Waaaay back in ninth grade, in Mr. Laeser's geometry class, I brought up angle trisection, knowing next to nothing about it. So, Mr. Laeser said he'd give extra credit to anyone who could trisect an angle using a compass and a straightedge. My attempts were dismal failures, and I think the only other person who attempted it was my friend George (of the famous GASUC) Wright, who spent some more time with it (I think he got nine red squares for his efforts).
Anyways, the summer of 1993 I was eating lunch with Bill Beyer, one of the Old Bulls at Los Alamos, and he suggested that I derive a method for angle trisection. Well, when the gauntlet is dropped like that there is simply no choice in the matter. So I sat down that afternoon and thought about it and by lunchtime the next day I had my trisection method all worked out. So then he said that what he really wanted was a method of n-secting an angle, so that afternoon I sat down and figured that one out (it is really a very simple extension of the trisection algorithm, so no big deal).
After this I did a little research into trisection methods. I don't remember much of what I found, except that most of the methods are clever geometric methods that are an awful lot easier to implement than my method. I wrote up my stuff in some little .tex papers: the trisection trilogy. I never submitted them to anyone, but I put them here in their original form, warts and all. This was done long before I knew about things like iterative maps, and there are all sorts of little errors, Way Too Much Information sections, foolish statements, etc.
One other thing -- I seem to have misplaced the bibliography!
For the impatient among you, here is the basic idea behind the trisection algorithm: Start with the angle. Bisect it twice. Take this new angle, and add it to the old one. Bisect than angle twice. (Angle is now 5/16 the original angle; next iteration will be 21/64). Continue until you get bored.
N-section (or really prime-section, since e.g. to 10-sect an angle you need to 2-sect it and 5-sect it) follows naturally.