It is possible to construct φ exactly with a straight-edge and compass. Would the approximation of 5π/6 - 1 be used because it's easier to calculate quickly?
Yes φ is a constructible number and more generally an algebraic number, solution of the polynomial equation x^2 = x + 1. However π is not, and so is my approximation of φ as 5π/6 - 1. Here non-constructibility (in the mathematical sense) translates to the fact there is no method to "straighten" an arc into a segment using a compass and a ruler. But bear with me because the 5π/6 - 1 approximation of phi has more to say.
First, the "conspiracy theory" that the meter is linked to Earth's dimensions and harmonizes with ancient measurement units through a shared reference actually predates the meter's definition. This idea was a thread of interest among the scientists who developed a universal measurement system – one that could be derived anywhere on the planet.
>One can well sense that it can only be through comparisons of measurements made in ancient times & in our days on monuments still existing, that I can determine to how many of our toises the Geometers of antiquity would have evaluated a degree of Meridian. Now I find, 1st. that the side of the base of the great pyramid of Egypt taken five hundred times; 2nd. that the cubit of the Nilometer taken two hundred thousand times; 3rd. that a stadium existing & measured at Laodicea in Asia Minor, by Mr. Smith, & taken five hundred times; I find, I say, that these three products are each of the same value, & that each in particular is precisely the same measure of a degree [of a Meridian], which has been determined by our modern Geometers.
Alexis-Jean-Pierre Paucton, Metrology, or Treatise on measures, weights and currencies, of the ancients and the moderns, 1780
>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.
φ = 2cos(π/5) lead us to this construction around a pentagon from which we can derive the "pige" or "quine" of cathedral builders (for now consider this is historically true) https://fr.wikipedia.org/wiki/Pige_(mesure)
What I mentioned earlier was that using a circle-based construction(diameter = 1m), one can derive a non-constructible approximation of φ, namely φ̃ = 5π/6 – 1, with the remarkable property that 0.2 × φ̃² = π/6, thanks to φ² = φ + 1.
But what’s truly elegant is that this process has a symmetric counterpart, where we approximate π using φ. This time, we begin with a constructible triangle, sometimes called the triangle of the builders (1, 2, √5), whose perimeter is:
t = (1 + 2 + √5)/10
This value is fully constructible with compass and straightedge, and numerically it approximates π/6 to four digits. If we treat this `t` as a stand-in for π/6 in the previous formula:
φ = 5t – 1
we recover the *exact golden ratio*:
φ = (5 × (3 + √5)/10) – 1 = (1 + √5)/2
And then, going full circle:
0.2 × φ² = t again
In both directions, 0.2 (i.e., 1/5) emerges as the key scaling factor, bridging the decimal system, φ, and π through geometry. It ties together:
- the constructible (t from the triangle),
- the transcendental (π/6 from the circle), and
- the algebraic (φ² = φ + 1)
^this is a new result I just found.
For the historically conservative, arguments can be made that these considerations are pseudo-historical, that the "quine of cathedral builders" is an unsubstantiated myth. See the wikipedia link above for the "pige"
>This greatly saddens those who have built an entire "operative" narrative around this kind of knowledge supposedly passed down in secret among the compagnons of the Tour de France for centuries… and have made it their pedestal. The question of how "tradition" is constructed among the compagnons (and incidentally among the Freemasons) remains a taboo that absolutely needs to be broken — and not just for the sake of advancing historical knowledge.
>Le Corbusier considered various sets of proportions, notably using a human height of 1.75 meters, before settling in 1947 on a single set based on a height of 1.83 meters. He chose this because the associated Modulor measurement of 226 cm corresponds to within less than a millimeter of 89 inches — 89 being a number in the Fibonacci sequence that provides some of the best approximations of the golden ratio.
>This system was intended to unite all nations around a universal standard, effectively casting aside the metric system, if not the decimal system entirely. We know how that turned out: the Modulor was essentially used for only one major creation — albeit a significant one — the Cité Radieuse in Marseille, completed in 1952, where all dimensions, down to the built-in furniture, are derived from the Modulor.
