25.15091915 as the major and minor radii respectively, we get 202.96077061132, and are even farther from familiar Lunar Year values, although Stonehenge may well have additional Lunar Year figures to suggest to us.

I’m a little worried about that, since 25.15091915 inches would be a form of the Palestinian or sacred cubit made from multiplying the standard Remen value of 1.216733603 by the likely true value for the Royal Cubit mentioned by Lepsius and Stecchini, as opposed to Morton’s Royal Cubit, which is more or less the consensus cubit of ~20.62 inches.

That would certainly give an interesting geodetic display to reconsider the equation using Morton’s cubit, but it might run afoul of reasonable accuracy to impose this value on the sarcen circle mean. That’s somewhat puzzling, since sqrt 60 may give the better display with such a sacred cubit than with 50.3018383, but we already know there turned out to be much merit in a value of 50.3018383 ft as the sarcen circle’s mean radius,

Sometimes it can take awhile to see the wisdom of what the ancients were up to, and we shouldn’t expect it to be perfectly clear on the first try.

For what it’s worth, I don’t know if I have ever seen anyone try to break these parts of Stonehenge down into a scheme that defines their logic. Like the Great Pyramid, Stonehenge may be an ancient monument where certain aspects try to command almost too much focus. Much has been said about the Great Pyramid’s perimeter and height; much less has been said about its apothems or edge lengths.

For what it’s worth, 203.0782797 is a reciprocal (1 / (486 / (Pi^2))) and might be best suited for use with reciprocal metrological units, or inversion prior to applying analysis – or something more obvious might be staring me right in the face and I’m not seeing it yet.

Maybe it will leap out at us soon. Meanwhile there is still much more to consider. Stonehenge certainly isn’t done talking to us yet.

— Luke Piwalker

More About Stonehenge, Pt 7

Hmmm… Perhaps there’s only enough to say here that it could have been a postscript at the end of the last post, but enough was probably said there, so this could end up a short post.

Several things I could swear I started catching glimpses of

1. Stonehenge beginning to make statements about alternative sets of calendar numbers without prompting, and

2. One or more largely unexplored alternate calendar number sets may have unsuspected potential to offer new insights into ancient geodesy

Could Stonehenge succeed where even the Great Pyramid has thus far failed to help better sort out the mess that ancient geodesy can become on account of the exclusion of the literal mile from this mathematics (even while this math utilizes the literal value for the mile for geodetic modelling by means of otherwise nonsensical ratios such as 1 foot: 100 Miles to bridge the perpetual gap between these disparate units)?

Since the Big Idea is now on table with that, why don’t we back up a little bit here?

First, in terms of looking for the Anomalistic Month at Stonehenge, I perhaps could have appendaged the last post with the observation that (1 / (1.451319051)) x 4 = 27.56113480 ((1 / 1125) x (Pi^3), which is significant because giving a max perimeter of (106.7438159 / 2) Megalithic Yards to the inner bluestone circle, we get in feet (106.7438159 / 2) x 2.720174976 = 145.1809284 ft and (106.7438159 / 2) x 2.719256444 = 145.1319046 – thus it’s already been found at Stonehenge, and if 27.56113480 is a valid figure for representing the 27.55454988 day Anomalistic Month, good old Stonehenge is already trying to give it away to us.

In this case, as with the outer sarcen circle, I would have to say that the “AE” Meg Yard 2.720174976 is quite likely the primary intended unit for interpretation, as the “Incidental” Meg Yard associated figure 145.1809284 may be virtuous, but it might also turn out to be a bit strange if we’re not lucky.

Honestly, the subject of four possible different sets of calendar numbers may be confusing enough without starting closer to the beginning. Thus far we have the “A” set of numbers, wherein the half Venus Cycle of “18980” is represented as 18983.99126, and the “B” set of numbers, wherein the half Venus Cycle is represented as 18997.72194, and the ratio between the two sets

18997.72194 / 18983.99126 = 1.000723277

We should be able to multiply a finished “A” set by 1.00073277 to get a finished “B” set, but there are some calendar numbers on the loose which are not separated by that ratio, and some counterparts separated by that ratio that seem to remain obscure.

Stonehenge’s 51.95151515 outer sarcen circle radius seems to belong to the “B” set as its take on the important calendar number 52 (as in 52 weeks in a year, etc), but the number that keeps trying to assert itself as second most important is 52.04568991, rather than the expected 51.95151515 x 1.000723277 = 5.198909049, which seems quite shy about making appearances somehow.

When we try to establish the formulas that connect a calendar set internally, we sometimes get odd results that exceed the limits of the expected and seem to point at the existence of other sets, and it’s hard to ignore these hints forever no matter how inconvenient or puzzling the matter may be.

The two other sets currently suspected of existing, hypothetical “C” and “D” sets obviously, 18976.67846 and 18990.40381, with some faint hints of a possible fifth.

I came up with something while exploring Rio Bec mathematically called “Rio Bec Equation #3” – it’s a formula that goes something like this

Earth Year / 1.62231147 = Venus Orbital Period / 1.62231147 = significant figure X; X / 1.62231147 = significant figure Y = 1 / Long Count (half Venus Cycle)

It’s a lovely equation and one inspired by working in situ, but it doesn’t seem to link the figures together according to their respective groups, but instead makes references that span the different groups.

One of the most interesting and coherent inputs is the (3600 / Pi^2) version of the Earth (calendar) Year, the output parameters are

364.7562611, 224.8373804, 138.5907605, 85.42796071, 52.65817464, 18990.40381

How many of those numbers have we already seen at Stonehenge? 85.42796071 x 2, by the way, is 170.8559214 from the last post concerning Stonehenge and Teti’s pyramids.

All of that goodness, and yet the Long Count/half Venus Cycle that comes out at the end is NOT of the “A” or “B” set, yet given the series it belongs to and what it’s made of, one is really tempted to take it seriously – and that may be what should happen, and this may be a strength of this system of numbers rather than a shortcoming to operate that way.

By the way, here’s a trick or two you can spot if you’re willing to reach in there for them, that reveal more of the inter-relatedness and inter-connectedness of certain numbers

2 / 170.8559214 = 11.70576931 = 15170.67702 / (360^2), with 15170.67702 inches the possible N-S distance between the summits of Chephren’s and Mycerinus’ pyramids = 600 Palestine Cubits of 2.107038476 ft.

5 / 170.8559214 = 29.26442326 (aka “The Real Mayan Annoyance”)

170.8559214 / 9 = 18983.99126, Long Count “A” is in there after all.

Alright, how about letting that stand for now as trying to bring the reader more up to full speed with this and leave it for another post to try to retrace my steps and be more sure of what I think comes next here?

For now, since this is a post where I talk about 145.1809284 and 18990.40381 — at least this part of my notes is relatively comprehensible, and they say

18990.40381 / 2^n = 6.89028370^2, and 6.89028370 = 1 / 145.1809284.

Also, in case I haven’t shared these yet either, (2160 / Pi) = 687.5493542 is still trying to be a representative of the Mars Orbital Period 686.971 d and 779.2727283 (already thought to have been found at Stonehenge), is still trying to be a representative of Mars Synodic Period 779.96 days

687.5493542 / 305.7985078 inner sarcen circle circumference, ft = 224.8373808

779.2727283 / 224.8373808 = 346.5939362, the reference source figure for the Eclipse Year again being 346.62 days

It was also observed that the ratio 346.62 / 27.55545 (Eclipse Year / Anomalistic Month) is ~12.57941523 = 50.3176609 / 4, which looks a lot like 50.30183830, or exactly 1/2 of the proposed 100.6036766 ft mean diameter for the sarcen circle, which would make ~12.57941523 to be exactly 100.6036766 / 80, and it was observed that “Eclipse Year / Venus Orbital Period” may also be as significant ratio at Stonehenge, although this item is currently less certain.

I will try to get to the point momentarily that this post was going to try to make until pre-empted my more data from a rapidly expanding model of Stonehenge at long last. It’s a wonder to behold and may even carry a risk of making ancient Britain seem even more interesting that ancient Egypt, where I am not sure I have ever seen the like in spite of some strong parallels.

–Luke Piwalker

Scrutinizing Stonehenge

This may be hard to explain, so please bear with me here. Having Stonehenge seemingly volunteer data on the Eclipse Year, I’m wanting to take stock of just how much planetary data we have already seen it expressing, and I’d like to see how it might express whatever is missing for that list.

One problem with this is that much of this planetary data for the visible planets is still under consideration as to which candidates might represent the “A” and “B” values that belong to the respective main calendar sets. Having to inspect Stonehenge looking for values that haven’t been clearly defined yet makes an already challenging project that much more challenging, even if in the final accounting we can hope that Stonehenge has enough to say about planetary cycles to assist us in being certain of any particular choices from the nominations.

I wouldn’t put it past the ancients to have data to offer on planets they were weren’t supposed to know about, but I don’t have any claims to make here. Rounding up any evidence related to their possibly being aware of Neptune or etc would be a massive project in itself of combing through history, and I’d rather work with the numbers, especially if I’m going to have any monopolies by default on certain perspectives relating to the numbers.

One nominee for representation of the 378.09 day Saturn Synodic Period is the 377.8020786 ft side length (unpaved) in my model of the Great Pyramid, which might then require from Stonehenge a demonstrable affinity for the “unpaved” figures for the Great Pyramid – which it may have, but it might take more than that to be more conclusive. It is an intriguing prospect, though, and a noteworthy one.

I’m also looking more at possibilities of representing multiple Lunar constants by their ratios, although it still looks pretty tangled to me. What’s interesting to me here are some recurrences in a particular range, which ideally might provide some small semblance of a frame of reference

Lunar Month ~29.53 d / Draconic Month ~27.2122 d = 1.085175032

Jupiter Synodic Period 398.88 d x 2.72 = 1.0849536

In a similar range are expressions like ~(360 / Lunar Year) x 1.067438159 or (2 / ~Phi) / 1.067438159^2. Also occurring in this range are figures such as 360 / (1 / (Perimeter Great Pyramid, paved) = 1.086519707 and, perhaps particularly interesting since I’ve given 892.9807632 as a possible max diameter on the Aubrey Circles, 892.9807632 x Remen in feet = 1.086519701.

