The Special Circumference of Pi diameter
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The Special Circumference of Pi diameter

Friends, the circumference seems to be a key geometric figure, especially essential in mathematics, since it contains the parameter (Pi) of interrelation between the rectilinear length and the closed circle.
This engenders properties, many of which have not been established up to now, among which we could consider and study the one referring to the property of the circumference of establishing measurement units dependent on it, and vice versa, establishing special circumferences dependent on the unit of length that we are using.
I will call this property for the moment "Circumference dimensional property, CDP".
To explain it, we will put a small indicative postulate:
Postulate:
"Every unit of measure (mm; cm; m; km, etc.) produces a unique special circumference whose diameter is Pi, and whose perimeter is Pi squared (Pi^2).
And vice versa;
Every circumference that we choose and take as a special circumference of Pi diameter, produces or engenders an interior unit circumference, whose radius becomes a special unit of measurement: m.s.u = diameter/Pi." (see drawing)
* Therefore, for each unit of measure there is a unique special circumference whose perimeter is the square of its diameter.

1.- (To obtain the Cyclic Pi ) Well then, this special circumference is very appropriate and can be used as a way of obtaining Cyclic Pi, applying the decimal reduction coefficient as seen in the drawing:
(P^2/10)^17 = 8/10, thus being the Cyclic Pi = 3.14159144414199265.....
* Let's remember that the cyclic Pi in a Pi number, which I understand is the correct Pi because it has all the required properties to be the true number Pi, and which I understand will be used in the future when its qualities are well studied.

The cycle of Pi Squared

2.- (Cycle to test the accuracy of the Pi applied)
Beside, we also use it as a Pi squared circuit to measure and appreciate the accuracy of the number Pi we are using:
Pi^2 = 8/[(Pi^2/10)^16] = Pi^2,
in such a way that if we use this circuit to rotate the value of Pi^2 in successive turns of the circuit, if the value of the applied Pi is correct, the value of Pi^2 will remain unchanged indefinitely, and if the applied Pi is not correct, the value and circuit will be destroyed quickly, the faster the greater the error made in the applied Pi.

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."

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.