Squaring de Circle with ruler and compass
The extended Pi
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Squaring de circle with ruler and compass.
Inside the low precision that the use of ruler and compass allows.
Before looking at how the circle can be squared in a simple and enjoyable way, let's expose the concept of extended Pi, which we will use for this quadrature of the circle.
An ancient geometer (I don't remember who), thought that if we bend a curved line, for example to build a circumference, it would lose a very small portion of length, that is, it would contract a little when bending, and vice versa, if we extend a curve this minimally increases in length.
Possibly if he saw this new concept of the extended Pi, he would be a little more satisfied with he thoughts and deductions.
The extended Pi is obtained by a composition of segments internal to the circumference, of Pi/2 each, which, as seen in the drawing, are repositioned together forming a capital A, starting with the central segment placed on the diameter and center of the circumference.
The other two go from the upper point (O in the drawing) of the axis and of the circumference, passing united by the lateral ends of the central segment, and ending in their cut with the circumference (point S of the drawing).
Each of these segments represents an approximate value of Pi/2 (0.0009), which would make us doubt a little if "that ancient geometer would be right or not."
In any case, the important thing here is to consider that the extended Pi is a property and element of all the circumferences, and that it can be used for example, to square the circle within the measurement margins that the ruler and compass can do.
From the arithmetic point of view, squaring the circle is not possible because the number pi necessary is irrational, and even an exact division by 2 cannot be done to find the necessary side of the square.
However, and since what is required is to do it with a ruler and compass, in this case it is possible to do it since these devices cannot exceed three decimal digits of precision (aPi-> 0.0009).
Well, the example of squaring the circle that I expose is within these limits of precision of the ruler and compass does not exceed.
As we see in the drawings, this process is done in two steps.
Drawing 1.- Find the approximate segment of Pi,
Drawing 2.- Find the approximate segment on the side of the square.