Ancient Geodesy
I almost hesitate to leap from metrology to geodesy (Wikipedia: geodesy, also named geodetics, is the scientific discipline that deals with the measurement and representation of the Earth) – I have enough unbelievable things to tell people about the mathematical prowess of ancient people as it is.
Even some of the most open-minded people seem to struggle to get past the idea that the ancients could only work in fractions. Perhaps in some backwards way it’s good fortune that I’ve seen what it looks like when a pyramidologist takes that premise to such extremes that the intrinsic folly of it starts to become obvious.
I must be about as open-minded as they get but I still cannot get my head around the idea of ancient Egyptians doing actual rocket science in fractions. That has to be going too far with what must have started as a relatively plausible premise.
Sooner or later, both Egyptology and pyramidology may have to face it that the surviving papyrii showing math using fractions may be simplified, generalist works, not mathematical treatises reflecting the state of the art. Some of these papyrii cover so many subjects that they very much seem to be generalist works, in fact.
It’s inevitable, though, that the next incredible thing I have to share is that the ancients seem to have been keenly aware of the size of the earth, and seemed to delight in finding ways to incorporate this knowledge into the proportions of ancient monuments and architecture with an insistence that seems to imply that once upon a time, this knowledge was lost with potentially disastrous consequences, and they were determined that it should never be lost again.
I should note that it’s remarkable that no matter how different the mathematics of individual proposals, this basic idea comes up time and time and time again in interpretive works on the pyramids, as if we are all intuiting the basic truth. Again, even Sir Isaac Newton is among those who seemed to have such a “gut feeling” that the study of Egyptian pyramids would reveal knowledge about the size and measurement of the Earth that exceeded that of his day.
How did they know this? The ancient Greeks were able to work it out fairly closely, depending somewhat on exactly how one chooses to interpret the historical record. It doesn’t seem that implausible for someone to have beaten them to it, and gone after the problem more diligently or creatively until they eventually achieved accuracy to rival modern standards.
https://en.wikipedia.org/wiki/Eratosthenes
“Geodetic 2 Pi Roots”
When working with Tikal intensively for the first time several years ago, it wasn’t long before I encountered a “geodetic 2 Pi root”. There are at least three of these, and I call them “2 Pi roots” because when they’re multiplied by 2 Pi to an unspecified power (in this case, the 3rd power), they generate significant figures, and the “geodetic 2 Pi roots” generate significant geodetic figures.
The system of mathematics I use allows representation of the Earth’s equatorial circumference (Wikipedia, Earth: 24901.461 mi) with great fluidity and resonance as accurately as 24901.19742 (miles), accurate to about 1/4 mile, notably better than a number of modern mapping datums of the 20th century.
24901.19742 belongs to a whole series of numbers linked together by the ratio 2 Pi. The Great Pyramid, which probably commands more attention than any other, has been reckoned by numerous researchers ever since the science of modern pyramidology began, to be an embodiment of 2 Pi, the ratio between the circumference of a circle and its radius, because of its apparent 2 Pi perimeter/height ratio.
In this “geodetic 2 Pi series” at 24901.19742 / 2 Pi to the third power, we find
24901.19742 / ((2 Pi)^3)) = 1.003877283
I seem to have found this ratio in the Great Pyramid, repeatedly. The ratio between its adapted apothem were it not concave on the sides, and its actual apothem which gives truth to ancient authors describing its apothem length as “one stadium” (alt. stadia or stadion). Taken as 500 Remens of 1.2166733603 ft, 1.216733603 x 500 = 608.3668015 ft, which requires slight indentation of the sides to achieve (this concavity of the Great Pyramid sides has been both photographed and measured; I refer to Flinders Petrie’s data on the matter).
Without this indentation, the apothem (again, the length of a line from the center of a side of the base to the very peak at the top) would be 610.7875012 ft. Because the use of addition, subtraction, or trigonometry often produce figures that require slight adjustment to be valid, this was adjusted to the valid figure of 610.7256118 ft ((1944 x Pi) / 10).
610.7256118 / 608.3668015 = 1.003877283
This is only one of MANY ways of referencing the size of the earth that has been found in my model of the Great Pyramid, which is an extension of Carl Munck’s basic model. The apothem values are part of my own contribution to it, since to the best of my knowledge he never published valid figures for them, nor do I know if he ever acknowledged the concavity of the sides as Flinders Petrie did.
