In the latter case we can switch to a direct execution of the much coveted quadrature, re-starting from the basic square, (side Φ of the radius |
Ca in the figure) bypassing the preliminary construction of the great triangle with its whole cohort, which in any case remains the keystone in translating the transcendence of the π into immanent form.
According to the procedure described above, from the base of the starting square with vertex in
C we will obtain the length
Ca in increasing golden proportion φ, of which we will make the radius of the primary circle, assigning it a value of
1,0 (instead of the diameter, whereby the Φ0 in the figure).
Therefore the side of the square
C will take on value Φ.
To obtain its square root it will be sufficient to extend it from
b, and then with the center in the middle of
ba to trace the semicircle
apb [green stroke] which will be intersected at the point
p by the primary diameter, as a vertical extension passing through
The Pythagorean relation is confirmed
bC:Cp = Cp:1, from which
√bC, side of ¼ of the square whose perimeter has the length of the circumference with radius
Similarly we proceed from this by adding it in turn on the diameter to the radius
If we then wanted to quarter the vision on a functional value of π as simple2√Φ, we would likewise see in it the side of the square with a perimeter equivalent to the circumference with radius = ½, and in the form4√Φ the side of the square with an area matching that of the circle with radius = ½: a kinship at the DNA level.
Ca thanks to the arc
pd with center in
C, then tracing the semicircle
dqa [orange stroke] with a center in the middle of
da, intersected by the primary vertical diameter at point
q, so that
Cq, the square root of
Cp, will be the side of the ¼ of the square with the same area as the circumference with radius
Ca = 1.
It will be enough to reproduce the last square obtained, superimposing it on the circle in the four quadrants to take note of its compliance without compromise.
I don't think you can ask for more for squaring the circle.
The writer will not take a position between the three perspectives, nor seek glory for himself, he has already collected enough of it in the course of previous incarnations. Nor does it expect profits: its seven Web domains, containing several
searches and advanced solutions (see below)
, do not sell services or advertisements of any kind.
As long as it is not said that "he came close"...
Initially he would have been satisfied if everyone had intended to agree that this sacred triangle was a subject of such importance as not be kept closed in a drawer, whatever the outcome and the risk might be; but now that his research has matured
, he no longer has any doubts.