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

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    The asnwer probably is no (or at least other methods must be used as well), compare with this MSE question. – Dietrich Burde Feb 16 '16 at 12:27
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    @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. – Jack D'Aurizio Feb 16 '16 at 12:35
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    Related : http://math.stackexchange.com/questions/201076/is-there-any-sans-calculus-proof-of-irrationality-of-pi?lq=1 – 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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Dietrich Burde
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