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
Luke Piwalker 