Sadly, this doesn’t quite match some other interesting candidates such as height Great Pyramid (after pavement) 480.3471728 x (1.177245771^5) = 1.086152820 but a silver lining here may be that 480.3471728 / (1.177245771^1) = 1.5 x 2.720174976 and 480.3471728 / (1.177245771^2) = 346.59399367, making this likely Eclipse Year figure quite easily found in the Great Pyramid.

There may be some tough decisions ahead making sense of all that unless the ancients found some ingenious and inclusive strategy so that nothing important gets left out there. We do already seem to see a lot of that kind of thing where we begin to interchange the “AEMY” and “IMY” Megalithic Yards in the equations, and it may be safe to say we have already seen Stonehenge being very inclusive when it comes to relationships of the Remen and 1.067438159

Another intriguing one may be Eclipse Year 346.62 / Lunar (Synodic) Month 29.53 = 1.173789367. That might be the often mentioned figure “Alternate Pi” (Munck) 1.177245771 by some reckoning, but in case that’s too much of a stretch, I’ve been having a new look at an old number — 1.174718783.

1.174718783 seems to be a number I announced first to the world, in early October of 2004, with nothing being posted after that, to the best of my ability to determine. I supposed I’d have to guess it quickly fell through the cracks for being uncommunicative, in spite of a robust affinity for the Radian.

Normally if something talks to the Radian, we’d expect it will be happy to talk to Pi, 2 Pi, or 360, but occasionally we get exceptions, and this wouldn’t be the first time that a number has been found that may be one. It almost seems typical for such numbers that will only talk to the Radian primarily, that they tend to do so at unusually high powers of the Radian. They can be good at ignoring the standard power probes like 1.177245771 and 1.62231147, which is another thing that can make a large contribution toward a number’s obscurity.

I also find it interesting that the ratio between Eclipse Year 346.62 and Draconic Month ~2.72122 is about 4 / Pi. Not only is that a possible calibration point for some of this, but it tries to give me a mental picture of someone squaring the circle over a side view of the Cheops pyramid and then placing the moon in an “orbit” above it (I’m not sure having seen diagrams like “thousands” of times is an exaggeration), and I suppose it might only make things worse that the 120 Meg Yard outer perimeter of the sarcen circle expressed in feet / the “unpaved” perimeter of the Great Pyramid = 1080 / 10^n since many such diagrams or their accompanying text makes 1080 miles to be the moon’s radius.

Examining the relationship Venus Synodic Period ~584 d / Eclipse Year 346.62 = 1.68484219 may raise some interesting possibilities. This figure is quite close to 8 Palestine Cubits of some kind, this particular equation pointing to one at about 2.1060 rather than necessarily at my favorite of about 2.1070.

A Palestine Cubit of this sort can be obtained from the equation Radian / Megalithic Yard = 57.29577951 / 2.720174976 = 2.106326983 although its constituency according to “Remen x Royal Cubit = Palestine Cubit” may be uncertain. It may be more the stuff of “Remen x sqrt 3” instead?

Perhaps I shouldn’t go off on such a tangent just now, but more is probably due the reader regarding the sorts of “sqrt 2, 3, 5” we might see with this style of math.

Note that if we retrofit that equation with the Incidental Meg Yard of 2.719256444, we get

57.29577951 / 2.719256444 = 2.107038475, my favorite Palestine Cubit.

In spite of its incidental origin (in modern times at least), the “Incidental Megalithic Yard” is not necessarily quite so incidental mathematically.

So I’m seeing more than several equations involving one of Stonehenge’s apparent favorite subjects, 1.067438159, since seeking the ratio between the Venus Synodic Period and the Eclipse Year, but I should note that the same may also be true for another number, which is one I’d like to emphasize again – 1323.891320.

I could start out reviewing again how this number came to light. It does have priors all the way back to about August 2004, when I decided it was the perimeter of a Thom Type A flattened ring at Callanish (and a month later it came up again trying to redeem an imposter version of the Square Root of the Volume of a Sphere) but it managed to be forgotten in all the turmoil over cartography.

More recently, it secured attention when I went to experiment with circumscription and inscription of the Great Pyramid and other major Giza pyramids in vertical planes through the center of opposing sides and through their diagonal length, experiments which gave some uncanny results including showing 1.067438159 and 1323.891320.

In August 2004, I pointed out that 360 / 132.3891320 = Incidental Megalithic Yard 2.719256444, so at Callanish we seemed to have already been looking at evidence that the application of “AEMY” vs “IMY” might be selective, if it’s even that limited at other sites since it apparently isn’t at Stonehenge.

It also came to light in the here and now that 5 / 132.3891320 = a number I still hesitate to talk about because even though it seems to be important, it also seems rather strange and perhaps too easily confused with other things that perhaps it would be better to have a better grasp of before tossing the unnamed number into the mix.

Interestingly, for the second time a hint that the design of Stonehenge managed to embrace functionally one of the odd variations on the Remen generated by Stonehenge’s vast ambition to talk about the Remen and 1.067348159 in great detail. I’ve yet to see what looks like a functional role for the (1.067438159)^3 = 1.216264895, but this may be twice that there are hints of one for (1.177245771 /1.067438159)^2 = 1.216322751.

More interestingly, a first casual search for more planetary data at Stonehenge has somehow also shed more light on the candidate Draconic Month figure of 2.721223218.

In addition to the classic formula 10313.24031 / (19.46773764^2) = 2.721223218, I’m starting to see things like 132.3891320 / (1.067438159^3) = 2.721223218 / 4 (or (132.3891320 / 4) / (1.067438159^3) = 2.721223218) and (Pi^3) / (1.067438159^2) = 2.721223218.

Depending on finer details, the conjunction of Venus Synodic Period and Lunar Ratio may also establish a rare opportunity to ply 1.067438159 at unusually high powers perhaps as high as 1.607438159^6, whereas even for how much I’ve worked with 1.607438159, the previous record may be 1.067438159^3. I’m sure Stonehenge would love to do that if it can find a way somehow.

One of the troubling things in all this has been how often these ~1.0849-1.086 numbers show likely relationship to the Indus Foot without being willing to show what looks like any proper one that I’m able to recognize so far. That’s still a mystery, and makes it less certain how to proceed with evaluating what’s in this range.

I did also run into several possible nominations during the latest inquiries for the Anomalistic Month of “27.55454988 days” (Wikipedia), which turn out to be AEMY / (Pi^2) — 2.720174976 / (Pi^2) = 27.56113483 / 10^n and IMY / (Pi^2) — 2.719256444 / (Pi^2) = 27.55182815. 

I don’t know if they actually work in practice, but they have nice pedigree, perhaps particularly at Stonehenge where there’s focus on the Megalithic Yard, and with such pedigree may indeed work well there, but I haven’t actually tried them out yet and there’s still a lot to consider.

Regarding the still somewhat surprising nomination of 138.6375748 as the perimeter of the bluestone “oval with corners” (Thom, perimeter 51.06 MY x 2.72 = 138.8832000 ft, one possible vote of confidence for it may be that compared to the 120 MY = 326.4209971 ft outer sarcen circle perimeter,

326.4209971 / 138.6375748 = 2.354491541 = 1.177245771 x 2 — there’s Stonehenge still upholding a great dedication to the expression of 1.177245771.

One of the other things it could have done to make such a gesture is for the perimeter to be 138.5907605 ft rather than 138.6375748 because 1.177245771^2 = 138.5907605 / 100, which the perimeter actually is when reconsidered after swapping in IMY for AEMY.

Thanks to this versatility of the Megalithic Yard, the designers of Stonehenge seem to have very deftly managed to write 1.177245771 x 2 and 1.177245771^2 with a single brushstroke!

Is that not absolutely remarkable?

Also surprisingly, 138.6375748 responds to the square root of a false Remen constructed as (1.177245771 / 1.067438159)

138.6375748 x (1.177245771 / 1.067438159) = 152.8992544, 1/2 of the inner circumference of the sarcen circle in feet.

Probably the most dramatic support for the proposed 138.6375748 value is its interaction with (10.67438159 / 2) = 5.337190795, 53.3719095 MY also being the apparent circumference of the inner bluestone circle once its “17 MY” max diameter is recognized as 2 / 1.177245771 = 1.698880598 MY.

138.6375748 / ( 5.337190795^1) = 25.97575757

138.6375748 / ( 5.337190795^2) = 48.6634438 / 10

138.6375748 / ( 5.337190795^3) = 9.118906603

138.6375748 / ( 5.337190795^4) = 1.708559231

138.6375748 therefore belongs to a series already running through Stonehenge that includes its 25.97575757 x 2 = 51.95151515 ft outer sarcen circle radius and its 48.6634438 ft inner sarcen circle radius, making it a possible next logical step in the series or progression.

At the lower end of the series, 9.118906603 = 364.75626211 / 4, with 364.75626211 (the “B” version of the Earth year) standing in for 365 days, and 1.708559231 is something I probably shouldn’t get into, other than that historically it’s one of two candidates for the theoretical height of a certain Egyptian pyramid without its pyramidion (Teti’s), a matter I have yet to resolve because both candidates seemed to show considerable merit and pedigree.

I also hate to bring it up for fear someone may mistake it for a cubit of some kind, when we already have would-be cubits coming out our ears. I’ve yet to encounter any appreciable incentive for it to be some kind of cubit, even if it might be near to some bona fide example of a short one, such as the cubit at Karnak mentioned by Berriman.

I would have to go over my notes carefully to say much more, but I think the essence of the debate must have been something to the effect of

1.5 x 1.067438159^2 = 1.601157232 x 1.067438159 = 1.709136340

VS

(57.29577951 / 2) / 1.067438159 = 19.46773764 / (1.067438159^2) = (103.903030 x 2) / (1.067438159^3) = 1.708559217, our “mysterious” number in the Stonehenge series just shown.

In spite of such apparent successes as with that (1.067438159 / 2) series, Stonehenge still holds many mysteries. Consider the “14 Meg Yard” minor diameter of the bluestone ellipse. 14 x 2.72 = ~38.03 ft. It could easily pass for the Venus Cycle (canonically, 37.960 = (18980 x 2) / 1000), but will that work in harmony with its mathematical environment?

