The Cyclic Pi Theorem
Of ferman: Fernando Mancebo Rodriguez--- Personal page. ----Spanish pages

Hello friends, allow me to start by setting up a small postulate:
"Pi is an exponential and cyclic number, which means that cycles of powers of Pi can adjust and measure all other parameters of the circumference."
And with this we can establish a slightly broader theorem, which would be:
"All the parameters of a given circumference (r=1), as well as all the parameters of its squares and circumferences inscribed and circumscribed on it, till infinity, can be adjusted and measured by means of powers of the Squaring Pi, with application of a decimal reduction coefficient (10^n) to place the value of the searched parameter at its appropriate level".
For example:
Pi^34 = 8.10^16, and therefore being the squaring Pi= 3.14159144414199265........
Where 8 is the square circumscribed to the circumference, and 10^16 is the decimal reduction coefficient.

Firts contact with the Cyclic Pi number

For a first contact with the cyclic Pi, and to observe its simplicity and interrelation with all the parameters of the circumference, (in this case, semi-square inscribed to the circumference) I give an example of similar procedure to the one used in the floating point, in which the decimal reduction coefficient is replaced by exponent E.
That is, to interrelate and measure all the parameters of the circumference (Pi, radius, diameter, inscribed and circumscribed squares, etc.) we can use the floating point form of expression in which there is a coefficient (c) of only one digit, followed by a decimal fraction formed by base (b) and exponent (e).
For example,
Cyclic Pi^17 = 2.8284271247461900...x 10^8 = 2.Sqrt2.10^8
Being therefore Cyclic Pi = 3.14159144414199265 .......
A this way with all other parameters.
E.G. Cyclic Pi^34 = 8 x 10^16

Summarizing:
Power cycles of Pi raised to 17, (Pi^17)^n, divided by the decimal reduction coefficient Dcf., can adjust and give us any rectilinear parameter of the circumference (radius, diameter, inscribed and circumscribed squares, etc.)
f[(Pi^17)^2n] / Dcf = A.P (any parameter)
and vice versa:
From any circumference parameter we can obtain the cyclic number Pi, applying the corresponding decimal reduction coefficient.

Logical philosophy of Pi: geometric and arithmetical similarity.

Similar to how geometrically the circumference represents a closed cycle inside and outside of which its construction and composition parameters are redistributed (radius, diameter, inscribed and circumscribed squares, etc.), equally arithmetically the number Pi is constituted in cycles or circuits of powers that measure and analyze all these construction parameters.
Measuring the first cycles to the parameters of the base circumference, and having and constructing cycles of Pi bigger circumscribed to these that are measuring the circumferences and circumscribed squares, that is, an arithmetic behavior similar to the geometric structure.
In this sense, it is not a transcendental number or value that tends towards a straight, indefinite and invisible horizon, but also arithmetically transforms into well-defined cycles around a central point.

* Dimensional loss when bending the straight lines.
"Every straight line that curves has a dimensional loss of length when all the points on the interior of the curve approach each other, and vice versa."
In the same sense, both the sum of the sides of the polygons inscribed in the circumference and the series produce us straight lines, and with that, with minimal increase of length.

* In the following diagram, we can see how power cycles of Pi can give us all the squares and circumference circumscribed between them to infinity (applying the appropriate decimal reduction), and how the current algorithmic Pi cannot.

* The mathematical (and perhaps philosophical) reason for the application of a decimal reduction in the powers of Pi is that the peculiarities of the decimal system make the high powers tend to take enormous values, when what the number Pi needs is that the cycles of powers that maintain values close to unity, an issue that is solved with the decimal reduction, (that is, without moving away from radius 1 that creates the parameters of the circumference).

Friends, I am frequently asked about the initial idea for Cyclic Pi (at first I called it Squaring Pi, 2009).
Well, the idea arose me thinking about the nature of the number Pi and observing a drawing similar to the one that I expose here.
And the question that assailed me was quite clear and forceful:
How is it possible that the geometry of the circumference and its Pi value go one way, and the arithmetic adjustments go the other very different way?: (by endless series of fractions)
How can arithmetic not be able to build a formula that directly interrelate all its construction parameters, including the number Pi?: In such a way that from any of them it can be extracted any other one, as in all other geometric figures.
I said me, "no, that's not possible, surely that formula exists but we haven't discovered it yet."

