Luke Piwalker

More About Stonehenge, Pt 5

I’m looking at Thom’s Stonehenge data again and I think the time may truly have come for the subject of symmetry breaking, although I haven’t mastered it and I’m not sure of the best way to present it.

However, as hopefully demonstrated once again, there is also the matter of min, max and mean for each circle or ellipse, relating to the thickness of each circle and the thickness of the stones that are used to mark them.

Hence it’s not perfectly clear to me yet whether some of the issues with equations should be filed under symmetry, or under the thickness of circular perimeter, or whether ultimately they are one and the same concern.

The implication is that each circle or ellipse is actually three, based on min, max and mean values, so that’s trying to juggle a lot of circles and ellipses at the same time, which naturally tries to get very messy – but it’s probably worse if the exploration hasn’t been done in order to properly provide estimates of min, max and mean values.

Perhaps I should just keep trying to take random looks at things and hope the pieces eventually begin to fall into place better?

It was omitted from my most recent notes, but I think I’ve retraced the origin of the entry of one of the unsolved mysteries into the equations.

346.62 / 8 = 4332.75 = 360 / 1.03860135

(For those who might be unfamiliar with it, 4332.75 = ~Jupiter Orbital period, 4332.59 d – it’s still under consideration how the mathematics in question prefers to represent this value)

If we take the proposed 779.2727283 ((57.29577951 / 2) x 2.720174976) ft diameter of the outer bluestone circle and divide by the Venus Orbital Period figure of 224.8373808,

779.2727283 / 224.8373808 = 3.465939363, compare to the raw eclipse years value of 3.4662 x 100 = 346.62.

However, I have yet to determine what value I think they’re using for the eclipse year exactly – at present I’m just pointing out possible mathematical relationships.

It might be possible that 779.2727283 is intended to be a slightly higher figure more closely approaching the Mars Synodic Period of 779.96 days. It’s still undecided what figures the ancients preferred for doing this.

If there’s a standard of accuracy on approximations for calendar figures, I’m not sure it’s been identified, and might be challenging to distinguish from a broader standard of accuracy inherent in simpler mathematics (where sometimes 364 needs to be accepted instead of 365, for example, such as the Mayan calendar figure 819 = 364 x 225 / 100).

At this point, this is still sheer speculation, but it may be possible that Stonehenge functions as a calendrical calculator, and could be a little bit loose around the edges mathematically deliberately, if such a thing is possible, in order to inform us of a small spectrum of different figures we can use to represent calendar numbers.

There are already two primary sets of such numbers have been identified but there may be more, and no definitive way of relating some of them to one another has been identified yet. Different equations can sometimes implicate different values because of their small differences, which may have contributed to a grasp of a larger number of working calendar sets being elusive.

Many experiments have pointed to 779.2727283 as a possible Mars Synodic Period value from one or other of these sets of calendar numbers, but still not quite conclusively.

I might note in this context that Stonehenge’s ellipses do tend to remind me of elliptical orbits, so it is perhaps interesting if we find any of Stonehenge’s circles or ellipses measured out in values that are suggestive of planetary cycles, which is why it’s quite remarkable that throwing around Petrie’s c.a. 225 inch unit at the inch value which so closely resembles the Venus Orbital Period, seems to be proving to be rather successful.

If anyone’s made it this far reading this, they’re probably rather understandably wondering if I’m just chasing my tail here by this point.

However, I believe that Stonehenge does slowly render up its secrets in the face of persistent, even if inept, inquiry.

The work of the last 24 hours seems to have shed some possible light on the inner bluestone circle and the curious shape formed by its intersection with the bluestone ellipse.

Professor Thom writes of this (Megalithic Remains in Britain and Brittany, pg 145)

“It will be noticed that the inner circle and inner ellipse produce a kind of oval with ‘corners’. It has a calculated perimeter of 51.06 MY which, when divided into 26 parts, produces a spacing of 1.96 MY”.

The inner bluestone circle has a diameter equal to the minor diameter of the inner Trilithon ellipse of 27 by 17 Megalithic Yards.

Presently, my determination is the same as what I have long suspected based on the prevalence of the 1.177245771 ratio (360 / inner circumference sarcen circle = 1.177245771) and its relevance – that the correct meaning of “17 Megalithic Yards” (17 does not belong to this system of numbers) is in fact 20 / 1.177245771 = 16.98880598 Megalithic Yards, whether we are reckoning it as a Meg Yard of 2.720174976 ft (Alternate e’ Meg Yard) , or a Meg Yard of 2.719256444 ft (Incidental Meg Yard).

That’s really splitting hairs in terms of actual field measurements of course, but it does matter to the mathematics, to which we can hopefully always look for clarification.

I’ve long held out for 2.720174976 being the primary intended Meg Yard used in the outer measures of the sarcen circle, because among other things this provides the outer/inner ratio of the truly amazing number 1.067438159 (Thom appears to have obtained 1.066666666 here if we use his Pi formula literally, which should remind us that his measurements essentially closely corroborate those of Petrie for the inner sarcen circle values).

Astute readers might spot this, btw: 1.066666666 x 1.000723277 = 1.067438159 and hopefully everyone should recognize the hugely important fine ratio of 1.000723277 by now.

So, when we take Thom’s “17 MY” diameter for the inner bluestone circle to be 20 / 1.177245771 = 16.98880598 Megalithic Yards, we get for its circumference in Megalthic Yards (and probably regardless of which Megalithic Yard we choose to use to convert the figure to feet)

Diameter 16.98880598 x Pi = circumference 53.37190808 ft

53.37190808 x 2 = 106.7438159 = 1.067438159 x 100

Returning to the passage from Thom

This too may have given way, at least partly, to persistent inquiry.

51.06 MY x 2.72 = ~138.8832 ft; 51.06 MY x 2.720174976 = ~138.8921348 ft

For much longer than I’d like to admit, I thought this was going to turn out to be 1 / 72 = 138.8888888 in spite of some mild misgivings about 72 being too simple of a number. 720 / 2 = 360 after all, and there’s nothing too simple about 360 to keep it from being of immense utility. 360 has become one of my standard mathematical probes, along with the Radian 57.29577951, 2 Pi, 1.177245771, 1.62231147, and others.

It’s finally come to my attention, however, that what we might really be looking at is 138.6375748 ft

Why do I think this may be so? There are many reasons that have already surfaced.

If we take the inner sarcen circle radius and divide by 4, we get

51.95151515 / 4 = 129.8787879.

129.8787879 x 1.067438159 = 138.6375742

Since we have already seen Stonehenge going on ad infinitum about the Remen, what is 138.6375742 ft in Remens of 1.216733603 (even if it may be a seemingly incomprehensible in Morton Royal Cubits)?

138.6375742 / 1.216733603 = 113.9421415 – and what is that?

113.9421415 = 10.67438159 x 10.67438159

Going back a third time to the same passage from Thom

1.96 MY x 2.72 = 5.3312; 1.96 MY x 2.720174976 = 5.331542953

10.67438159 / 2 = 5.337190795

The number 26 does not belong to this system of numbers, but in accordance with the premise of using illegal numbers as the number of structural components to suggest nearby values (see El Castillo at Chichen Itza, and probably any recently cited passages from Robert Bauval’s observation on structural features at Djoser’s pyramid complex),

51.95151515 / 2 = 25.97575758 might be what they meant, even if it posed an obvious challenge to otherwise write 25.97575758 via the number of stones in a circle

138.6375742 / 25.97575758 = 1.067438159 / 2

I at least like to think this latest effort might be yielding a little bit of unprecedented progress, even if Stonehenge is not near to finished with giving up its complex mysteries.

–Luke Piwalker

The Model of Silbury Hill Last I Checked…

This is due for a review – it goes all the way back to before the “Giza standard” of accuracy on forced approximations was quite recognized – but for what it’s worth I’ll try to piece together a more complete picture of Silbury Hill instead of just referring to its base measures and projected slope length exclusively.

Carl Munck declared that the perimeter at the top is 100 x Pi feet, a claim I can neither confirm nor deny since after all this time I still don’t know his data source and can only look the model for itself for support.

He gave 550 feet for Silbury’s radius, which is more data I still don’t know the origins of but it seemed to be corroborated by the author or authors of the Celtic NZ site (except that they had the perimeter at 1728 feet), so I ran with it and haven’t regretted it since.

So, we have a radius of 275.2186571 ft, a diameter of 275.2186571 x 2 = 550.4373142 ft, and a circumference of 550.4373142 x Pi = 1729.49823 (1728 x 1.000732377).

The projected slope height (from the base to the non-existent apex) was given as 317.8563582 (540 x 1.177245771) ft.

Here’s one the part I’m having trouble finding in the data, but from the looks of it, the actual slope height from the base to the actual summit should probably have been 259.7575758, 1/2 of 10 times the 51.95151515 foot outer radius for the Stonehenge sarcen circle.

This is from Feb 2003 when I worked on it

Base 275 / cos 30 = Hypotenuse ~317.5426481,
Hypotenuse ~317.5426481 x sin 30 = Height ~158.771324
Diameter 275 x 2 = 550

Base 225 / cos 30 = Hypotenuse ~259.8076211,
Hypotenuse ~259.8076211 x sin 30 = Height ~129.9038106
(Diameter 225 x 2 = 450)

Circular top: diameter 100 feet, radius 50 feet, circumference 100 Pi feet, area 2500 Pi sqrt ft…

2500 Pi, 7853.981634…
7853.981634 / 1.351926225 = 5809.475021
7853.981634 / (1.351926225^2) = 4297.183466
7853.981634 / (1.351926225^3) = 317.8563583
1.315926225 being the reciprocal of the Squared Munck Megalithic Yard

The height of the missing “cap” or the difference between presumed actual height, and full project height, was reckoned at Absent Conical Section, Silbury ht 28.82083036 ft.