Hopefully we will continue to learn more about the amazing Stonehenge.

— Luke Piwalker

Some Random Weirdness

I often like to let the numbers steer a conversation although that may make it more difficult for some to follow the logic, but the logic of it is exploring relationships between numbers and particularly those which allow certain numbers to be grouped in informative ways – in ways that store the most data within the combinations, or sometimes in ways that permit the expression of limited but specific and important data.

Let’s start in the vicinity of “Not A Remen”, my working name for 1.214121857 x 10^n. I’d rather not go there, I tried to avoid it when it came up in ancient Mesoamerican matters, but it seems to want to follow me lately, along with a sidekick of 1.213- something that may be 1.214121857 / 1.000723277 = 1.213244345.

How did these become an issue lately?

Partly because of Newton. Several sources inform us that Newton determined the “profane” (Royal) cubit at 20.6284 inches and the “sacred” cubit at 25.0265 inches. They are related by 25.0265 / 20.6284 = 1.213206065.

Of course that looks like 1.216733603 and I’ve tried correcting it to that with some pleasing results, but I think “Not A Remen” is probably going to prove to be data retrievable from Stonehenge, and 1.213-1.214 numbers have also been turning up in Egypt at Giza.

Additional, Ernest Moyer made a remark that’s enough to send me racing back to the ~1.7 ft Nippur Cubit to double check a few things

https://wordpress.com/block-editor/post/pijedi.home.blog/196

Moyer: Take the defined radial distance and divide by the theoretical relationship between the Egyptian system and the Greek system, 20.626/1.7, and we get 12.133, the Greek foot.

http://www.egyptorigins.org/happinessinterrupted.htm

If we take Maragioglio and Rinaldi’s data for the distance from the base of Chephren’s pyramid to the base of its subsidary pyramid G2a (14.80 m) and to the inside of G2a’s enclosure (12.2 m), we get 14.80 / 12.2 = 1.213114754

I really haven’t much dossier on “Not A Remen” 1.214121857 / 1.000723277 = 1.213244345. Back in the day we must have missed thinking it was cool, perhaps for being a bit mystified, that the coordinate 4523.893421 Munck plastered on the “Face on Mars” / 3.728757071 a coordinate he placed on the similar “Monument to Humanity” at Marcahuasi, Peru gives the ratio 4523.893421 / 3.728757071 = 1.213244345.

To make things even more weird, Moyer’s page features geodetic data at the bottom of the page from the WGS84 global mappng datum that includes the figure 1213.3205 in the context of the polar radius. Did Berriman and I both miss the boat on something here?

Thankfully that gave me the chance to bring up 3.728757071, which has become one his more obscure contributions. Sometimes I forget why Munck picked that for anything, which is right about the time I get reminded that even if it’s fallen far out of favor, it’s hard to really get rid of and tends to pop up again if we haven’t seen it for too long. It is, get this .1234567901 x (Pi^7) / 10^n.

A bit strange but these days, the “Thoth Remen” and the side length of the Great Pyramid without pavement are part of that series.

I’ve already written a bit about “Not A Remen” although I haven’t yet compiled material into a single dossier or post, I think I’m hoping to learn more about it before I get to that. I get the feeling that there’s more that we’re invited to know about it, but I don’t know what that is.

Every now and then it’s good to not only follow the numbers, but to follow the odd clue. Quite recently I was pointing out how we can find values for both Lunar and Solar Years in the Chephren Pyramid (1 / (ht in Royal Cubits) = Solar Year b) and 1 / Perimeter = Lunar Year b.

With Chephren, we also have Side / 2 = Lunar Year x, by the way.

Chephren’s pyramid is also on about those most famous right triangles, the Pythagorean kind, according to some consensus.

What if we made a right triangle with sides of Lunar Year, Solar Year, and x?

Depending on which of these two years we make the hypotenuse, x = ~360 / Lunar Year or ~2 / Venus Orbital Period. That might have seemed kind of cool to people who were this into astronomy to have a triangle of “Sun, Moon, and Venus”.

Maybe something, maybe nothing. Just following a clue…

I’ve previously written about how the data in Miroslav Verner’s The Pyramids for some of the Giza subsidiary pyramids gives what looks like different values for each side, which I’m quite tempted to take seriously because of the display of fine ratios in question.

What shall we use for the base perimeters, however? The total of the four sides, or four times the average of the four?

I went back to look at that question again, and for G1a where a height figure is given, noticed a) that if we use each side as if it belonged to its own equal-sided pyramid, as many as four different significant perimeter/height ratios might be obtained, and b) if I take the two means of the most similar sides of G1a according to Verner’s data as describing two separate equal-sided pyramids of same height, I get ratios looking like 2 Pi and 1 / (sqrt 240), a potent combination that would be an admirable thing to see combined in a single pyramid that way.

I’m still not entirely sure the best way to interpret this data from Verner, but I seem to still be finding more reason to take it seriously.

It’s a shame I didn’t start this post soon enough to capture some of the random weirdness that inspired it, that I couldn’t write down on paper fast enough. Apparently there are new revelations to be discovered about the Great Pyramid’s apothem without pyramidion, and about a very similar number that surfaced in the past year. Suffice it to say that both numbers and their real place in things are quite likely still poorly understood.

That apothem without pyramidion value is a strange one, it’s sometime quite the silent type, but among other things it’s proven to be responsive to the Squared Munck Megalthic Yard.

That’s probably enough meandering for now. For anyone who made it this far, a bonus – I can at least name one “can opener” that works on a “can” of 3.728757071, it’s 1.676727943, which should perhaps be a standard part of anyone’s toolkit for this number system for sometimes being able to resolve a tough case just such as this one.

1.676727943 works to at least the 3rd power in this particular application.

One doesn’t have to throw much Pi at 1.676727943 to see why ancient Maya mathematicians with skills unrecorded by history would have been rather fond of it, but 1.676727943 pulls 5 / Venus Orbital Period out of 3.728757071, so 3.728757071 could indeed have been of interest to the Maya, which might be why I seemed to find it the Venus Platform at Chichen Itza, if it wasn’t an artifact of consolidation.

I certainly hope to get back to these subjects after I’ve done more following where some new numbers lead.

–Luke Piwalker

Some Half-Baked Stonehengery

Just in case there’s anyone following this blog but not my postings to the GHMB, this is something I wrote there today. Some of it is part of my posting here on Asian pyramids, but the questions continue to try to embrace new concerns, which most recently include new work on Stonehenge and the question of how my “school” of math might represent an Eclipse Year of ~346.62 days.

http://grahamhancock.com/phorum/read.php?1,1131508,1217014#msg-1217014

“> Real Pi 3.141592654 / 0.816 = 3.849990997 rounding 
> to 3.85 
> 
> 3.85 x 0.816 = 3.1416. 
> 
> This unit is hidden on page 47 of Thom’s book 
> ‘Megalithic Sites in Britain’ 

You’ve probably figured out by now that offhand 3.849990997 is possibly an alternate take on that dreaded metrological mystery known in my work as “Dos Ellifinos” (my figure is 3.842777382 ft), and that of course the “Ellifino” is trying valiantly to be identified as a bona fide ancient metrological unit because it relationships to known ones very strongly imply that it would have been accepted as one, independent of any other evidence or lack thereof. 

The standard “Ellifino” I work with is 19.21388691 and the double “Ellifino” is of course 19.21388691 x 2 = 384.2777382. 19.21388691 registers on my radar as (16 Remens x (Pi^2)) / 10^n. 

Numbers in the range of the double “Ellifino” tend to resemble the reciprocal of the very important calendar number (this is not just the Maya here) 260, and at least some of them should probably be treated as if that’s what they are. 

There seems to be something very similar to the “Ellifino” that may be some kind of long form of it that has additional calendrical function, specifically (half Venus Cycle) / (Pi^2) / 10^n. To use the most obvious form of that in my work, that’s 

18983.99126 / (Pi^2) = 1.923480465 x 100. 

The other main form of the half Venus Cycle that I use gives the reciprocal the outer sarcen circle radius: 

18997.72194 / (Pi^2) = 1.924871674 x 100 = (1 / 51.95151515) x 1000 = (1 / ((120 x 2.720174976) / Pi)) x 10^n 

Now there are still a lot of problems here, and one may be that if we choose to grapple with things at such a scope, we may not only need to sort out numbers into whose “school” they belong to, but what subdivisions of a particular “school” they belong to. I’m already working with two sets of calendar numbers here but there are constantly signs of several more that I still haven’t mastered, and every formula I try to come up with tries to segregate these numbers in different ways rather than in harmonious ways. 

Also, I can use the main value I use for the “Ellifino” as case in point as to some of what we’re in for using the classic Vesica Piscis roots to define relationships between metrological units. 

19.21388691 x sqrt 2 = 2.717253945 as roughly the Megalithic Yard, while 19.21388691 / sqrt 3 = 1.109314278, roughly the Indus Foot, but 19.21388691 / sqrt 5 = 1.718642290 / 2, almost exactly the consensus Royal Cubit of ~1.718873385. 

You see why I think they kicked sqrt 2, 3, and 5 in a literal sense right straight out of the math I use? 

What I think is interesting is that, for example, if for a minute we humor Thom with his projected division of the bluestone “oval with corners” into 26 parts, and furthermore take him very literally that he has provided us with 260 / 10 thereby, and retrofit (260 x 2) / 10 into the equation for “large Ellifinos” 

(1 / (260 x 2)) x (Pi^2) = 18980.00846, which I’m certain everyone must recognize as the canonical half Venus Cycle to remarkable accuracy. 

Almost makes me want to beat someone gently about the head with a cubit rod and ask if they can do that with 22/7, but apparently that someone no longer wishes to be named? 

Numerous people’s work may not be mutually exclusive when it comes to things like that, but again it may be a matter of successfully sorting things out into working systems. IMHO, ANY working system is nothing to take lightly because of the inherent difficulties, not to mention that most of us are making what must be some fairly insane demands on numbers in the first place in trying to create such all-encompassing grand schemes as we do – but anyone with a working system must have proven that such things are possible, the question now perhaps being just how many working systems are there? 

Creative ambiguity, another mathematical art perfected by the ancients? 

