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

arXiv comment26 pages, color figures (For crisper figures, please contact the second author)