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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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).
3. added radian marks (up to 6, only in normal direction).
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.

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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.

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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.

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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.

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8 Different methods of getting the 10 parts of a cubit

I’ve rearranged the formulas in Cubit parts and more, updated to be sorted by the general method used. This shows the patterns between the different divisions better.

cubit-parts-grouped.pdf

Changelog:
2018-12-16:
Added some formulas for Royal Cubit and Great Span (half royal cubit)
2018-12-18: Added formulas based directly on π, and also √2/π
2019-01-15: Added formulas for “double pole”, standardised a lot of the formulas to use same denominator, to better show link between formula and number of digits, added page for formulas based on e/π∛3, added tau-equivalent formulas to pi page, added inch, and assorted minor tweaks and corrections.