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.
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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. If N is in the interior IV. CLASSIFYING REGULAR of the hyperbolic plane, then the table is POLYGONAL TABLES small. If N lies on the boundary circle, then the table is neutral. And if two such orbits lie Using area properties of triangles in on the boundary circle, then the table is the hyperbolic plane, a geometric necessarily large. Ideally, we would like to description of large triangles was developed be able to classify a polygonal table simply in terms of its sides, a1, a2, a3, or its interior using characteristics of the table itself, angles, α1, α2, α3 [2]. For example, a triangle without having to search for and analyze any is large if and only if
2 2 2 cos 1 cos 2 cos 3 2cos1 cos2 cos3 1 sin1 sin2 sin3 .
As was mentioned in the previous section, consider F(x)s s . Since we have three the location of a particular type of orbit, 1 2 supplementary angles, denoted N, is indicative of the classification of an n-gon table. It will be helpful to know precisely where to find such an orbit under F(x)s s , our condition of interest—regular polygonal 1 2 2 2 tables. The following lemma provides the specific location of such an orbit, if it exists, and thus xs s F(x)s s (see Figure relative to the table. 1 n 1 2 4, for example). By side-angle-side we have Lemma. Let F be the outer billiard map congruent triangles, xs1sn F(x)s1s2 . defined for a non-large, regular n-gon table (Note that SAS is a valid congruence P with interior angle . Let s1, s2,…, sn be condition in hyperbolic geometry.) the vertices of P labeled in order according Since x lies on the perpendicular to the orientation of F. If x is the point from bisector of , it must be the case that the exterior side of sn s1 that |x s1| = |x sn| which implies via congruence (i) lies on the perpendicular bisector of that |F(x) s1| = |F(x) s2|. By the isosceles sn s1 , and triangle theorem, we have
F(x)s2 s1 /2 . We can now (ii) forms xs1sn /2 , mimic the argument for congruent triangles then x is an element of the unique n-periodic 2 orbit N with rotation number 1/n. with F(x) and F (x) in place of x and F(x), Furthermore, N forms a regular n-gon N with respectively. Continuing around the table, i interior angle . we find triangles xs1sn F (x)si si1 for 1 ≤ n–1 By the definition of the outer billiard i < n. This means that the points F (x) and map, |x s1| = |F(x) s1|. Since P is a regular x will be equidistant from sn. Furthermore, polygon, |sn s1| = | s2 s1|, and by our since assumption, . Let us
F n1 (x)s s xs s , n n1 n 1 2 2
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Figure 4. A regular 7-gon table and the point x.
n–1 we know that F (x), sn, and x are collinear. triangle will have two angles of / 2 and n Therefore, F (x) = x and the orbit N is n- one angle of 2 / n . (Note that in hyperbolic periodic with ρ(N) = 1/n. Since P is a non- large table, F must be an elliptic or parabolic geometry (n 2)/ n .) Letting a be the isometry, either of which guarantees that N length of a side of P, we have from the dual is unique. The rest of the lemma can be hyperbolic law of cosines verified forthwith. cos 2 / 2 cos2 / n We have determined that the cosh a periodic orbit N, which for regular polygonal 2 sin / 2 (1) tables is significantly described in the lemma, can serve as an indicator of the table's This side of P also defines a triangle classification (see Figure 5). It should also inside N (which exists because P is small). be noted that the neutral case serves as the Let ϕ be the interior angle of N (see Figure key to the general classification—a bi- furcation point, in a sense—because small 6). By the lemma we have / 2 as two and large tables can be considered with angles of this triangle, with ϕ as the third. regard to the neutral condition. In other Thus, within N we get words, the small and large cases can be revealed through knowledge of the neutral cos 2 / 2 cos case. This observation will play a significant cosh a 2 . role in the argument below. sin / 2
Now let us take an even closer look at the relationship between a regular n-gon We can simplify the right-hand side by using table P and its orbit N. Viewing this orbit as the trigonometric rule of complements the closed figure N, we know from the wherever we have a sine or cosine of lemma that N itself is also a regular n-gon. / 2 /2 /2 , leaving Let P be a small regular polygonal table. We can construct a triangle by joining 2 the center of P with two consecutive vertices sin / 2 cos cosh a . of P. By the regularity of the table, this cos 2 / 2
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Figure 5. Small, neutral, and large regular 4-gon tables in the Klein-Beltrami model.
Figure 6. Two triangles sharing the side a.
We substitute this expression into Equation 1 and find that
sin 2 / 2 cos cos 2 / 2 cos2 / n cos 2 / 2 cos2 / n . cos2 / 2 sin 2 / 2 sin 2 / 2 sin 2 / 2
Since ϕ is dependent upon , it makes sense to simplify so that
cos 4 / 2 cos 2 / 2cos2 / n cos sin 2 / 2 sin 2 / 2 sin 2 / 2
cos 4 / 2 sin 4 / 2 cot 2 / 2cos2 / n. sin 2 / 2
We have the difference of perfect squares in the numerator, so
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Figure 7. Small, neutral, and large regular 4-gon tables in the Poincaré model of the hyperbolic plane
cos 2 / 2 sin 2 / 2 cos cot 2 / 2cos2 / n sin 2 / 2 cot 2 / 21 cot 2 / 2cos2 / n cot 2 / 21 cos2 / n1.