And there is also in the notes

So let me skip ahead here a bit… when we go to calculate the Volume of an untruncated Silbury Hill… trying for a moment to put up with Pythagoras, which makes the height exactly 1/2 of the slope height… 317.8563583 / 2 = 158.9291782

1/3 * Pi * (R^2) * h

.333333333333 x Pi x 275.218657^2 x 158.9291782 = 126062312.1

Notice what we hit on the way up…

.333333333333 x Pi x 275.218657 = 288.2083036… 100 x times my suggestion for the “true and intended” height of the missing section of a truncated conical Silbury…

And then

Warning: Student driver. Education in progress.

Here we go… Let me try to re-write these more to my liking…

2.752186362 / cos 30 *[substituted by Alternate e’ / Pi]* = Hypotenuse 317.8563583,
Hypotenuse 317.8563583x sin 30 [subsituted by x, blame Pythagoras] = Height 159.1549431
Diameter 2.752186362 x 2 = 550.4373139

Base 225 / cos 30 [both substituted by x and y, blame Pythagoras] = Hypotenuse 259.757576,
Hypotenuse 259.757576 x sin 30 [substituted by x, blame Pythagoras] = Height 129.878788
(Diameter 225 x 2 = 450)

317.8563583- 259.757576 = ~~58.0960823~~ [killed by Pythagoras] substitute 57.74615025

If I remember correctly, I might have just barely got away with keeping the cosine function of a 30 degree slope angle while substituting 2.72017496 / Pi for 2.720699046 ((sqrt 3) / 2) x Pi

Area of Silbury’s Base:

If C = 1729.249822, D = 550.4373139, R = 275.218657, A = 237960.9067

Which is not yet necessarily the most talkative of figures either…

Silbury’s Base, In Remens
If C = 1421.223034, D = 452.3893421, R = 226.1944671, A = 160736.5383…

452.3893421 again appears to be the Great Pyramid’s height without its missing capstone, which has some profound positive consequences for other parts of the pyramid such as the apothem.

Silbury’s Base, in M. L. Morton Royal Cubit
If C = 1006.036766, D = 320.2314485, R = 160.1157242, A = 80541.1527…

237960.9067 / 2.72014796 = 874800 = dh 1.6785 x 25920 x 2…

If we use A = Pi R S for the surface area, that is Area = Pi x Radius x Slope Height, Pi x 275.218657 x 317.8563583 = 274826.5255

But as you can see, it’s been a long time since this work took place. It might be a good idea to the check the accuracy on any approximations involved before taking it too seriously.

–Luke Piwalker

An Experiment with a Putative Indus Foot

Jim Wakefield managed to bring the Indus foot to my attention – it’s about 13.2 inches or 13.2 / 12 = 1.1 feet. I’d rather shy away from the subject because it could have been tough determining just how many feet that was supposed to be (1.111111111 for starters?), but I finally noticed that 13.2 x 1.5 = 19.8, and I’d long been curious about a unit of measurement called the Pole that was mentioned in one of Henry Lincoln’s books, of 198 inches.

I’d already figured out that the Pole might be suitable for geodetic modelling at the ratio of feet:mile x 10^n, because 24901.19742 / (4 Pi) = 198.1574329 x 10, so 198.1574329 / 15 = 13.21049553 inches / 12 = 1.100874627 became my nomination for the Indus foot and is still my favorite of several.

The other nomination is, hopefully not surprisingly

(1.100874627 / 1.000723277 = 1.100078965)

So, having recently done some revisiting the concept of describing the origins of, or relationship between, metrological units via the geometry of squares and rectangles such as 2 Remens being the diagonal to 1 Royal Cubit, or the Palestine Cubit being the diagonal to a rectangle of 1 Remen by 1 Royal Cubit and etc, it finally occurred to me that maybe an Indus Foot should have a place in such proceedings.

To use my favorite putative value of 1.100874627 ft for the Indus foot, along with my other standard values

A rectangle of 1 Indus Foot by 1 Remen has a diagonal of

sqrt (( 1.100874627^2) + (1.216733603^2)) = 1.640842956 = 3.281685911 (1 meter in feet = ~3.28084)

A rectangle of 1 Indus Foot by 1 Royal Cubit has a diagonal of

sqrt ((1.100874627^2) + (1.718873385^2)) = 2.041188541 ft

Berriman’s value for a “Karnak Cubit” (so to speak) is 20.412 inches

A rectangle with width of 1 Indus Foot with a diagonal of 1 Remen has a length of

sqrt ((1.216733603^2) – (1.100874627^2)) = 1.320076047 = .1100063370 x 12

A rectangle with width of 1 Indus foot with a diagonal of 1 “Ellifino” of 1.921388691 feet has a length of

sqrt ((1.921388691^2) – (1.100874627^2)) = 1.57471108

A value of ~1.57471108 has been appearing in the Stonehenge data in this series. 1.57471108 is roughly near to the square root of the polar circumference in miles (sqrt 24860 = 1.576705426 x 10^n).

The “Ellifino” (a play on the “ell” or “elle”, an old name for cubit) is called that because ‘ell if I know what it is – it appears to be a lost metrological unit that is about sqrt 5 to the half Royal Cubit, and about sqrt 2 to the Megalithic Yard (hence 1.921388691 is close to the square root of 1/2 of a squared Megalithic Yard such as Munck’s), but even after tentatively placing 1.067438159 ft as a possibly much earlier appearance of the pied du roi, I still cannot seem to match 1.921388691 ft to a known metrological unit.

Given its relationship to other units as shown, it should likely have been a legitimate metrological unit in its own right, but I still cannot place it.

Also, I might note that 13.2 x sqrt 3 = 22.86307066. It might not be a match, but recently I’ve had some cause to be looking at figure near to about 22.88-something.

Perhaps we will learn more about these things in the near fugure.

–Luke Piwalker

More About Stonehenge, Pt 4

At the minute, I’m sort of scraping the barrel here and this post may also end up happening a half at a time, but sometimes insights happen while trying to explain something and insights would certainly come in handy right about now.

I suggested that the major diameter/minor diameter ratio of the inner Trilithon ellipse could be

“The inner trilithon ellipse measures 27 by 17 Megalithic Yards according to Thom. Taking this literally (not that we necessarily should), 22 / 17 = ~1.588, which might indicate constants such as 15.88133131 or 15.88669582”

This also came up again because someone mentioned the Great Pyramid’s base measurements in inches once too often, and this also plays a role in expressing its proportions in inches in my model – it looks something like this:

Specifically, ((1 / 9) x 2.720174976 x 12) / 360 / 32) = 3.148350667 (it’s not Pi, don’t even think about it, lol) = 1 / 317.6266261 = 1 / (158.8133130 x 2) and 3.148350667 / 2 = 1.574175333 which may match a number I’m catching fleeting glimpses of in diverse places, one of them being in other people’s work, and the other being some experiments I attempted with the Indus foot and square geometry.

Thus I’m curious if this really exists at Stonehenge in the form of 22 / 17 = ~1.588.

If we track the ratios across minor radius/diameter of the Stonehenge ellipses starting at a minor radius of “10 Meg Yards”, we will probably want the minor radius of the inner bluestone ellipse to be (1 / 1.177245771) x 2 x 10
Meg Yards, and the minor radius of the outer bluestone ellipse to be 10 MY of 2.720174976, at least as a starting point.

((1 / 1.177245771) x 2 x 10) x 1.588133130 = 2.698048562 MY for “27” and 2.698048562 x 2.720174976 = 7.339164183

That may be acceptable, because 7.339164183 / 224.8373808 = 326.4209960 = 120 x 2.720174976.

Perhaps I should expect to do with these aspects of Stonehenge what I did a long time ago with the sarcen circle, and compare them relentlessly among themselves and among the specifications for important numbers for significant neighbors like Avebury or Silbury Hill, a remark that may take some inspiration from 317.6266261 being the Silbury Hill projected slope length that almost was ((540 / 2) x 1.177245771 = 317.85863582 having finally won out).

In more recent years, 317.6266261 seems to have made a surprisingly strong show for itself at Tikal, and not surprisingly at sites close to Tikal. Perhaps the time is coming when I should re-examine the case for 317.6266261, and for that matter I should probably hunt down the complete model for Silbury Hill and get it back into circulation.

It may likewise be time to reconsider the role that a possible “Egyptian” unit of 1.700109360 might have been afforded at Stonehenge, in spite of any likelihood that when it comes to Thom’s “17 Megalithic Yards”, it may or may not have been 2 / 1.177245771 = 1.698880598 that was intended in cases like that.

Apologies for how complicated this tries to get, but ancient or not, rocket science is rocket science so don’t feel bad. It’s not always that complicated, part of the discussion here is really already a very familiar one

317.85863582 / 317.6266261 = 1.000723277

1.700109360 / 1.698880598 = 1.000723277

See, nothing that fancy there really, and that’s just how it is a lot of the time.

Regarding the previously estimated thickness of the Aubrey circle that almost looks like 2 Royal Cubits of some kind, I did experiment with a “1.725” cubit at various stages but I could never get it to behave like a good choice for a metrological unit. Probably more likely it’s just a mathematical constant and not a unit of measurement too, which may be related to an interesting but somewhat secular constitution

It’s not that hard to find in the Great Pyramid – and note the involvement of the apothem and pyramidion here (italicized) –

1.725028748 = (248.0502134 / 575.1793150) x 4 = (Pi / 1944) x 1.067438159 = (1 / 66052.47736) x (1.067438159^2) = 9.31515524 / 54

Even if the question of what it’s doing at Stonehenge were otherwise barely answered, so it will be something for future experiments. It should have some good reason for being at Stonehenge besides simply being the remainder in a subtraction problem.

It’s still tempting to think that an about 3.448961344 remainder (1.724480672 x 2) could indicate the 346.62 day Lunar eclipse cycle, but having obtained this remainder from two fixed values, we may simply not have the leeway that would allow that identity for the remainder value.

The exact calculated value of 1.724480672 is 1.725028748 to an accuracy of .999682280, still meeting the proposed universal ancient standard of >.9995, so if it doesn’t turn out to be some other surprise, an explanation is still in order.

