The Non Transcendental, Exact Value of π and the Squaring of the Circle

[omedam777]

I will keep on exploring order and balance:

i found this demo

The Coxeter's loxodromic sequence of tangent circles.

Coxeter_circles.png

With 4 adjacent circles (1,2,3), the last one in the center of the three others (0).
If the radius of the circles are in geometric progression, the reason of this progression is k = ɸ + √ɸ = 2.89...
The angle between consecutive triples of centers is atan(√ɸ) = 51.827° the same angle as the the Kepler triangle.

Explanations

This come from Descartes theorem

You see. I'm not trying to be preferential here, but you're the reason I'm staying here, Arthur. You bring forth food for thoughts. Thanks for sharing the demo.

[Arthur]

Thank you. I saw you use a lot of desmos site/software and found this one. (It looks like fractals too.)
I asked AI (DeepSeek) to give me the step by step geometric construction for these 4 circles. (i am on it and will give the result later)
So in fact instead of going from phi or square root of phi to a circle, this one do the reverse. From circles to phi ( the reason of geometric progression (like Kepler triangle) for tangent circles).

[Arthur]

TangentCircles.jpg (66.92 KiB) Viewed 1407 times

Circle 1: Center O1(0,0), radius ( 0.5 ).
Circle 2: Center O2(1.9327265874521373 ,0.21839703590760529), radius ( 1.445026819131982 ).
Circle 3: Center O3(−2.0156180265279415 ,4.219499653156852) , radius ( 4.176205016021387 ).
Circle 4: Center O4(−4.053860620518499 ,−11.8977918452241) , radius ( 12.069456500688826 ).

Angle <O4O3O2> is atan(√ɸ) = 51.827°... to the 10th decimal places.

To plot these coordinates in geometry software, we obtain four circles whose radii are in geometric progression with a ratio of ɸ+√ɸ.

[Hush]

Why C.B.,

You Nailed it,

Because math is a dangerous business if you don’t do it properly. You can slip into madness without even noticing it. And I’m serious about that.

We should celebrate C.B.'s revelation having finally made a correct and valid statement but if you are truly sincere, then please take your own advice and work hard to understand the flaws in your own 1/2-page faked Pi=3.1446 logic nonsense. This requires putting your pretend math concepts aside and trying to understand real mathematics. Can you do that C.B., are you up for the challenge to learn real mathematics for a change?

You may have to take back those math books you advised 777 to read or go back to school but suggest starting with grade school where kids first learn about traditional Pi=3.14159 however please be polite and don't interrupt the teacher and share your foolish belief True Pi=3.1446, just be a good student and listen to the teacher otherwise you will probably fail grade school math and end up exactly where you are today, no smarter or wiser still pushing your 1/2-page faked Pi=3.1446 nonsense - ok.

My dear C.B. and all other Pi-Radicals, all of whom foolishly believes Pi can be solved for as a variable, derived using silly equations based upon the golden ratio and other such nonsense or, as in your case C.B. foolishly attempting to solve for Pi using a 1/2-page's worth of pretend math equations where you magically transfer variables defined as area into distances and just plop them into the legs of a right-triangle (talk about naive), where in fact Pi is NOT a variable but rather a constant taken as the ratio of the circumference to the diameter of a circle and therefore cannot be treated otherwise, but you Pi-Radicals cannot get this simple concept through your brainwashed skulls that Pi is NOT a VARIABLE - jeppers-creepers you guys.

So if we respect the definition of Pi, and not treat it like a variable, then we must find a way to calculate the circumference of a circle because we already know the diameter. Earth humans currently only know how to precisely calculate straight line distances and all curves are approximations but very precise using traditional Pi=3.14159. Therefore traditional Pi is also calculated from straight-lined polygons to precisely estimate the circle circumference. Do any of you understand that NOBODY of EARTH origin actually knows the true circle circumference curved line length. Earth humans only know how to calculate straight line lengths precisely so, in order to obtain the circle circumference, which is a curved line, we use polygons as a clever way to estimate the length of the circle's curved circumference.

Lets repeat as necessary, by definition Pi is NOT a variable, it's a constant defined as the circumference divided by diameter of a circle, so to solve Pi requires finding the circle circumference one-way or another. There is no other method that adheres to the definition of Pi to calculate Pi.

