Squaring the Circle (part II)
In the first part we saw how to get the perimetric squaring of the circle and we saw how it correlates with the golden section. For the quadrature of the area we must use the root of five (√5) which is always constrained by Phi (Φ = 1/2 + √5 / 2 = 1.61803398...).
When I started this study I was surprised how much geometry and mathematics were entangled (I'm sorry I'm just a novice student!), But all this made me passionate and led to believe that they are universal languages, of which we still know very little.
Let's go back to the circle, that for simplicity of calculation we do it with radius 1, and with the four squares that are formed by tracing the symmetry axes of the square that circumscribes the circle. Here we can note that the diagonal of the inner squares has value √2 = 1.414213562...., while the diagonal of the rectangle formed by two inner squares has value √5 = 2.236067978.... as we could also verify with the Pythagorean theorem
At this point we may already have finished because the solution is already under our eyes, but for lovers of geometry and mathematics I want to show you some things that I hope you will find interesting...
I make a premise, I am neither a mathematician nor an engineer and I I do not pretend to prove anything, but rather to show what my passion for geometry and computers has brought to the surface. In my naivety I find these crazy things!
In fig.2 you will find two blue squares, the inner one has the perimeter similar to the blue circle (as shown in the first part), while the outer one has the area similar to the circle. In fact the area of the circle is r² * π = 1² * π = 3.14159265358979, while the square l² = 3.20000000000003 with a difference of 0.05840734641024 or 99.94159265358980% of accuracy.
As for the perimeter I tried to perform the same operations with the golden π (see www.jain108.com) and the result is slightly closer different since the area of the circle is 3.14460551102969, the difference is 0.05539448897034 or 99.94460551102970% accuracy.
The key to solving the problem is the √5, in fact the point where the semi-circle of radius 1 intersects the diagonal with value √5 is the point where the straight line is born which, intersecting the circle, finds the exact point where it will pass the side of the square.
One thing I noticed is that reporting the point of intersection between the projection of √5 / 2 (the one used to build the golden rectangle) and the semicircle of radius 1 with an arc whose center is the center of the circle and of radius √2/√5 create on √2 the point needed to find the line that intersects the circle for the quadrature.
When I started studying this design I noticed a series of facts that I called "a series of strange coincidences" of which I report some measures. This helped me to see the beauty that is the basis of this design, visually already evident, with the perfection with which these lines intersect in precise points, but to understand that there are very precise rules governing these lines was really exciting!
formulas governed by √5 and Φ shown in fig.2:
Φ/(Φ²+1) = 0.447213596
1/Φ = 0.618033989
√(Φ-1)*2 = 1.572302756
√(√5-1)/2-(√5-1)/2 = 0.168117389
√2/√5 = 0.632455532
3/√5 = 1.341640786
4/√5 = 1.788854382
In fig.3 the complete drawing in which all the intersections can be admired.