Perhaps it’s significant that if we use all three major diameters for Stonehenge ellipses in a single equator using Thom’s values, we get 30 / 27 / 22 = 1 / 198, looking like perhaps someone was thinking of the pole or rod of 198 inches, and the minor diameters give 20 / 17 / 14 = 1.680672269 / 2, since I’ve recently been talking about mysterious numbers on the low end of 168-something, but perhaps things won’t work out that way in the long run.

Interestingly, perhaps, in the course of several hours I’ve noticed a couple of cases where the proposed Petrie unit in inches may make itself integral to Stonehenge’s mathematics in a way that’s both striking, and reminiscent of the mathematics of Tikal’s pyramid temple, Temple V.

I’d begun working with the 224.8373808 figure at Tikal, and curious if the exceptional N/S orientation of Temple V in contrast to the E/W orientation of the rest of Temples I-VI might hint at a special affinity with polar circumference through major calendar numbers such as the Venus Orbital Period when represented as 224.8373808.

Sure enough, I was treated to a rather deliberate looking display wherein 224.8373808 at slightly higher powers was linking to a polar circumference expressed as 24858.38047 miles.

Observe how we can start with the standard Lunar Year value I use, and get to the polar circumference via Venus.

353.9334578 x (224.8373808^3) = 1 / 24858.38035

If I take the mean diameter for the Stonehenge sarcen circle, 316.0557714 feet / Pi = 100.6036766 ft

100.6036766 x (224.8373808^2) = (360 / 353.9334567) / 2, and 100.6036766 / 224.8373808 = 24858.38047 x (360 / 2)

We can look back here again

“Specifically, ((1 / 9) x 2.720174976 x 12) / 360 / 32) = 3.148350667 (it’s not Pi, don’t even think about it, lol) = 1 / 317.6266261 = 1 / (158.8133130 x 2) and 3.148350667 / 2 = 1.574175333″

And see that it isn’t far removed from the same discussion about the Venus Orbital Period of 224.8373808 days and the Lunar Year of 353.9334567 days.

3.148350667 x 224.8373808 = 353.9334567 x 2

If the outer circumference value for the Aubrey holes is 892.9807632 ft,

892.9807632 x 224.8373808 = 200.7754599 = 1.003877283 x 2, even though that should be expected because the minor circumference value for same was suggested to be 2 / 224.8373808 = 889.5317998 giving the ratio 892.9807632 / 889.5317998 = 1.003877283

What may be surprising is that we can still go

889.5317998 x 224.8373808^3 = 1.011036956, which is (1 / 1.574175326) / 2 Pi, so we are still not far from the same question even using just the Venus Orbital Period and 2 Pi here.

I’ll try to get back to this post shortly, I have a couple more things I want to try to post about today.

–Luke Piwalker

More About Stonehenge, Pt 3

To continue with our experiment in progress, again just a scouting mission for now…

We noticed some important looking ratios between major and minor diameter of some of Stonehenge’s ellipses in Pt 2.

Here’s where it may start to get a little tricky?

Thom’s data gives us 17 Megalithic Yards for the diameter of the inner bluestone circle. We can see in his diagram that the diameter of the bluestone circle is equal to the minor diameter of the inner Trilithon ellipse, also 17 Megalithic Yards. I’m skeptical that’s its really 17 (which does not belong to these numbers), as opposed to something close.

17 x 2.720174976 = 46.24297458 ft. What that’s supposed to represent is a very good question.

46.24297458 x Pi = circumference 145.2765893 ft. What that’s supposed to represent is also a good question.

Perhaps we’ll set that aside for now and move on? I will note in passing however that 145.2765893 could be 145.2368735 ((sqrt 540) / 16) if that will actually fit into the “mathematical environment” of Stonehenge. There may also be at least one more candidate, including 145.1809284 which offhand seems a rather strange number.

The minor diameters of the outer and inner Trilithon ellipses are 20 Megalithic Yards and 17 Megalithic Yards according to Professor Thom. 20 / 17 = 1.176470588, and that’s probably really going to want to be Munck’s “Alternate Pi” 1.177245771, so I hope that’s what it gets to be.

The minor diameter of the inner Trilithon ellipse divided by the minor diameter of the bluestone ellipse is 17 / 14 = 1.214285714. Did they mean the Remen as a ratio, or did they mean “Not a Remen” aka 1.214121857?

It’s “Not a Remen” because I can’t quite bring myself to believe it is. Particularly if the Remen is a geodetic value, there may be no need to drive the value that low, Berriman’s value of 1.215 may even be somewhat questionable geodetically.

Would it be advantageous to pair 1.214121857 with 1.177245771 in this manner?

1.214121857 / 1.177245771 = 1.031324031, whose resume includes 1/2 of a Royal Cubit in inches, generic area of a circle, and Solar Year/Lunar Year thus far in this mathematics.

That’s still not for certain if the ellipses haven’t been solved, but it might be a start.

It is somewhat tempting because the series continues

1.214121857 / (1.177245771^2) = 8760.481958 and

1.214121857 / (1.177245771^3) = 7441.506416, the inner sarcen circle area of Stonehenge

1.214121857 / (1.177245771^4) = 6.321115436, a “Mayan” wonder number

Ratio between a Remen and a Not A Remen?

1.216733603 / 1.214121857 = 1.002151140 “2 Pi root” of polar circumference.

There are possibly some more new things I don’t yet understand

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

An early attempt to analyse the positions of the Aubrey holes was undertaken by Gerald Hawkins a professor of astronomy at Boston University in the 1960s using an IBM 7090 computer. In his book Stonehenge Decoded, Hawkins argued that the various features at the Stonehenge monument were arranged in such a way to predict a variety of astronomical events. He believed that the key to the holes’ purpose was the lunar eclipse, which occurs on average about once a year on a 346.62 day cycle. Lunar eclipses are not always visible as the moon may be below the horizon as it moves across the sky, but over 18 to 19 years (18.61 years to be precise) the date and position of a visible eclipse will return to its beginning point on the horizon again. As the motion of the moon’s orbit also causes it to work its way across the sky on an 18.61 year cycle in what is known as the journey between major and minor standstill and back, the theory that this period was both measurable and useful to Neolithic peoples seemed attractive.

Lunar movements may have had calendrical significance to early peoples, especially farmers who would have benefited from the division of the year into periods which indicated the best times for planting. 18.61 is not a whole number and so it cannot be used to predict an eclipse without precision equipment, using only crude marker stones or timber posts in a circle instead. Hawkins’ theory was that three 18.61 year cycles multiply out to 55.83, which is much closer to an integer and therefore easier to mark using 56 holes. Hawkins argued that the Aubrey Holes were used to keep track of this long time period and could accurately predict the recurrence of a lunar eclipse on the same azimuth, that which aligned with the Heel Stone, every 56 years. Going further, by placing marker stones at the ninth, eighteenth, twenty-eighth, thirty-seventh, forty-sixth and fifty sixth holes, Hawkins deduced that other intermediate lunar eclipses could also be predicted…

On astronomical symbolism several analysts from Gerald Hawkins^[6] to Anthony Johnson^[7] have noted that Plutarch^[8] reported that Typhon / Seth in Egyptian and Greek myth was identified as the shadow of the Earth which covers the Moon during lunar eclipses. Plutarch further records that the Pythagoreans symbolically associated Typhon with a polygon of 56 sides, hence the connection of 56 to lunar eclipses is explicit, at least for the Hellenistic era. Although less complex and romantic than Hawkins’ ‘stone age calculator’ such a technique is certainly feasible if only in theory.

I’m just going to point out a few curious things here for now for whoever can make sense out of them.

891 / 56 = ~1 / (2 Pi)

24901.5 / 56 = ~1 / 224.8

346.62 / 8 = 4332.75 = 360 / 1.03860135

5 / 89298.07632 = 5.599224738

I will perhaps have to continue tomorrow.

Breaking Symmetry

I apologize for the complexities of this, but on the other hand ancient monuments aren’t necessarily simple to interpret. Quite the contrary, they’re suggestive of prodigious ancient mathematicians whose enthusiasm for expression challenges the very laws of mathematics.

One need only consider the business at Stonehenge where 1.067438159^3 = ~1.216733603 and (1.177245771 / 1.067438159)^2 = ~1.216733603 to see a splendid example of that.

Sometimes we seem to be invited to remeasure monumental architecture – for example, remeasuring the Great Pyramid in a Royal Cubit of (1 / (360 / 2)) x (Pi^3) x 10 = 1.722570927 feet, even if the Great Pyramid is designed on a Royal Cubit of 1.718873385 feet, and even if 1.722570927 turns out to never have been accepted by ancient people as an actual Royal Cubit.

At Stonehenge, there is the business of the Squared Munck Megalithic Yard (SMMY). Again, this unit is clearly related to the Remen (9 / 1.216733603) = 7.396853327, the “SMMY” in feet, but the Meg Yard that is its square root (sqrt 7.396853327 = 2.719715671) doesn’t belong to this system of numbers.

Consequently, if we go to obtain the area for the sarcen circle with a perimeter of 120 Megalithic Yards, with a Megalithic Yard of 2.720174976, the area figure we get is a little strange, and less than ideal by some standards.

We can obtain a more ideal area figure by substituting 2.719715671 into the equation (this will be squared as the Radius in units of 2.719715671 is squared), but it’s mainly if not exclusively for the sake of the area value. The resulting radius, diameter and circumference values will be invalid of themselves, just as 2.719715671.

This is also true at the Great Pyramid, particularly since I’ve given it a base perimeter value of (1 / 9) Megalithic Yards of 2.721074976 x 10^n. If we aren’t content with the base area value, we can conditionally “remeasure” the perimeter as (1 / 9) Megalithic Yards of 2.721074976 x 10^n in order to obtain a more pleasing value.

A Megalithic Yard of 2.720699046 ((sqrt 3 / 2)) x Pi has similar problems, requiring squaring before it’s valid within the normally high precision system of numbers in question. Hence the adaptation of 2.720699046 into 2.720174076 ((1 / 1125) x (Pi^5)) which doesn’t have the problem of not belonging until its squared.

