I'm so bad at billiards that I ended up in the hyperbolic plane [video]

TL;DR: Billiard trajectories may seem simple, but in an outer billiards system where the ball “just grazes a corner of the table and then travels the same distance again,” even a polygon can produce cycles, singularities, stable islands, and even unbounded orbits—and regular polygonal tables ultimately lead you into the hyperbolic plane. ## From Inner Billiards to Outer Billiards Hit a billiard ball, and it repeatedly bounces off the edges of the table. On a rectangular table, its trajectory is orderly; but once the table is no longer rectangular, the trajectory becomes difficult to predict. Mathematicians call these systems “inner billiards.” But what the video really wants to show is “outer billiards.” The rules are simple: 1. Hit the ball so that it just grazes a corner of the table; 2. Continue past the point of contact for the same distance again. Repeat this, over and over, and we get an attractive pattern. This is essentially a “reflection” jump: reflect the current position through a vertex to the opposite side. ## Cycles: Starting with a Parallelogram Table A parallelogram table is the easiest example to start with. Like a rectangle, it can tile the plane without gaps, so its trajectories behave quite regularly: - Starting from certain positions, we get a **cycle of 8 positions**; - From a different starting point, the period is **4**; - More generally, these 4 regions follow a 4-step cycle, those 8 regions behave a little more intricately, and another 12 regions all have period 12. Instead of counting each step until the ball returns, we can give every point its own color and see how each starting position evolves: after one step, two steps, four steps... The changing colors directly reveal the structure of the orbits. ## Singularities: The Trouble with Landing on a Boundary What if the ball lands exactly on the boundary between two regions? The first step still looks fairly normal, but on the second step, the ball’s path runs along an edge of the table—we cannot tell which corner to jump around. This is called a **singularity**. Singularities are organized by “depth”: - Some starting positions are **immediate singularities**, such as any point on those dividing lines; - There are also **depth-2 singularities**, **depth-3 singularities**, and so on. In other words, singularities are not isolated troublesome points, but a structure that propagates forward in layers. As the iteration depth increases, the distributions of singularities become noticeably different for different table shapes: - For **irregularly shaped** tables, singularities are packed closely together; - The greater the iteration depth, the denser the singularities; - Yet some regions remain unaffected. This raises a question: **What determines the shapes of the stable regions?** ## Islands: Isometries Within Cycles After many balls have been set in motion, the overall picture quickly becomes chaotic. But within each “island,” the billiard operation is actually an **isometry**, preserving shapes. This leads to an elegant conclusion: every “island” in the same cycle has exactly the same shape. The video gives these examples: - One periodic cycle contains **35 tiny decagons**; - Another cycle is likewise made up of identically shaped islands. These islands are the “tame” parts of the system—the mappings between them are rigid, simply moving the same shape to different positions. ## A Counterexample: Unbounded Orbits on the Penrose Kite Does every starting point produce a cycle? No. The video presents the first shape proven to have **unbounded orbits**, one of which was **first discovered by Richard Schwartz in 2007**. This shape is called the **Penrose kite**, commonly used in aperiodic tilings of the plane—in other words, it inherently carries the seeds of “never repeating,” and its outer billiards orbits indeed never return, traveling farther and farther away. This is the classic “bad news” of outer billiards research: rules simple enough to state in a single sentence can still permit unpredictable behavior that never closes into a loop. ## Reversing the Question: Fix the Ball, Move the Table’s Corners So far, we have mapped where the ball goes based on its initial position, with the trajectory depending on the table’s shape. What if we reverse the question? **Fix the ball’s position and observe where it eventually goes as we vary the location of an additional corner on the table**—this gives us a colorful map. Several features of this map are worth noting: - **There is always a black region**: adding a vertex there would place the ball directly inside the table; - **A uniformly colored ring surrounds the table**, fitting precisely with the ball’s original trajectory; - Adding the new vertex inside this ring does not affect the path, which is why the ring has a uniform color; - The farther away the ball is, the more boundary lines appear in its trajectory to constrain this ring. In other words, distant perturbations are “filtered” through successive layers, and only certain placements of an added corner affect where the ball ultimately goes. ## Toward the Hyperbolic Plane If squares can tile the Euclidean plane, what about **pentagons**? The answer is to tile the **hyperbolic plane**. In this space, all the interior angles of these pentagons are **90 degrees**, just like those of squares in Euclidean space. This step moves the entire discussion from flat geometry to hyperbolic geometry. In the hyperbolic plane, regular polygons can tile the space as neatly as square tiles do in Euclidean space, while the trajectories, cycles, and singularity structures of outer billiards take on new forms. The video creator’s favorite tables happen to be **regular polygons**—because they naturally connect the beauty of outer billiards with hyperbolic geometry. Beyond polygonal tilings, the study of hyperbolic outer billiards still holds many unanswered questions. ## Acknowledgments and Behind the Scenes - Video creator: **2swap**. Background music by **6884**; you can support his Bandcamp through the link in the description. - Simulation code and an introduction to the topic were provided by **David J. Romano**—a doctoral student in chemistry at Brown University who also wrote the simulation code used in the video. - Expert review: **Professor Richard Schwartz** (the author of the paper on the Penrose kite) and **Dr. Lael Edwards-Costa** (who has published related papers and created several interactive outer billiards webpages, linked in the description). Outer billiards is a classic example of how **simple systems can produce unpredictable results.** ## Sponsorship This episode is sponsored by **NordVPN**: it encrypts traffic and routes it through servers around the world. Its built-in antivirus protection works at the network boundary—scanning downloads, detecting and removing malware, flagging fake online stores, blocking known phishing links, and optionally blocking ads. One account can protect up to ten devices. Visit `nordvpn.com/twoswap` for an exclusive offer and 4 extra months, with a 30-day money-back guarantee. Source: https://www.youtube.com/watch?v=kL9BTbIGxLg