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.

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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.