For my usual purposes, no doubt I’ll generally want to take Thom’s 26 divisions as meaning “25.7575758” (51.95151515 / 2) divisions, but see what can happen when I don’t. 

Numerous people’s work may not be mutually exclusive when it comes to things like that, but again it may be a matter of successfully sorting things out into working systems. IMHO, ANY working system is nothing to take lightly because of the inherent difficulties, not to mention that most of us are making what must be some fairly insane demands on numbers in the first place in trying to create such all-encompassing grand schemes as we do – but anyone with a working system must have proven that such things are possible, the question now perhaps being just how many working systems are there? 

I’ve gone back to the unsolved question of what an Eclipse Year is supposed to look like to someone who used the same math I do, I have two main candidates thus far but since they aren’t separated by the 1.000723277 as my two primary sets of calendar numbers are, then I’m apparently still missing something. What’s interesting, though, is that I think Stonehenge may be able to readily provide me with both figures were I not already working on it. 

At this point I’m still completely unsure what to make of the following – when I had a go at the specks of data I have on Asian Pyramids, the numbers I get looked steeped in Megalithic Yards and/or Lunar calendar values, and something came up that has not only puzzled me ever since, but has started recurring in my experiments, for no good reason that I know of. 

I have a side length value in my data for the “Zangkunchong” pyramid of “29.34 m” (I’m still trying to figure out if it may be a misplaced diagram of the “Tomb of the General”). 

https://www.crystalinks.com/pyramidchina.html

That’s 96.2598430 ft. It’s half of an “Ellifino”-like number, 96.2598430 x 2 = 192.5196860 but is one of those numbers which goes “over the top” and exceeds 19000 when we try to multiply it by Pi^2 to make it into the half Venus Cycle/Mayan Long Count: 192.5196860 x Pi^2 = 19000.9314 / 10^n. Does anyone want to declare that the half Venus Cycle went that high? I already feel terrible that even a ratio as fine as 1.000723277 tries to add 17+ disposable days onto the canonical 18980 value. 

Let’s let the Cubiteers of Petrie’s legacy have a minute to catch up and make 96.2598430 ft into 56 Royal Cubits of 1.718925760 ft each, bring on the controversy… 

Now let’s look at that diagonal – assuming symmetry, 96.2598430 x sqrt 2 = 136.1319755 ft = 272.2639510 / 2, which is why I’m also talking about Megalithic Yards in the context of Asian pyramids, ok? 

What most interests me here may be that 1 / 96.2598430 = 1.038854801, and this number from raw data is suspiciously close to not only the 103.9030303 ft outer sarcen circle diameter I use for Stonehenge (51.95151515 x 2), but it’s also suspiciously close to at least one more additional number, 1.038679451, which is the one that’s started recurring. 

It makes 1.038679451 look like it belongs in there somewhere that 360 / 1.038679451 = 346.5939363, which is one of the two current contenders among the numbers I use for the honor of representing a “346.620076 day” Eclipse Year. If we use 1.039030303 instead, it finds the other current nomination, 360 / 1.039030303 = 346.4769015. 

I could only hope that maybe if I had a little more data on those Asian pyramids or more than this pea-sized brain of mine, maybe it would help to sort out what they mean by “272.2639510” – the Megalithic Yard, the Draconic Month, or are they wanting to give us an important lesson in both? 

See, when I started thinking about Lunar themes there, I came up with the Meg Yard / reciprocal of the Lunar Year, which in my usual terms would be 

2.720174976 / (1 / 353.9334578) = 96.27609351 = 1 / 1.038679452 to use the “Alternate e’ Meg Yard” of 2.720174976 

2.719256444 / (1 / 353.9334578) = 96.24358376 = 1 / 1.039030303 to use the “Incendental Meg Yard” of 2.719256444 

And we might be able to retrofit 2.721223218 (Draconic Month = 27.212220817 days) into that successfully. What I’m thinking is basically that if the numbers I use are able to approximate the Draconic Month that well, why wouldn’t they do so, even if the figure can be a little awkward to work with? 

Pretty sure I’m looking at some working systems here, and that they are enthusiastically conveyed by at least some ancient architecture, including Stonehenge, but they look complex and hard to prove when there are things that haven’t quite fallen into place yet like that. 

I guess in summary what I mean to say is ‘Ellifino how many “Ellifino”-like numbers are at large and what functions they are trying to serve, so be careful of them? 

Very happy if you have a system of numbers that’s a done deal. Still working on mine, lol. 

What do I mean by “I think Stonehenge may be able to readily provide me with both figures were I not already working on it”? 

Again, Prof. Thom’s data for the bluestone “circle with corners” gives 51.06 Meg Yards as its perimeter and 51.06 x ~2.72 = ~138.8832 ft. With much trepidation, I decided to nominate 138.6375748 as one possible correct interpretation of this.

138.8832 or 138.8888888 ft seem to overshoot the mark when we try to do this, but 138.6375748 x 2.5 = 346.5939370.

If I wanted to cop just a couple more inches on Petrie’s raw data for a still somewhat ambiguous structure here, I could have nominated 138.3399854 ft, or exactly 1/10 of the perimeter of the Mycerinus pyramid following its recent revision, but if we’re aiming at about 346.62 with this, that would probably undershoot the mark.

If Stonehenge really is attempting to strike up a conversation about the Eclipse Year like that, that may help keep estimates and nominations on the right track.

Why do I like the idea of 346.5939370 as one of possibly more than one valid figure that may expressed by Stonehenge describing the Eclipse Year? Well, for one thing, Stonehenge is sufficiently circular that no doubt we have a license to throw 2 Pi, as if we needed one.

346.5939370 / (2 Pi)^4 = 5 / 224.8373808, a common variation on the Venus Orbital Period, which is perhaps partly inspired by the notion that the occultation of Venus traces a 5-sided figure.

346.5939370 / (2 Pi)^5 = 353.9334582, main value for the Lunar Year

So we would have a set of numbers wherein the Lunar Year, the Eclipse Year, and the Venus Orbital Period are all conveniently linked together by the 2 Pi ratio. Find one, find all three – we need only know enough about calendars to recognize what we’re seeing, and that ancient people may have gravitated toward number systems that can be interlaced like that.

Things are still uncertain here – for example, 360 / 138.6375748 = 2.596698626 – an obvious reference to the mighty calendar value 260, but not in an expected flavor – but this may be because of the use of an unspecified number of multiple Megalithic Yards having been incorporated into Stonehenge’s design which may correspond to the multiplicity of calendar number sets in general.

I haven’t even had the chance yet to get back to what Stonehenge may be insinuating about Jupiter’s Orbital Period and it’s proper representation in the particular system.

I’ll try to get right back to this subject when I can get just a little firmer grip on it. It’s hard to think sometimes what experiment to try next (another old adage among scientists, “If we knew what we were looking for, we’d have already found it”) so mainly things have been proceeding here by me stumbling forward. Frankly, Stonehenge has been amazingly generous with new possibilities in recent work and it’s a lot to take in.

I will add this before closing for now: Be forwarded that Stonehenge may (or may not) be nominating an additional figure to represent the hugely important Venus Orbital Period, as if there weren’t enough ways to do that – even when I’ve been doing a lot of unveiling how well Petrie’s ~224.8 inch Stonehenge unit does within Stonehenge’s math when taken as 224.8373808, that may not be the only variation on the Venus Orbital Period that’s on Stonehenge’s mind.

It’s within the mathematics that I’ve proposed for the Aubrey Circle.

Hint: for 224.8373808, it’s 2 / 224.8373808 = 889.5317998 (ft).

–Luke Piwalker

Apologies…

Back when I was an avid patron of science publications, I often heard scientists comment about how scientists were like blindfolded men examining an elephant. One grabs its leg and declares it’s a tree, another grabs its trunk and declares it’s a snake, and so forth…

Not a good way to start the New Year, but I’m feeling a bit like one of these blindfolded scientists today, after going over WMF Petrie’s measures between major Giza pyramids again.

Very easily possible I’ve tried to do too much too fast with too many distractions, and come up with the wrong answers, in spite of what I can hope is a very good track record of not having to submit revisions.

Something doesn’t seem quite right with that work, and especially not if the math is going to show that Petrie was almost perfectly accurate here but off by three or four feet over there.

Hence, I’m not sure what there is to, for example, stop his figure of 15170.4 inches from being something rather different than my first attempts at interpretation say it may be. It could be, say, 15198.17755 (((15 / (Pi^2)) x 10^n) for all I really know, in spite of the very strong appearance that more than one of these measures displayed the use of the Palestine Cubit.

For that, I’d have to say that everything I’ve had to say about these measures so far needs to be taken with a very large grain of salt.

I’ll leave the posts up for now, but please be advised that the work needs to be reconsidered before resting much confidence in it. Under the circumstances I certainly shouldn’t have declared that any of it has been solved. I’m very eager to make and share new discoveries on a regular basis and to see some of this work finished at long last, but that’s no excuse for being hasty.

Apologies for any confusion I might have caused concerning the orientation of Giza’s pyramids, it’s certainly my intent to cause less confusion, not more.

I suppose there’s some irony in that if I’ve erred here, it may well have been for trying to stick too close to the data, for trying to be too respectable and too accurate. There is such a thing, and I still don’t know of an accurate “yardstick” by which to measure Petrie’s accuracy for large measures such as these, even if we can rightfully consider him to generally be rather impeccable when it comes to smaller measures.

For that, yes it is a lot like trying to practice zoology while blindfolded, as tired as the story of the blindfolded men and the elephant may be trying to get.

Speaking of elephants, btw, elephants are part of my standard explanation of why I don’t pay much attention to accurate placement of decimal points.

We used to make up fancy expressions for this like “decimal harmonic” and “digit string”, but the point is very simple – if math is being used as language here and digits are like letters and numbers are like words, it really doesn’t matter if I write “ele.phant” or “elephan.t”, either way the reader is hopefully thinking of a large gray animal with big floppy ears.

Sometimes I just put the decimal point where I think it will be easiest for a search to spot, and maybe I shouldn’t even be using decimal points in the first place.

At any rate, when I talk about the critical nature of “mathematical truths”, I’m certainly not talking about accurate placement of decimal points.