Hence 2.720174976 has come to be known as “AEMY”, the Alternate e’ Megalithic Yard, since 2.720699046 is not only ((sqrt 3 / 2)) x Pi, it’s the tetrahedral “e'” constant, the true ratio between the surface area of a sphere and the surface area of the largest tetrahedron that it can effectively contain.

We see symmetry breaking when we try to use addition and subtraction to determine mean values (see details of the sarcen circle’s mean values), and we see symmetry breaking when we try to use valid approximations of invalid numbers such as sqrt 2, sqrt 3, sqrt 5, etc. Some equations give us things like (1 / (6 x 1.177245771)) = 1.415733832 for sqrt 2 (sqrt 2 = 1.414213562), hence 2 / 1.415733832 = 1.412694925 gives us a second rather useful approximation of sqrt 2.

It’s not unusual to see “false square root pairs” like that when attempting to approximate invalid numbers like that, and while it adds to the complexity of things, more importantly it adds to the data storage and retrieval capacity of the monumental architecture in question, and data storage and retrieval seems to be at the heart of every proposal that gives the ancients credit for knowing anything.

For the Squared Munck Megalithic Yard, the symmetry breaking for its square root, since we’re using “AEMY” 2.720174971 instead of its true square root, gives us the false square root pair AEMY/IMY

(2.719715671^2) / AEMY 2.720174976 = IMY 2.719256444, the “Incidental Megalithic Yard”. It’s not really purely incidental, it’s an important figure, and if we have equations whose symmetry depends on (2.719715671^2), then to preserve that symmetry we may at times have to substitute both AEMY 2.720174976 and IMY 2.719256444 in order to maintain it.

It may sound confusing but in the long run it keeps us from littering the place with invalid numbers generated from invalid square roots, so the reality is probably that it actually works out to be much less confusing than it could have been.

This may be what I get for picking up my calculator before my coffee cup this morning, but in trying to comprehend the meaning of the ~145.2765898 ft circumference of the Stonehenge bluestone circle (it’s not likely to be a singular figure either, but we have to start somewhere), I noticed that in relation to Petrie’s unit which (in inches) is resplendent of the Venus Orbital Period of ~224.7-8 days, that using the idealized figure of 224.8373808, 145.2765898 x 224.8373808 = 326.6360794 x 10^n, which is rather suggestive of the 326.4209971 (120 x 2.720174976) ft outer perimeter established for the sarcen circle.

In the course of both trying to make these equations come out their best, and trying to turn some of Thom’s values like “17 MY” and their ratios into more realistic figures, we may end up looking at more breaches of symmetry involving the Meg Yard and Squared Munck Megalithic Yard.

Rather than divide 326.4209971 (120 x 2.720174976) by 224.8373808, it may be more advantageous to “remeasure” it as 120 x 2.719256444 before using the formula, partly because the applicability of 224.8373808 may not be limited to the first power.

Just as with the false square and false cube roots of the Remen at Stonehenge, we may be looking at more examples of ancient math prodigies who tried to bundle and unify math and metrology into a neater package than nature would actually allow.

Math can’t always give us everything we’d like to see, but the ancients may have found ingenious ways to try to bend the rules just a little sometimes.

The actual details of all this may yet have me second-guessing where we actually see “Not A Remen” 1.214121857 as a ratio between the proportions of Stonehenge’s ellipses.

It might also be possible that we are seeing breaches of symmetry that point to additional forms of the Megalithic Yard. I did manage to stumble over some equations that may point in that direction.

As if three forms of the Megalithic Yard weren’t enough, we are dealing with a mathematical system that seem to be ambitious enough, and versatile enough, to address the subtle difference between figures resembling ((sqrt 3) / 2) x Pi such as 2.720174976, and figures suitable for representation of the lunar Draconic Month of 27.212220 days (27 d 5 h 5 min 35.8 s).

Figures resembling 27.212220 or 27.212220 / 10 do appear in my calculations often enough, but I have yet to master their use even after calendars became central to this discussion a few years ago. It still remains to be seen if the mathematical messages of Stonehenge or the Great Pyramid or any other ancient monuments managed to effectively encompass this concern.

Again, though, the data I have on Asian pyramids suggests they may be an excellent place to look for clarification on such matters, and that does imply some good possibilities that someone mastered the problem long ago and left teachings written in stone, if we can rise to the challenge

For what it’s worth, the main valid figure in question is 2.721223218 – compare to the “textbook figure” of 27.212220 for Draconic Month. Particularly if ancient metrology originally derives from timekeeping as my work suggests it did, there could have been such a variation on the Megalithic Yard.

Why put up with the inconvenience of different variations on metrological units co-existing side by side? Perhaps because what we’re trying to quantify doesn’t lend itself to convenience easily.

There’s sufficient evidence of these ancient units of linear measurement having geodetic usefulness in both geodetic measurement and geodetic modelling, but either one can be a complicated matter because of the way the earth is proportioned.

In contrast to the quasi-religious overtones of authors like John Michell or Bonnie Gaunt concerning a solar system showing some sort of intelligent design, the reality is awkward mess which our ancestors may have barely managed to effectively describe by seizing on a few fortuitous coincidences and milking them for all they were worth.

Geodesy is enough to challenge modern man’s abilities because the earth or the planetary cycles don’t necessarily lend themselves easily to the neatly bundled idealized quantification that some hope for (i.e., an earth circumference of 4 x (10^n) meters or an Earth Yard / Venus Orbital Period of “Phi” as some authors suggest).

In reality, the polar and equatorial circumferences are different and hence both also differ from the mean. Do we use three different values for circumference in unit x, or three different values for unit x because of it? It’s a perennial question. Some seem to get around it by sticking to the mean circumference value, but on might want to question how much geodetic or navigational value this would actually have.

Getting just close enough to the New World to holler “Land ho!” from the crow’s nest is one thing, navigating around dangerous shoals or reefs may be another.

It’s why to this day even after being inspired by people like John Michell for decades now, I’m more likely to want to recommend pages or chapters from their books rather than recommended the books themselves – and don’t get me started on gematria.

Gematria may be a fascinating subject and it may have actually been used historically to “encode” significant mathematical information, but peppering a work containing serious metrological inquires with geometria must be a great way to get disregarded by the mainstream as a mere numerologist no matter how many very real metrological breakthroughs might be present.

Someday I will have to get back to the subject of whether John Michell railing against the metric system was more genius, or more folly. There’s evidence that the metric system may also be older than the hills, but somehow the pieces still have trouble fitting together.

It’s still difficult to sort out whether the ancients regarded (for example) 57.29577951^2 / 1000 = 3.28280635 (as feet) as some kind of meter, and it may matter to the discussion since the other data indicates 3.28280635 ft as very likely to be the intended thickness of the Stonehenge sarcen circle.

Obviously this belongs to Stonehenge’s discussions of circular math, but does it belong to Stonehenge’s discussions of metrology?

I’ll add this to the discussion for now since I already got into the subject of “false square root pairs” – the difference between 2.720174976 and 2.721223218 by ratio is 2.721223218 / 2.720174976 = 1.000385358, which may be a “false square root” of the ubiquitous and indispensable 1.000723277 ratio.

One reason 1.000723277 may have entered the picture is because of trying to simultaneously origins for various metrological units that lie in the geometry of circles, and in the geometry of squares (as in the relationship of different units as diagonals to one another), but that may be be a bigger matter for a separate discussion another time.

From here I will have to give this more thought, it’s still rather new to me. I must say I’m quite surprised to see just how integral that something amazingly like Petrie’s Stonehenge unit seems to be to the mathematics of Stonehenge. More than ever I think he was very much on the right track with that.

–Luke Piwalker

“At Sixes and Sevens”

Maybe there’s a reason that expression came to be. If you ask me, the number 7 is the work of Satan personally, and I like to think I have enough quaint old siting legends for stone circles to back me up on that just a bit.

Why is that? Well, many in Egyptological circles like to insist that the ancient Egyptians used 22/7 as Pi, and many like to insist that the decimal point wasn’t invented until when Wikipedia says so, basically. However, especially lately looking at some very impressive ancient math theory involving 22/7, it becomes more apparent that using numbers like 11 (11 x 2 = 22) or 7 may indeed try to exclude the applicability of the true Pi ratio.

I wrote recently,

I used to try and explain what numbers I do (and don’t use) as that I only use whole numbers that can be made out of 2, 3, 4, 5, 6, 8 or 9 by multiplication or division, which is true enough, but I think a better answer is here

http://grahamhancock.com/phorum/read.php?1,321308,321531#msg-321531

i.e, the math I’m using has possible roots in the sexigesimal system and only uses whole numbers that can be made by multiplication or division out of numbers from 1-10 by which 360 is evenly divisible. That may be the best reason that 7 and 11 (and 13, 17, 19, 21 etc) got kicked to the curb which made room for Pi.

Naturally people ask me (they actually don’t but I sometimes wish they would), if your crazy ancient mathematicians didn’t use the number 7, what did they use?

Good question.

One possible answer of long standing is 7.008385550. (If it’s a correct answer, it many not be the only one).

What is this number? 7.008385550 = 19.46773764 x 360.

7.008385550 / 360^2 = 5.40770490, an interesting number that’s usually good to see. 7.008385550 belongs to an important if brief series constructed by multiplying or dividing by (2 Pi)^n. 360 / 7.008385550 = 1.027340740 / 2.

The inner sarcen circle circumference of Stonehenge x 4 = 305.7985078 x 4 = 1.223194031. 1.223194031 x 57.29577951 = 7.008385550.

7.008385550 x 57.29577951 = 401.5508514

1 / 401.5508514 = 24903.44614 / 10^n, 24903.44614 being the other expression for equatorial circumference that this mathematics frequently provides us with.

If 7.008385550 is going to be 7 (at least sometimes), 22 might be 7.008385550 x Pi = 22.01748917. That’s the Meg Yard 2.720174976 x this form of pyramid Phi 1.618829140 x 2, among other things.

