The Cycle of Pi Squared
Of ferman: Fernando Mancebo Rodriguez--- Personal page. ----Spanish pages

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The Cycle of Pi Squared

Meaning in brief:

"The Pi squared cycle is a circuit or mathematical cycle to verify the accuracy of the number Pi that we apply.
Geometrically we suppose that we make circulate the value of Pi squared (little ball in the roulette wheel), increasing exponentially in each turn the possible inaccuracies of this value, in such a way that if the applied Pi is exact, the circuit remains invariable; but if the applied Pi is not accurate, the circuit will wobble and be quickly destroyed."

First Part.

The Pi Squared Cycle: Roulette or Pi Trap.

Accuracy test.

Preamble:

For many years I have been surprised by the apparent persistence of the number Pi in "hiding and escaping" from a clear sample of its definition and mathematical demonstration through formulas that show us its true value and situation.
However, and from this cyclical formula or function, my feeling has changed and my doubts have been cleared and I understand that the number Pi can tell us:
"Well, here I am, enclosed, located and measured by my own construction parameters. Do you see me now"?
Friends, as some of you already know, I have my own point of view and proposals for the number Pi.
I understand that it has an irrational value (but not transcendental) and that it is totally related, integrated and measured by the construction parameters of the circumference (diameter, inscribed and circumscribed squares, etc.)
During these years of considering it like this, I have been able to verify that it has more logic and with greater mathematical and geometric properties than the algorithmic Pi that is currently used.
So let me present one of these particulars that I find interesting, and that would also represent a proof of its validity.
It refers to what I call the Cycle of Pi squared, and which relates the square circumscribed to the circumference (8) with the number Pi of its inner circumference, and in which the operating decimal number (10) is used to maintain the level of operations (of powers and roots) close to unity, that is, at the level of the dimensions we are using.
This cycle function is simple, and as the name implies, it is cyclic since the input result of the mathematical function is the same as the output result, if the correct number Pi is applied.
It is represented geometrically by a cycle or roulette on which we can rotate or maintain the circumference of the square of Pi, without its value changing with the cyclical function that is applied to it.
However, if we add a non-exact value of Pi to this loop, then the circumference of P squared is quickly distorted and destroyed.
Simplifying: The cycle would be like a tour in continuous rotation around a circle of value Pi squared, in which tour this value of Pi squared is analysed and "tuned" or (increased in error) at each turn by the function exposed cyclic, in such a way that if we have applied the correct value of Pi, the cycle and the value of the circumference remain unchanged indefinitely.
But if the applied Pi is not correct, the circle wobbles and is destroyed quickly: or faster, when greater the error in the applied Pi.
* In the same sense, with a variable of this cycle, and with a correct number Pi, we must measure (through powers and roots) all the squares and circumscribed circles on a first given circumference.

Demonstration and Testing

As we can see in the drawing, the effectiveness of this Pi squared cycle is very high, since although the algorithmic Pi currently used is very approximate, when we submit it to the Pi squared cycle test, already for the fourth return to the cycle is destroyed and out of the loop.
And if we use only the approximate value of Pi = 3.14, the value of the cycle (Pi^2) is destroyed in the first loop.
On the other hand, the Squaring Pi remains intact, which in my point of view is a clear demonstration of its validity.

Second Part.

Bending squares to get their inscribed circumferences

As some of you tell me that you are interested in the philosophy and logic of my formulas to obtain Pi, I will summarize the fundamentals:
My idea and formulas to obtain the number Pi consist of doubling rectilinear parameters of the circumference (diameter, inscribed square, circumscribed square, etc.) by means of powers and roots of these parameters, until obtaining the appropriate curve of the number Pi.
Now, another practice way to proceed is to choose an upper square with side Pi (Pi squared) and reduce it to the square circumscribed to the circumference (8), and reversing this process applying a simple formula to adjust the number Pi exactly.
In the drawing I show you a way of explaining it, where a larger square with side Pi is supposedly subjected to powers and roots until it becomes the square circumscribed to the circumference, and from here, doing the inverse operation we can obtain the number Pi.

The interrelation module Pi -- 2.Sqrt 2

From my point of view, the currently used algorithmic number Pi is superseded and outdated by multiple tests and application of mathematical logic.
In other words, it is not acceptable in our time (in my opinion, of course) that formulas for obtaining and constructing the number Pi based on its construction parameters are not accepted, and yet continue to use an algorithmic series with the certainty that they are correct.
For me the evidence is compelling due to its quantity and clarity of application.
For example, the drawing that I put, as well as the video that I send you so that you can review it if you have time.
In the drawing, it is expressed that the relationship between the number Pi and the diagonal of its inscribed square (interrelation module) must be obtained from each other by mathematical logic.
If you observe this obtaining, you will see that the Pi quadrant that I expose does comply with these perspectives. Instead the algorithmic Pi fails.
Therefore and for me the question of mathematical logic that we should ask ourselves can be:
Can a false number (mine) united, structure and interrelate with formulas all the construction parameters of the circumference, while the current algorithmic Pi cannot?

(*) From the Circumscription Theorem:

In this diagram we see how all the circumferences and squares circumscribed on a first given circumference can be accurately measured by exponential formulas of the squaring Pi, and how the adjustment with the algorithmic Pi progressively deviates.

Here another simple way of obtain the Squaring Pi in function of its construction parameters.

---> Video

Thank you friends