The only time decimal placement really matters here is when we are talking about square roots, which operate at every other decimal place of the square, and cube roots, which operate at every third decimal place of the cube, and so forth, something I’m temped to call the “root rule” and stick with it.

That’s why for example the highly fundamental values of sqrt 2.4, sqrt 240, sqrt 24000, and etc belong to this math, and sqrt 24, sqrt 2400, sqrt 240000 and etc don’t.

–Luke Piwalker

The Interpretive Use of Royal Cubits and the Mysteries of Chephren

I come to an interesting place in my adventures with ancient architecture and metrology, where I’m seemingly asked to give up my beliefs without actually giving them up. It may not be a difficult choice, but it does seem something of a strange one. Have I gotten lost? How did I get to such a point?

Well, I have thus far tried to maintain as the standard for the ancient Royal Cubit, Michael Morton’s Royal Cubit of precisely .03 Radians in feet. Metrology of necessity seeks out standards, and I can think of a better standard for this consensus cubit.

Even Petrie for a long time entertained this thought, even being no doubt aware that it implied that the ancient Egyptians were somehow aware of, and availed themselves, of the modern foot, and that the modern foot has to be much older than we know.

For years now, I’ve avoided endorsing any additional values as actual cubits, even though I have some extras myself and one which matches an actual rod described by Karl Lepsius.

I experimented with trying to design a pyramid whose four different sides were a simple number of four different cubit-like values, whose disparities would generate fine ratios between adjacent sides, and at its most detached from know reality, it still represented at least an apparent composite of values actually found in Flinders Petrie’s data.

I was discontented that if we were to call them actual cubits, we’d be calling something as esoteric as 1.726276422 ft a Royal Cubit, even though such a figure might be an inevitability of geodetic modelling – if we treat the polar circumference of earth in miles as 24858.38047, and we want to represent it in architecture as 144 x 10^n units at a ratio of unit:mile, we get a unit of 24858.38047 / 14400 = 1.726276422.

What shall we do with that and how shall we distinguish it from other cubits? Shall we call it a polar cubit?

Of course, that was only my first attempt at a hypothetical model of something like that, and it may easily be possible still to build a better one.

I wanted only to get any pyramid whose fine ratios between sides were the 2 Pi Roots of the equatorial, polar, and mean circumference, and one more useful feature, such as 1.000723277 or the Great Pyramid’s proposed “unpaved/paved” ratio (a hypothetical layer of missing pavement having affected its finished height and width) of 1.001426825.

My hypothetical pyramid even had some crosstalk between sides that gave us the ratio for equatorial circumference in miles / polar circumference in miles as a nice bonus

I’m still not sure what to think of this hypothetical model, whether it would have been worthwhile enough that we should expect we might already be looking at one, or not.

However, there may be another possibility in this neighborhood, which is that given an actual absolute standard for the Royal Cubit of presumably .03 Radians, we have something to take ratios against to obtain these same kind of largely geodetic ratios when using fundamental units similar to the Royal Cubit without them necessarily being Royal Cubits themselves.

In terms of interpretive use of Royal Cubits – to give an example, the success that can be had “remeasuring” the Great Pyramid in Royal Cubits of (1 / (360/2)) x Pi^3 x 10 feet = 1.722570927, a good match for one of the rods described by Lepsius, even though the Great Pyramid was seemingly designed on a Royal Cubit of .03 Radians = 1.718873385.

We may be able to broaden our list of examples. I had to file Chephren’s subsidiary pyramid G2a as unsolved, because the better cubit for its design for once didn’t seem to be Morton’s if it were going to enjoy the best mathematical rapport with the architecture of its immediate environment, but rather the so-called “Silbury Cubit” of 1.720116607 feet.

“Silbury Cubit” = Base Perimeter Silbury Hill 1729.349823 / (320 Pi) = 1.720116607

It usually looks to be some sort of artifact of the difference between relating units to one another through circular geometry and simple fractional relationships, and through the geometry of squares and rectangles, such as the sqrts 2, 3, 5 array of the Vesica Piscis. The later carries with it its own advantages and disadvantages and inexactitudes.

The so-called Silbury Cubit is none other than Morton’s Cubit multiplied by the undisputed king of all fine ratios, the ubiquitous 1.000723277.

1.718873385 x 1.000723277 = 1.720116607

This value is also more properly and realistically the side length of a square with a diagonal of one 1.216733603 ft Remen.

Thus is has reason for being, but it’s having a hard time so far proving itself in the field, so to speak, because Morton’s cubit keeps trying to prove itself adequate as the fundamental cubit of most Egyptian pyramids.

Enter Chephren and its satellite pyramid G2a.

Let me tell you a few things about the Chephren pyramid. I still use Munck’s value of 225 Pi ft for a side because I’ve never had sufficient cause to stage a rebellion, even if I’m more than half convinced that the Chephren pyramid’s parents wanted a pyramid of 600 x 1.177245771 ft per side instead.

Then there’s its diagonal, what a messy business! One of the still unsolved mysteries of Giza. Why is it so mysterious?

Well, I’ve contented myself that Munck’s measures are close enough to any possible consensus value, and given what they are

Or, they might have thought that writing the number “1” was the biggest waste of “paper” (architecture) they’d ever seen, and preferred that interpreters should understand this, and take the mystery value to mean something much more useful, such as the reciprocal of the ruling fine ratio, 1 / 1.000723277 = .9992772450, which isn’t within the standard for approximating 1000.00000, but it is within the standard for interpreting the raw, rigidly determined geometric value of 999.6486611 ft.

225 Pi = 706.8583471, side 706.8583471 x sqrt 2 = diagonal 999.6486611 feet. Within the established “Giza standard” of accuracy for this work, that could either be 1000.000000 modern feet, a chance to offer evidence to those skeptical that the ancients knew the modern foot…

Perhaps this was a in a way a missed opportunity for a clear demonstration of the foot that influenced the decision that the Mycerinus should be a very uncharacteristically round 216.0000000 feet in height.

Nonetheless, once again an assessment of 1000 feet is still within the standard of accuracy, and within parameters still constitutes a mathematical truth, as is at the same time 999.2772450.

Given that the ratio between what we’d expect from G2a and what best serves it purpose = the ratio between the standard Royal Cubit and the unit seemingly in use in G2a’s design = the ratio between what the Chephren pyramid was and what it could have been = the ratio between what may be two correct interpretations of its diagonal = 1.000723277.

They might have spanned that difference with a paving layer of about .34058 ft thickness, but that somehow seems fairly thin (the hypothetical pavement layer for Cheop’s pyramid is more than twice that) but these ancient geniuses may have found another way, by placing 1.000723277 in our hands at every turn.

Thus can we take a “Silbury Cubit”, or simply Morton Cubit x 1.000723277 = 1.720116607 and go back and remeasure Chephren’s pyramid with it at the invitation of G2a, to obtain pleasing alternate values for its basic parameters of perimeter and height.

That would be us remeasuring Chephren’s pyramid with a value of 1.720116607 ft because it’s advantageous, the same as we would remeasure Cheop’s pyramid with a value of 1.722570927 ft because it’s advantageous, even though it’s designed on a Royal Cubit of 1.718873385 ft.

We thus get a very metrological expression

225 Pi = 706.8583471 / 1.720116607 = 4.109362959 x 100 = (5 / 1.216733603) x 100

To go with Chephren already having a height of (1.216733603^2) / Pi x 1000 ft = 471.2388982 = 150 Pi feet, which divided by 1.720116607 provides

471.2388982 / 1.720116607 = 273.9575307 = 1 / 365.0200808, a reference to the Solar Year to go with the references to the Lunar Year that lurk in Chephren’s pyramid

BTW, yes you caught that, the height of the Mycerinus in this model in Royal Cubits is 1 / Solar Year – or symbolic solar year, or calendar year, I should say.

Perimeter Chephren Pyramid = 706.8583471 x 4 = 900 Pi = 2827.433388 ft = 1 / 353.6776513, which is one of the major representatives of the ~354 day Lunar Year (the “B” value, to use my usual descriptive).

The 1.000723277 ratio we’re provided with will turn the B value into the A value, so that it’s right in there without having to actually be physically present, so that we get both instead of having to be puzzled whether it’s one or the other.

The Mycerinus seems to have been designed with a unit of 1.729249823 feet, even if it was designed on a cubit with a different value.

Relative to Morton’s geometric standard cubit, this is 1.729249823 / 1.718873385 = 1.006036767, one of the more important larger fine ratios, as evidenced by an apparent 100.6036767 mean diameter for the Stonehenge sarcen circle and may other possible appearances, including ratio between sides of Giza subsidiary pyramids (see Miroslav Verner, The Pyramids, and etc).

All of this of course goes on independently or what was or wasn’t called a cubit, as opposed to what might have simply been a cubit-like value.

Anyway, I could hardly think of a more satisfactory answer to the mystery of the Chephren diagonal, and the mystery of G2a, and it seems very much in the spirit of Lepsius’ cubit at Cheops’ pyramid, so there seems to be very significant precedent for a “re-measuring” at Giza.

Maybe sometime soon I’ll get around to explaining what these proposed proportions for G2a do for its relationship with its mathematical environment, but I want to look at the data from Maragioglio and Rinaldi for awhile longer and try to make sure I have the very best perspective on things that the system of numbers I use (and the available data) will afford.

There may be additional rigors we can subject these proposals to in order to try to be more sure.

–Luke Piwalker

What Else Is New?

On the one hand, it’s a precariously (or perhaps preposterously) ambitious thing to do to try to work out finer details of Stonehenge and Giza at the same time; on the other hand, the two developing views may complement and illuminate one another as discoveries seem to begin happening at a sometimes almost alarming rate.

Some old discoveries return to light as well.

This may be hard to believe after I’ve gone on ad infinitum about Stonehenge being fairly well saturated with references to 1.067438159, but 1.066666666 is in there somewhere too.

1.066666666 x (Pi^2) = 1.052757803 and I will hopefully be able to explain momentarily why this is important, and

1.066666666 x (Pi^4) = 103.9030303 / 10^n, 103.9030303 being the standard value for the sarcen circle’s outer diameter in feet, based on an outer perimeter (circumference) of 120 Megalithic Yards of 2.720174976 feet. (For what it’s worth, 1.066666666 = 240 / 225, relating to the “B” version of the Venus Orbital Period, where the “A” version is 224.8373808.

