I am well aware of the analytical proofs. I was wondering if a purely geometric one was/can be found.
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see here https://en.wikipedia.org/wiki/Proof_that_π_is_irrational – Dr. Sonnhard Graubner Feb 15 '18 at 20:28
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Another good ref https://projecteuclid.org/download/pdf_1/euclid.bams/1183510788 – Donald Splutterwit Feb 15 '18 at 20:29
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2@Dr.SonnhardGraubner the OP asked for a geometric proof – idok Feb 15 '18 at 20:30
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Geometric proofs of irrationality usually relies on the fact that some continued fraction is heavily structured, like $\varphi=[1;1,1,1,\ldots]$ or $\sqrt{2}=[1;2,2,2,\ldots]$. The continued fraction of $\pi$ does not exhibit any nice structure, hence I am partial to believing that a purely geometric proof of $\pi\not\in\mathbb{Q}$ cannot be designed. On the other hand, it is also true that $\arctan x$ has a nice generalized continued fraction. – Jack D'Aurizio Feb 15 '18 at 20:44