(You don't know me, but I came across this, and I can't help but reply; and hopefully with a couple interesting ones.) As someone who did a degree in maths, I could hopefully do quite a few. But the two I *like* (of the ones not previously mentioned) are:
Construction by straightedge and compass giving algebraic extensions of degree power-of-2, and then proving the non-constructibility of certain numbers, giving you three unproven-for-thousands-of-years theorems at once: The Impossibility of Doubling the Cube (http://en.wikipedia.org/wiki/Doubling_the_cube), Trisecting the Angle (http://en.wikipedia.org/wiki/Trisecting_the_angle), and Squaring the Circle (http://en.wikipedia.org/wiki/Squaring_the_circle). (More (http://en.wikipedia.org/wiki/Constructible_number))
Not-Burnside's Lemma (http://en.wikipedia.org/wiki/Burnside's_lemma). I don't know why, but I love this theorem. (And it's got the interesting weirdness with its name. See the "History" section.)
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timeasmymeasure
More proofs
Date: 2009-02-23 02:43 pm (UTC)