I also seem to have “discovered” 1.003877283 as the ratio between the size of the Great Pyramid’s base and the size of the platform it sits on, although even with the data from Lehner and Goodman via the impeccable Glen Dash, it’s hard to be certain if this involves the size of the platform at the top or the bottom of it.
The three main “geodetic 2 Pi roots” I recognize are 1.003877283, 1.002151142, and 1.003151727
1.003877283 x ((2 Pi)^3) = 24901.19742 (miles) / 100, 1/100 of the Earth’s equatorial circumference (Wikipedia: 24901.461)
1.002151142 x ((2 Pi)^3) = 24858.38047 (miles) / 100, 1/100 of the Earth’s polar circumference as best represented by the system of numbers I use (Wikipedia: 24859.73)
1.003151727 x ((2 Pi)^3) = 24883.20000 (miles) / 100, 1/100 of the Earth’s average (mean) circumference as best represented by the system of numbers I use. Some other pyramid researchers use this mean figure also.
We may find another member of the equatorial 2 Pi root series at Teotihuacan, where the complementary data retrieval tool, 2 Pi, may be posted repeatedly and rather obviously. This is 24901.19742 / ((2 Pi)^2)) = 6.307546992 x 10^n, which may be the ratio between the size of the Quetzalcoatl pyramid and its surrounding enclosure. 6.307546992 may also be the perimeter/height ratio of an unspecified major Egyptian pyramid, perhaps one that is presently being taken as 2 Pi.
Thus far, the pyramids that seem to have been solved at Giza seem to have their own unique perimeter/height ratios, which is a trend that may continue so that we may yet find there (or elsewhere in Egypt), important figures like 6.307546992 and 6.115970155 as the perimeter/height ratios of various pyramids.
Geodesy at Tikal
Both Carl Munck and Michael Morton share in the credit for “my” “discovery” (re-discovery) of the figure 24901.19742. Both got within a hair’s breadth of it but stopped, so to the best of my knowledge I am the first to ever publish it. It was first discovered working with the Great Pyramid, originally according to Munck’s “geomathematical” (map-based) data on the Giza pyramids.
Munck also observed that according to Teobert Maler’s data, the platform width of Tikal Temple I was, when converted from meters to feet, 24.901, which he astutely recognized as 1/100 of the earth’s equatorial circumference at a modeling ratio of modern feet to modern miles. You can see this drawn and labelled in Munck’s own hand just before halfway down the page here
http://www.viewzone.com/bigpicture/bp112311.html
Lest anyone think this is mere coincidence, Maler’s data for the El Castillo pyramid at Chichen Itza shows that they seem to have done the very same thing there too, only they wrote it “backwards” as its reciprocal. Several years ago I managed to extract a second figure for the earth’s equatorial circumference from the platform length of Tikal Temple I. It’s not as good as 24901.19742 but it is valid and it is sometimes a mathematical fact of life, and since it rivals the accuracy of some 20th century mapping and geodesy, it’s hard not to accept it.
So by all appearances they were doing at Tikal exactly what Munck thought they were doing at Giza, posting geodetic data at the summits of important pyramids.
My first geodetic discovery at Tikal happened many years ago. I’d just come in the door with photocopies from a college library of Maler’s study of Tikal, and in the first five minutes I discovered that the data gave us a diagonal for a pyramid base that looked like 111.5419203 ft (an important number, for a number of reasons).
As it turns out, 111.5419203 is the square root of ten times 1/2 of the earth’s mean circumference as expressed here
111.5419203^2 = (24883.2 x 10) / 2
As with Giza, it seems rather insistent (and rather creative) the way these geodetic figures are expressed at Tikal.
The Pyramid of Meidum and A “New” “2 Pi Root”?
I’m filing this under “2 Pi Roots” for now since I don’t know quite know what to do with it.
Several years ago working with Flinders Petrie’s data for various Egyptian pyramids, in which the sides of these pyramids often appear to be somewhat unequal, I noticed that some of the ratios between uneven sides resembled some of the “geodetic 2 Pi Roots”. I began to wonder if these pyramids were deliberately made irregular in order to express these “geodetic 2 Pi Root” as ratios between sides, and designed a hypothetical pyramid that could express the three main geodetic roots as well as the main equatorial / polar circumference ratio.
