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Is there a geometric proof for irrationality of $\pi$? That would be neat.

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    $\begingroup$ The asnwer probably is no (or at least other methods must be used as well), compare with this MSE question. $\endgroup$ – Dietrich Burde Feb 16 '16 at 12:27
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    $\begingroup$ @DietrichBurde: I am not so sure of that. The continued fraction of $\pi$ does not have a nice structure, but there are plenty of generalized continued fraction related with $\pi$ that have a nice structure, and maybe they can "rendered" with a geometric construction. $\endgroup$ – Jack D'Aurizio Feb 16 '16 at 12:35
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    $\begingroup$ Related : math.stackexchange.com/questions/201076/… $\endgroup$ – Watson Feb 16 '16 at 12:35
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There is the beautiful short proof by Ivan Niven, A simple proof that $\pi$ is irrational, by elementary calculus. I am not aware of a pure geometric proof, but this does not mean, of course, that there might not be one. But it seems that the methods needed to prove the irrationality of $\pi$ always require also calculus, see here.

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