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

TL;DR

This paper explores the complex dynamics of billiard trajectories within the fractal boundary of the Koch snowflake, using computer simulations of approximating prefractal billiards to formulate conjectures about periodic orbits.

Contribution

It introduces a novel approach of studying fractal billiards through prefractal approximations and computer experiments to develop conjectures on orbit behavior.

Findings

Computer experiments reveal patterns in billiard trajectories on prefractal approximations.

Formulation of conjectures regarding the existence of periodic orbits in the Koch snowflake billiard.

Proposed methods for proving properties of billiard orbits on fractal boundaries.

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.