The beast rose
A river of red inside
No longer swinging from trees
But through affine space
Pushing between the atoms
Into a place of imagination
Orthogonal to reality
I am somewhat wary based on the previous try, but I have another, better, solution to trisecting an angle. There is no active search on my part, but it just popped into my head.
The problem as stated is generally impossible to solve, as shown by Pierre Wantzel (1837). Wantzel's proof relies on ideas from the field of Galois theory—in particular, trisection of an angle corresponds to the solution of a certain cubic equation, which is not possible using the given tools.
The Greeks had a real problem with many things, including related rates as well as infinities, and the sqrt 2. I promised an explanation of the irrational nature of sqrt 2, so here it is first. No matter how you choose a number to define distance units, even if it is the Planck scale, when you travel orthogonal to that in 1 unit then rotate that vector it is between units and is thus imaginary by the common math. It would imply that the very definition of the number system is incomplete. I suppose that http://en.wikipedia.org/wiki/Dedekind_cut is intended to patch one hole, but many others exist.
So, back to trisecting an angle with nothing more than a compass. The Zeno's paradoxes demonstrate the misunderstanding of related rates. An angle drawn by any means, if it be consistent would require a specific time to complete an arc, exactly as a clock and by dividing the time you have the exact relationship to the curved space.