As soon as we wonder why it’s not 1.067438159, we realize that 1.067438159 / 1.066666666 = 1.000723277 and discover the ever-important fine ratio 1.000723277 if we have not already become aware of it, and learn more about why the mathematics prefers multiple versions of planetary cycle values.

Recently I stumbled over some materials by Don Barone, I was rather impressed with the spirit and content of them even for being of a different school of thought, and I though adding Giza pyramid sides together was a rather creative and astute thing to attempt.

I tried it with my numbers, and for the Mycerinus and Chephren pyramids with respective baselength values fixed at 345.8499635 and 706.8583471 ft,

345.8499635 + 706.8583471 = 1052.7083106

That’s the recently noticed number we saw found in Stonehenge 1052.7083106, to an accuracy of

1.0527083106 / 1.052757803 = 0.9999529878, an uncommonly high level of accuracy

The difference between the Cheops and Chephren Pyramids is my standard model, which mainly uses Munck’s Giza data

754.5275745 – 706.8583471 = 47.669227400

That’s another number recently reckoned as being important, 47.66008749, to an acceptable accuracy of 47.66008749 / 47.669227400 = 0.99980826393.

One reason 47.66008749 has become recognized as important is because it’s up top in a very important 2 Pi series

47.66008749 / (2 Pi^1) = 758.5338512, proposed platform length Great Pyramid

47.66008749 / (2 Pi^3) = 192.3188691, the lost metrological unit known only as the “Ellifino”

47.66008749 / (2 Pi^4) = inner sarcen circle circumference of Stonehenge / 10^n

and etc, further down through an impressive series, also including two particularly powerful data retrieval tools, sqrt 60 and 2 / 1.622311470.

Also worth pointing out, the combined sidelengths of the Mycerinus and Cheops pyramids amounts to 754.5275745 + 345.8499635 = 1100.377538 which is 1000 x 1.100874629, my favorite candidate for the main absolute value of the Indus foot, to an above standard accuracy of 1.100377538 / 1.100874629 = .999548457

Their difference, 754.5275745 – 345.8499635 = 408.677611, approximates the simple multiple of the “Mayan” wonder number 1.021521080, 1.021521080 x 4 = 4.08608432, to an acceptable accuracy of 4.08608432 / 408.677611 = .99983072476.

I should probably also note that the proposed 7200 Pi = 22619.46711 inch East-West distance between Cheops’ and Mycerinus’ pyramids, might perhaps be seen as a macrocosm of part of the Great Pyramid’s interior, given that Petrie gives for the length of the Queen’s Chamber a value of 18.87250000 ft, which is likely enough to turn out 18.84955592 ft = 226.1946711 inches, in spite of some obvious and widespread temptation to take it to be “11 cubits” (~18.92 feet, depending on our idea of a cubit).

Getting back to 1.052757802, it has some interesting aliases, and it may be a metrological unit in its own right, given that it’s situated like one.

Upon further inquiry, it seems to have appeared as an estimated “1.052960599” several years ago in the midst of my experiments with different metrological units, applying the sqrt 2, sqrt 3, and sqrt 5 values of the Vesica Piscis geometry, wherein it’s also pointed out that it appears to be 1/2 of some type of Palestine Cubit.

1.052757802 x 2 = 2.105515604, a variation on the Palestine Cubit with possible geodetic value, and one which may have already been found in at least one Egyptian pyramidion as the particular diagonal to a rectangle measuring 1 Remen by 1 Royal Cubit.

Many if not all of the pyramidia considered so far seem to put a very clear focus on metrological units that way, sometimes rearranging them in creative ways.

As with my favorite Palestine Cubit 2.107038475 ft, which is linked to the “A” value for the half Venus Cycle by the equation 4 / x = VC/2, 2.105515604 is also linked to a primary half Venus Cycle value, the “B” value, by the same equation.

I haven’t unravelled all of it yet, but I am indeed indebted to Don Barone for inspiration here that has led to some rather interesting results.

Postscript/Addendum:

Perhaps looking at the ratios between the parts of this design as described by Petrie could offer a little additional guidance?

The ratio between 36857.7 (center of Cheops to center of Mycerinus, direct) and 17873.2 (center of Chephren to center of Mycerinus, direct) is 36857.7 / 17873.2 = 2.062176891, compare to Royal Cubit in inches / 10 = 2.062648062

The ratio between 22616.0 (Cheops to Mycerinus, East-West) and 13165.8 (Cheops to Chephren East-West) is 1.717783955, compare to Royal Cubit in feet, 20.62648062 inches / 12 = 1.718873385 ft

Royal Cubit, in feet, as ratio as well as actual measure. Quite impressive. Well done, ancient architects!

That is already what I get, by the way, for having chosen 72 x Pi x 10^n inches as the possible distance from Cheops to Mycerinus East-West and 13159.47254 inches as the possible distance from Cheops to Chephren East-West: 22619.46711 / 13159.47254 = 1.718873385

36857.7 / 19168.4 = 1.922836543 and

29102 / 15170.4 = 1.91834098

We appear to be near to the realm of reciprocals of various ways of representing the number 52, half Venus cycle / Pi^n, and the fierce and dreaded Ellifino

Also a curious thing, 22616.0 / 94502 looks at first glance much like the reciprocal of sqrt 174960 (486 x 360 = 6^5 x 225), while the number 36857.7 resembles 174960 / 10^n Palestine Cubits.

29102 / 13931.6 = 2.088920153, just below some possible Palestine Cubits on the low side, 2.091411128 being the one formed from Royal Cubit 1.718873385 x Remen 1.216733603 = 2.09141128 ft = 25.09693353 in.

Lastly, 19168.4 / 17873.2 = 1.072466039, which certainly reminds me of a very important 1.073519416.

Maybe it will get sorted out somehow exactly what this most mysterious 29102 is supposed to represent?

–Luke Piwalker

What’s New? (Petrie’s Pyramid Rectangles, Pt 1)

Happy New Year! I hope 2020 is kinder to us all than the past year has been to many of us. I’d love to start the New Year off with a bang with some new discoveries, but I think I’ve nearly bitten off more than I can chew.

Probably not a smart thing to do in the middle of trying to gain new ground in understanding Stonehenge, but the question of the layout and orientation of Giza’s main pyramids has always been enticing – it is merely coincidental, or is there meaning to it?

With the observation that the North-South distance between the pyramids of Chephren and Mycerinus (Khafre and Menkaure) may be directly related to the length of the Great Pyramid’s platform, the question has become even more enticing.

If it’s what it looks like, it’s like figuring out a single word in an undecipherable language for the first time. It opens the door for high hopes that more will follow.

There do appear to be some possible patterns in Petrie’s data for the Giza layout that seem worth pursuing.

For example, the East-West distance between the pyramids of Cheops (Khufu) and Chephren is given by Petrie as 13165.8 inches. That’s 1097.15 feet. The East-West distance between the pyramid of Cheops and Mycerinus is given by Petrie as 22616.0 inches. That’s about 1095.736434 Royal Cubits.

Such things could just be coincidence, but already a value of 1096.622712 (((1 / 9) x (Pi^2)) x 10^n) appears to fit neatly for both of these.

In the case of the Great Pyramid’s platform length, one of the things that’s attractive about my interpretation is the metrological value

On June 15, 2017, I posted to the Internet concerning data from Glen Dash, “New Angles on the Great Pyramid”

http://www.aeraweb.org/wp-content/uploads/2015/01/aeragram13_2.pdf

Here’s a tidbit (I hope) – for the data from Lehner & Goodman via Dash’s paper, the average of the mean figures for each side come to 231.175 meters = 758.441627 ft

I’m quickly seeing a lot of good arguments for 758.5338511 ft as the “Intended Figure By Original Design” – including that it may have been their only shot at repeating the fine ratio (a geodetic constant when multiplied by prominent Giza constant 2 Pi, at the third power) that appears across proposed untruncated apothem / truncated apothem lengths, as

Sidelength Great Pyramid Platform 758.5338511 ft ? divided by
Sidelength Great Pyramid (proposed unpaved model) 755.6041600 ft
equals 1.003877283 = Earth Circumference in miles (24901.19743 / 100) / ((Pi^2)^3))

Looks a lot like 360 of the Persian Cubit Candidate 2.107038475… 

758.5338511 ft / 360 = 2.107038475 ft

Such a Persian Cubit gives a pleasing and meaningful measurement of the perimeter of the Great Pyramid (with pavement aka Munck Model) at

3018.110298 ft / 2.107038475 ft = 1432.394488 = 1/4 Radian x 100

(I seem to be mixing up the names Persian Cubit and Palestine Cubit here as I still sometimes do. I don’t know why, I like to think I know better than that)

Petrie’s data gives for the North-South distance between the pyramids of Chephren and Mycerinus 15170.4 inches; 758.5338511 ft x 2 = 15170.67702 / 10. Thus, 15170.67702 inches = 1264.223085 ft = 600 Palestine Cubits of 2.107038475 ft

However, the material remains challenging. The Palestine Cubit wasn’t likely the sole metrological unit used in Giza’s layout, and there doesn’t seem to have been any singular form of the Palestine Cubit, quite possibly for mathematical reasons, wherein different forms satisfy different equations.

I may (or may not) have solved one part of the puzzle, wherein the distance between the centers of the pyramids of Cheops and Chephen represent the diagonal of a rectangle of, according to Petrie

Centre of First to centre of Second Pyramid 13931.6 and 13165.8  = 19168.4 at 43º 22′ 52″

http://www.ronaldbirdsall.com/gizeh/petrie/c13.html

Which would be, interpretively, 13942.74005 and 13159.47254 = 19172.15996 inches.

This would mean an error of about 11 inches on Petrie’s part for the North-South Distance.

I regret having to report that, but it doesn’t strike me as terribly implausible. There doesn’t seem to be enough consensus on the Great Pyramid’s base measures to demonstrate that errors of some 1/2 foot on Petrie’s part can be excluded from possibility (see Dash’s compilation of data)

13931.6 inches = 1160.966666 ft, making the possible error proportional to about 9-10 inches on the basis of scale alone, and there is also the still somewhat contentious matter of whether the apices of Cheops’ and Chephren’s were exactly centered.

