Today I gave a (brief) talk for the Graduate Seminar in the math department here at UVA. This is an informal setting, which is only for grad students (no professors heckling the speaker). They're generally an hour, but I only went slightly over half of that time. All the same, it was a pretty comprehensible talk. At the very least, it was comprehensible material - who knows how the talk went.
The title for my talk was Origami Numbers. I walked through the first 5 of the Huzita-Hatori(-Justin-...) axioms [wikipedia], which are enough to do standard straight edge and compass constructions. The first 4 are pretty straight forward, and it's with the 5th that you start making interesting things - parabolas. Then with the 6th axiom, we are able to obtain cubic roots, showing that origami is more powerful than a straight edge and compass. A great reference, and my starting point, for all of this, is the book Project Origami by Thomas Hull (which has lots of other goodies).
One thing I found while preparing my talk was identified as Lill's method in this paper by Alperin and Lang. It's a graphical method for finding roots of polynomials, and I'd never seen it before. The paper 'Geometric Solutions of Algebraic Equations' by Riaz talks about it, as does this site, which I just found (so I should take another look at). Lill's method draws a piecewise linear path (starting, say, at O, and ending at T), where the lengths of pieces correspond to coefficients of your chosen polynomial. Then you are supposed to draw another path from O that bounces around off the lines you made and ends at T. This line shows you one of the roots for the original polynomial (see the above references for more details). It doesn't seem particularly practical if you want to find roots (I'd go for Newton's method, if all else failed), but it's fun to have a new graphical way to think about things.