Pyramids of squaring Pi

by Fernando Mancebo Rodríguez.
Malaga.Spain.

The Squaring Pi consists on functions (exponentials) of the inscribed and circumscribed squares to the circumference.

Squaring Pi=  3,141591444141992652182488412553.....

The pyramids of squaring Pi are numeric tables developed in pyramid or triangle form, which show us as successive powers of Pi go approaching to successive decimal powers of the inscribed and circumscribed squares to the circumference, to end up coinciding at certain level.
With the values of these levels of coincidence we can obtain the squaring Pi by means of root of these values.

Below is showed two pyramids that relate the squaring Pi with the perimeters of the inscribed and circumscribed squares to the circumference.
Firstly the relative to the inscribed square, where we observe that the Pi powers go approaching to the decimal product of the inscribed semi-square to the circumference, till get to (Pi^17) and  (2 x Sqrt2 x 10^8) where is produced the coincidence of values.
Being this way in this level-point Pi^17 = 2 x Sqrt2 x 10^8

Inscribed square to the circumference.
  

Circumscribed Square to the Circumference

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.

As we see, the odd powers of squaring Pi drives us to the inscribed square to the circumference, and the even powers drives and gives us the circumscribed square.

Observations.

In the Pyramids of the Squaring Pi we observe as the Pi powers are approximately the double that the decimal powers (x10^n) applied to the perimeters of the squares, and it is due to get any decimal value applied to the sides perimeter is necessary the square of the number Pi (Pi^2 = 9.8696....)

We also observe that the powers of Pi in relation with the squares perimeters are the order of 2n+1 and 2n+2 due to for starting the pyramids of powers we need of +1 or +2 the powers of Pi to get the first term in the powers of the squares' perimeters.

Reasoning the number n of powers

The number of decimal powers n (10^n) that multiply the sides of the inscribed and circumscribe squares to the circumference is the number of powers applied to the triangles legs that form these sides when they are obtained by the Pythagoras theorem.

It seems to be that the coincidence numbers in powers (n=8 and n=16) for the perimeters of the inscribed and circumscribe square to the circumference are produced to this level due to these n-numbers are the numbers of times that we must to multiply the sides (legs) of the triangles to build the perimeters of the squares, as for the Pythagoras theorem.
Say, to form a side of the inscribed square (hypotenuse) it is necessary to elevate any leg to the square, what gives us as result 4 powers of legs for any square-side and 8 powers to the both square-side inscribed to the semi-circumference (Pi)

* For the pyramid of the circumscribed square the result will be double because of here it is not a semi-square, but a complete square.

Vision of alignment on the units' column.

 Other vision or geometric perspective is the alignment of the powers of Pi on the column of units.
This is gotten dividing the powers of Pi (Pi^2n+2) by 10^n, and with this we go observing clearer as these Pi powers go drive us to 8, the value of the perimeter of the circumscribed square to the circumference.
Getting this value (8) for n=16. (Remember, the number of powers that we must subject to the legs of the triangles component of the circumscribe square to the circumference)

Antecedents: The birthday of an idea.

 The first idea for searching the Squaring Pi was born from the observation of the curve functions in the Cartesian coordinates.
 If we look at the function y=x^2, this function gives us a curve, which in values between 0 and 1 is similar to a quarter of circumference. 
So, if the perimeters of the inscribed and circumscribed squares to the circumference are straight lines, and the inscribed circumference is a curve, (having both the same basic parameters of construction: circumference diameter and squares sides), then it should be possible that adequate powers and roots of these perimeters give us any function that unites both parameters. 
Later on, alone I must to practice and operate till find the "Circle's squaring":  The Squaring Pi.

Observation on the current Pi number

With the current algorithm method for obtaining Pi what we make is the addition of the semi-circumference points to build with them a straight line*, but Pi is an arc of circumference and not a straight line.
* Because here we are uniting and adding in a continue way the n-gon sides of the polygon in that we divide the circumference.
In this case, we forget a property or geometric principle that could say us:

"Any straight line that goes being curved endless, also goes losing dimension or longitude till disappear in a central point when this is curved indefinitely (endless) in symmetric or circumferential shape." 

And this is due to when we curve a straight line, the points that form the same go closing progressively among them by the interior side of the curve, till join together in a central point if the curvature is symmetric and endless.
Inversely, in the case of the algorithmic Pi, to the component points of the circumference we go adding them in straight line, and with that, we go extending them till form a straight line with more longitude (although in minimum value) than Pi in curved line.

Title: Pyramids of Squaring Pi.
Ferman: Fernando Mancebo Rodríguez.
Malaga.Spain.
2009-08-01