GoKawiilby 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:
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 T K 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.
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