It’s also 2 Indus feet, using my favorite candidate for a figure that is valid within this system of numbers, Indus foot = 1.100874458 ft = 13.21049360 inches.

22.01748917 x 25 = radius Silbury Hill, 550.4372293 feet, a specification that makes Silbury Hill’s base measures not only directly conversant with those of the Great Pyramid, but with the equatorial circumference of the Earth as well.

There is more, (I don’t know if I’m capable, but I’m trying to write a brief post for once) but I thought this might be something to mention for anyone who didn’t see it coming when Silbury Hill found its way into the discussion

7.008385550 x (Pi^2) = 69.16999287 = 24901.19742 / 360

–Luke Piwalker

More About Stonehenge, Pt 2

I’d like to write this down for posterity but it is very much a work in progress and I don’t know what it means yet, if I ever will. I hope the reader will keep in mind that what is new here is only a preliminary look, merely a scouting mission, if you will.

As I’ve lamented repeatedly in recent years, while Stonehenge’s data continues to dazzle us with new things even now, all of the data we have so far comes from the sarcen circle of 60 stones. To the best of my knowledge, Munck only ever published a figure for its inner radius, diameter, and circumference and the rest of what we know about it was largely my doing, building directly on Thom’s data for the outer circumference.

Hence after all these years, we still know very little – tragically little – about the rest of it.

I went back to Prof. Thom’s text on Stonehenge (Megalithic Remains in Britain and Brittany, Chapter 11. Apparently being unable to access my other materials by Thom has likely put me out of touch with the nature of his geometry.

I’ve reported “errors” in Thom’s descriptions of triangles on which the measures of his ellipses are based, but his data may be correct if it’s taken into consideration that he may be deriving his triangles from focii of ellipses rather than the more straightforward triangles that I was constructing – not that I’m not necessarily obtaining integral values for triangle sides myself, but that question is not really my concern and the matter may furthermore be subject to metrology.

In my realm, Thom’s measures don’t have to indicate integral values of metrological units, and some of them may not, particularly when converted to some of the other units we have seen in evidence at Stonehenge like the Remen and Royal Cubit.

In apparent absence of clarifying remarks in Thom’s text, I took up the matter of the measures of the bluestone ellipse with pixel measurements taken of his diagram (MRBB pg 143)

I reckon the innermost elipse (the bluestone ellipse) to have dimensions of about 22 by 14 Megalithic Yards. 22 / 14 = ~Pi / 2. The inner trilithon ellipse measures 27 by 17 Megalithic Yards according to Thom. Taking this literally (not that we necessarily should), 22 / 17 = ~1.588, which might indicate constants such as 15.88133131 or 15.88669582, or perhaps the ratio might go all the way to slightly over 16 (16 x 1.000723277) when all things are considered.

The outermost trilithon ellipse measures 30 by 20 Megalithic Yards according to Thom, providing a ratio of 1.5 which may or may not correspond to the 15 stones that ideally comprise the trilithon ellipses.

Munck suggests that the 15 stones in the unusual “horseshoe” display (framed by the two trilithon ellipses of 30 by 20 and 27 by 17 Megalithic Yards of about 2.72 feet according to Thom) and the 60 stones in the trilithon circle (sarcen circle) indicate the applicability of the square roots of 15 and 60 to the mathematics.

I am still inclined to support Munck in this, not only on the basis of the strength of the numbers and the results, but on the possible basis of folklore that may indicate when unusual mathematical gestures may be applicable to the interpretation of Megalithic circles or other Megalithic architecture (see Stith Thompson Motif-Index of Folk-Literature https://en.wikipedia.org/wiki/Motif-Index_of_Folk-Literature)

Regarding the Aubrey Holes, I’ve long suggested a circumference of 892.9807632 based mainly on the strength of the numbers. 89298.07632 was Carl Munck’s “Grid Latitude” for the Great Pyramid, a number very carefully chosen in order to receive that high honor from Munck. Regardless of the status of Munck’s geography, it’s an important and often remarkable number mathematically.

Some other researchers use a lower figure but I’ve pointed out before that the Aubrey circle should have a thickness, just as the sarcen circle does. With the sarcen circle, we have now significant outer, inner, and mean values, and a proposed thickness for the circle itself.

Hence, if 892.9807632 ft seems overly large as the circumference of the Aubrey circle, it may be the maximum circumference value, or the outer circumference value.

According to Professor Thom (MRBB, page 146) the mean radius is 141.80 +/- 0.08 ft. Radius 141.80 ft x (2 Pi) = circumference 890.9556766 ft. 141.88 ft x (2 Pi) = 891.4583314 ft.

For the outer bluestone circle, Thom gave a value of 28.65 Megalithic Yards (MRBB, pg 145). 1/2 Radian = 57.29577951 / 2 = 28.64788976; 28.64788976 x 2.720174976 = 77.92727283 = (.8 x Pi^4).

892.9807632 / 77.92727283 = 11.45915584 = 5.729577951 x 2

Converting 892.9807632 to Megalithic Yards of 2.720174976 ft, we find

892.9807632 / 2.720174976 = 32.82805848 = 57.29577951^2

3.282805848 ft being the apparent thickness of the sarcen circle based closely on our data sources.

What does this suggest so far? Obviously a circular monument that is among other things, something of a treatise on circular mathematics in multiple metrological units including modern feet.

Comparing the 30 Megalithic Yard maximum diameter of the largest of the Trilithon ellipses to the ~28.65 Meg Yard diameter of the outer bluestone circle,

30 / 28.64788976 = Pi / 3. We already have likely found Pi / 2 as the max/min value for the bluestone ellipse, and now we’ve found Pi /3, which is an even better find than Pi / 2 when it comes to data retrieval from ancient monuments. Relatively speaking Pi / 3 works much better at higher powers, increasing greatly the amount of data that can be stored and retrieved.

Let’s back up a moment to review a little archaeological history.

At the age of 19, Flinders Petrie surveyed Stonehenge and produced the work Stonehenge: Plans, Description, and Theories

https://books.google.com/books?id=rUUIAQAAMAAJ

First of all, we are indebted to this work (and that of RJC Atkinson) for data that affords us confidence as to what the inner sarcen circle measurements should be.

SDPT pg 23 “Taking up now the sarsens and inner bluestones, the outer sarsens are 1167.9 ± .7 diameter, and the inner bluestones 472.7 ± .5 inches diameter; these quantities are very nearly as 10:4. The former has been recognised as 100 Roman feet, the latter is therefore 40 feet.”

1167.9 inches / 12 = 97.325 ft. The figure that Munck and I use is 97.33868824, or 80 Remens of 1.216733603, or 100 Roman feet therefore of .9733868824 modern feet. Petrie’s measurement is remarkably close to this mathematical ideal.

In a noteworth paper, The Acropolis Width and Ancient Geodesy, author Nicholas Kollerstrom observes

Precision of the Parthenon… The temple was constructed to be 225 feet long and 100 feet wide. In 1882, Penrose carefully ascertained this length and breadth, “measured on its upper step,“ and thereby estimated the Greek foot to have been between 12.160 and 12.167 inches. He added rather casually, in a footnote, “The breadth, 101.34 is exactly a second of latitude at the equator.“[6]…

The Circle of Stonehenge… The monument was surveyed by the British archaeologist Flinders Petrie, who had a specially made lightweight surveying chain of his own design, that could be pulled taut across uneven ground for greater accuracy. The inner diameter of the Sarsen circle he found to average 1167.9 inches. This, he affirmed, was “recognised as 100
Roman feet.“[22, 23]… Then, in 1956, the archaeologist Ronald Atkinson estimated the inner diameter of the Sarsen circle as averaging 97 1/3 feet. [24]

Penrose’s remarks, acknowledged in Berriman’s Historical Metrology, are what gave rise to my determination of a 1.216733603 ft Remen.

Returning to Petrie’s survey, there are several other interesting things at least that emerge

SPDT, page 22-23 Applying then the 225-inch basis of the bank and ditch obtained above, we find these are 16, 18, and 20 of the unit, or radii of 8, 9, and 10, numbers very likely to occur. Taking 4046 / 18 as practically the best defined, we obtain 224.8 ; but the most accurate result will be from the intermediate point, of 17 units diameter, on the crest of the bank; this is 3820 ± 2, -7-17 = 224.70 ± .12. Next, how far is this applicable to other parts? The tumuli centres 92 and 94 are about 3391 apart, and the inner faces of the stones 91 and 93 estimated at 3376 (originally) ; the latter is better than the tumuli by far, and / 15 = 225.1.’.. On trying the sarsen circle neither the inner nor outer diameter agrees to this unit, and the trilithons and inner bluestones are equally intractable…

Taking up now the sarsens and inner bluestones, the outer sarsens are 1 167.9 ± .7 diameter, and the inner bluestones 472.7 ± -5 inches diameter; these quantities are very nearly as 10:4. The former has been recognised as 100 Roman feet, the latter is therefore 40 feet* The foot by this would be 11.72, or 11.68 by the sarsens alone. The arrangement of the trilithons is obscure ; each of the pairs have their inner faces in a straight line, but there is no scheme sufficiently consistent and distinct to be worth entering on here. To sum up, there are two units shown. One in the earth circle and its mounds and stones (91-94), and also in the line of barrows, and perhaps the parallel banks. This is, by the best elements, 224.8 ± .1 inches, excluding the barrows, which give 226.7. In “Inductive Metrology” I have shown that there is evidence, from entirely different materials, for a prehistoric mean unit of 22.51 ± .02 inches, probably of Phoenician origin. The close agreement of this with 10 x 22.48 ± .01 is striking. The other unit, the foot of 11.68 or 11 .72, is as closely accordant with the Roman foot, which, though 11.64 Rome, had a mean value of 11.68 ± .01 in Greece, Africa, and England. Not that this shows Stonehenge to be post-Roman, as the unit was the great Etrurian and Cyclopean unit, originally – o derived from Egypt, and it may have been introduced at any date into Britain.

