  We introduce and prove numerous new results about the orbits of the
$T$-fractal billiard. Specifically, in Section 3, we give a variety of
sufficient conditions for the existence of a sequence of compatible periodic
orbits. In Section 4, we examine the limiting behavior of particular sequences
of compatible periodic orbits and, more interesting, in Section 5, the limiting
behavior of a particular sequence of compatible singular orbits. The latter
seems to indicate that the classification of orbits may not be so
straightforward. Additionally, sufficient conditions for the existence of
particular nontrivial paths is given in Section 4. The proofs of two results
stated in [LapNie4] appear here for the first time, as well. A discussion of
our results and directions for future research is then given in Section 6.
