The Meter, Golden Ratio, Pyramids, and Cubits, Oh My | Hacker News Reader

The Meter, Golden Ratio, Pyramids, and Cubits, Oh My

iforgeiron.com

39 pointsindigodaddy28 comments

I really enjoyed this and I plan to do some figuring on this to verify...

I was struck by the idea that understanding the arc of movement in the heavens is one of the few natural round things I can think of. The arc can also only be proven through observation and measurement over great time.

Finally I see the cubit as intersection human scale ergonomics with celestial movement.

When I set up my shop a posted a scale based on Nippur Cubits. Which I have defined as 518.5 mm. I'm thinking I should make some adjustments.

https://en.wikipedia.org/wiki/Ancient_Mesopotamian_units_of_...

I got a lot out of this article but I'm cubit curious.

    Length of Nippur cubit: 51.84 ± 0.02 cm
    Ratio to Egyptian cubit: 99/100
    Divided into 28 fingers (each finger = 1.8514 cm)
    
    
    Relationship to Other Measurements
    One Roman foot = 24/25 of a Greek foot
    One Roman foot = 35/36 of an English foot
    One small egyptian cubit = 44.45cm = 6/7 of a Nippur cubit
    One Roman foot = 16 fingers = 1.852 x 16 = 29.63 cm
    One Nippur cubit = 28 fingers = 1.852 x 28 = 51.84 cm ±0.02 = 4/7 of one Roman foot
    
    
    Pyramid Measurements in Nippur Cubits
    Great Pyramid base: 440 Egyptian cubits = 444.4 Nippur cubits
    Great Pyramid height: 280 Egyptian cubits = 282.8 Nippur cubits
    
    
    Mathematical Connections
    Pyramid height (282.8 Nippur cubits):
    - √8 = 2.828
    - 200*√2 = 282,8 (the height of the Great Pyramid) = 100*√8
    Base diagonal:
    - 628.43 cubits = 200*π
    Finger measurement:
    - 1/100,000 of a mean meridian minute (99.95% precision)
    
    
    The silver number: 1 + √2
    Some properties:
    1.414 + 1 = 2.414
    1 / 2.414 = 0.414
    1.414 – 1 = 1 / 2.414
    
    
    Using the Nippur cubit as measurement unit, the height of the Great Pyramid measures 200 × the silver ratio, and the diagonal of the pyramid's base measures 200 × π!
    
    The perimeter of the Great Pyramid measures half an arc minute of Earth's meridian with 99.98% accuracy, which is equivalent to the measurement made by Méchain and Delambre at the end of the 18th     century, after 7 years of measurements and 9 years of calculations…

Source: https://www.messagedelanuitdestemps.org/de-la-coudee-de-nipp...

The article's claims about the dimensions of the Great Pyramid relating exactly to pi and to the golden ratio can't both be right.

Let's assume that the side of the pyramid has length exactly 1, for a base perimeter of 4. Then according to

> if you divide half the length of the Pyramid’s base perimeter by its height, you end up with π

the height h equals 2/π. While according to

> If we take the surface area of the four sides, and divide that by the area of the base, we come to the value of Ψ,

Ψ = 2 * sqrt(h^2+1/4) = 2 * sqrt(4/π^2+1/4) ~ 1.618993187

That's close to Phi = (1+sqrt(5))/2 ~ 1.618033989, but certainly not equal, already differing in the 3rd decimal.

The 4 sides of the Great Pyramid are a bit concave and consisting of 8 flat parts, not 4. You can see this for example in the pictures here: https://www.scirp.org/journal/paperinformation?paperid=12343...

The linked paper argues, that this is for stability, but it might also be intentional and it might close the error to Phi as you calculated.

https://fr.wikipedia.org/wiki/Pige_(mesure) (this is also called a quine)

The measurements (paume, palme, empan, pied, coudée) follow a Fibonacci-like progression:

    Cubit = Foot + Span (52.36 = 32.36 + 20)
    Foot = Span + Palm (32.36 = 20 + 12.36)
    Span = Palm + Thumb’s breadth (20 = 12.36 + 7.64)

Each larger unit is approximately φ times the previous one:

    7.64 cm × φ ≈ 12.36 cm (paume → palme)
    12.36 cm × φ ≈ 20 cm (palme → empan)
    20 cm × φ ≈ 32.36 cm (empan → pied)
    32.36 cm × φ ≈ 52.36 cm (pied → coudée).

