Zep Tepi Mathematics 101

A few weeks back I released another paper, entitled Zep Tepi Mathematics 101, which I thought was appropriate, but it llooks like it was a bad choice and is not appealing to the target market.

The first part is actually a simple explanation of how the Giza site was laid out, using mostly √2, √3, and √5. No Orion required.

You can find the latest version at  Zenodo (current version is 1.1.0), and also at Academia.

Here are some of the more important images discussed:

 

Six pyramids aligned, 55.5k BCE

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Somewhat cryptic message

I haven’t post anything in a while… got caught up with keyboard layout optimisation again, but am now back detangling Giza.

My guides nudged me to stumble across this last night.

At first I was flabbergasted, I was stunned…
Kept thinking I could not be the first …

Meawhile I got the distinct impression my guides were doubled over with laughter …

Rechecked the calculation on Wolfram Alpha (full values, rounded values) but it’s right… Using the rounded values, the answer is 1618.0049443… or 1618 rounded.

Asked Google who else has seen this but could not find anything … if you know, please leave a comment …

So this is a somewhat cryptic image … those “skilled in the art” will immediately understand what it is about and what it means.

The dimensions of the rectangle are thanks to John Legon.

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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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