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Author Lapidus, Michel L. ♦ Niemeyer, Robert G.
Source arXiv.org
Content type Text
File Format PDF
Date of Submission 2009-12-19
Language English
Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics
Subject Keyword Mathematics - Dynamical Systems ♦ 65D18 ♦ Secondary 37A99 ♦ 74H99 ♦ Primary 37D40 ♦ 37C27 ♦ 37D50 ♦ 65P99 ♦ 37C55 ♦ 58A99 ♦ math
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.
Description Reference: "Gems in Experimental Mathematics", T. Amdeberham, L. A. Medina and V. H. Moll, (Eds.), Contemporary Mathematics, American Mathematical Society, 517 (2010), pp. 231-263
Educational Use Research
Learning Resource Type Article
Page Count 26


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