Illustration: https://imgur.com/a/ZqprfAd

This is where the golden number intervenes.

With the egyptian cubit defined as π/6 meters we have the following configuration:

    - a circle with diameter 1 meter
    - perimeter is  π meters
    - 1/6 of the perimeter = π / 6 = 1 cubit

Surprisingly though:

    - the remaining perimeter, 5/6π = 2.617993877991494
    - but φ^2 = 2.618033988749895

See this picture for a visual reference: https://imgur.com/a/HmnfDKR

Seems like we could construct an approximation of φ. Difficult to compute a square root like that though. Not quite. This is where the golden number shines:

    φ^2 = φ + 1

Hence φ = φ^2 - 1, approximately 5/6π - 1

In other words, draw a circle of diameter 1, remove 1/6th of the perimiter, substract the length of the diameter (1) and you get 1.618..., about φ

This means that if you use a random stick for the diameter of the circle, you can use this trick to build a system of measuring units organized as a quine/pige. If the diameter is the "palm" of your quine, the arc segment of length φ palms represents the span. To build bigger units, just use addition. palm + span = foot. To build smaller units, use substraction: span - palm = thumb's breadth.

That's it. In a few operations we actually multiplied a length by (5/6π - 1)

This is 0.99997521018 of φ

144/89, often cited as an approximation of phi, is 0.99996510539 of phi, and its not clear that multipying by this ratio is actually simpler to compute than using the geometry described above, since it will actually carry out the multiplication, not just indicate the right factor. And frankly, this seems simpler to do than dividing by 10 to construct other lengths in the metric system.

On top of this approximation, you can build measuring rods like this up to 4 levels above or below the initial rod and still get 4 nines of accuracy. You can go up or down 20 levels and still have 3 nines of precision with respect to the value you'd get with the actual value of phi.

------

Having said that, let's not overlook the fact that in the french imperial measuring system, what the introduction of the meter put an end to, a span is 20cm, namely, the metric system in disguise... Scaling by phi, it turns out that

0.2 [the span] * phi [to get to the foot] * phi [to get to the cubit] = 0.5235..., approximately π/6 at 4 nines of precision

The equation:

    0.2 * phi^2 ≈ π/6

Substituting with phi ≈ 5/6π - 1 and (ab)using the property that phi^2 = phi + 1

    = 0.2 * (phi + 1)
    = 1/5 * (5/6π - 1 + 1)
    = π/6

In other words the error in approximating the golden ratio gets "corrected" when relating it back to π, but this only happens when we "derive" π/6 from 0.2 as 1/5*phi^2 (here phi is our 5/6π - 1 approximation). And this is exactly how the span relates to the cubit and its historical value is 20cm.

Really makes you think. By anticipation, I go check wikipedia's page on the span, knowing that there are many variants and not all agree on 20cm as reference.

https://fr.wikipedia.org/wiki/Empan

    In the very particular system of the cathedral builders of the Middle Ages,
    the span measures exactly 89 king's lines.

I follow the link

https://fr.wikipedia.org/wiki/Ligne_(unit%C3%A9)

    144 lines in a foot (= 4 palms)

palm / span = 144 / 89 = an another approximation of phi.

Damn.

>One sixth of π is 0.5236 [one cubit]

To get one sixth of π, first get one sixth of 2π. This can be done by using only a compass, drawing one circle of the same radius on the original circle, then a second circle where this circle and the original circle intersect, and so on...

https://www.wikihow.com/Divide-a-Circle-Into-6-Equal-Parts

Then you can divide the angle in two using the compass.

The French definition of the meter can also be described as "such that the distance from the equator to the North Pole, through Paris, is 10,000 km." This is a handy number to know!

It means, for example, that the circumference of the Earth is four times as big, or 40,000 km (roughly speaking; I know that the Earth is smooshed due to its rotation etc.). And then to estimate the diameter, you just divide that by pi:

40,000 km / 3.14159... ~= 12,732 km

and half that is the radius of the Earth:

r_Earth ~= 6366 km

One use case is to estimate the minimum latency for communications between two points on opposite sides of the Earth, which would be how long it takes light to travel halfway around the Earth:

= 20,000 km / (speed of light)

~= 66.7 milliseconds

Not quite the minimum. :-)

Beams of neutrinos can be sent through the Earth and detected on the other side.

https://www.livescience.com/19075-neutrino-particle-communic...

