Squaring the Circle (part I)
Updated: Jun 12, 2019
Even if it is impossible to reach perfection, getting closer and being able to see its beauty is a privilege.
In reality it is not yet possible because on this level we do not yet have the means to understand the nature of certain irrational or transcendental phenomena and consequently we fail to "see" beyond. Only with the evolution of consciousness can we connect to new knowledge that will open the door to future discoveries.
My personal attempt to find the squaring of the circle started from the search for a square that had the perimeter similar to the given circle, during this process I ran into some solutions, even quite precise (99.9% of difference between the perimeter of the circle and the square) whose execution, however, did not seem to me to be linear. So I continued in the research and after several attempts it appeared to me this elegant, simple and very precise solution. The fact that it was connected with the golden ratio Φ (Phi) made me realize that it was the right path. I want to show you how easy it is to find the perimeter squaring of a circle
Let's take an example to see the simple steps to do to square the circle. Let's start by drawing a circle of radius 1 (to simplify the calculations, but it works with any diameter) and divide it into four squares, in this case of 1x1 (fig.1).
Now we take the first square at the top left and we build the golden rectangle.
Draw a straight line from the midpoint of the base of the square to the opposite right corner, this line is length √5 / 2 = 1.11803398 ... With the compass turn that line so that it runs along the square's side (fig.2).
Now we can see the formula to obtain Φ, the number that defines the golden ratio, or 1/2 + √5/2 = 1.61803398875....
We can notice that there is a side of this rectangle that intersects the circle (fig.3). The square whose side passes through this intersection has a perimeter closest to the circle as we will see in a moment.
Now we must trace the diagonals to determine the angles of the square and then draw a straight line parallel to the base of the rectangle through the point found and get the other points on the diagonals to obtain the square (fig.4) or do some other steps to make it easier...
To identify more clearly the points to create the square we can draw all eight possible golden rectangles, so as to have two points for each side (fig.5)
Let's analyze the numbers found: the perimeter of the circle is π * d, or 3.141592654 * 2 = 6.283185308.
The perimeter of the square is l * 4, or 1.572302756 * 4 = 6.289211024.
The difference obtained between the two results is 0.006025716 or 99.993974284% accuracy!
If we wanted to embrace the theory that π could be obtained from Φ (fascinating theory that I fully share), the golden π which has a value of 4 / √Φ = 3,144605511.... is excellently explained by the mathematician Jain 108 (www.jain108.com).
With this value the difference between the perimeters would become 0.000000002 or 99.999999998% accuracy!!
I was very surprised on how the golden section allows with its simplicity and beauty to obtain such precise results.
The drawings were made with CAD software (Fusion 360) which guarantees very efficient drawing accuracy and provides a 9-digit approximation, so I feel confident and satisfied with the values obtained. Obviously it is possible to reproduce this drawing even with a compass and a ruler as the ancient Greeks did. So, in the end, I can say that Φ squares the circle (at least for the perimeter ;)... (fig.6)
In the second part I will integrate this drawing with the construction of the square with an area equivalent to the circle.