So put those silly solving for Pi equations into the garbage where they belong, and stop treating Pi as a variable and just focus on finding the best method to calculate the circumference of a circle, and when you're done thinking about it, don't think too long, you'll realize polygons is the only way to best approximate the true curved line circle circumference and this respects the definition of Pi which is circumference divided by diameter. You have to adhere to the strict definition of Pi to solve Pi, not those dumb ass equations where you Pi-Radicals always try to solve for Pi likes its a variable.

There is only one credible method to derive Pi, which is by definition, find the best method to calculate the circumferential distance around a circle (polygons) and divide by the diameter. Nothing else will work ok. Do you dimwits understand?

Again you can't solve for Pi as a variable, you have to find a way to calculate the circle circumference using straight-lines and polygons works, and is a very accurate approximation and good enough for now, good enough for all round and circular calculations in both math and physics (Ptaah CR #712). It will take a few centuries before Earth's brilliant scientists will figure out how to precisely calculate true curved lined distances while trying to save our dying Sun - CR #251.

[C.B.]

Bildschirmfoto 2025-09-17 um 19.14.49.png (198.96 KiB) Viewed 1388 times

This is another nice way to derive π:

We have the series:

1; b; b^2; b^3; b^4; b^5; b^6…..

where b=π/4.
having these values we can take whatever row of three consecutive b values, simplify if needed and apply Pythagoras.

The sum of all Areas together results in Φ + √Φ = 2.890

[C.B.]

Again you can't solve for Pi as a variable, you have to find a way to calculate the circle circumference using straight-lines and polygons works, and is a very accurate approximation and good enough for now, good enough for all round and circular calculations in both math and physics (Ptaah CR #712).

What do you mean by solving π as a variable?
in 4b=π there is nothing variable.

[C.B.]

as in your case C.B. foolishly attempting to solve for Pi using a 1/2-page's worth of pretend math equations where you magically transfer variables defined as area into distances and just plop them into the legs of a right-triangle (talk about naive), where in fact Pi is NOT a variable but rather a constant taken as the ratio of the circumference to the diameter of a circle and therefore cannot be treated otherwise,

Check up the other method. In the diagram you’ll find the same magnitudes for lengths and areas. Take your pick to construct the triangle.

[C.B.]

Pi is NOT a variable but rather a constant taken as the ratio of the circumference to the diameter of a circle and therefore cannot be treated otherwise,

The polygon method doesn’t hold to this principle. The result of the polygon method is not π= P/D.

4b=π holds to that principal of the quotient btw Perimeter and Diameter, because I can measure the circumference directly, whereas you with the polygons can only approximate and then fake
π= P/D.

[Arthur]

This is another nice way to derive π:

We have the series:

1; b; b^2; b^3; b^4; b^5; b^6…..

i am not sure

b^5 is the circumference of the 1st circle
b^4 is the perimeter of the 1st square.
b^3 is the circumference of the 2nd circle
b^2 is the perimeter of the 2nd square.
b is the circumference of the 3rd circle
1 is the perimeter of the 3rd square.

That's it?

[Hush]

Well C.B.,

What do you mean by solving π as a variable?

As I've tried to explain you're a lost cause, beyond hope, too naive, arrogant, bull-headed to understand real mathematics otherwise I wouldn't have to explain to you over-and-over why you foolishly, truly are attempting trying to solve for Pi as a variable using pretend math nonsense. Your silly equations make no attempt to estimate the circle circumference, a distance required to solve for Pi, which means you're treating Pi as a variable like every other Pi-Radical dimwit. Perhaps someday, not likely in your brainwashed lifetime, you may realize that you have to actually find the circumferential distance of a circle, to solve for Pi, and stop treating it as a variable. Anyway, this is fundamentally why nobody, I do sincerely mean nobody, believes your 1/2-page Pi=3.1446 proof is credible. Convincing anyone your 1/2-Page Pi=3.1446 is credible, is NOT going very well is it. As they say, the proof is in the pudding, meaning you're not convincing anyone your 1/2-Page Pi=3.1446 Proof is valid, you're like a Jehovah Witness, whose told over-and-over again, their beliefs are crap but just wont go away (LOL)