STRUCTURES Blog | Mathematical Billiards

by Lael Shelton Costa on January 10, 2024 |   Discuss on: What's that structure? Do you enjoy a game of pool or billiards? I certainly do; I find it very satisfying to pocket a ball with a complicated shot following several bounces. I’m not a very skilled player, though, so I miss a lot of shots. Luckily, I can blame these failures on friction and inelasticity in the collisions, or on the unexpected interference of a different ball. You can imagine my relief when I learned of the existence of mathematical billiards! In this post, I will describe several varieties of mathematical billiards and discuss how I use computer experiments to make progress in studying these games. The images you see partially come from Geogebra and from a piece of software I created and host on my website. More on that later; let’s dive into the first game. Inner Billiards Imagine you’re playing billiards, but instead of a standard rectangular table, you’re using a custom-made one with no pockets and in a shape of your choice. We can draw the shape of an elliptical billiard table for instance (see Fig. 1). This choice is just an example, however! You can use any shape at all as long as it is convex, meaning that you have a clear shot from any point on the rails to any other: Example of a billiards bounce in an ellipse according to the regular inner billiard map. Mathematicians describe this process as a mapping, in which the directed chord “xy” is sent to the directed chord “yz”. Now pick two different points at the ellipse’s boundary – let’s call them “x” and “y”. Imagine that you strike a point-like ball at x in the direction of y. It will traverse the cyan path in Fig. 1, bouncing off the wall towards a new point “z”. In an idealized, frictionless world, we can follow the path of the ball through as many bounces as we wish. The resulting sequence of chords is called a trajectory. Mathematical billiards is the study of the trajectories of balls in these idealized billiards games. For Advanced Readers: Billiards as dynamical systems Mathematicians like to describe “games” like this as dynamical systems. A dynamical system is a description of a “state” (in this case, a pair of points like x and y above, such that at some time the ball travels from x to y) together with a way in which that state evolves over time (in this case, the physical notion that when the ball bounces at y, it will change course and head for z). Those familiar with the theory of dynamical systems might describe the game in the following way: billiards is a dynamical system on the space of directed chords in the body K that defines our table. The billiards map TK is the transformation that does the operation described above: it “eats” the directed chord xy and “spits out” the chord yz, such that the chords obey the law of reflection at the point y. Geometers who study these kinds of systems ask questions like: does K admit a periodic trajectory? That is, can one choose an x and y such that resulting trajectory retraces its steps exactly after some time? A periodic billiards trajectory. Several iterations of a nearby trajectory which is not periodic. Many mathematicians have studied this form of billiards and much is known, but there are also fundamental questions which remain open. For instance, it is not known whether all triangles admit periodic trajectories. A good reference for the topic can be found here. Outer Billiards Occasionally a novice billiards player like me might strike the cue ball a bit too forcefully and knock it entirely off the table. If you’ve had this experience, worry not! Mathematics is once again there for us with a game called “outer billiards.” This game no longer follows the “ordinary” rules of physical billiards (i.e. the physics of collisions on a table), but is an abstract game describing the motion of an object around a given geometric shape according to an entirely different set of mathematical rules. For illustration, we choose again the simple example of an ellipse. Consider a point “x” outside the ellipse and construct its two tangent lines to the ellipse’s boundary. As sketched in the animation in Fig. 3, pick one of the two tangent points (called “p” in the figure) such that the interior of the ellipse is on the left side of the blue line segment, as seen from a player located at x. Finally, reflect the player’s position x through that tangent point to obtain the new point “y”. The resulting path from x to y then corresponds to one “strike” in this modified billiard game. One step of the outer billiard map. The point x is sent to the point y by what mathematicians call the outer billiards map. One step of the outer billiard map. The point x is sent to the point y by what mathematicians call the outer billiards map. Outer billiards is a new dynamical system, where a state is a point in the plane outside the table, and the evolution of the system is the map taking x to y as described above. Having arrived at y, we can repeat the same procedure again and again. The sequence of all places the “billiard ball” is going to visit in this way, when starting at x (in other words the set containing the point x, the point x is sent to, the point that point is sent to, and so on) is called the orbit of that point. Outer billiards provide a rich field of study because there is a variety of behaviours on display. Depending on the choice of geometric shape of the body being studied, one may see very predictable behaviour, or something chaotic. The outer billiards system is very well-behaved in the case of an elliptical table: every orbit lies on an ellipse which shares its foci with the table. But tables have been demonstrated for which orbits diverge to infinity (reference). Caveat: Definition of Outer Billiards for Non-Smooth Shapes Note that in the definition of outer billiards, in the case of a non-smooth body, “tangent” may simply mean “passes through a vertex.” Singularities of the Outer Billiard Map For the rest of this post, I will predominantly be discussing cases in which the geometric shape that is being studied is a regular polygon. The attentive reader may have noticed a hole in our definition of the outer billiard map: what happens if the tangent line along which we wish to reflect x lies along a flat side of that shape? Points on one side of this line are reflected through one vertex, while points on the other side are reflected through another. There is no way to define the map for a point on the line that bridges that gap, so we will say that the map is not defined here, or that it has a “singularity at x.” When our table is an n-gon, this means we must exclude n rays (the extensions of the sides of the polygon) from the domain of the map. Apart from these singularities, however, the map is very nicely defined. But… could it happen that a billiard ball which starts at a non-singular point eventually reaches a singular point and gets stuck? This can indeed happen. We have to call all of those starting points singular too, because we are only interested in points whose entire orbits are defined. So what does the complete singularity set look like? Here is where the computer really shines. Singularity diagrams for an equilateral triangle, a square, and a regular hexagon. In each case, the billiard table is the white polygon in the centre, the orange lines are the singularities, and the teal lines connect points in a single orbit. If our table shape is a regular tiling polygon (i.e., triangle, square, or hexagon), the singularity pictures are tilings of the plane (and every nonsingular point lies on a periodic orbit). But for regular polygons with other numbers of sides, the pictures we get can be much more complicated (see, e.g., Fig. 5). Some cases have been studied in detail. The pentagon has well-described fractal behaviour and non-periodic orbits, for example. For all regular polygons, no matter how many sides they have, we can make the following observations: there are some polygon-shaped regions in the plane where there are no singularities (the dark regions in the images). If a ball starts in one of the regions, it will be sent to another and another. Sometimes, the ball may come back to the region it started in, and if that happens, then it turns out that every starting point in that region will come back to its starting point! For these reasons, we sometimes call these regions “periodic islands.” For tables of almost any shape, it is typical to observe a number of these islands together with some messier regions in between them. Singularity diagrams for the pentagon, heptagon (7 sides), and dodecagon (12 sides), and a long orbit in the pentagon case. For Advanced Readers: Hyperbolic Outer Billiards One variation of this problem I have found particularly interesting to explore is the translation into hyperbolic geometry (for a primer on this topic, see e.g. this website). A key difference for our purposes is that there is no longer a single regular n-sided polygon for each n, but an entire family of them with different side lengths. Let’s start with equilateral triangles. A typical singularity diagram appears to have circular periodic islands of various sizes, as well as smaller connected components with unknown descriptions. Perhaps these regions comprise periodic orbits? Perhaps they have some fractal structure? Typical behaviour for triangles and pentagons. Again, orange lines indicate the singularity sets and teal points belong to a single orbit.