Towards the Koch Snowflake Fractal Billiard: Computer Experiments and Mathematical Conjectures

In this paper, we attempt to define and understand the orbits of the Koch snowflake fractal billiard KS. This is a priori a very difficult problem because ∂(KS), the snowflake curve boundary ofKS, is nowhere differentiable, making it impossible to apply the usual law of reflection at any point of the boundary of the billiard table. Consequently, we view the prefractal billiards KSn (naturally approximating KS from the inside) as rational polygonal billiards and examine the corresponding flat surfaces of KSn, denoted by SKSn . In order to develop a clearer picture of what may possibly be happening on the billiard KS, we simulate billiard trajectories on KSn (at first, for a fixed n ≥ 0). Such computer experiments provide us with a wealth of questions and lead us to formulate conjectures about the existence and the geometric properties of periodic orbits of KS and detail a possible plan on how to prove