\centerline{Trisections, $k$-Sections, and Borg Processes} \centerline{(The Trisection Trilogy)} {\sl Stephen Judd Los Alamos National Laboratory / Northwestern University judd@sgt-york.lanl.gov} \vskip 1cm {\bf Introduction} Angle trisection is a problem which goes back at least as far as the ancient greeks. Several methods for doing so are mentioned in, for instance, Euclid's Elements [1]. A more interesting problem is trisecting an angle using only a compass and a straight-edge. Clearly some angles (such as $45^\circ$ and $108^\circ$) can be trisected exactly, but as many texts on modern algebra will demonstrate it is impossible to find a general solution to this problem without resorting to curves other than circles (parabolas and hyperbolas, for instance) [2][3]. The next problem is then how to approximate a trisection using a compass and a straight-edge. Over the past 2000 years several methods have been proposed; some of these are in references [3]-[9]. I offer another, iterative, method for trisection, and then generalize it to $k$-section, of an angle. This method may be, in the words of [4], a "slow iterative procedure of a naive kind", but if such is the case I wish I could write more slow and naive numerical algorithms!