Elegance defined

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.

where
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.

Approximating the Fine Structure Constant (again) aka There’s Something About φ

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!

137.0799391

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.

Approximating the Fine Structure Constant

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:

\frac{1}{\alpha} \approx \sqrt{{137^{2} + (\frac{\rho^{9}}{4}})^{2}}

 

The WolframAlpha version of the formula is √(137² + ((plastic number^9)/4)²), which produces

137.0359992970725102075551820936909082242952278384186387671

We can check the percentage accuracy as well:

100/137.0359992970725102075551820936909082242952278384186387671 / (fine structure constant)

produces 99.9999998% accurate. Not bad.

 

 

437

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.

But the oddness does not stop there.

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34 Ways to calculate the Royal Cubit

Updated versions of the different ways of approximating the cubit. Also includes separate table for Grand Metre (1 plus royal cubit) == 1.5236…. == 1.524.

These are done with pi, phi, e, roots and powers (usually of basic primes), as well as ln, log, sin, cos and tan.

See square roots, cube roots and ln(4) for formulas not shown below.

Changelog at the bottom.See also The Magical Mystical Royal Cubit for the Pretty Picture version.

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The Spooky Stuff

To borrow a phrase from Robert Bauval, this falls under the Spooky Stuff category.

It is a very strange connection between the Grand Metre (1 + royal cubit), the base of the natural logarithm ⅇ, and the royal cubit as measured in inches.

Changelog:
2018-11-29: added Spooky Stuff 7 and 8
2018-12-03: added Spooky Stuff 9
2018-12-04: added Spooky Stuff 10
2019-04-24: added Spooky Stuff 11

Royal cubit, e and inch

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:

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The royal cubit and the stars

There are still some things that bother me about the Royal Cubit, in particular, why did they choose π/6, and why did they use it in preference to the metre.

Something interesting related to the first question has surfaced. Which may just be another random co-incidence like all the others, or maybe not.

First up, a reminder that the Royal Cubit is ⅙ of the circumference of a circle with diameter 1 metre, in other words the arc on a 360/6 = 60° segment, as follows:

Circle, 60 degrees, royal cubit

As you may know, the ancient Egyptians were very fond of their ankh, which looks like this:

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Playing with numbers

Playing around with the great pyramid.

Length of base + height = 440 + 280 = 720 royal cubits.

720/360 = 2 (save this for later)

If we convert to digits by multiplying by 28 (because 28 digits in royal cubit), we get

720 x 28 = 20160

If we divide by 360, we get

20160/360 = 56, which is the number of digits in 2 cubits (see saved value of 2)

As you may know, I’m developing a theory that whoever came up with the cubit system also divided the circle into 336 Zeps. So if we divide by 336 instead….

20160 / 336 = 60

Which is um, all sorts of things, and maybe a hat-tip (or the finger) to the Sumerians and their sexagesimal system.

But what the 60 also equates to, is if you had a wheel of radius 1 metre, then 60 rolls of the wheel would equal the sum of the side and the height.

If your wheel had a diameter of 1 metre (like a chariot), then it would be 120 turns of the wheel. Exactly.

Another large link to the metre

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.

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