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

Abstract

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 $\partial(KS)$, the snowflake curve boundary of $KS$, 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 $KS_n$ (naturally approximating $KS$ from the inside) as rational polygonal billiards and examine the corresponding flat surfaces of $KS_n$, denoted by $\mathcal{S}_{KS_n}$. In order to develop a clearer picture of what may possibly be happening on the billiard $KS$, we simulate billiard trajectories on $KS_n$ (at first, for a fixed $n\geq 0$). 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 such conjectures.