I think a few metrological discoveries might have fallen through the cracks the past year. Early last November, I posted an incomplete compilation of which units belong to which metrological unit families, but at least several things are notably absent from it.
One has come to light again while working with projecting hexagonal figures onto Stonehenge, following what I believe to be rather important observations about geometry and metrology by drew at GHMB.
Essentially I think what drew is showing us is that Stonehenge is proportioned in places like a generic reference to hexagonal geometry, thus I’ve recently been experimenting to see what else hexagons might teach us about geometry and metrology if we project them onto the circles of Stonehenge.
“Not-Phi” = Egyptian Royal Foot
When we get to the Stonehenge outer bluestone circle, we see 1.618829140 (“Not-Phi”) showing up prominently, and showing how removed it is not from more conventional metrological figures such as the Harris-Stockdale Megalithic Foot
As before, the inner hexagon is made of six triangles with edges equal to the Radius, so Radius being 77.92727283 ft, the perimeter of the inner hexagon would be 77.92727283 x 6 = 467.5636369 ft.
This is an important number that was prominent in Munck’s work although it has been rarely seen of late, but it does have its reasons for being.
The triangles would have height of 77.92727283 / sqrt (4/3) = 67.48699792 and this intrigued me that we should see what looks like the Radian x Megalithic Foot (57.29577951 x 1.177245771 = 67.45121412 ft again so soon in this exercise.
The outer hexagon would thus have perimeter of 67.45121412 x 6 = 404.7072847, and that’s where things got very interesting, because 404.7072847 x 1618.829139.
…a long time ago I worked with tetrahedra, and adapted the surface area of sphere / surface area of circumscribed tetrahedron, 2.7201688946, to the Megalithic Yard value 2.720174976, such that for a generic sphere of surface area obtained for A=4Pi*r2 by making the Radian value the Radius value so that 4 Pi x 57.29577951^2 = 41252.96124 and 41252.96124 / 2.720699046 = surface area of tetrahedron 15162.63304; with the substitution of 2.720174976 for 2.720699046 we obtain 15165.55428 as a generic surface area for a circumscribed tetrahedron.
It’s been noted that 1.618829140 / 1.067438159 = 15165.555428 / 10^5, but perhaps it tends to get overlooked that 1.618829140 x 1.067438159 = 1728/1000.
So there is what it is – although it may keep trying to fall off the radar because it comes with different rules that a 1.622311470 ft Assyrian Cubit, 1.618829140 may be a real ancient metrological unit value, and what it really is would be the Hashimi Cubit / Egyptian Royal Foot.
As such, the value should be added to the list of metrological unit values under the Egyptian Royal Foot.
“Shaved” Meter = Palestinian Cubit
Also, awhile back it was observed somewhere in my writings that a meter of (57.29577951^2) / 1000 / 1.000723277 ft would be the second known member of the Palestinian Cubit family
3.280433687 x 2.107038476 = 6912 / 10^n is our proof of this.
Long Royal Cubit = Megalithic Foot
Apparently it has avoided mention, but because we know that the Long Royal Cubit (1.718873385 x 1.000723277) is synonymous with the Long Indus Foot, and the Long Indus Foot is in turn synonymous with the Megalithic Foot, then the Long Royal Cubit belongs to the Megalithic Foot family.
The Short Remen / Thoth Remen Family
Yes, dear reader, you heard that right. The Short Remen and Thoth Remen belong to the very same family of ancient metrological units, which seems to have been recently realized in essence back when the two related meter values were equated with one another.
Naturally, this family includes shorter unit versions based on the Remen such as the Short Greek Foot (10 / (Pi^2)) ft, but it also includes one of the shorter forms of the Palestinian Cubit, 2.105515605 ft, as opposed to the standard value of 2.107038476 ft.
Still Unaccounted For?
Two of the most important ancient units of measure that are still unaccounted in conventional historical terms for outside of several important variations on the Megalithic Yard would of course be 1.676727943, and the Stonehenge outer sarcen circle radius/diameter unit.
We know they exist because of all the pointers to them, but putting a name to them and determining the optimum unit values involved remains challenging, even when we are able to project the exact unit values.
