\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!