Whatever we might make of all that, 11.72 is a little bit tantalizing if we consider previous findings on Stonehenge. One result of an 80 Remen inner sarcen circle diameter is a circumference of 305.7985078, which against the 360 degrees of a circle, is 360 / 305.7985078 = 1.177245771, Munck’s “Alternate Pi”, a major, indispensible constant. 11.72 is about a mere 5/100 (1/20th) of an inch from expressing 11.77245771 in modern inches.

Also, Petrie’s observance of a possible unit of 22.48 x 10 = 224.8 is indeed striking because of the resemblance of this figure to the Venus Orbital Period in days. The most fluid and useful value for this in the math I use seems to be in fact 224.8373808, rather than things more closely resembling the “textbook” figures of about 224.67 to 224.701.

As I’ve often demonstrated, the ratio between the outer sarcen circle circumference reckoned as Thom’s 120 Megalithic Yards and using a Megalithic Yard as 2.720174976 ft = 120 x 2.2720174976 = 326.4209971 ft is

326.4209971 / 305.7985078 = 1.067438159

This number seems to be hugely important and fundamental to a great deal of this mathematics.

I’ve even pointed out how Stonehenge tells us about how it is ALMOST the cube root of the 1.216733603 ft Remen

1.067438159^3 = 1.216264898

and how 1.177245771 / 1.067438159 = 1.102870233 is ALMOST the square root of the 1.216733603 ft

1.102870233^2 = 1.216322751

I am still asked to believe that ancient people who knew that didn’t know what a decimal point was…

If we convert this proposed unit of a possible 22.48373808 inches to feet, 22.48373808 / 12 = 1.873644840 ft

1.873644840 = 2 / 1.067438159

For what it’s worth, I interpret 11.68 as 11.68064258 which of course equals .9733868820 x 12.

Now, let’s return to Professor Thom’s book for a minute where he gives the mean radius for the Aubrey circle as 141.8 +/- 0.08 ft. = circumference ~141.8 x (2 Pi) = 890.9556766.

890.9556766 / 2 = 445.4778383 x 2

1 / 445.4778383 = 224.478058 / 10^n.

If we adjust this to give 224.8373808, 1 / 224.8373808 = 444.6658999 ft; 444.6658999 x 2 = 889.5317998 ft;

889.5317998 / (2 Pi) = 141.5733830 = (1 / 60) / 1.177245771.

Thom would only be off by several inches with this no doubt more challenging measurement, if this is correct.

141.5733830 x 2.5 = 353.9334575, still the most useful representation of the “354” day Lunar Year.

Intriguingly, if we take the established inner sarcen circle circumference of 305.7985078 ft and convert it into Megalithic Yards of 2.720174976 ft, we get

305.7985078 / 2.720174976 = 112.4186902 which is half of

112.4186902 x 2 = 224.8373808

Carl Munck was fond of talking about about ancients who knew their mathematics forwards and backwards…

1 / 112.4186904 = 889.5317998

If 892.9807684 and 889.5317998 are the maximum and minimum circumference for the Aubrey circle repectively

892.9807684 – 889.5317998 = 3.448968593 = 1.72448296 x 2

I know, it looks like 2 Royal Cubits, but let’s not be hasty.

Meanwhile, Wikipedia says

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

It was found that the pits were an average of 0.76m deep and 1.06m in diameter… The holes are in an accurate, 271.6m circumference circle, distributed around the edge of the area enclosed by Stonehenge’s earth bank, with a standard deviation in their positioning of 0.4m

271.6 m = 891.0761155 ft

1.06m = 3.477690289

Should we compare that to this possible reckoning of the width of the holes?

892.9807684 – 889.5317998 = 3.448968593 = 1.72448296 x 2

If we take 892.9807684 and 889.5317998 to be the possible max and min respectively, the mean would be

(892.9807684 + 889.5317998) / 2 = 891.2562841

The ratio between the possible min and max values and the estimated mean would be about

892.9807684 / 891.2562841 = 891.2562841 / 889.5317998 = 1.001938643

I’m going to refrain for now from delving too deeply into that one, it may take some thought. In spite of some obvious candidates, this one (actually a pair of numbers near to the raw estimate 1.001938643) might have been particularly well chosen by the ancient architects of Stonehenge.

How about for now we suffice it that the possible Aubrey circle min and max values 892.9807684 ft and 889.5317998 ft themselves possibly well chosen?

In yet another possible “emulation” of the Great Pyramid (or vice-versa)

892.9807684 / 889.5317998 = 1.003877279

The same ratio contained in the Great Pyramid’s apothems, and thought to be the ratio between its base and its platform, because the Great Pyramid loves to talk about the Earth’s circumference, and in the presence of 2 Pi (as in the Great Pyramid perimeter/height ratio = 2 Pi, or a circular monument like Stonehenge), that’s exactly what the “equatorial circumference 2 Pi root” 1.003877279 does

1.003877279 x ((2 Pi^3)) = 24901.19742 / 100

–Luke Piwalker

A Few Musings on Petrie and the Orientation of the Giza Pyramids

Twelve years ago after long having to take matters into my own hands with missing details from Munck’s model of Giza, I decided that once and for all I was going to take all of Giza apart piece by piece and leave no uncertainty about their being details that Munck’s ideas couldn’t account for.

That was more or less the beginning of the end – when it came to the distances between the major Giza pyramids, I couldn’t reconcile data extrapolated from Munck’s geographic coordinates with Petrie’s data, and I certainly couldn’t reconcile Petrie’s data with that obtained from Lehner’s materials.

Here is me 12 years ago asking the resident experts at GHMB what was going on that there should be such major differences between Lehner’s data and Petrie’s as over 100 feet when it comes to how far the pyramids of Cheops, Chephren, and Mycerinus are from one another.

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

In the midst of this, I learned from Lehner’s Giza Plateau Mapping Project publication something I hadn’t known – that the Great Pyramid was equipped by an official cartographic agency with a geographic benchmark, such that we should know right where it is, versus the risky guesswork of trying to extract the coordinates for Giza’s pyramids from a 1:50,000 scale map. A piece of 1:50,000 scale Giza mapping was the only base data I could ever find in one of Munck’s publications, although my collection of his works is probably only about 60% complete due to the obscurity of a number of them.

In the case of Tikal, Munck had gone so far in the name of data quality and accuracy as to purchase data from a professional cartographic company, and Morton and I both must have managed to assume that certainly he’d done the same thing in the case of Giza, as central as Giza is to his entire model.

I decided it was time to throw away the maps and come back to this work later when I’d gotten the mapping part of it out of my head. It would be ten years until seeing a Great Pyramid thread on a fringe forum and starting to reminisce, that I ended up back at this without the maps this time – keep the math, ditch the maps – and instead focusing on metrology and ultimately astronomy, at least as far as astronomy is concerned with calendars.

Jim Wakefield having recently brought up the subject of whether Petrie’s measurements for the distances between the major Giza pyramids are meaningful (Petrie didn’t seem to think so, but there many things Petrie may not have realized for being too preoccupied with actual field archeology) the matter is trying for my attention again, and perhaps some interesting things have come up because of it that are worth sharing.

http://grahamhancock.com/phorum/read.php?1,1202170,1215565#msg-1215565

No theories, hypotheses, or proposals here – this is just a first scouting mission, just me looking over yonder to see what I can see and writing down what comes to mind when I see it, right or wrong with a good possibility of being wrong. I’m not crunching the numbers here to see how much sense things really make, just some initial impresions and musings.

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

	N	E	Direct
Centre of First to centre of Second Pyramid	13931.6 and	13165.8 =	19168.4 at 43º 22′ 52″
Centre of First to centre of Third Pyramid	29102.0 and	22616.0 =	36857.7 at 37º 51′ 6″
Centre of Second to centre of Third Pyramid	15170.4 and	9450.2 =	17873.2 at 34º 10′ 11″

13165.8 reminds me of an overly large Earth circumference in feet figure but perhaps that’s just a passing coincidence. In my mind, 29102.0 really wants to be 29201.60646, a very purposeful and geodetic number of Remens of 1.216733603 if we want to look at it that way, but perhaps something else is afoot here. 36857.7 would like to be equal to the number of feet in 1/2 of one Squared Munck Megalithic Yard if it can, but it may not be so lucky. 22616.0 is reminiscent of 72 Pi = 22619.46711 / 100.

15170.4 is probably enough to get me reminiscent about tetrahedron-spotting. Back in the day I tried to give Hoagland’s (and Harleston’s) suggestions about the importance of tetrahedra a fair trial, and pursued the mathematics of the Platonic Solids hoping to learn if it had been incorporated into the ancient mathematics we seemed to be looking at. After searching everywhere for intelligible formulas, I finally ended up having to take two of the Platonic Solids apart piece by piece and come up with my own formulas.

By the way, I think Hoagland was right (long story for some other time), someone was very keen on tetrahedra – I’m just not sure it was ancient architects.

15162.63304 is the surface area of a circumscribed tetrahedron if the surface area of the sphere which contains (circumscribes) it is reckoned generically using the Radian value of 57.29577951 as the radius in the formula A=4πr2

(57.29577951^2) x (4 Pi) = 41252.96124 “generic” Surface Area of a Sphere

41252.96124 / 2.720699046 = 15162.63304 “generic” Surface Area of Tetrahedron

Because 2.720699046 does not belong to Munck’s system of numbers (its square does but is constantly being overshadowed by more dynamic and useful things which are very close to it), I adapted this using the valid expression

41252.96124 / 2.720174976 = 15165.55428, which seems to have turned out to be a rather secular non-conversant number in spite of one notable claim to fame for being the ratio of two very important numbers

1.618829140 / 1.067438159 = 1.516555434

This being the case may still leave room to wonder about some other possibilities, such as 1.517067701 or 1.517204467 (or perhaps even 1.517652318) x 10000, although I think in general pursuing the Platonic Solids only had marginal success. Rather than entry level into geometry, the ancient math at Giza and elsewhere seems to attempt to make a language of circular and spherical geometry, and utilize it to encode statements about geodesy, timekeeping, and useful relationship between the numbers involved.

