# The Douglas Triangle

As 2021 drew to a close, I uploaded a new paper about the Douglas Triangle, and its use at Giza. It provides the missing link between the base sizes of Khufu and Khafre,

The triangle looks like this:

The triangle has some interesting properties, which are discussed in the paper, available at Zenodo or Academia.

As usual, soon after, I realised a few more things, and now need to update it, but that may have knock-on effect on Zep Tepi Mathematics 101 (ZTM101), so I need to run some checks first.

I have not found a direct use of the triangle (as triangle) at Giza, but have now found the three side ratios. Consider: (click images for larger view)

# Zep Tepi Mathematics 101

A few weeks back I released another paper, entitled Zep Tepi Mathematics 101, which I thought was an appropriate title, but it looks 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.

Here are some of the more important images discussed:

# The inscription on the Great Pyramid

There is an inscription carved into the rock near the actual entrance to the great pyramid. There seems to be only one decent photo available, but it is copyrighted, so you will have to accept my hand-drawn version instead:

Various people have offered their interpretation of what it means, usually based on a reading of Proto-Sanskrit or Proto-Libian or similar.

My own interpretation is much simpler. It’s basically what is arguably the most well-known mathematical formula in the world, taught to every primary school child.

How do we typically indicate “area” on a drawing? Usually by some form of shading or hatching, and indeed, Unicode includes some characters indicating exactly that, for example Today, we use the Greek letter π to represent the ratio of a circle’s circumference to its diameter. A different culture would have used a different symbol … perhaps a symbol with a diameter and a circumference, like this?

And thus the mystery is solved …

# 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.

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.

# 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)

# The Secret is as Simple as e

I updated my paper on the alignment of the pyramids again, and included some new diagrams. The most important of which is this stunning bit of simplicity. As a reminder. ⦦e means “360/e”, which is 132.437°. The pure elegance of this may be disruptive to existing theories about the alignment of the pyramids. You read it here first 🙂  (Always wanted to say that.)

You can see the paper itself for the accuracy table. Two are accurate to less than 0.1° and the other is below 0.3°.

# 408, 199 and 159

A follow-up to the 437 post, this time applying the same methodology to Khafre and the other two pyramids.

If you have seen my papers then you will know that Khafre is often connected to the number 3. This same idea pops up here.

So …. Khafre has a base length of 411 ₢. 411 – 3 is 408.

If we had a square with side 408, then the diagonal would be 408√2, or 576.999.

If we divide 1000 by that (a process which effectively takes the inverse, and fixes the location of the decimal point), then we get

1000/576.999 = 1.733104858, which is a close approximation of √3 … it differs by 0.0010540… Accuracy is of course limited by having to start with whole-cubit dimensions.

I have tried the same approach with the other two pyramids, as follows. SInce we are using smaller dimensions to start with, accuracy does suffer a bit:

# 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.