November 21st, 2018, 01:53 AM  #1 
Newbie Joined: Nov 2018 From: Bulgaria Posts: 3 Thanks: 0  circle squared
Who constructed nearest approximation of squaring circle and what geometrical method had been used (only with compass and straightedge)?

November 21st, 2018, 02:34 AM  #2 
Senior Member Joined: Oct 2009 Posts: 629 Thanks: 192 
It is very easy to give an approximation as close as you wish, since the constructible numbers are dense in $\mathbb{R}$. Hence you can find values as close to $\pi$ if you wish, which you can then construct and make it into a square.

November 21st, 2018, 02:46 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,641 Thanks: 625 
Start with a diameter and cut angles in half as long as you want. Form triangles with the radii and get areas of triangles, adding up to approximate circle area.

November 21st, 2018, 11:05 PM  #4 
Newbie Joined: Nov 2018 From: Bulgaria Posts: 3 Thanks: 0 
Thank you for replies. As far as approximation of Pi I found one approximation such as sqrt(2)+sqrt(3) app=Pi (doubtfully calculated by Plato) which could be used for squaring the circle using arcs or similar triangles. Have you ever heard of these methods?

November 21st, 2018, 11:22 PM  #5  
Senior Member Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
The entire question is whether $\pi$ is constructible by straightedge/compass and this has long been proved impossible. But there are constructible numbers arbitrarily close to $\pi$ so the distinction of constructibility becomes meaningless if you just want to approximate it.  
November 23rd, 2018, 03:55 PM  #6 
Senior Member Joined: May 2016 From: USA Posts: 1,206 Thanks: 494 
Among the ancient Greeks, Archimedes found a lower and upper bound for pi that is more intuitive than what you refer to. https://itech.fgcu.edu/faculty/clind...rchimedes.html By the 17th century, Wallis had found more powerful methods of approximation. https://en.m.wikipedia.org/wiki/Wallis_product 
November 26th, 2018, 06:54 AM  #7 
Newbie Joined: Nov 2018 From: Bulgaria Posts: 3 Thanks: 0 
I fully agree with your arguments. But based on those arguments it means that we could not calculate exact area of circle because of transcendence of Pi. On the other hand many mathematicians lived after Linderman had have tried to approximate both areas. many of then are prominent mathematicians. I asked for the best approximation just to compare different gemetrical methods but received no answer. Does it mean that theoretical studies do not follow the practical ones (even just for "educational purposes")?
Last edited by Oldy; November 26th, 2018 at 07:38 AM. 
November 26th, 2018, 07:57 AM  #8  
Senior Member Joined: Oct 2009 Posts: 629 Thanks: 192  Quote:
Nobody does this anymore though: approximating the circle with straightedge and compass. Historically it was an important question. Now it's solved and absolutely nobody has been working on this question for decennia.  

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