For example, and observing the drawing:
If 2 is the diameter that measures the extension or length of the circumference, and 2 Sqrt.2 the parameter of tension or amplitude of the Pi curve (semi-circumference of radius 1). necessarily in mathematics there must be a simple and direct formula that expresses and measures Pi as a function of these two parameters (extension-diameter and amplitude of the Pi parameter).

And then I thought and ask myself:
How should that formula be?
Logically by powers and roots, which is like curves are built in Cartesian coordinates. eg. y=x^2
And what parameter of the circumference seems to be the one that has been subjected to roots (by its irrational results)?
Of course, the number Pi seems to be.
Then logically, I need to proceed in the opposite way: I will subject Pi to continuous powers until I find some other rectilinear value of the previous parameters.
And not too late after, a strong approximation to a supporting parameter of the circle appeared to me, which is its semi-inscribed square 2.Sqrt.2
And since then I have been building the connections, formulas, cycles, properties, etc. of this cyclic number Pi.
And of course, coming to the conclusion that the cyclic Pi is the right Pi number.
I don't think the other Pi by series has any chance of being correct, because it is complicated, disconnected, endless, ethereal; with methods (series) unconnected and far from the value of the construction parameters; and why not, "transcendental" and not regular.

The Special Circumference of Pi diameter

Postulate:
"Every unit of measure 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.
Well then, this special circumference is appropriate and 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.....
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.

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."

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.

Logical and philosophical discussion and proof on the cyclic Pi
Geometric and arithmetic similarity in the Cycle of Pi squared.

Let us hypothetically suppose that a person that has never made a circumference begins to build it by turning the compass and perhaps wondering if, once turned around, the return line or point will completely coincide with the starting line or point.
But, once he checks the total coincidence and closure of the circumference, he will rest and realize that all the devices and construction parameters are correct and have not varied throughout the drawing.
Well, the same happens arithmetically with the formula of the Pi Squared Cycle, that we verify that the output value and return of the cycle coincides with the input value (Pi squared), and therefore that all the elements and parameters (Pi) are correct and suitable for the construction of the perfect circumference (Pi squared).
* And as we can see, this value of Pi corresponds exactly to the Cyclic Pi.
In the case of geometric construction, the devices would be mainly the compass, and the interrelation parameter would be the value of Pi.
And in the case of the arithmetic adjustment, there would be no physical devices, but rather the formula of the Cycle of Pi squared and the parameters of Pi as well as the value of the square (8) that contains, delimits and joins the number Pi and the diameter.
Then:
Geometrically, with the compass and the value of Pi around, we complete or close a perfect circle.
Arithmetically, with the Cycle Pi^2 and correct Pi value, we match and close a perfect circle.

Some examples

Next I put drawings of how to use the Squaring Pi cycles to obtain simple parameters of the circumference; although they can be extended to obtain values of all the squares and circles circumscribed between them.
Likewise, some curious examples such as the Squaring Pi Pyramids.

Following the initial idea.

Before see more examples, let me first expose a function of the initial idea (as a set of powers and roots of the parameters of the circumference, including Pi) and that would soon prove to be a basically Pythagorean composition.
The mathematical function would be carried out through a formula that flexes, bends, expands and curves the semi-square inscribed in the circumference (2.Sqrt.2, as seen in the drawing), until it becomes the value Pi, when applied to it the appropriate sum of exponents (n=8).
In this formula we see that there is an exponent (n) that, giving it successive values, expands and curves the semi-square (2.Sqrt.2 =2.828427...) from its minimum value (2.828427...), to a higher value at 3.16....
Well then, when this sum of exponents (n) is equal to the sum of exponents necessary to obtain (2.828427...) by the Pythagorean Theorem, that is, n=8, then the value of this bending function is the Cyclic Pi, which I understand must be the correct value of Pi.
That is, when this function of powers and roots acquires the character of a Pythagorean function, it gives us the value of the number Pi.
For n=8; Pi = 3.14159144414199........

More examples

In this second pyramid, it is shown the power Pi^34 in relation with the perimeter of the circumscribed square to the circumference (8) by the decimal powers 10^16.

To finish, a consideration: On the adjustment of Pi by the sum of sides of inscribed polygons:

Logical postulate?

"The sum of sides (when these tend to infinite) of a polygon inscribed to the cirumference will represent an expanded or extended staight segment from the circumference and must measure a little more than it, even if it is minimally."

Postscript:

Friends, I hope you understand me and know how to forgive me, but in my lucubrations and mental fantasies I think that this new method and Cyclic adjustment can represent a new renaissance of the number Pi due to its logic, simplification, development, expansion and cyclical interconnection, etc.