  We describe the periodic orbits of the prefractal Koch snowflake billiard
(the nth inner rational polygonal approximation of the Koch snowflake
billiard). In the case of the finite (prefractal) billiard table, we focus on
the direction given by an initial angle of pi/3, and define 1) a compatible
sequence of piecewise Fagnano orbits, 2) an eventually constant compatible
sequence of orbits and 3) a compatible sequence of generalized piecewise
Fagnano orbits. In the case of the infinite (fractal) billiard table, we will
describe what we call stabilizing periodic orbits of the Koch snowflake fractal
billiard. In a sense, we show that it is possible to define billiard dynamics
on a Cantor set. In addition, we will show that the inverse limit of the
footprints of orbits of the prefractal approximations exists in a specific
situation and provide a plausibility argument as to why such an inverse limit
of footprints should constitute the footprint of a well-defined periodic orbit
of the fractal billiard. Using known results for the inverse limit of a
sequence of finite spaces, we deduce that the footprint (i.e., the intersection
of the orbit with the boundary) of a piecewise Fagnano orbit is a topological
Cantor set and a self-similar Cantor set. We allude to a possible
characterization of orbits with an initial direction of pi/3. Such a
characterization would allow one to describe an orbit with an initial direction
of pi/3 of the Koch snowflake billiard as either a piecewise Fagnano orbit, a
stabilizing orbit or a generalized piecewise Fagnano orbit. We then close the
paper by discussing several outstanding open problems and conjectures about the
Koch snowflake fractal billiard, the associated 'fractal flat surface', and
possible connections with the associated fractal drum. In the long-term, the
present work may help lay the foundations for a general theory of fractal
billiards.
