  A key point in Douady and Hubbard's study of the Mandelbrot set $M$ is the
theorem that every parabolic point $c\ne 1/4$ in $M$ is the landing point for
exactly two external rays with angle which are periodic under doubling. This
note will try to provide a proof of this result and some of its consequences
which relies as much as possible on elementary combinatorics, rather than on
more difficult analysis. It was inspired by section 2 of the recent thesis of
Schleicher (see also Stony Brook IMS preprint 1994/19, with E. Lau), which
contains very substantial simplifications of the Douady-Hubbard proofs with a
much more compact argument, and is highly recommended. The proofs given here
are rather different from those of Schleicher, and are based on a combinatorial
study of the angles of external rays for the Julia set which land on periodic
orbits. The results in this paper are mostly well known; there is a
particularly strong overlap with the work of Douady and Hubbard. The only claim
to originality is in emphasis, and the organization of the proofs.
