Cubit parts and more, updated

[Note: for older version and discussion, see No body parts were harmed producing this post. ]
[Note 2: for newer version, see 5 Different methods of getting the 10 parts of a cubit. ]

[Changelog:
2018-11-26:
My gut had been insisting for a long time that there was another formula involving pi, phi and e involved with these cubit parts, and eventually today I found it. Not only is it incredibly simple (in basic form) but also incredibly accurate. The basic form is simply (phi x e)/pi = φe/π, which comes out at 1.400013584. If we divide this by 7, we get 0.2000 m, and from that can scale to 0.3000 m, and from that, to the other subdivisions. Have updated the formulas in the PDF attached. I’m particularly chuffed with the formula for the foot…. 🙂
2018-11-30: Added formulas based on √(π²+φ²)/e
2018-12-02: Added formulas for great span and remen based on φπ/e construction.]

While playing around with my calculator in recent times, manipulating pi, phi, e, and the cubit e.t.c., I sometimes ended up with an answer of 1.199981615 which is extremely close to 1.2, and I thought that was interesting. Eventually I took note of the formula to get there, and then realised that 1.2 m is 4 Egyptian feet … and then the penny dropped.

From the foot, we can double to the pole, or half to the double handbreadth. A bit of playing around revealed reasonably easy adjustments to give other fractions of the cubit, all based off the same ratio, which is simply pi over (phi squared)  = π/φ².

So here’s an updated list of cubit parts, plus some other measurements like assorted feet, in PDF form. Posting screengrabs from LibreOffice Math is getting tedious 🙂

cubit-parts.pdf

The closer you watch, the less you see.

Continues from Part 1.

The second lemma needs to be deduced from reading these pages from chapter two of Christopher Milbourne’s Illustrated History of Magic published in 1975. (Dear publisher: please treat this as free advertising for your wonderful book.). Yes it’s all relevant, and it starts off with a story about Khufu. Click to enlarge for easier reading.

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If it’s not a tomb, then what is it?

Further to my thoughts (which covers some ground covered by others already, so they were not all new ideas) about why the great pyramid is not a tomb, let me speculate on what I think it really was.

But before we get there, it is possible that it may be a tomb, but not in the parts that we know about. There may be another entrance on the west wall (i.e. the one away from the Sphinx), with a separate tunnel and room system like the Bent Pyramid. This possibility is strengthened by the fact that there are different stone types used in the lower courses, which make a triangular pattern, with the current known entrance at the apex of the one on the northern side.

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No body parts were harmed producing this post

[Note: the formulas below have been superseded by Cubit parts and more, updated ]

It is the mainstream academic consensus that ancient people based their measurements on body parts. As discussed in Some thoughts on the cubit (and foot) , I don’t think that’s necessarily so, despite at least two (the Persian and Roman feet) at one point being based on the length of the foot of their leader (or his over-sized statue).

I still think the names that these measurements became known as (cubit, foot, palm, etc) were backnyms to create a easy-to-use-and-remember name for a length.

Even something simple like measuring the width of a hand is not exact because fingers move, flesh compresses, etc., and fingers are not all the same width.

So… here’s a little exercise to show that we can get the same list of parts of the cubit just with maths. In some cases two different methods produce values on either side of the target.

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The Mile. Not the Green Mile or the Last Mile or even Mile 22

Just the British mile.

Which, bless them, they defined as being 8 furlongs long. Yeah, what the heck is a furlong… you need to be a British horsey type to be familiar with such things. For the rest of us, a furlong is 660 feet. So that makes a mile 8 x 660 = 5280 feet.

In the early hours of this morning I had an embarrassing “light bulb” moment. As previously shown, a foot is simply ⅕ of a Grand Metre ( 1 + π/6). This morning, the flip side of that hit me … that the Grand Metre (especially in the slightly rounded for practical purposes form of 1.524m) is exactly five feet. I suppose we could call that a fiveet.

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The Persian Foot

The Persian foot was slightly larger than the well-known English foot.

According to The Lost Science of Measuring the Earth by Robin Heath and John Michell, it was based on the foot of Darius at exactly 1.05 British feet, which converts to 32.004 cm, while Howard Crowhurst, in his video (and presumably book) pegs it at 32 cm. Wikipedia of course is happy to claim a mere 300 mm (citation required…).

We can relate this to the cubit with reasonable accuracy. Remember that the normal British foot is 1/5th of Grand Metre (1 metre plus 1 royal cubit).

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Introducing the wroft

School children around the world today generally use the same length ruler, being 30cm long. For the Yanks and maybe some British, they may still use the foot ruler.

I remember when  I was in junior school around Std 1, we did the whole switch from imperial measurements to metric, and I had to stop using my nice wooden ruler with the embedded brass strip, in exchanged for this new 30 cm plastic ruler with miniature “inches”.

My first thought of course was “why 30cm?” … that’s a strange number. It’s not called anything, it doesn’t divide nicely into 1 metre, so what’s the story?

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Why the Great Pyramid is not a tomb

Some thoughts on why the Great Pyramid is not a tomb… we’ll get to what it may be in another post.

Various other people have pointed out things about the Great Pyramid which indicate that it was not a tomb. These include:

1. The sarcophagus was empty.
2. It’s a bit small for receiving a king in full burial kit … mummified with layers of cotton wrapping, then assorted nested boxes, gold death mask, etc.
3. The sarcophagus (and the entire pyramid) are devoid of any carvings or inscriptions. There’s absolutely nothing along the lines of “I came. I saw. I built.” or “Here lies King Khufu with his six wives and 97 children. He was The Man” or even “Warning: no tomb robbers (or Englishmen) allowed.” Nil. Nada. Unlike your typical tomb.

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Cubit with squares

Okay, REALLY last one for tonight. Found an approximation for the cubit using square roots. Not very good, in fact worst so far.

Formula is

Cubit in squares

Comparison table is now:

MethodValueDifference from π/6Abs difference
π/60.523598775598300.000000000000000.00000000000000
φ²/50.523606797749980.000008022151680.00000802215168
Ave Year0.523607797087850.000009021489560.00000902148956
π – φ²0.52355866483991-0.000040110758390.00004011075839
10²/φ²π²ⅇ²0.523764440990030.000165665391720.00016566539172
cube roots0.523600350171940.000001574573650.00000157457365
square roots0.523403736847960.000195038750330.00019503875033