Even though I’ve been staring at these numbers and relationships since 2018, I only tumbled onto this sequence this week. Those Egyptians loved mixing their units of length into irrationals.
The sequence is this, it’s a set of ratios:
F : ₢ :: 1 : é :: ℳ : φ² :: ₷ : π.
F = foot 0.3047 m (probable original length, not 0.3048 as declared now)
₢ = royal cubit 0.5236 m
1 = 1 metre
é = e – 1 = 1.71828…
ℳ = 1 + ₢ = 1.5236 ( == 5 feet)
φ² = golden ratio squared
₷ = six feet = metre + cubit + foot
π = pi.
While poking around Giza, I stumbled on this approximation for the inverse of the fine structure constant. It’s not as good as my previous one, but since it relies on cubit -> metre conversion, it qualifies as “interesting.”
It’s just 100φ² ₢, expressed as metres, So we just multiply that by π/6, and voila!
Which according to Wolfram Alpha,
100 / (100φ²π/6) / (fine structure constant)
is 99.9679457% accurate …
Well, according to the current known value, which is a moving target.
The fine structure constant is described as one of the most fundamental physical constants. I don’t pretend to understand anything about it apart from its value, which is close to 1/137. We’ve only been using it for around 100 years.
So the fact that the Khafre pyramid uses 137 as a size multiplier (base is 3 x 137, height is 2 x 137) is annoying, especially since the Khufu pyramid references the speed of light in metres per second, twice. The ancient Egyptians are not supposed to know those things.
Since the fine structure constant itself is a messy decimal number (0.0072973525693…) it is usually referenced via its inverse, as 1/137, which provides a reasonable approximation.
However there have been attempts to find a better approximation, and Scott Onstott has a page with some approximations. Which naturally was a challenge I could not resist….
So I saw that one of the better ones was based on 137²+π²…. and since the royal cubit is π/6, and I have lots of approximations for that, it was simply a matter of finding the right one that produced a better result. That turned out to be based on the plastic number… so without further fanfare, here we are:
While playing around with the calculator, looking at relationships between the numbers at Giza, I took another look at the diagonal of the great pyramid.
The great pyramid has a base of 440 ₢. This means the diagonal is 440√2, which is 622.25 ₢. This is about 4 ₢ more than 1000/φ, which is 618 ₢. The error percentage is about 0.6%, which is annoyingly close.
So I got to wondering what base size would produce a diagonal more or less exactly 1000/φ.
The answer turns out to be 437, which is 3 ₢ less the existing size.
437√2 = 618.011 which is as close as you are going to get using whole-cubit dimensions.
I have no explanation for this. It just highlights again the ancient origins of the metre, inch and royal cubit, and how they mysteriously link together with π and ⅇ. But what about φ you ask?…. here you go:
So I’m watching a video from 2012, featuring a young-looking Graham Hancock
Which, before they got all wooshie about the Mayan 21 December 2012 end of the world thing, had some interesting info. In particular, the observation concerning the difference between two sides of the great pyramid, and its height, which goes like this:
two side of pyramid = 440 x 2 cubits. In metres, that’s 460.7669 m.
height of pyramid = 280 cubits = 146.6077 m
Difference = 460.7669 – 146.6077 = 314.1592 which you may recognise as 100 times pi.