That margin of error may invite numerous alternative theories, but all things considered, I’m rather comfortable with it under the circumstances. Nothing wrong with alternative theories either, I’m skeptical that any researcher has a monopoly on truth and often the work of other researchers helpful and inspiring to look at even when it can be a little hard to grasp for subscribing to a different fundamental school of thought.

Dash’s data, by the way, may allow room for a slightly irregular base for the Great Pyramid, as opposed to the geometric ideals that most of us work with.

One possibility is that in reality, the Great Pyramid may have four sides, none of which quite match, in order to accomodate or illustrate as many as four different models or hypotheses or ancient mathematical styles.

Another possibility is that Great Pyramid was built in a slightly irregular manner at the base so that one of the base diagonals can express a slightly different mathematical constant than that expressed by the geometric ideal of a regular pyramid.

The diagonal I customarily give for the Great Pyramid’s base, extrapolated from Munck’s model, is 1067.077725 ft (calculated at 1067.063129 ft, showing unusually high accuracy of approximation at .9999863215).

Being able to generate a second Great Pyramid base diagonal of 1067.348159 ft by means of an slightly irregular base would have likely been an extremely enticing prospect to the Great Pyramid’s designers.

The two propositions of four different sides and two different diagonals may not be mutually exclusive.

Meanwhile, I think I’ve stumbled upon something at Stonehenge, which really shouldn’t have surprised me, after what I’ve had to say about Stonehenge recognizing that 1.067438159 is almost the cube root of the Remen, and so forth.

Occasionally there may be mathematical facts so enticing that a system may choose to honor them even if it cannot quite embrace them completely, and the same is true of the present system.

While I could rattle off a list of examples of this, why don’t we just skip ahead to the latest case in point?

Being that Petrie seems to have identified a metrological unit at Stonehenge of ~224.8 inches

The cube root of 224.8 is 6.080399327 = 5 x 1.216079865, very close to the standard Remen value I use of 1.216733603, and the cube root of 224.8 x 100 is 28.22271362, very close to the figure of 28.25389852, which is the reciprocal of my standard figure for the Lunar Year. The square root of 224.8 x 10 = 1 / 2.109122793, intriguingly close to my favorite Palestine Cubit value of 2.107038476 (ft).

Meanwhile, back at Giza, for the rectangle I think I might have solved

Centre of First to centre of Second Pyramid 13931.6 and 13165.8  = 19168.4 at 43º 22′ 52″

Which would be, interpretively, 13942.74005 and 13159.47254 = 19172.15996 inches.

Such a rectangle would have a length/width ratio of 13942.74005 / 13159.47254 = 1.059521193

Among other things, 1.059521193 = (9 x 1.17725771) / 10.

Morton referred to 1.059521193 as a “hunab-meter” in reference to Hugh Harleston’s work, and while I may not necessarily agree with this interpretation, curious readers may be able to use the term “hunab meter” to help locate more of Morton’s findings about this number still surviving on the Internet.

Naturally, being placed in the vicinity of the world’s greatest monument to the 2 Pi ratio, i.e., the Great Pyramid, we’d expect this number to give up some important data if we apply the number 2 Pi.

One piece of unexpected data that came from this might benefit from me going further forward with this “progress report” on the Giza Layout.

For the distance between the pyramids of Cheops and Mycerinus, Petrie gives us

Centre of First to centre of Third Pyramid 29102.0 and 22616.0  = 36857.7 at 37º 51′ 6″

I have a fairly good confidence level that 22616.0  inches is going to turn out to be 22619.46711 (72 Pi x 10^n), but 29102.0 is perplexing. It of course very much wants to be the same figure as 240 Remens — 240 x 1.216733603 = 292.01606447, for intents and purposes the cube root of the Earth’s circumference — but I doubt if an error of some 100 inches on the part of Petrie is anywhere near believable.

So what is this figure if it isn’t 29201.60646 inches? That’s a very good question!

Let’s go back one more time to the figures from first Petrie then myself

Centre of First to centre of Second Pyramid 13931.6 and 13165.8  = 19168.4 at 43º 22′ 52″

Which would be, interpretively, 13942.74005 and 13159.47254 = 19172.15996 inches.

The area of such a rectangle would be 13942.74005 x 13159.47254 = 183479104.8203 square inches, or 183479104.8203 / (12^2) = 1274160.45014 square feet, which are both very pleasing numbers to see and at the possible correct identification of a well-chosen set of numbers.

For diagonal / width (d/w) and diagonal / length (d/l)

d/w = 19172.15996 / 13159.47254 = 1.45690945451 = ?

d/l = 19172.15996 / 13942.74005 = 1.37506400400 = (8 x 1.718829140) / 10

What is 1.45690945451? Another very good question.

1.456909454 x 2 = 2.913818908

It turns out this number has some interesting Giza-related aliases like “Giza Vector squared divided by 1.067438159” = 29138.92457 / 10^n and (360/2) x 1.618829140 = 291.3892452, and I’m sure there must be more, let me find one…

224.8373808 x (360^2) = 29138.92455, how’s that?

Someday I’m going to have to write a piece on “Not A Remen” ( 1.214121856) not that “Not A Remen” is likely to be a a singular entity (there’s also a 1.213-figure currently at large, and it may have been in Newton’s calculations whether he was aware of this or not).

Maybe someday we can straighten it out once and for all whether it’s supposed to any ancient person’s idea of a metrological unit or not, but it was clearly a mathematical constant they were likely to be aware of.

Whereas 292.0174976 / 240 = 1.216733603, my standard Remen value, 291.3892455 / 240 = 1.214121856, “Not A Remen”.

As we might recall, 1.214121856 bears the noble pedigree of (1 / 1.067438159) x 360^2 = 121412.1856

29138.92455 bears as part of its pedigree 2.825389856 x 10313.24031 = 29138.92455

2.825389856 being 1 / Lunar Year in my work, and 10313.24031 being the generic area of any regular circle or the number of inches in 500 Morton Royal Cubits.

I have to admit, though, the ancients are really putting me through my paces here, and it’s really rubbing my nose in how much still seems to be lost from ancient history.

Disclaimer: If I’ve gotten things wrong so far, 29138.92455 is probably the most suspect, it presumes about 3 feet on Petrie’s accuracy, and falls short of the Giza standard of accuracy if the accompanying parameters are fixed, but this is the first time I worked with something that measures almost 1/2 mile.

I’m really not sure what the rules are but slightly relaxed standards of accuracy could be part of the bargain at this scale, and I’m hoping we get to learn what might be typical. I will in the meantime continue to search for a more accurate but equally pleasing answer.

I wish I knew where to turn for more clarification, but some of the Giza Plateau Mapping Project’s data turned out to be a disaster that left me asking people why there should be discrepancies between the datasets of Petrie and Lehner on the order of 100 feet or more, and I don’t think that question has even been answered.

http://grahamhancock.com/phorum/read.php?1,237549,237865#msg-237865

This is also where it gets slightly confusing since 29138.92455 is probably best measured in Palestine Cubits of 2.106326971 feet rather than of 2.107038476, and in “Incidental Meg Yards” of 2.719256444 rather “Alternate e’ Meg Yards” of 2.720174976 ft.

So were there multiple versions of the same metrological unit in use in the design of the placement of Giza’s pyramids, or are these possibly additional clues to the location of a small error?

All I can really say is that 29138.92455 makes a good deal of sense mathematically speaking, and that it is an interesting number worth knowing about.

Anyway, Happy New Year – there are some new thoughts on ancient things.

–Luke Piwalker

More About Stonehenge, Pt 6

Something I should mention is that in interpreting Prof. Thom’s “17 Megalithic Yard” diameter for the inner Trilithon ellipse and the inner bluestone circle as 2 / 1.177245771 = 16.98880598 Megalithic Yards is that 16.98880598 x 2 Pi = 106.7438159, one of the numbers that Stonehenge seems quite fond of talking about, achieved this time by dragging in the circumference / radius ratio of 2 Pi for any circle, and applying it to the mathematics of this circular monument.

Regarding the ellipses, I don’t think I have data from Thom on their perimeters except for the smaller (27 by 17 MY) of the two Trilithon ellipses, which Thom gives in “Megalithic Rods” of 2.5 Megalithic Yards as 27.99-28.03 (Megalithic Remains in Britain and Brittany, pg 144, Table 11.1)

Using a flat 2.72 ft per Megalithic Yard for the estimates, that’s ~190.332-190.684 ft

This seems to be a relatively good match for the data according to https://keisan.casio.com/exec/system/1223289167

Where Perimeter = 70.010455524686 MY = 190.4406261672 ft  for an ellipse of 27 by 17 Megalithic Yards.

This might be where the symmetry breaking kicks in and we can assess the perimeter in both linear Megalithic Yards, “AEMY” 2.720174976 ft and “IMY” 2.719256444 ft. Both of these values give significant figures, although for the one I think is more obvious, 70.08385550 x 2.720174976 = 190.6403500, I find it quite fascinating that if we take the 1.067438159 that Stonehenge seems fairly infatuated with and divide it by the 56 holes of the Aubrey circle

1.067438159 / 56 = 190.6139570 / 10^n. That’s pretty good for using an illegal number like 56, the accuracy of approximation here is .9998615559, well above the inferred .9995 minimum.

This may or may not be a hint as to a preference whether the “AEMY” or the “IMY” is to be considered the primary intended unit, just as AEMY 2.720174976 appears to be the primary intended unit for the exterior of the sarcen circle.

If we use 70.08385550 x IMY = 2.719256444 = 1905.759757, which has some interesting aliases including 1.177245771 x 1.618829140. 1.618829140 isn’t “Phi”, but it’s an important adaptation of number natural to a “2 Pi” pyramid (perimeter / height = 2 Pi) such as the Great Pyramid seems to be.

Anyone thinking in the same math might be expected to get that one in there somewhere as it seems important to the math in general.

I might have to give a lot more thought to 190.6403500 and 1905.759757 before I decide which one reflects the primary Megalithic Yard for this ellipse perimeter – AEMY, IMY, or neither?