Makes you think... The fact we don't have documents isn't surprising given that the campagnons (or later freemasons) communicated practical (then mystical) knowledge esoterically for political reasons (See https://fr.wikipedia.org/wiki/Compagnonnage).
Nonetheless, the same motivations and the same quest for harmony (in the obsessive, symbolic sense) can be observed in Le Corbusier. As if the situation follows a geometric progression: in this sense, the "ancients" were as puzzled as we are by unexpected harmonies and actively sought them, and if you look at the historical sequence that lead to the definition of the meter, this is what you find.
Compare the length of a greek foot with a roman foot: 30.87cm vs 29.62cm. The ratio matches 24/25 with 3 nines of precision. 24•25•7 forms a pythagorean triangle. As if the definition of some measurement units were retrofitted to facilitate conversion. If this kind of behavior leads to the formation of a strange graph of quasi-conversions or numerical coincidences, then maybe we could explain the emergence of patterns such as the 5π/6 - 1 approximation of φ without needing to argue for (or against) someone's intention behind what appears as a design choice.
Alternatively the measures of the tools or geometric constructs that drive these conversions are idealized/approximated with a ratio, hence the delusion of the conspiracy theorists. But as I said, "ancients" had the same attitude, in particular with irrational numbers they wished to express as a ratio. Imagine the kind of problem the pseudo-phi <-> pseudo-π/6 complex I desribed above posed to people who where attempting to construct a straight line of length pi using only a compass and a straight-edge and establishing mathematics more rigorously. That's quite a nasty trap. Surely they found themselves in a mindstate that must not be that different from ours. Put in other words, the situation is hyperstionnal, and if we want to understand what is happening (whether this is an illusion or not) I think we should try to tackle this from a cognitive angle and model surprise explicitly.
Xmd5a|9 months ago
First, the "conspiracy theory" that the meter is linked to Earth's dimensions and harmonizes with ancient measurement units through a shared reference actually predates the meter's definition. This idea was a thread of interest among the scientists who developed a universal measurement system – one that could be derived anywhere on the planet.
>One can well sense that it can only be through comparisons of measurements made in ancient times & in our days on monuments still existing, that I can determine to how many of our toises the Geometers of antiquity would have evaluated a degree of Meridian. Now I find, 1st. that the side of the base of the great pyramid of Egypt taken five hundred times; 2nd. that the cubit of the Nilometer taken two hundred thousand times; 3rd. that a stadium existing & measured at Laodicea in Asia Minor, by Mr. Smith, & taken five hundred times; I find, I say, that these three products are each of the same value, & that each in particular is precisely the same measure of a degree [of a Meridian], which has been determined by our modern Geometers.
Alexis-Jean-Pierre Paucton, Metrology, or Treatise on measures, weights and currencies, of the ancients and the moderns, 1780
more context: https://anonpaste.pw/v/71abb0f8-5a03-4cb5-879a-d4f44ad6d57c#...
original: https://gallica.bnf.fr/ark:/12148/bpt6k55491755/f126.item
>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.
https://www.theguardian.com/science/2020/dec/06/revealed-isa...
Having said that(-1 downvotes!), let's recap:
This is how we can construct a royal cubit from a circle of diameter = 1m:
https://imgur.com/a/HmnfDKR
φ = 2cos(π/5) lead us to this construction around a pentagon from which we can derive the "pige" or "quine" of cathedral builders (for now consider this is historically true) https://fr.wikipedia.org/wiki/Pige_(mesure)
https://imgur.com/a/ZqprfAd
What I mentioned earlier was that using a circle-based construction(diameter = 1m), one can derive a non-constructible approximation of φ, namely φ̃ = 5π/6 – 1, with the remarkable property that 0.2 × φ̃² = π/6, thanks to φ² = φ + 1.
