What you are describing is an approximation of a quadrature of the circle. The problem that goes by the name of "quadrature of circle" is a question of equality, not approximation. Your approximation is correct (I guess, I did not bother to calculate it). There are probably an infinite number of approximations including the classical ones. But you have failed to construct the exact quadrature. It has been proven that no such exact quadrature is possible. Pi is not commensurate with any algebraic solution (excepting infinite series).
You should take the time to read my article before drawing a conclusion, the calculations are exact there is something going on that is very difficult for me to explain with words at the current moment but if you would take the time to use the formula you would come to the conclusion of exactness.
You start with an approximation of pi. All you can ever get is an approximate answer. Pi is a transcendental number. Here is a better approximation. 1 million digits of pi. One Million Digits of Pi · Pi Day
You don't even bother to read my article but keep making assumptions how can you come to a correct conclusion of what am trying to express? its as simple as using my formula you are missing the point.
I tried to read your article but ... let's just say it was incomprehensible. Same with your videos. As I said, it has been proven that no such construction exists.
Ok fair enough but did you use the formula? because if you did you would arrive at the conclusion c^2= pi*r^2
Ok fair enough but did you use the formula? because if you did you would arrive at the conclusion c^2= pi*r^2... or... "c^2 - pi*r^2= 0"
Let's see. \(\sqrt{\pi/3.125} \approx 1.0027\). \(180/\pi=57.2958\) \(57.2958*1.0027*1.0027 = 57.6056\) I can't see where the 115.2 comes from. A circle is a 2 dimensional mathematical figure. When you're talking about "reality", you sound like you're not doing mathematics any more. And yet this whole thing about quadrature of a circle is a mathematical thing, isn't it? Also, I can't understand what you mean when you say that treating a circle as 3D will bridge a gap between the 2nd and 3rd dimensions. That makes no sense to me. You're not explaining clearly right now. I can say that much. Can you explain why you believe that your constructed square has the same area as the initial circle?
Well, you are a very special person. Incredibly smart people have attempted this construction for thousands of years. And here you have a supposed solution. A very simple supposed solution.
If I have a square of side length L then it has area \(L^2\). And if I have a circle of radius r, then its area is \(\pi r^2\). So, if I want a square with the same area as the circle I need \(L^2=\pi r^2 \rightarrow L = \sqrt{\pi}r\) and if \(r=1\) then \(L=\sqrt{\pi}\). That's all fine. So you say that you have geometrically constructed a square from a circle of radius r, with sides that have length \(\sqrt{\pi}r\). Is that correct? And \(\pi=3.1415926535...\)?
James.R said "You're not explaining clearly right now. I can say that much." Yes I am aware of that I have not figured out the best way to explain it. Can you explain why you believe that your constructed square has the same area as the initial circle?[/QUOTE] Well if you use the rational version "360/115.2" its way easier to understand substitute this ratio for pi then everything becomes simplified so this follows a^2+b^2= c^2 ..."( r/4+r )^2 + ( r/4+r )^2 = a^2+b^2"...= 3.125*r^2 since I stated in my theorem that pi*r^2 and 3.125*r^2 are Quantum entangled doppelgangers then what ever results you conclude on one circle will also be concluded on the other circle because they are both perfect circles actually the only circle that are not perfect are polygon so for the circle pi*r^2 you just have to adjust the formula ..."( r/4+r )^2 + ( r/4+r )^2 = a^2+b^2" to the Euclidean form to get the exact measurement but it doesn't matter what perfect circle you use to make the construction the measurements will always be the same the only thing that changes is the ratio of pi, then you would just need to manipulate the difference but the ratio used is irrelevant and not even that necessary the solve the problem.
Let's check this one. The left-hand side reduces to \(\left[ \left( \frac{5}{4}\right)^2 + \left( \frac{5}{4}\right)^2 \right]r^2 = 2\times \frac{25}{16}r^2 = \frac{25}{8}r^2 = 3.125r^2\) That is correct. However, 3.125 is not \(\pi\), because \(\pi = 3.1415926535...\). So, if your geometrical construction corresponds to the left-hand side of the above equation, then it doesn't generate a square with area equal to the circle, but a square with a smaller area.
the square with equal area to the circles area when r=1 the side length of that square is equal to sqrt{pi}
Yes James am aware of this that is why you have to manipulate the formula all that's happening is you are changing the size of the diameter of a perfect circle when you change the ratio of pi that is used.