The calculator gives the ellipticity of this ellipse as
0.62962962962963 = 1 / 1.588235294, which would easily pass for a proposed possible value of 1.588133131 = 3.176266261.

We might get a lucky break – a bit of “Rosetta Stone” as it were, a reference point in the form of a calculated 57.24898132038 Meg Yard perimeter for an ellipse of 22 by 14 Meg Yards. That’s probably going to much want to be 57.29577951 the Radian, regardless of which Meg Yard it’s reckoned in.

57.29577951 x AEMY 2.720174976 = 311.70090913 / 2

The 3119.345432 square foot surface area calculated for the Great Pyramid’s missing upper section nearly wanted to be 3117.0090913, and there would have been considerable merit in it, were the height of this missing section not very sharply defined by the proposed height of the Great Pyramid without it.

57.29577951 x 2.719256444 = 155.8019176 = 311.6038353. Not the Great Pyramid data 311.9345432 we could have hoped for, so that may take some consideration whether AEMY could have been the primary Meg Yard here, and whether 311.6038353 might be a much under-appreciated number we are being alerted to.

For the smallest of the three ellipses, the innermost bluestone ellipse, the calculator gives linear eccentricity f = 8.4852813742386 which I’ve identified as sqrt 72. The accuracy here if this is to be taken as 1 / .1177245771 = .9989261614, but hopefully we don’t have to worry yet.

The “Giza Standard” of >.9995 doesn’t have to apply to raw data, and since neither 22 nor 14 belong to this math, there’s apparently some adjusting to do that may improve the accuracy on the geometry when the true proportions are applied.

Taking the ancients to be perfections with math whenever possible as is hopefully being demonstrated here, we should probably expect the true Pi ratio since 22 and 14 being meant literally seems unlikely.

For the largest of the three ellipses, the outer Trilithon ellipse, the data gives

ellipticity c = 0.66666666666667
linear eccentricity f = 11.180339887499 (sqrt 125) = ~1.117818629

(Pi/3) x 1.067438159 = 1.117818626, and Pi/3 is a wonderful thing to see here, where it’s possibly being repeated along with another repetition of 1.067438159 next door to a possible (Pi/3) in the linear eccentricity of the inner Trilithon ellipse of 27 by 17 Meg Yards

linear eccentricity f = 10.488088481702 (sqrt 110) = ~ (Pi / 3) x 10 = 10.47197551?

Pi/3 would probably be a powerful thing to incorporate into Stonehenge’s design to help extract additional data from its numbers, just as it’s exactly that at Tikal and helps Tikal’s architecture make remarkable mathematical statements in series. If I would call anything a key to understanding Tikal’s pyramid temples, it’s (Pi/3).

I don’t have extensive data on sites near to Tikal, but what I have seen suggests proportions and features that may also be designed with (Pi/3) in mind at surrounding sites, Topoxte probably being my favorite.

I think at the moment, the circumference of the outermost of these three ellipses of 30 by 20 MY still eludes me. It’s calculated at 79.327197946453 MY = 215.783787375 ft, and I’m not sure what to think of that. Could be something elegantly simple like 216 (60^2) Meg Yards, or it could be a reference to the earth’s proportions (24901.19742 / Pi = 79.26297317 x 100).

For the ellipticity 0.66666666666 = 1 / 1.5, if not 1.5, several close numbers that are of importance are (1.5 / 1.00073277) = sqrt 224.6748781 / 10, being a more accurate but less useful representation of the Venus Orbital Period. If there turn out to have been more than two sets of calendar numbers, it may belong to one of them.

An important number just over 1.5 might be the ~1.512-1.52064 (John Michell) Greek Cubit value in feet. I’m likely to take it to be

(12.16733603 in / 12 = 1.013944669 Greet Foot in feet; 1.013944669 x 1.5 = 1.520917704

Although not necessarily as I think there was a bit of divergence in my last experiment with the Greek Cubit, which may have been related to trying to adapt Michell’s ratio between “long” and “short” forms of ancient units of measures (Dimensions of Paradise, pg 109).

About the raw number 56 from the Aubrey Circle, we’ve already done some very interesting things with it just like it is, but when it comes to trying to find a definitive interpretation of something valid within the same system of numbers, (if there is one), I’m still puzzled.

Here’s something I learned just now that I find interesting – if we take the 138.6375748 perimeter for the “cornered circle” formed from the bluestone ellipse and the inner bluestone circle, and divide by the inner diameter of the sarcen circle, we get

138.6375748 / 97.33868822 = 1.424280287

I have some confidence we’ll see more of 1.424280287 other favorite “Mayan” wonder numbers from Tikal at Stonehenge, just as we’ve seem a number of them already, and after all, one might think of 1.424280287 as 1.676727943 / 1.177245771, because that’s how I found it and that background is part of the introduction, but 1.424280287 is also (1.067438159 / 4) x 1.067438159 or 1.25 x (1.067438159^2) or (1.067438159^2 / 8).

Why shouldn’t we expect 1.424280287 at Stonehenge therefore?

I assume the ancient Maya were among the ones aware of this trick

360 / 1.42428087 = (224.8373808^2) / 2, so there is Petrie’s Stonehenge unit value yet again in a different way.

Since Stonehenge is also seemingly quite enthusiastic about the Remen, we have to ask

1.42428087 / 1.216733603 = 1.170577410. I won’t give a lecture on that unless a subject at hand such as Stonehenge gives me more reason to, but I’ve certainly been meaning to bring up 1.170577410 on a slow day for some time now.

I’ve also been doing a little work on Petrie’s distances between Giza Pyramids wherein we find a measure of 15170.4 inches. I’ve observed that this looks to be related to the platform width I determined for the Great Pyramid based on contemporary data, which further boosts the Great Pyramid’s powers of geodetic expression.

Platform width / pyramid width = 758.338512 / 755.6041600 = 1.003877283 = (24901.19742 / 100) / ((2 Pi)^3)

What I should like readers to be aware of is that 758.338512 / 360 = 2.107038475, my favorite candidate for the Persian Cubit, which is essentially the diagonal of a rectangle of 1 Remen by 1 Royal Cubit. It talks to the “B” version of Venus’ Orbital Period, the canonical figure 225 days, to say

225 / 2.107038475 = 1.067438159

This discussion on Stonehenge meandered back to the Great Pyramid because 15170.4 = ~1517.067702, and 1517.067702 / 2 = 758.5338512. 15170.67702 inches would be 1264.223085 ft, but we can see the relationship betwen this and the pyramid platform better with the figure in inches, just as we can see the inch value 224.8373808 for Petrie’s Stonehenge unit better at Stonehenge by leaving it in inches for now.

1.61882914 / 4, 1.718873385 x 2 (2 Morton Cubits in feet), 24 Remens 2.9201606467 = cube root of 249.0119742 and others are all directly connected to 2.107038475 by 1.177245771 at various powers of multiplication or division.

This is one possible point of divergence in the Greek Cubit in that along with the previously shown formula (12.16733603 in / 12 = 1.013944669 Greet Foot in feet; 1.013944669 x 1.5 = 1.520917704 there are also some things like

2.107038475 / (1.177245771^2) = 1.520331129

to lend a bit of uncertainty sometimes.

The Great Pyramid also gets into the discussion here because at Stonehenge we observed that 1.42428087 / 1.216733603 = 1.170577410, and 1.170577410 x 360 x 360 = 15170.67703

Here’s a neat little trick I just stumbled over, possibly not for the first time

(1.618829140 x 10^n) / 15170.67703 = no, not 1.067438159, but 1067.077717, the adapted Great Pyramid diagonal for the Munck model.

I might have saved some of that for a discussion on the orientation of the Giza Pyramids if not for wanting to write down several observations as they were in progress.

One final one, though – 15170.67703 / 360 = 421.4976953, and 421.4976953 x Pi = 1323.891320

That’s a number I ran into during experiments with circumscription and inscription of my pyramid models that generally went so well that I’m half convinced that such considerations went into the selection of their proportions. These experiments also readily yielded 1.067438159 built into ancient architecture – in this case Giza’s pyramids – in yet another manner.

More in the context of Stonehenge, perhaps, one thing that 1323.891320 is, is 360 / IMY 2.719256444, and (2.720174976 x 4) x 1.216733603.

Several other interesting things are that the suspected max circumference of the outer bluestone circle 892.807632 / 138.63757488 = 6.441116443, another “Mayan” wonder number found at Tikal that’s part of 2 Pi series that leads to the equatorial circumference in miles (6.441116443 x ((2 Pi)^7) = 24901.19742).

Also, the relationship between the possible 138.6375748 ft perimeter for the “cornered” circle and 1/2 of the 305.7985077 foot inner circumference of the sarcen circle is

(305.7985077 / 2) / 138.6375748 = 1.1177245771 / 1.067438159

Another runner up would be 138.5907605 (11.77245771^2), and depending on how which Megalithic Yard we use to interpret the outer sarcen circle perimeter, we can have both across a significant ratio

120 AEMY (120 x 2.720174971) / (1.177245771 x 2) = 138.6375747

120 IMY (120 x 2.719256444) / (1.177245771 x 2) = 138.5907605

Another thing about 138.6375747 is that 138.6375747 / Pi = 4.412971064 — that’s 2.720174976 x 1.622311470, and

2.720174976 x 1.62231147 x 1.177245771 = 51.95151515, outer radius of the sarcen circle, thus that radius is linked to its primary unit, a Meg Yard of 2.720174976, by the combination of two major constants 1.622311470 and 1.177245771

In turn, 51.95151515 x 1.177245771 = 611.5970160, the adapted slope length of the Great Pyramid, but more in context, 611.5970160 / 2 = 305.7985080, inner perimeter of the sarcen circle.

Essentially, Stonehenge seems to display utterly mind-boggling mastery of the numbers 1.177245771, 1.067438159, and 1.216733693, and we should probably include 224.8373808 while we’re at it.

“Not a Remen” may be in there too, just as it may be at Giza, even if it appears at Stonehenge somewhere other than originally suspected.

More later I very much hope, as there are some major ancient architectural models that are still incomplete.

–Luke Piwalker

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