This is true. Using a neutrino beam, the minimum latency would be approximately:

(diameter of Earth) / (speed of light) ~= 42.5 milliseconds

Some high-frequency traders might actually be able to justify the investment needed to enable such low-latency communications!

Come to think of it, you could also use gravitational waves...

the earth is flat everyone knows that it's how our lord and savior jesus christ made it

The numerological mysteries of the Great Pyramid thing is BS, of course. Eco has a passage in Foucault’s Pendulum where he skewers this exact notion:

“He threw open the shutters dramatically and pointed. At the corner of the narrow street and the broad avenue, stood a little wooden kiosk, where, presumably, lottery tickets were sold.

“Gentlemen,” he said, “I invite you to go and measure that kiosk. You will see that the length of the counter is one hundred and forty-nine centimeters—in other words, one hundred-billionth of the distance between the earth and the sun. The height at the rear, one hundred and seventy-six centimeters, divided by the width of the window, fifty-six centimeters, is 3.14. The height at the front is nineteen decimeters, equal, in other words, to the number of years of the Greek lunar cycle. The sum of the heights of the two front corners and the two rear corners is one hundred and ninety times two plus one hundred and seventy-six times two, which equals seven hundred and thirty-two, the date of the victory at Poitiers. The thickness of the counter is 3.10 centimeters, and the width of the cornice of the window is 8.8 centimeters. Replacing the numbers before the decimals by the corresponding letters of the alphabet, we obtain C for ten and H for eight, or C10H8, which is the formula for naphthalene.”

If anything that just proves the point more. Either the kiosk was conveniently built with all of those coinciding measurements on accident, or someone built them into it intentionally. One is more likely than the other.

Or there are no “coinciding measurements”, and someone intentionally selected some random dates and numbers based on the sizes of the existing kiosk?

>One of our readers, John Frewen-Lord, speculates that the metre may be the modern version of a measure that was familiar to the Pharaohs.

https://www.theguardian.com/science/2020/dec/06/revealed-isa...

>Newton was trying to uncover the unit of measurement used by those constructing the pyramids. He thought it was likely that the ancient Egyptians had been able to measure the Earth and that, by unlocking the cubit of the Great Pyramid, he too would be able to measure the circumference of the Earth.

Every building in the US is filled with dozens of lengths factoring into integer multiples of 12 inches. The Great Pyramid is only one of many pyramids, which are only a small fraction of the Egyptian world, much of which has been uncovered. If the Egyptians were using the metric system, it would be evident in every temple, monument and home.

also if u line up pyramid to orion’s belt it would line up perfect to around 12,000 years ago with 26,000 year cycles…so it could be hundreds of thousands of years old nobody really knows for sure

We have carbon dated the Great Pyramid. It is about 4000 years old, right about the time of the reign of the pharaoh Khufu (who was allegedly buried inside).

Probably too mant details, but how?

The stone itself would be older, so why isn’t it carbon dated to be older?

It used to be covered by another layer of stone, why isn’t 4000 years ago when this layer was removed?

So many questions..

Carbon dating is performed on once-living matter. Thanks to cosmic rays, the carbon in the atmosphere is radioactive at above-equilibrium levels. Once fixed into a plant, it decays. That makes it possible to measure the time since the carbon was last in the atmosphere.

Plant matter made its way into the pyramids both in the stuff that was put inside them, and also in the mortar, which was baked over wood fires.

Cool. Didn’t know that. Thanks!

We know for sure it isn't [1], from multiple sources. Stop spreading nonsense conspiracies.

What does it even mean "line up to Orion's belt"? Wouldn't everything pointing up also "line up"?

[1] https://en.m.wikipedia.org/wiki/Great_Pyramid_of_Giza#Age

The Truth About The Bass Pro Shops Pyramid https://youtu.be/jtiDOU--zmw

So ancient Egypt had more modern units than today's US. No surprise there.

Lots of meaningless numerology, so what’s the point?

I think the point is that if the meter didn't exist we would have to invent one.

AI can produce a lot of wasted time.

Article is from 2018

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