# Phi redux

I started poking around Giza and ancient Egypt after watching a video on the Nebra disk, and trying to find a similar circle-split-by-phi in the alignments of the Giza pyramids.

I didn’t find what I was looking for at the time (in terms of phi being in the alignment of all three pyramids) but did find lots of other things, including phi between P1 and P3.

In the mean time, the so-called Genetic Disc from Peru came along, and I immediately noticed a similar division there. Here’s the two sides of the disk:

Genetic disk from Peru

# 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.

# Poly wants a cracker

Some updates to the zeptractor, probably done for now until I figure out if this was just a humongous waste of time or not. Well at least I got to learn something about svg drawing.

Changes:
1. added assorted famous regular polygons: triangle, square, pentagon, hexagon, heptagon, octagon, dodecagon.
2. resized inner circle to divide radius in golden ratio. Also marked where circumference would be divided in golden ration (both directions, measuring from zero).
4. added tau divisors … they’re more logical than the pi divisors. Arial has a sucky tau symbol.
5. put phi symbol on bare radius to avoid confusion.

The zeptractor, the tool for people who really know their way around a circle.

# It’s New! It’s Improved! But does it Have a Point?

Well actually it does have a point, right there in the middle.

I’ve made some cosmetic improvements to the zeptractor, so it now looks like this:

Improved zeptractor

# Wheely, guys?

After making the Zeptractor, which looks like this:

The Zeptractor

I felt that it had something to tell me…. I just didn’t know what.

So as per usual when my intuition is being coy, I resolved to stare at it until the secrets revealed themselves.

# Introducing the Zep

[As per usual I’ll probably revise this over time, changelog is at the bottom.]

This post introduces a new way of dividing the circle. We already have the normal 360° method inherited from the Babylonians, as well as the French misguided attempt to use 400 Gradians, and even binary degrees that divide a circle into 256 parts.

As I delve into the Egyptian way of measuring things, it seems possible that they divided the circle into 336 parts, which I will call Zeps (from Zep Tepi, the first times). I don’t think the classical Egyptians used this, but maybe the people from Zep Tepi who most likely built the Great Pyramid, and came up with the cubit and related measures, did. Reasons why I think so will become clearer during this article.

# Inching along

The British/Imperial inch as we know it today has a long and varied history, stretching back to ancient times, as does the foot. Different countries and empires and ages had different values, which you can read about here, here and here.

The inch is currently “defined” at 2.54 cm “exactly”, but there is no record of them measuring any king’s thumb to get that figure, so I’m guessing other factors convinced them to pick that value.

But as it happens, it is probably a good choice, because there’s a nice relationship between the Royal Cubit and the inch, via the Egyptian digit.

If we take, as I believe was the original case, a cubit as being 1/6 of the circumference of a circle with diameter one metre, then a digit will be one 28th of that.

Then we simply multiply by e/2, where e is the base of the natural logarithm.

That gives us 0.0254 m, or one inch.

From the digit to the inch.

We can simplify that down to the following, noting that both 12 and 28 were important numbers to the Egyptians.

pi and e inch formula

# The Descent of Egyptian Rulers

The problem:

There is a lot of confusion about the actual lengths that Egyptians used to measure things.

For example, the Royal cubit is given as 0.51m to 0.55 m, which is quite a range. The cubit is quoted at “about 18 inches” or 44 or 45 cm, while the poor remen jumps between 0.3701 and 0.3750 m.

Then some sources claim that a cubit is 24 digits and the royal cubit 28 digits, while the best example of an actual cubit rod, in the Turin museum, clearly shows that the digits in the cubit are shorter than the extra four needed to make the royal cubit.

So why so much confusion?

Possible causes:

Having played around a lot with the numbers, in the process discovering or rediscovering assorted formulas both for the royal cubit and the parts of the cubit, with varying degrees of accuracy, some ideas bubbled up.

# How Cubit and Royal Cubit are connected

The strange system the pyramid builders and/or the ancient Egyptians used for measuring has been bothering me for a while. On the one hand, we have the logical divisions of the cubit, into nice 15cm chunks and smaller, while on the other hand we have the royal cubit as π/6 which is a horrible irrational number.

So on the face of it it seems odd that there should be some relationship between a cubit at 0.4500m and a royal cubit at 0.5236+ m. The difference is 0.07356+, and trying ratios does not get us very far.

But my intuition was insisting that there must be some easy connection between the two. So I played around a bit, and noticed that cubit/royal cubit was around 0.8594, which rang a bell, as 0.5236 x phi is 0.8472.