I’ve mostly left it alone since then because I probably want to retract the idea that this was achieved by using different values for the Royal Cubit for each side, even though this idea was heavily influenced by some of Petrie’s comments during his struggles to make simple numbers of Royal Cubits out of everything.
It’s been so long since I looked at it then, I was curious what pyramids might have originally inspired this hypothetical pyramid model, and working afresh with the step pyramid of Meidum, I again encountered several ratios between sides resembling the “geodetic 2 Pi Roots”. I re-examined the ratios between adjacent sides for additional Egyptian pyramids examined by Flinders Petrie, hoping to see if there was one that actually matched my hypothetical pyramid model.
Sneferu’s step pyramid at Medium is strange and intriguing – today its remains stand as a step pyramid, even though it’s supposed to have been finished into the same sort of pyramid we see in the major pyramids of Giza.
The step pyramid of Meidum represents a number of research opportunities, including the chance to examine the mathematics that might have been concealed beneath the surface of an Egyptian pyramid in the proportions of the individual steps and the layers used to build them.
It also represents the possibility opportunity to examine relationships between interior and exterior mathematics and proportions of an ancient pyramid.
Since I began working with this subject, I’ve held the view that ideally, any data concealed in a pyramid should also be presented on the exterior – it seems rather senseless to have to tear a pyramid apart to retrieve its data – but even now this is still only a hypothesis that this is what we will eventually find.
The step pyramid of Medium is a little strange in its exterior however. I’m not absolutely certain if these remarks are still valid today after over 100 years of additional discoveries, but Petrie stated that it was built just before the Great Pyramid, and probably unbeknownst to Petrie his data for the pyramid of Meidum makes it look like exactly that mathematically.
Petrie and other sources suggest that like the Great Pyramid, the finished pyramid of Meidum was a 2 Pi pyramid (perimeter/height ratio = 2 Pi), but the resemblance between the two may go deeper than that. Petrie’s measurements and calculations give 3619 inches as an approximation of its original height.
3619 inches / 12 = 301.5833333 ft, which is strikingly close to 1/10 of the Great Pyramid’s 3018.110298 ft perimeter as determined by Munck.
In other words, at least at first glance, Sneferu’s Medium pyramid looks a lot like the Great Pyramid with its mathematics and proportions re-arranged. Thus far, it literally tends to look like the “father” of the Great Pyramid mathematically.
In re-examining some of Petrie’s data for various pyramids and the ratios between unequal values for their sides, a pattern may emerge (data from Petrie in inches)
Sesostris II (El-Lahun)
At pavement N 4161.4 E 4174.5 S 4168.9 W 4169.3
4169.3 / 4161.4 = 1.0018983995
4168.9 / 4161.4 = 1.0018022780
Sesostris II subsidiary (Queen’s) pyramid (El-Lahun)
N 1071.2 E 1069.7 S 1072.3 W 1073.2
1073.2 / 1071.2 = 1.0018670649
Out of three additional pyramids, figures starting “1.0018…” appear at least twice already.
So is there a figure in this range that is significant?
It turns out that there actually is – it’s 1.001812743, and it’s significant as a “2 Pi” root
This is about where it starts to become recognizable
1.001812743 x ((2 Pi)^6) = (1 / 1.622311470) x 10^n
1.001812743 x ((2 Pi)^7) = sqrt 15 x (10^n)
1.001812743 x ((2 Pi)^8) = (1.216733693 x 2) x 10^n
1.001812743 x ((2 Pi)^9) = 152.8992541 x 10^n (half the inner circumference of the Stonehenge sarcen circle in feet
1.001812743 x ((2 Pi)^10) = 9606.943469 x 10^n – that’s 3018.110298 or 301.8110298 / Pi, which were just discussed up the page
The series continues to be useful to at least as high as (2 Pi)^13, 2 Pi to the 13th power. I’m rather impressed.
You can also see in Petrie’s pyramid data from El-Lahun that there may be a second ratio that is very similar to 1.001812743. I don’t have a guess yet what it might be.
It seems as if the ancients still have a lot to teach us. They probably should, they had a very long time to get that good at math, believe it or not.
–Luke Piwalker
Luke Piwalker 