Geometric definition of Squaring Pi. From Archimedes to Squaring Pi.
I would like here to revise the methods for obtaining Pi, always observed from my viewpoints. To revise the main methods used till now to obtain Pi, we will divide them in three types according to the used basic principles. These types would be: Convergent, Parallel and Integrated.
Convergent (and orthodox) will be the methods that use of the geometric reality to get the number Pi. As example, we have the method of Archimedes that starting from the triangulation and use of successive polygons he was getting to come closer more and more to the number Pi. As we see, this method uses of the geometric reality to obtain Pi, although its difficulty is in the almost infinite number of operations that it is necessary to make to obtain important decimals of Pi.
The Parallel method (heterodox) is not direct method for the strict geometric reality, but rather we use operations with factors that we know they can go near to Pi number, but they are not intimately related with the geometry, but rather they are simple mathematical fractions that we ahead of time already know they will drive us near to Pi. In this example we have the number series that at the moment we use for obtaining Pi.
This method, also a philosophy of Pi, it is the one that takes notice of the properties of the inscribed and external squares to the circumference for propitiating formulas and operations with powers and roots which should drive to Pi. (See drawing in web page.) * We can remember that roots and powers are the base in the triangulation of squares.(Pythagoras) This option or method sustains that if the circumference depends and it is built on its inscribed squares, then this circumference should be defined by power and roots of the sides of these squares. And as in the practice we can observe an apparent coincidence between powers of Pi and the powers of the sides of the squares, because we understand that this method should be correct in its configuration and results. http://ferman.topcities.com/pi_direct_formula.html