It’s difficult to see a number like Petrie’s 15174.0 and not think of tetrahedra or other Platonic Solids, since whatever this number turns out to be, it’s like to have a history of being a suspected tetrahedal surface area value.

For the distance between the pyramids of Cheops and Mycerinus, I don’t think it’s quite possible to construct a rectangle with sides of 22619.4611 and 29201.60646 with a diagonal of 36984.26666 within the current established “Giza Standard of Accuracy” for forced approximations.

For the distance between the pyramids of Cheops and Chephren,

13931.6 and 13165.8 = (diagonal, the direct distance between) 19168.4 in

We can apparently construct the meaningful rectangle 13942.74005 and 13159.47254, with accuracy on a diagonal of 19172.64382 of .999947627, although other meaningful rectangles might also be made of this.

13159.47254 x 10^n seems to make a good symbolic value for the Earth’s circumference in feet because it’s able to accommodate a number of ancient metrological units functioning to actually be able to measure the earth in neat numbers, rather than being used to merely model it across a feet:mile ratio

19172.64382 is interesting because it’s yet another we can derive from circular geometry and a Remen value of 1.216733603 (ft)

(1 / (1.216733603 x 2)) x 360 = 200 Squared Munck Megalithic Yards, a great example of how the Squared Munck Megalithic Yard is related to the Remen, and

(1 / (1.216733603 x 2)) x (360^3) = 19172.64382

The 19168.4 inch distance given between the pyramids of Cheops and Chephren is about, in Remens of 1.216733603 ft, 1312.8318119 Remens which resembles another expression of the Earth’s circumference in feet, possibly a symbolic value for the polar circumference.

Back at 15170.4 pertaining to the distance between the pyramids of Cheops and Chephren, I can say a few things about the candidate 15170.67701.

15170.67701 / (360/2) = 2.107038474, my favorite Palestine Cubit candidate as discussed in a preceding post.

15170.67701 / ((360/2)^2) = 2.926442324, something I’ve misnamed the “Real Mayan Annoyance” – it IS the real figure in question, the Maya didn’t have a monopoly on it, and it’s not a bug, it’s an unadvertised feature, which I will talk about in more detail some other time. Originally the annoyance was that it wasn’t the expected 2.920160646, which seems rather ubiquitous. Suffice it for now that we can find this “annoying” number at Giza – and Stonehenge – if we’re aware of its existence.

Perhaps more importantly, 15170.67701 / 2 = 758.5338505 x 100.

758.5338505 is what I’ve had for the Great Pyramid’s platform length for several years now, it’s quite a powerful thing to combine with the Great Pyramid’s 2 Pi perimeter/height ratio – talk about neatly bundled data!

The other thing about this number is that given my proposed mean base measures for the Great Pyramid (in its present state, lacking a hypothetical pavement layer that afforded with a second dataset in its measurements),

Platform length 758.5338505 / Pyramid Length 755.6041600 = 1.003877282

This is the same ratio between the ideal apothem of the Great Pyramid (paved) – what the apothem value would have been were the pyramid not indented slightly on the sides as indicated by Petrie and others – and the actual apothem value of 500 Remens of 1.216733603 ft each after the indentation.

If anyone’s been following my posts awhile, they may recognize the Equatorial 2 Pi Root Ratio 1.003877282. We simply take this number and multiply by the Great Pyramid’s 2 Pi perimeter/height ratio several times and

1.003877282 x ((2 Pi)^3) = 24901.19742; earth’s circumference on miles = ~24901.4

The Great Pyramid is telling us the equatorial circumference of the Earth – again!?!

This would apparently also mean that we don’t have to tear up the Great Pyramid’s hypothetical missing pavement to get the message underneath it because it’s already been built into the sides, courtesy of the indentation. (500 Remens as the apothem value also lends truth to ancient Greek historians describing the apothem length).

There still is a missing data point, though, because I haven’t been able to determine whether 758.5338505 is the width of the Great Pyramid’s platform at the top or the bottom of the platform. I did make an attempt to sort that out but I’m not 100% sure I succeeded.

http://grahamhancock.com/phorum/read.php?1,1198957,1208583#msg-1208583

http://grahamhancock.com/phorum/read.php?1,1198957,1208733#msg-1208733

I am getting ahead of myself here, though – I would like to look at Petrie’s data more carefully, including what his figures might look like in Remens or Royal Cubits.

However, the thing that most inspired me to try to write something up even at this stage of the proceedings is probably this – it’s the ratios between pyramid distances, which again are 36857.7, 19168.4 and 17873.2 inches

36857.7 / 19168.4 = 1.922836543 = ~ half Venus Cycle / (Pi^2)

36857.7 / 17873.2 = 2.062176890 = ~1/10 Royal Cubit in inches, 20.62648062

19168.4 / 17873.2 = 1.072466038 = ~1.073519414

1.073519414 is part of a powerful (Pi / 3)^2 series that was found at Tikal by the author several years ago. It may not be the right answer here, but it easily comes to mind and it’s very close to the target figure. One candidate for 17873.2 might be 17891.99025 (5 / height Great Pyramid capstone = 1.789199020 / 10) but that too is only another possibility. The pieces should fit together, not just look good on their own.

I should say that at least one other possible “1.07-” number on the low end may have surfaced since looking more closely at data from Petrie, and from Maragioglio and Rinaldi recently. They might have a surprise for us, and may have written it several times to be sure we get it?

It’s of course the appearance of the Royal Cubit in inches as a ratio here that’s my biggest motivation for writing this. It’s enough to inspire some confidence that at least we’ve gotten one right, and yes there is order and meaning to the placement of Giza’s pyramids.

–Luke Piwalker

Some Musings on the Nippur Cubit and the Persian Cubit

Some of this I’ve posted elsewhere in various forms, but I don’t know if I’ve ever written anything very comprehensive.

I don’t have much to say on the Nippur Cubit, I haven’t paid it enough attention although I did recently nominate a value of 1.689907782 ft as being one possibility, in spite of the temptation offered by 2 / 1.177245771 = 1.698880598, which is in turn near to and at possible risk of confusion with a putative Cubit value of 1.700109360 that I proposed years ago right after first reading Algernon Berriman’s book.

Berriman is one of my major influences mainly on account of his recognition of relationships between the Remen and other metrological units, and on account of his pointing them out, within several days of reading his book I had forged my nominations for the true values of many of the metrological units he discusses – much more quickly that I could believe that the 1.216733603 ft value I’d based most of them on, could truly be a primary value for the Remen used by the ancients.

Berriman in Historical Metrology described a Cubit of 20.412 inches = 519 mm = 17.01 ft and pointed out such a cubit being reported by J. H. Breasted at Karnak in Ancient Times (Breasted, 1935).

I could be wrong about 1.689907782 being a good suspect for a Nippur Cubit, the subject is admittedly a little new to me and I have NOT yet explored it very carefully, but nonetheless like Munck I do try to nominate things for good reason or else refrain, and new metrological units can often be simple and straightforward to grasp.

Given that Wikipedia says of the Nippur Cubit,

“In 1916, during the last years of the Ottoman Empire and in the middle of World War I, the German assyriologist Eckhard Unger found a copper-alloy bar while excavating at Nippur. The bar dates from c. 2650 BC and Unger claimed it was used as a measurement standard. This irregularly formed and irregularly marked graduated rule supposedly defined the Sumerian cubit as about 518.6 mm (20.42 in).^[11] “

20.42 / 12 = 1.701666666

Perhaps I should spare myself some grief and declare the Karnak Cubit and the Nippur Cubit one and the same, but while there may be a few actual instances of a Cubit of 1.700109360 ft even in prominent Egyptian pyramids, the efficacy of 1.700109360 or even 1.698880598 is another matter entirely. They may look good on paper, but do they really fly and have they ever flown well enough to be awarded stripes?

1.689907782, if it doesn’t turn out to be a big disappoinment, it might yet prove to the be the last great hope for something in this range that actually makes a good metrological unit.

All the same, 20.62648062 (the number of inches in a Royal Cubit) / Remen in feet 1.216733603 = 1.695233909 (144 x 1.177245771).

Will it be any wonder if this mystery is not quite solved yet?

This brings us now to the Palestinian Cubit.

Berriman described it as 25.25 inches = 641 mm, and sqrt 3 to the Remen. Authors like John Michell, Bonnie Gaunt and others have pointed out the simple geometric relationships between some important metrological units, adding to the foundation of Petrie who pointed out the Remen being half the diagonal of a square with sides of 1 Royal Cubit, even if he may have grossly underestimated the significance of this.

25.25 inches = 2.104166666 ft. Using 1.216733603 ft as the Remen, Berriman’s formula gives us 1.216733603 x sqrt 3 = 2.107444420, and the Persian Cubit value I prefer is 2.107038463 ft both because it may be intimately related to the mathematics of the Great Pyramid’s platform, and because it’s easily linked to a premiere value acknowledging the half Venus Cycle (aka the “Mayan” “Long Count”)

4 / 2.107038463 = 18983.399137 / 10^n, representing the canonical 18980 days in the half Venus Cycle.

4 / 2.105515606 = 18997.72205 / 10^n, the second-string premiere representation of the half Venus Cycle. What it lacks in accuracy – not that that is what matters here, this is the realm of calendars where sometime people chop 5 days off of 365 or add a day to every fourth year and etc – it makes up for in many other ways.

It’s possible that 2.10703846 might be one of those numbers suitable for geodetic modelling at a typical feet:miles ratio, rather than geodetic measurement. In my work so far, the nominations for Palestinian Cubit that might be able to actually measure the earth without going outside of the system of numbers to do it seem to be Palestinian Cubits of 2.105515606 and 2.102515666 ft, measuring possible circumference values at 675 x 10^n units.

It’s also possible there’s a small swarm of worthy and valid candidates for Palestinian Cubit values (there is also a value of 2.10632699 ft that is worth looking at and maybe the same is even true of 216 x (Pi^4) = 2.104036366).

