Phi and Pi, as twin parameters.
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Phi and Pi, as twin parameters

Curves lose dimension or length

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Friends, from my beginnings to review the peculiarities of the number Pi, (with the first work on the squaring Pi: Pi direct formula) I realized that if we bend a straight line to form a curve, this line loses length as be in a curved line since all its points are closer to each other on the inside of the curve.
But once observing the large number of internal segments of the circumference that were very close to being directed to the Phi number, I began to think and deduce that perhaps these two parameters (pi and Phi) were not only parallel in their components, but perhaps the same parameter, but measured in a different way, or if we want, in different dimensions: straight dimension and curved dimension.
After many attempts to square the circle and measure multiple interior segments of the circumference, this idea became established and the real reason why an easy squaring of the circle is not possible:
Because when the segments relative to Pi lose dimension, it is not possible to square them with any linear segment within the circumference.
Nevertheless, segments of Phi yes, they have multiple squares and interrelation between them.
For this reason, the current reasoning that it is not possible to square the circle because Pi is transcendental, for me, they face consistency since in no geometric construction we get to use thousands of decimal digits.
And also, there are other segments with infinite decimal places and we can construct them, such as the square root of segment 2, or the diagonal of the square.
We also do it with the segment 1/3 = 0.333333 .... which have infinite decimal places.

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Therefore and in summary, we can say that:

Phi and pi are two twin elements that represent the same parameter, but measured in a different geometric dimension:
Phi is the parameter in its rectilinear form, and Pi is this same parameter once curved to form the circumference.

And as a geometric principle:

All straight lines lose dimension or length when they are curved.
In the circumference, its curvature represents a negative increment in length, which we will measure by means of a coefficient of expansion (Phi/Pi) whose value we estimate as:
Dc (Phi/Pi) = 1.0009590223 .....

For example the segmento b,
at Pi = 0.785398163.....
x 1.0009590223...
at Phi = 0.78615136.....

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Squaring the circle: Small discussion (author's opinion).

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The use of square and compass is a practical way of drawing and not a theoretical one, that is, measurements, transports and traces have an action limit that does not exceed one hundredth of a millimeter of accuracy in any case, for each measurement taken.
Therefore and in practice with ruler and compass, in no case will we be able to execute a figure with an error of less than 0.01 m/m.
Therefore, to make an excuse that the circle cannot be squared with a square and a compass because Pi is transcendental, they do not have practical sense, but rather theoretical fantasy.
Thus, by means of the Phi number in this example, we can squaring the circle (theoretically) with an approximation of six decimal digits, (side of the square = 1.77245 | 9139), but as we have said, in the practice of ruler and compass we will only achieve an accuracy of (only 1,772 ... at most) because our vision and drawing tools do not allow us more.
But as the accuracy depends on the fewest number operations with arcs and segmenta we have to do to get the quadrature, and then, the more operations we do, the less approximation we will have.
Therefore, I understand that this method (phi) must be more exact than is exposed by Ramanujan (355/113), simply because with Phi only 5 operations are carried out, while they are the Ramanujan method, many more are carried out (more than 20).

Thanks all you.