Even if that were true, none of the proofs I’ve offered here rely on calculators or computer values for the constant of Pi. My proofs all based on applying the Pythagorean Theorem of a²+b²=c² to the circle’s formula of x²+y²=r². The same results could be achieved with just paper and pencil. Using Excel and/or Desmos allows the calculations to be done quickly, easily and with far more precision that any of the methods being advocated by those who want to promote that π=4/√φ, which simply cannot be proved mathematically or by physical measurement. It all relies on flawed geometric constructions, as revealed here in the analysis of Lear’s construction.

The radians that are programmed into most scientific calculators are based upon the fake value of pi = 3.141592653589793.

The correct value of 𝝿 = Pi = 4/√φ = 3.144605511029693144.

The correct radian is 45 times /√φ = 57.240884228133103 degrees.

His calculations for an angle = 1.8° show the inner perimeter at 3.1426260 and the outer at 3.1418480. These bounds show that with a simple understanding and application of trigonometry that Pi cannot possibly be 3.1446. That’s the key point here. And by the way, trigonometry is not dependent on a particular value of Pi. It’s based solely on the formula of the circle as x²+y²=r² and the application of the Pythagorean theorem as a²+b²=c². Nothing in mathematics supports a value of pi other than 3.14159…, and that includes not just the polygon method but calculus, infinite series, Monte Carlo simulations and other methods.

Please take a closer look at your proof: The Non Transcendental Value of Pi and Squaring the Circle.

You’ve made the same mistake made by Lear in his proof. You do an elaborate construction that proves nothing about the value of Pi and then come to a single line whose dimensions embed the golden ratio.

In your case, it’s line b which is the side of a hypotenuse of a Kepler triangle. The lengths of the sides of a Kepler triangle with a hypotenuse of 1 are 1/√φ and 1/φ. No magic there, and the b^2 is to b as b is to 1 is not needed to get there. That’s just the formula that defines the golden ratio.

So when you designate the side of length 1/√φ as b and then say 4b=π, you just “letting” 4/√φ=π and you’re done. You can’t “let” a term equal something else. You have to prove it.

Yes, it’s possible to have a square whose perimeter is the same as a circle, but for this circle mathematics tells us b must be π/4 with a value of 0.785398163. Your b based on the 1/√φ is 0.78615138, and there is nothing in your proof to show that this value is correct. It’s an unvalidated assumption and its wrong.

There are any number of right triangles with a point opposite the triangle’s hypotenuse on the circumference Here are a few:

https://www.desmos.com/calculator/azftymwxhk

By your logic I could construct a 3-4-5 triangle with a side of 0.8 and say 4b=π to prove that Pi is 3.2.

Just as with Lear’s “proof,” an equality is arbitrarily assigned to let the circumference of a circle, and thus Pi, equal a line with a length 4/√φ. The value of Pi is then proclaimed to be 3.1446… with flawed logic and no mathematical validation of the results.

Every “proof” of Pi as 3.1446… uses this same false step to get to its false conclusions.

But this averaged value 3.1415926 is still the perimeter of a polygon now intersecting the perimeter of the circle.
By this method if we get the π value already with 100 sides, which means that every side represents 3.6° of the circle and is a gross approximation, it shows also that the polygon intersecting the circle with a perimeter of 3.1415926 Diameters of the circle, is still larger than the circle. And this situation remains even if we calculate a trillion more decimal places, it doesn’t become more accurate as the first six decimal places remain the same.
In this case the real π value would be even less than the averaged value.

It would be btw 3.14107 and 3.141592, hence 3.141331 approximately.

So:
n = number of facets
Alpha = (360/n) / 2
((tan(Alpha)-sin(Alpha))÷3+sin(Alpha))×n
= apProximated Pi()…

Example:
n = 100
Alpha = 1.8°
((tan(1.8)-sin(1.8))÷3+sin(1.8))×100 = 3.141592…

Every child understands that the circumcircle is undoubtedly larger than the circle. And if n is already 100, the circumcircle is already less than 3.1446, Namely:

tan(1.80)×100 = 3.142626, which is less than 4/sqr(Phi())

And the inner circle results here: Sin(1.80)x100 = 3.14107…

And the averaged value for the 3rd part then gives: 3.141592….

Just think simply and logically.

Best regards, Gerhard Daniel Kadisch.

Translated with DeepL.com (free version)

ORIGINAL GERMAN MESSAGE:

Pi() kann unmöglich 4/sqr(Phi()) als 3.1446.
sein.
Die einfache Innkreis – Umkreis Methode zeigt ganz klar, dass sich Pi() an 3,14159265359…aproximiert.
Diese Methode kann allein mit Pythagoras ermittelt werden, wobei sich auch die Sinus-Kosinus-Tangens Werte bestätigen.

Wenn anstelle des Mittelwertes von Umkreis und Innkreis nur der 3.Teil genommen wird, aproximiert dieser Wert zu Pi() bei einem n-Eck von nur 100 Facetten, bereits an 3,141596.

Also:
n = Anzahl Facetten
Alpha = (360/n) / 2
((tan(Alpha)−sin(Alpha))÷3+sin(Alpha))×n
= aproximiert Pi()..

Beispiel:
n = 100
Alpha = 1,8°
((tan(1,8)−sin(1,8))÷3+sin(1,8))×100 = 3,141592..

Das der Umkreis wohl zweifelsohne größer als der Kreis ist, versteht jedes Kind.
Und bei n von bereits 100 ergibt der Umkreis somit schon weniger als 3,1446,
Nämlich:
tan(1,80)×100 = 3,142626. Somit kleiner als 4/sqr(Phi())

Und der Innkreis ergibt hier:
Sin(1,80)x100 = 3,14107…

Und der gemittelte Wert eben beim 3.Teil ergibt dann: 3,141592….

Einfach simpel und logisch denken…

Liebe Grüße, Gerhard Daniel Kadisch.

4b(the perimeter of the square) = π

is not a proven equality, which is like saying that 1=1 is not a proven equality either.
Come down to Earth. The exact value of π is 3.1446………