Now we let the size of P vary until We have essentially proven the N is precisely on the boundary of the following result. hyperbolic model. This would occur when the interior angle of N reaches 0. Theorem. Let P be a regular n-gon table in Substituting 0 in for ϕ yields the hyperbolic plane with interior angle .
2 Then P is cos0 cot /21 cos2 / n1 small when 2 and subsequently cos2 / n 2tan / 21,
neutral when 2 cot 2 /21 cos2 / n. , and Simplifying so that n is on the left-hand side large when and is on the right gives 2 . 2 cos2 / n 2tan / 21 cos2 / n 2tan /21, which indicates the angle measure of a We can look for agreement of this theorem neutral n-gon table. with the previously discussed method for classifying triangular tables from [2]. If the table is a triangle with interior angles α1, α2, and α3, then the table is neutral when
2 2 2 cos 1 cos 2 cos 3 2cos1 cos2 cos3 1 sin1 sin2 sin3
This triangular method overlaps with the 2 3 3 theorem above when the table is a regular 3cos 2cos 1 sin . (2) triangle, that is, when = α1 = α2 = α3. So using the triangular classification method, By the theorem, a regular triangular table the table is neutral when with interior angle is neutral when
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cos2 /3 2tan2 /21. Solving for gives = 2 arctan(1/2), which can be substituted into Equation 2, yielding
3cos2 2arctan1/2 2cos3 2arctan1/21 sin3 2arctan1/2.
This simplifies to program at Grand Valley State University for its generous support. 2 2 3 3 3 4 3 2 1 REFERENCES 5 5 5 1. A. Baxter and R. Umble, ―Periodic orbits and we see that the two classification of billiards on an equilateral triangle‖ methods correspond exactly. Amer. Math. Monthly 115 (2008) 479– As an extra treat, let us apply the 491. theorem to regular 4-gon tables. By solving 2. F. Doǧru and S. Tabachnikov, ―On for we see that the table is small when polygonal dual billiard in the hyperbolic plane‖ Reg. Chaotic Dynamics 8 (2003) t 2arctan 2 /2, neutral when 67–82. t 2arctan 2 / 2 , and large when 3. E. Gütkin and N. Simanyi, ―Dual billiards and necklace dynamics‖ Comm. Math. t 2arctan 2 / 2. Physics. 143 (1991) 431–450. This case is depicted in Figure 7. 4. B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge Univ. V. CONCLUSION Press, 2003. 5. R. Liboff and M. Porter, ―Quantum In this article we have presented a chaos for the radially vibrating spherical modest introduction to outer billiards and billiard‖ Chaos 10 (2000) 366–370. review of work related to the Moser- 6. A. Lopes and R. Markarian, ―Open Neumann problem. We also developed a billiards: Invariant and conditionally method for classifying regular polygonal invariant probabilities on Cantor sets‖ tables in the hyperbolic plane, with large SIAM J. App. Math. 56 (1996) 651–680. tables having the property that all orbits 7. J. Moser, Stable and Random Motions escape to infinity. Thus the Moser-Neumann in Dynamical Systems: With Special problem, which had previously been solved Emphasis on Celestial Mechanics, for Penrose kite tables in the Euclidean Princeton Univ. Press, 2001. plane [11] and triangular tables in the 8. J. Moser, ―Is the solar system stable?‖ hyperbolic plane [2], has been solved for Math. Intell. 1 (1978) 65–71. regular polygonal tables in the hyperbolic 9. B. Neumann, ―Sharing ham and eggs‖ plane as well. Needless to say, much work Iota, Manchester University, 1959. remains. For example, is there a way to 10. S. Otten and F. Doǧru, ―A lesson classify an arbitrary polygonal table in the learned from outer billiard derivates‖ hyperbolic plane? We hope that interested Amer. J. Undergraduate Research 9 readers have recognized in outer billiards (2010) 1–7. the beauty that we see and have realized 11. R. Schwartz, ―Unbounded orbits for that, although it can get mathematically outer billiards I‖ J. of Modern Dynamics heavy quite quickly, it is a viable area of 1 (2007) 371–424. research for undergraduates. 12. S. Tabachnikov, Billiards, Soc. Math. de France, 1995. ACKNOWLEDGMENTS 13. F. Vivaldi and A. Shaidenko, ―Global
This work was supported in part by a stability of a class of discontinuous dual professional development grant from the billiards‖ Comm. Math. Physics 110 National Science Foundation Advance Paid (1987) 625–640. grant at Grand Valley State University. We 14. W. Weaver, ―The mathematical thank the Student Summer Scholars manuscripts of Lewis Carroll‖ Proc. Amer. Phil. Soc. 98 (1954) 377–381.
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