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Sizing up Outer Billiard Tables AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 10, NO. 1 (2011) Sizing Up Outer Billiard Tables Filiz Doǧru Department of Mathematics Grand Valley State University 1 Campus Drive Allendale, Michigan 49401 USA Samuel Otten Division of Science & Mathematics Education Michigan State University 116 North Kedzie Hall East Lansing, Michigan 48824 USA Received: September 15, 2009 Accepted: March 16, 2010 ABSTRACT The outer billiard dynamical system models the motion of a particle around a compact domain, such as a planet orbiting a star. When considering outer billiards in hyperbolic space, an interesting problem is to determine precisely the conditions in which an orbiting particle breaks orbit and escapes to infinity. Past work has classified triangular and Penrose kite billiard tables according to whether or not their orbiting particles escape. This article presents a classification of regular polygonal tables. I. INTRODUCTION Whereas inner billiards can be thought of as a crude model for the motion of gas particles If you mention billiards in casual within a closed region, outer billiards was conversation, you will likely elicit thoughts of proposed as a crude model for the motion of a cue game that involves caroming balls on celestial objects within a gravitational field. a cloth-covered table. If you happen to be Jürgen Moser popularized this idea when he conversing with a mathematician, however, related outer billiards to the stability problem the term may instead bring to mind the well- [8]—will the planets of our solar system ever known dynamical system involving the escape the gravitational pull of the sun? motion of a particle within a closed domain. Correspondingly, are there any outer billiard The billiard dynamical system has a great systems from which things escape to deal of intrinsic beauty (e.g., see Figure 1), infinity? This is known as the Moser- is a source of mathematical connections Neumann problem. (e.g., to number theory [1] and Cantor sets The current paper, after a formal [6]), and has applications to phenomena like definition of the outer billiard transformation the classical motion of gas particles in a and a summary of a few known results, will closed container [12], optics, and quantum continue the work on the Moser-Neumann chaos [5]. Even non-mathematicians may be problem by classifying regular polygonal interested in the mathematical version of tables in the hyperbolic plane. billiards when they learn that Lewis Carroll investigated billiard trajectories within certain II. THE OUTER BILLIARD polyhedra [14]. TRANSFORMATION What if we move the game of billiards outside the table? This is, in fact, Let P be a convex billiard table, that not a new idea but was first put forth fifty is, a compact domain with piecewise smooth years ago by Bernhard Neumann [9]. boundary. For inner billiards, one considers 1 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 10, NO. 1 (2011) corner of the table). The map F can be iterated and the set of all iterations of a particular point is called an orbit. If a point can be iterated so that it arrives back upon itself we say that the point and its orbit are periodic; that is, if Fk(x) = x for some natural number k, and k is the smallest such number, then x and its orbit are k-periodic. With regard to the Moser-Neumann problem, early progress was made by Moser himself who found that all orbits were bounded for sufficiently smooth tables [7]. Later, it was determined that quasirational polygonal tables also had the property that Figure 1. An elliptic billiard table with an all orbits were bounded [3], [13]. A elliptic envelope. significant breakthrough was made by Schwartz who found that Penrose kite polygonal tables had unbounded orbits [11]. a ball in the interior of P that reflects off the As in [2], we will be working boundary curve. For outer billiards, we specifically with polygonal tables in the consider a ball in the exterior of P that hyperbolic plane where the outer billiard reflects through points on the boundary transformation can be defined by simply curve. (Outer billiards is referred to by some taking all geometric notions to be in their as dual billiards.) To be precise, given a hyperbolic sense (e.g., reflecting points table P and a ball x in the exterior, there are using hyperbolic distances). We employ the two points on P that form support lines Beltrami-Klein disk model of the hyperbolic through x. The outer billiard transformation, plane because of its intuitive representation denoted F, has either a clockwise or of collinear points, though it should be noted counterclockwise orientation and is defined that angles are non-conformal in this model. two-dimensionally as the map that sends x Within the Beltrami-Klein disk F can be to its reflection through the support point in extended so as to be defined on the the given direction (see Figure 2). boundary of the disk. The restriction of F to This definition holds whenever the the boundary will be denoted f and will play support point in the given direction is unique; a large role in the work to follow. (Properties F(x) is not defined when x lies on what is of the derivative of f were previously referred to as a discontinuity line (which is explored in this journal [10].) similar to the undefined nature of the inner billiard transformation when the ball hits a Figure 2. Outer billiard mappings with counterclockwise orientation. 2 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 10, NO. 1 (2011) III. TABLE SIZES rotation number of the function, denoted ρ(f). The rotation number can be thought of as The outer billiard map F is an area- the average portion of the circle that is and orientation-preserving isometry (and in traversed by one iteration of the trans- the case of polygonal tables, a piecewise formation. Within outer billiards we are often isometry). Furthermore, since we are concerned with the rotation number of a working in the hyperbolic plane, F can be particular orbit, rather than of the classified as an elliptic, parabolic, or transformation itself. This moves us to utilize hyperbolic isometry depending on the the following notion of the geometric rotation number and location of any periodic points. number for a q-periodic orbit O, which will be An elliptic isometry has one periodic point in denoted ρ(O) and will refer to quotient of the the hyperbolic plane. A parabolic isometry winding number ω of the (possibly self- has one periodic point at infinity, and a intersecting) polygon formed by O and the hyperbolic isometry has two periodic points period of O; that is, ρ(O) = ω/q (see Figure at infinity. 3). This holds both for orbits of F in the The extended outer billiard map f is hyperbolic plane and orbits of f at infinity. a circle map and constitutes a rotation of the What does all this have to do with circle at infinity; more specifically, f is an the Moser-Neumann problem? The answer orientation-preserving circle homeo- is that by using the mathematical framework morphism [2]. Therefore, by a well-known above we can classify polygonal tables in result in dynamical systems, if a periodic the hyperbolic plane, and a table’s orbit exists on the circle then all other classification will tell us whether or not its periodic orbits of f necessarily have the orbits escape to infinity. The classes—small, same period [4]. It is also the case that the neutral, and large—are defined using existence of a single orbit on the circle (a geometric rotation numbers and the parabolic isometry) implies that the points derivatives of periodic points at infinity. (The within this orbit are attractive on one side derivative of the outer billiard map in the and repulsive on the other side. If two hyperbolic plane is not central to the periodic orbits exist on the circle (a purposes of this article and so will not be hyperbolic isometry), then the points from developed here. See [2] and [10] for more.) one orbit attract while the points from the A small table corresponds with an elliptic other orbit repel. These descriptions, along isometry and so has periodic points in the with the elliptic case mentioned above, hyperbolic plane. A neutral table corre- exhaust all possibilities for f because it can sponds with a parabolic isometry and so has have at most two periodic orbits on the circle a single periodic orbit on the circle (which [4]. (This limit to the number of periodic can be called a neutral orbit). A large table orbits at infinity is especially important in the corresponds with a hyperbolic isometry and next section.) so has two periodic orbits on the circle. When dealing with a rotation on the Since one of these orbits on the circle is circle such as f, it is common to consider the necessarily attractive, all orbits around a Figure 3. Orbits with rotation numbers 1/4 and 2/5, respectively. 3 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 10, NO. 1 (2011) large table escape to infinity. This final fact n-periodic orbits. Toward this end, we move is, of course, the motivation behind the on to the next section where we will briefly classification. review the existing classification method for The class of an n-gon table can be triangular tables and then introduce a new determined by locating a particular n- classification method for regular polygonal periodic orbit, namely, one with rotation tables. number 1/n. We will use N to refer to this particular type of orbit.
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