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Nathan Tenny sent in this comment.
From: Nathan Tenny
Date: 2006 Jul 21
I'm assuming that the "true value of pi" story is serious, not some sort of highly oblique allusion to Ramanujan or something. I may be wrong.
The proofs that
(1) pi is transcendental and
(b) all transcendental numbers are noneuclidean,
while ironclad, aren't particularly easy. The Wikipedia entry on "Proof of impossibility" gives page numbers in Hardy & Wright's number-theory textbook---I had a course out of that book when I was a junior, but I screwed off pretty badly in that class, so I don't know if I should have understood that part or not. I doubt if the proof can be framed in terms that make it accessible to the typical squaring-the-circle kook.