The Squaring Pi: principles and foundations
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The Squaring Pi: principles and foundations

The Squaring Pi: Principles and foundations

Principles:

1.- Theorem of structural logic:

"If geometrically and on a given circumference, and relying on it, we can structure, inscribe and circumscribe squares and circles to infinity; also, and arithmetically, we will have to structure and measure all these elements as a function of each other to infinity"
And since Pi is a basic structural parameter, it must also have an arithmetic interrelation and structuring with all the other parameters, to infinity.

2.- As the circumference is a regular curve, and the curves are built by means of powers and roots, also Pi (semi-circumference of radius 1) must be interrelated with the other parameters by means of powers and roots of itself and of the other parameters. Simple example of curvature: y=x^2

3.-Practical result:
"With the appropriate powers of Pi we can measure each and every one of the parameters of the circumferences and their inscribed and circumscribed squares from a first given circumference (r=1) to infinity"
For examples:
Pi^17 = (2.Sqrt.2) x 10^8
Pi^34 = 8 x 10^16

* As we can see, and remembering the initial relationship of the simple curvature (y=x^2), there is a relationship of power and roots between the rectilinear elements and the powers of Pi that relate them:
y^n = Pi^2n , with a small variation in the powers of Pi, depending on the parameter we are looking for.
However, and to facilitate the operations of high powers and roots, we use the decimal number 10 to reduce the powers of Pi and keep them at a level close to unity.
An example of this is the most used basic value (Pi^2/10)^n,
9.8696.. /10 = 0.98696....
And then (0.98696..)^17 = 0.8, which multiplied by the reduction of 10 that we made, gives us the value of the square circumscribed to the circumference = 8

Foundations

Now, the question that we can ask ourselves can be:
There is some reason that the reduction powers of 10 for rectilinear elements are 8, 16, 32,... etc., and 17, 34, 51, 68,... etc. for the power cycles of Pi?
Well, it seems so, that everything is based on a basic structure of the Pythagorean theorem in relation to the radius of the circumference.
As we see in the drawing, to begin a construction of the squares inscribed and circumscribed to the circumference, and starting from the radius of the same, the total powers that must be applied to this radius to obtain the squares, would be the same that must be applied to the squares and number Pi, to interrelate them with each other.
Thus, to get (2.Sqrt.2 ) we need to square the (radius 1) four times, that is, a sum of exponents equal to 8.
Well, this 8 is the reduction exponent of 10 to be used with the rectilinear parameters, and double for the powers of Pi, increasing by 1, 2, or more, depending on the parameters that we want to obtain from the powers of Pi.
Summarizing: with the appropriate powers of Pi, we obtain all the parameters of the circumference, such as radius-diameter, and its inscribed and circumscribed squares, etc., to infinity.

Thank you friends