Consider that recently, I’ve been running into rather often in my more recent “travels”, numbers that are suggestive of being “hybrids” of various established metrological units – Remen x Indus Foot and that sort of thing. I haven’t explored this more fully, but part of it could be ancients being creative with their basic building blocks, and part of could be the basic geometric relationships between units.

I had my attention drawn back to this recently when I wanted to start rounding up historical materials to explain Isaac Newton’s thoughts on the Great Pyramid and its metrology and noticed that his “Sacred Cubit” in inches was about 1.213 of his Royal Cubit in inches, i.e., “Remen x Royal Cubit” if you want to see it that way.

http://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00276

Because of the geometric relationships between the Remen, Royal Cubit, and Palestinian Cubit, we can also define a Palestian Cubit as the diagonal of a rectangle with sides of 1 Royal Cubit by 1 Remen

Using my standard values here

sqrt ((1.216733603^2) + (1.718873385^2)) = 2.105935985, so really this can be attributed to these geometric relationships between units rather than the ancients playing “mix and match” with different units, although creative metrological expression has been by no means ruled out by this example).

To do the formula “mix and match” style,

1.216733603 x 1.718873385 = 2.091411007 which is a little different, but if geodesy or any other route leads us to a global circumference of 131407229.1 ft…

In fact, if I adjust John Michell’s formulas just slightly – tweaking his 0.9732096 ft Roman foot to 0.9733868822 ft is all that is required here! – I get 131407229.1 ft repeatedly for a global circumference value.

And the associated Earth radius value (or circumference / (2 Pi) at least)?

131407229.1 / (2 Pi) = 20914110.07 = 1.216733603 x 1.718873394

In other words, the Remen and Royal Cubit used in tandem geodetically if you want to put it that way, or simply another variation on the Palestine Cubit being a functional geodetic unit. This one may not be a very long series strung together by the common linking ratio of 12, but it may be a potent series, and includes a constant, 3.011631850 that was recently found upon review to have apparent importance in spite of some obscurity or elusiveness.

I’m seeing a lot of “1.213-” and “1.214-” numbers lately (including Margioglio and Rinaldi’s data on the features surrounding Chephren’s pyramid, but if the Remen has geodetic function in the manner suggested by Algernon Berriman or Jim Alison http://home.hiwaay.net/~jalison/blu5.PDF we may want to go ahead and take the liberty of rounding them up to ~1.215-1.217.

There IS a number in the 1.214 range and it has a fairly impressive pedigree in (1 / 1.067438159) x 360^2, VERY easy to build out of things lying around at Giza therefore, but I have yet to believe this is a valid value for the Remen, and the same goes for anything in the 1.213- range even if some were to be named.

The Palestine Cubit also appears in Egyptian pyramidia (pyramid capstones), and probably not limited to the fact that these pyramidia often seem metrologically-oriented thus that there are actual reports of at least one that measures at the base 1 Remen x 1 Royal Cubit (Hawass, Treasures of the Pyramids) hence a base diagonal of 1 Palestine Cubit.

Expect that we will find a pyramidion whose edge length or apothem length (line from center of one side of the base to the tip on top) will be a Palestine Cubit, if we haven’t already.

This now brings us to some “trivia” which I hope is interesting. Knock just a little off of 1.689907782 (not that we should actually do this as they are probably different numbers) and you’re at about 1.68.

Recently I was reading an old post by Robert Bauval at the “Hall of Ma’at”

http://www.hallofmaat.com/forum/read.php?6,462034,462034#msg-462034

“The enclosure or boundary wall of the Step Pyramid complex of Djoser is one of the most intriguing architectural edifice of the Old Kingdom. The wall is 10.5 meters high and its length is 1645 meters (544.9 x 277.6 meters) But this is not just a simple wall, though. It has 192 ‘bastions’, 14 ‘false doors’, and 1680 recesses. The recesses must have had a very important significance, for as Lehner explains:

‘…the builders did not form the recesses of the huge stone enclosure wall before they laid the blocks, as modern masons would. Instead, they hand-carved each recess into the face of the already laid masonry, an enormous task since there were 1680 recessed panels on the bastions and dummy doorways, each panel more than 9 m tall‘…

If one counts the recesses and also the protrusions of the great boundary wall, the following curious ‘calendrical’ values thus emerge:

West side: 1461 (sothic period)
East side: 1459 (sothic period)
south side: 366 (sothic Quadriennium)
north side: 366 (sothic Quadriennium) “

He goes on to say

http://www.hallofmaat.com/forum/read.php?6,462034,462133#msg-462133

“1680 recesses means nothing. But when you add to this figure the ‘flush’ pannels you get a total of 4383.

4383 is 12 x 365.25 (sothic Year) or 3 x 1461 (Sothic Period). Thus the result of adding ‘recessed pannels’ and ‘flush pannels’ is too striking to ignore”.

To the 22/7 Squad, 1680 should rapidly register as 240 x 7; in my own realm where we try to ignore numbers like 7, 11, 13, 17, 19 etc etc, I’m not sure what the definitive representation would look like. Those involved in metrology and geometry might perhaps also take 1680 to be (1 Megalithic Yard / Phi) x 10 but we don’t do Phi proper at “my school” (Pi Jedi Academy) either. In my own terms, 1 Meg Yard / Phi might look like 2.720174076 / 1.61882914 = 1.680332970.

Numbers in this range began to become just a little more familiar to me with the relatively recent observation that

“(Wikipedia figures) Saturn Synodic Period 378.09 days / Venus Orbital Period 224.701 days = 1.682636036 “

At this stage, I’m still only speculating, but some notes I made:

“Here’s the part I’d like to get to but it’s gotten a little muddled on my diagram – M&R at least have 128.10 m on the north side as the distance between the base of Chephren’s pyramid and the outer side of an enclosure wall

128.10 m = 420.2755906 ft; at the middle of the N base of Chephren they still have the distance at 58.72 + 69.42 = 128.14 m, so this appears to be consistent.

If I’m not mistaken, Petrie got 420.25 feet for this expressed as “5,043 inches” (71., paragraph 3) http://www.ronaldbirdsall.com/gizeh/petrie/c8.html (perhaps M&R borrowed Petrie’s data, I’m not entirely certain yet).

What I think I find most intriguing here is that 420.2755906 / 25 = 168.1102362.

As part of the set-up here, the diagram suggests they take the length of the enclosure wall on the North side to be 388.0 m long. That’s suspiciously round, and yet 388 m = 1272.965879, and 4 / Pi = 1273.238545, and the Chephren pyramid’s massive nearby neighbor seems to be more or less built on that ratio as 8 / (2 Pi), if one wishes to see it that way…”

So why am I affixing this trivia to a discussion on the Persian Cubit of about 2.09-2.10 or so feet?

One more from my notes:

“Here is another thing, however: 420.2755906 / 2 = 2.10377953.”

I’m not sure what exactly, but besides Bauval’s posts being an excellent exposition of numbers embodied by numbers of architectural features (and what sort of numbers they are, i.e. the thread title, “Djoser’s ‘calendrical’ wall?”), perhaps Bauval was onto something here?

Happy Holidays!

May the Fours be with you!

–Luke Piwalker

Lessons With the Master

While Carl Munck’s printed materials may get harder and harder to find, some of his video lectures are circulating via Internet and are easy to locate, and I’d like to recommend them to anyone who might be even remotely interested.

Even if we end up having to ignore his conclusions about maps, his conclusions about math and numbers and the way they were used are very much worth some time and consideration.

After considerable investment in the prospect, I did go through a period of disappointment and withdrawal from the subject for discovering that maps might not be the thing to be looking at after all, but it hardly seems fair to expect trailblazers to get everything right on the first try, whoever they may be.

I can just about guarantee that any numbers Munck wants to suggest are important really are, and if they don’t turn up on maps courtesy of his “Code”, they will turn up in the proportions of noteworthy ancient architecture.

Even now, the proportions of Giza’s monuments continue to give the suggestion of being as if part of some grand plan to make sure we don’t miss out on any important numbers.

I can also just about guarantee that any numbers Munck wants to talk about, they’ve likely been subjected to almost preposterously thorough cross-referencing with other important parts of the same number system to verify a significant place in the scheme of things – metrologically, if not cartographically.

I also enjoy hearing Munck’s lectures because he has a seemingly infectious sense of confidence, and it isn’t false confidence if we remember the parts of his work that withstand even after throwing the maps away.

I’m going to wait until I find my misplaced copy of John Michell’s Secrets of the Stones before I go on about urdummheit, but the most enjoyable thing about Munck’s lectures may be the vote of confidence he invests in our ancestors, the credit he is willing to give them for being as intelligent as we are in spite of any differences in culture or customs (not to mention many of our customs having been handed down from them).

Words can hardly describe how refreshing it is to hear someone express that much respect for the ancients without turning around and insinuating (out of habit I suppose) that the ancients were so dumb they had to do their rocket science in simple whole numbers, and in fractions, and couldn’t comprehend Pi or a decimal point.

Since there really may be tremendous metrological support for the idea that the ancients beat us and Erastothenes to figuring out the size of the earth, many authors who write about or acknowledge ancient mysteries or metrology, may have conceded unrecorded advancements in geography or geodesy to the ancients, an admirably open-minded stand against which any unwillingness to concede comparable undocumented advancements in mathematics somehow seems uncharacteristic.

Another thing that’s appealing about Munck’s work is his knowledge of archaeological sites. Internet followers of ancient mysteries may know some of these sites these days, but in 1998 or 1999 when I was first introduced to Munck’s work, I’d never heard of many of the fascinating sites he mentions, even for the way-too-many books on ancient mysteries and archaeology that I own.

After all the fuss about the pyramids of Egypt or Mexico and Guatemala, it was quite an eye-opener to find out that North America had so many pyramids of its own, for example.

If you enjoy Munck’s lectures, please consider that official versions of these may be available for purchase.

https://www.amazon.com/Code-Ancient-Advanced-Technology-Complete/dp/B0007US86Q

–Luke Piwalker