Mathematical Billiards (2024)

Originally published on the STRUCTURES Blog by Lael Shelton Costa on January 10, 2024.

I'll admit it: there is a profound sense of satisfaction in sinking a ball via a complex path involving multiple rail bounces. However, my actual skill level is quite high rather lacking, leading to a frequent number of missed shots.

Usually, I justify these failures by citing:

• Surface friction
• Inelastic collisions
• The annoying interference of other balls

Fortunately, I discovered the realm of mathematical billiards, where such frustrations vanish! In this article, I will explore various types of mathematical billiards and explain how I utilize computational experiments to investigate these systems. The visuals accompanying this discussion are derived from Geogebra and custom software hosted on my personal site.

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🎱 Inner Billiards

Imagine a game of billiards played on a bespoke table. This table has no pockets and can be any shape you desire. For example, we might choose an elliptical boundary.

To keep the game functional, the shape must be convex. In simple terms, this means that if you are standing at any point on the rail, you have an unobstructed line of sight to any other point on the rail.

The Mechanics of the Bounce

In this idealized, frictionless environment, we track a point-like ball. The process works as follows:

1. Select two distinct points, $x$ and $y$, on the boundary.
2. Launch the ball from $x$ toward $y$.
3. The ball bounces at $y$ and travels toward a new point $z$.

Mathematicians view this as a mapping, where a directed chord $\vec{xy}$ is transformed into a directed chord $\vec{yz}$.

For Advanced Readers: Billiards as Dynamical Systems

To a mathematician, this "game" is a dynamical system. This consists of:

• A State: A pair of points (e.g., $x$ and $y$) representing the ball's current trajectory.
• Evolution: The rule that determines the next state (the physical law of reflection at the boundary).

Specifically, billiards is a dynamical system acting on the space of directed chords within a body $K$ (the table). The billiards map $T_K$ "consumes" the chord $xy$ and "produces" the chord $yz$, ensuring the angle of incidence equals the angle of reflection at point $y$.

A primary question for geometers is whether $K$ allows for a periodic trajectory—a path where the ball eventually retraces its exact steps. While much is known, some mysteries remain. For instance, it is still unknown if every possible triangle admits a periodic trajectory.

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🌌 Outer Billiards

Sometimes, a clumsy player (like myself) hits the cue ball so hard it flies right off the table. In mathematics, this isn't a mistake—it's a different game called outer billiards.

Unlike inner billiards, this is an abstract game. It doesn't follow the physics of collisions but rather a specific set of geometric rules for moving an object around a shape.

How to Play (The Ellipse Example)

If we use an ellipse as our central body, the movement of the "ball" (point $x$) is determined by these steps:

• Step 1: Identify a point $x$ located outside the ellipse.
• Step 2: Construct the two tangent lines from $x$ to the ellipse's boundary.
• Step 3: Select the tangent point $p$ such that the ellipse's interior remains to the left of the segment $xp$ from the perspective of the player at $x$.
• Step 4: Reflect the point $x$ across the point $p$ to arrive at the new position $y$.

This transformation is known as the outer billiards map.

def outer_billiard_step(x, table):
    p = find_left_tangent_point(x, table)
    y = 2 * p - x  # Reflection of x across p
    return y

Orbits and Behaviors

In this dynamical system, the state is simply a point in the plane outside the table. By repeating the map, we create an orbit. The orbit of $x$ is the set: $\text{Orbit}(x) = \{x, T(x), T(T(x)), \dots\}$

The behavior of these orbits varies wildly depending on the shape of the table:

1. Elliptical Tables: Very stable. Every orbit stays on a larger ellipse that shares the same foci as the table.
2. Other Shapes: Can be chaotic. Some tables have been proven to have orbits that diverge to infinity.

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⚠️ Caveat: Non-Smooth Shapes

When dealing with shapes that aren't smooth (like polygons), the term "tangent" is interpreted more broadly: the line simply needs to pass through a vertex.

Singularities of the Outer Billiard Map

For the remainder of my research, I focus primarily on regular polygons. However, an observant reader might notice a flaw in the definition: what happens if the tangent... [Text ends abruptly]