Friday, September 12, 2003

Things to Read

Prodded by a statement made in a recent entry at Crooked Timber about impossibility theorems, I'm planning to dig out my old textbooks and reacquaint myself with Galois Theory, with the intention in mind of being able to rigorously lay out a proof of the impossibility of one of the classical challenges of antiquity - the trisection of any angle, using nothing more than an unmarked ruler and a compass.

The statement that spurred me to this undertaking actually referred to another problem, namely the task of squaring the circle, i.e, constructing a square equal in area to a given circle, using, as with angle trisection, only an unmarked ruler and a compass. Unfortunately, the proof of this impossibility hinges on the transcendentality of Pi, the proof of which is a lot more difficult than in the angle trisection case. That said proof is also an involved exercise in analytic number theory (as opposed to the algebraic sort, which I like), hardly helps matters.