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 ∂(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 KSn (naturally approximating KS from the inside) as rational polygonal billiards and examine the corresponding flat surfaces of KSn, denoted by S KSn . 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 such conjectures.

The Current State of Fractal Billiards

“…We have shown that in the case of Ω(KS), Ω(T ) and Ω(S a ), we can determine a sequence of compatible periodic orbits. We will see that in each case of a fractal billiard, under certain conditions, a sequence of compatible periodic orbits (or a proper subset of points from each footprint F n (x 0 n , θ 0 n )) will converge to a set which can be thought of as a true orbit of a fractal billiard table (or such a sequence will yield a subsequence of basepoints converging to what we are calling an elusive point in [LapNie2,LapNie3]). …”

“…This paper constitutes a survey of a collection of results from [LapNie1,LapNie2,LapNie3] as well as the announcement of new results on the T -fractal billiard table Ω(T ) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table Ω(S a ) (see [CheNie]). …”

Nontrivial paths and periodic orbits of theT-fractal billiard table

“…In [CheNie1], the fractal billiard table is a self-similar Sierpinski carpet billiard table. In [LapNie1,LapNie2,LapNie3], the fractal billiard table under consideration was the Koch snowflake fractal billiard table. In [LapNie4], recent results on the Koch snowflake fractal billiard table and a self-similar Sierpinski fractal billiard table are surveyed and the T -fractal billiard table is introduced; preliminary results regarding the T -fractal billiard table were presented without proof.…”

“…For the reader's easy reference, we provide in this section various definitions appearing in previous joint works; see [CheNie1,LapNie1,LapNie2,LapNie3,LapNie4]. However, the notation in all of the definitions will reflect the fact that we are discussing the T -fractal billiard table, which will significantly simplify the exposition.…”

“…We also note that Definitions 2.14-2.16 below are specific to the T -fractal billiard. However, it is plausible that one may be able to appropriately generalize them so as to account for what have been called periodic orbits of the given fractal billiard table(s) in each of [CheNie1,LapNie1,LapNie2,LapNie3,LapNie4]. Moreover, in some of the aforementioned works, we have indicated that a periodic orbit of a fractal billiard table may be the suitable limit of a sequence of compatible periodic orbits, namely the Hausdorff-Gromov limit.…”