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November 21st, 2018, 01:53 AM   #1
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circle squared

Who constructed nearest approximation of squaring circle and what geometrical method had been used (only with compass and straightedge)?
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November 21st, 2018, 02:34 AM   #2
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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.
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November 21st, 2018, 02:46 PM   #3
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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.
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November 21st, 2018, 11:05 PM   #4
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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?
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November 21st, 2018, 11:22 PM   #5
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Quote:
Originally Posted by Oldy View Post
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?
I'm not sure what your goal is here but the classical approximation 22/7 is already closer than $\sqrt{2} + \sqrt{3}$ and has the benefit of being rational. Its also worth mentioning that you can't square the circle by approximating $\pi$.

The entire question is whether $\pi$ is constructible by straight-edge/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.
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November 23rd, 2018, 03:55 PM   #6
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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
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November 26th, 2018, 06:54 AM   #7
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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.
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November 26th, 2018, 07:57 AM   #8
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Quote:
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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")?
You can make your own best approximation. Just cosmpute pi to as many digits as you wish, then use this number to create a square with sides this number. Done.

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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