Things were similar with the Egyptian Royal Foot. I used to think there was a separate unit that I jokingly called the “Ellifino”, which turned out to be 1.8 Hashimi Cubits, and then the Hashimi Cubit of 1.067438159 ft turned out to be 100/108 of the Egyptian Royal Foot, making them synonymous. Now we know that these units all belong to the same family along with Petrie’s Stonehenge Unit.
I have recently demonstrated that the height of the Great Pyramid would also be based on this later unit, whatever the correct quantity of units would be.
Our proof of this is that 10 times the outer sarcen circle diameter / height Great Pyramid (unpaved) = whole number or whole number x or / 10^n.
519.515151515 ft / 481.0325483 ft = 108/100.
I have proposed that the base unit may be an oversized Assyrian Cubit of 1.622311470 x 1.000723277 that comes with its own set of rules, as would an undersized Assyrian Cubit of 1.618829140 ft also apparently, but somehow this seems to remain purely speculative for lacking stronger proof.
(1.622311470 x 1.000723277 = 1.623484851 ft)
1.623484851 x 32 = 51.95151515
(1.623484851 x 1000) / 481.0325483 = 3375/10^n, but both of these conversion values seem a little bit strange. 32 as a conversion factor isn’t unheard of but 3375 is one of those numbers that it is so rare to see that it’s most likely best avoided.
As far as I was able to tell, we can probably file the Megalithic Cubit of Athanasios Angelopoulos under the standard Palestinian Cubit as well.
Unit Interactivity
There’s a great deal of interactivity between standard unit values, and while this may not help to identify units, it may help to identify their legitimacy and perhaps even sometimes their origins. I’ve written about this before as the “unit x times unit y” phenomenon and so forth as it became more noticeable when adding a few new units to the repertoire.
(So far I think the total number of true ancient units of measure may not be likely to exceed 12; I find it remarkable that we can reduce such a list of ancient units down to a figure this small).
I mention this here because it was was noted that Royal Cubits x Megalithic Yards = Stonehenge sarcen circle radius/diameter unit.
It was also noted that the inner sarcen circle circumference of Stonehenge (technically in units made from Megalithic Feet) / Thoth Remen = value in Egyptian Sacred Cubits.
We can reduce this to Megalithic Feet / Short Remen = Sacred Cubits.
Additionally, it’s recently been observed that Long Royal Cubit / Palestinian Cubit = 3 Draconic Megalithic Yards, which reduces to Megalithic Foot x Palestinian Cubit = Draconic Megalithic Yards.
Thus the pointers to the legitimacy of some of these units continue to accumulate.
We might wish to notice
As before, the inner hexagon is made of six triangles with edges equal to the Radius, so Radius being 77.92727283 ft, the perimeter of the inner hexagon would be 77.92727283 x 6 = 467.5636369 ft..
1.718873385 x 2.720174976 = 467.5636369 = 9 x 51.95151515.
Thus 46756.36369 has managed to receive honorable mention twice in the course of a single day. I am happy to see this because I’ve generally had a rather hard time finding 46756.36369 although we knew it was a rather important number because it was Munck’s “Grid Latitude” for the Mycerinus Pyramid.
Thus we now begin to find it has some metrological value – perhaps it could even prove to be a “Unifying Value” in the sense of Harris and Stockdale
A Rarely Mentioned Metrological Mystery
It might be time to start having another look at this because I don’t think it’s been discussed in awhile, but the base diagonal from Munck’s Great Pyramid model keeps trying to be some sort of metrological anomaly.
The value has been specified in my work as 1067.077718 ft, and yes it’s sort of a heartbreak that it couldn’t come out 1067.438159, but that’s the way it works out. We know some of why 1067.077718 is important mathematically, but metrologically it thus far seems rather enigmatic.
Since the discovery of the C value for the Venus Orbital Period at Stonehenge, I have wondered how many valid versions of the Petrie Stonehenge Unit there might actually be.
This doesn’t of course actually mean that the VOP C value or the Great Pyramid base diagonal at the pavement are metrological values, but what is intriguing here is that (24 x 10^n) / 1067.077718 = 224.9133272, Venus Orbital Period C, just as (24 x 10^n) / 1067.438159 = 224.8373808, Venus Orbital Period A.
Apparently then the Great Pyramid base diagonal can at least lay claim to being an astronomical value.
–Luke Piwalker
Luke Piwalker 