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We describe the periodic orbits of the prefractal Koch snowflake billiard(the nth inner rational polygonal approximation of the Koch snowflakebilliard). In the case of the finite (prefractal) billiard table, we focus onthe direction given by an initial angle of pi/3, and define 1) a compatiblesequence of piecewise Fagnano orbits, 2) an eventually constant compatiblesequence of orbits and 3) a compatible sequence of generalized piecewiseFagnano orbits. In the case of the infinite (fractal) billiard table, we willdescribe what we call stabilizing periodic orbits of the Koch snowflake fractalbilliard. In a sense, we show that it is possible to define billiard dynamicson a Cantor set. In addition, we will show that the inverse limit of thefootprints of orbits of the prefractal approximations exists in a specificsituation and provide a plausibility argument as to why such an inverse limitof footprints should constitute the footprint of a well-defined periodic orbitof the fractal billiard. Using known results for the inverse limit of asequence of finite spaces, we deduce that the footprint (i.e., the intersectionof the orbit with the boundary) of a piecewise Fagnano orbit is a topologicalCantor set and a self-similar Cantor set. We allude to a possiblecharacterization of orbits with an initial direction of pi/3. Such acharacterization would allow one to describe an orbit with an initial directionof pi/3 of the Koch snowflake billiard as either a piecewise Fagnano orbit, astabilizing orbit or a generalized piecewise Fagnano orbit. We then close thepaper by discussing several outstanding open problems and conjectures about theKoch snowflake fractal billiard, the associated 'fractal flat surface', andpossible connections with the associated fractal drum. In the long-term, thepresent work may help lay the foundations for a general theory of fractalbilliards.

Families of Periodic Orbits of the Koch Snowflake Fractal Billiard