Periodic orbits for square and rectangular billiards

Abstract

Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then completely characterised by the number of elastic collisions. The rules of mathematical billiards may be simple, but the possible behaviours of billiard trajectories are endless. In fact, several fundamental theory questions in mathematics can be recast as billiards problems. A billiard trajectory is called a periodic orbit if the number of distinct collisions in the trajectory is finite. We classify all possible periodic orbits on square and rectangular tables. We show that periodic orbits on such billiard tables cannot have an odd number of distinct collisions. We also present a connection between the number of different classes of periodic orbits and Euler's totient function, which for any integer $N$ counts how many integers smaller than $N$ share no common divisor with $N$ other than $1$. We explore how to construct periodic orbits with a prescribed (even) number of distinct collisions, and investigate properties of inadmissible (singular) trajectories, which are trajectories that eventually terminate at a vertex (a table corner).

Figures (10)

• Figure 1: Examples of periodic orbits on the square billiard table; the orbits in panels (a), (b) and (c) have periods two, four and six, respectively.
• Figure 2: Examples of period-six orbits on the square billiard, generated by the initial pair $\langle P_0, \alpha_0 \rangle = \langle 0.2,\, \tan^{-1}{\! (2)} \rangle$ in panel (a), the shifted initial pair $\langle P_0, \alpha_0 \rangle = \langle 0.25,\, \tan^{-1}{\! (2)} \rangle$ in panel (b) that is from the same equivalence class, and the pair $\langle P_0, \alpha_0 \rangle = \langle 0.6,\, \frac{\pi}{2} - \tan^{-1}{\! (2)} \rangle$ obtained by rotation in panel (c), which we consider part of a different family of period-six orbits.
• Figure 3: The unfolded period-six orbit from Figure \ref{['fig:equivalence']}(c) on the square ${ {\color{blue} \hbox{\color{black}{\sf ABCD}}}}$. The periodic orbit has type $(2, 4)$, which means that the unfolding requires two reflections about a horizontal side and four reflections about a vertical side before the trajectory repeats on a translated copy of tables with the same orientations.
• Figure 4: Equivalent representation of the period-six orbit from Figure \ref{['fig:equivalence']}(c) on the torus. Panel (a) illustrates the trajectory on the large square table comprising all four orientations of the square ${ {\color{blue} \hbox{\color{black}{\sf ABCD}}}}$. The left and right sides ADA are identified to form a cylinder in panel (b), and the top and bottom sides ABA are subsequently identified to form the torus in panel (c), on which the period-six orbit forms a closed curve.
• Figure 5: Hypothetical period-three orbit on the square ${ {\color{blue} \hbox{\color{black}{\sf ABCD}}}}$ shown as a triangle composed of the three points of collision $P_0$, $P_1$, and $P_2$ at which the billiard trajectory makes angles $\alpha_{0}$, $\alpha_{1}$ and $\alpha_{2}$ with the table, respectively. The incoming and outgoing angles at these points are intended to be equal, and labelled so accordingly.

Theorems & Definitions (21)

• Proposition 2.1
• Definition 2.2: Equivalence class $\mathcal{C}_K(p)$ of period-$K$ orbits
• Remark 1
• Definition 3.1
• Remark 2
• Proposition 3.2
• Theorem 3.3
• Definition 3.4
• Proposition 3.5
• Corollary 3.6