But what’s truly elegant is that this process has a symmetric counterpart, where we approximate π using φ. This time, we begin with a constructible triangle, sometimes called the triangle of the builders (1, 2, √5), whose perimeter is:
This value is fully constructible with compass and straightedge, and numerically it approximates π/6 to four digits. If we treat this `t` as a stand-in for π/6 in the previous formula: we recover the *exact golden ratio*: And then, going full circle: In both directions, 0.2 (i.e., 1/5) emerges as the key scaling factor, bridging the decimal system, φ, and π through geometry. It ties together: ^this is a new result I just found.For the historically conservative, arguments can be made that these considerations are pseudo-historical, that the "quine of cathedral builders" is an unsubstantiated myth. See the wikipedia link above for the "pige"
or this recent article: https://classiques-garnier.com/aedificare-2021-2-revue-inter...
this one too: http://compagnonsdudevoir.fr/?p=790
>This greatly saddens those who have built an entire "operative" narrative around this kind of knowledge supposedly passed down in secret among the compagnons of the Tour de France for centuries… and have made it their pedestal. The question of how "tradition" is constructed among the compagnons (and incidentally among the Freemasons) remains a taboo that absolutely needs to be broken — and not just for the sake of advancing historical knowledge.
Also this blog post traces the confabulation of the quine to Le Corbusier's Modulor system based on the golden ratio: https://blogruz.blogspot.com/2007/12/en-qui-quine.html
>Le Corbusier considered various sets of proportions, notably using a human height of 1.75 meters, before settling in 1947 on a single set based on a height of 1.83 meters. He chose this because the associated Modulor measurement of 226 cm corresponds to within less than a millimeter of 89 inches — 89 being a number in the Fibonacci sequence that provides some of the best approximations of the golden ratio.
>This system was intended to unite all nations around a universal standard, effectively casting aside the metric system, if not the decimal system entirely. We know how that turned out: the Modulor was essentially used for only one major creation — albeit a significant one — the Cité Radieuse in Marseille, completed in 1952, where all dimensions, down to the built-in furniture, are derived from the Modulor.
Makes you think... The fact we don't have documents isn't surprising given that the campagnons (or later freemasons) communicated practical (then mystical) knowledge esoterically for political reasons (See https://fr.wikipedia.org/wiki/Compagnonnage).
Nonetheless, the same motivations and the same quest for harmony (in the obsessive, symbolic sense) can be observed in Le Corbusier. As if the situation follows a geometric progression: in this sense, the "ancients" were as puzzled as we are by unexpected harmonies and actively sought them, and if you look at the historical sequence that lead to the definition of the meter, this is what you find.
Compare the length of a greek foot with a roman foot: 30.87cm vs 29.62cm. The ratio matches 24/25 with 3 nines of precision. 24•25•7 forms a pythagorean triangle. As if the definition of some measurement units were retrofitted to facilitate conversion. If this kind of behavior leads to the formation of a strange graph of quasi-conversions or numerical coincidences, then maybe we could explain the emergence of patterns such as the 5π/6 - 1 approximation of φ without needing to argue for (or against) someone's intention behind what appears as a design choice.
Alternatively the measures of the tools or geometric constructs that drive these conversions are idealized/approximated with a ratio, hence the delusion of the conspiracy theorists. But as I said, "ancients" had the same attitude, in particular with irrational numbers they wished to express as a ratio. Imagine the kind of problem the pseudo-phi <-> pseudo-π/6 complex I desribed above posed to people who where attempting to construct a straight line of length pi using only a compass and a straight-edge and establishing mathematics more rigorously. That's quite a nasty trap. Surely they found themselves in a mindstate that must not be that different from ours. Put in other words, the situation is hyperstionnal, and if we want to understand what is happening (whether this is an illusion or not) I think we should try to tackle this from a cognitive angle and model surprise explicitly.
Some more links:
https://www.messagedelanuitdestemps.org/les-principales-unit...
https://martouf.ch/2021/03/le-metre-une-matrice-universelle-...