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Pyramids of squaring Pi
by Fernando Mancebo Rodríguez.
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.
Below is showed two pyramids that relate the squaring Pi with the perimeters of the inscribed and circumscribed squares to the circumference.
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.
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.
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.
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.
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.
"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.
Title: Pyramids of Squaring Pi.
"Chance favors the prepared mind." - Louis Pasteur