The reason billiards is so difficult to analyze mathematically is that two nearly identical shots landing on either side of a corner can have wildly diverging trajectories. The key idea that Tokarsky used when building his special table was that if a laser beam starts at one of the acute angles in a 45°-45°-90° triangle, it can never return to that corner. Adjust the original point slightly if the path passes through a corner. Aim your cue stick accordingly, keeping in mind that you’ll need to strike the cue ball with enough force to achieve your desired outcome while maintaining control over its path. 6. When folded back up, the path produces a periodic trajectory, as shown in the green rectangle. If you reflect a rectangle over its short side, and then reflect both rectangles over their longest side, making four versions of the original rectangle, and then glue the top and bottom together and the left and right together, you will have made a doughnut, or torus, as shown below. Since each mirror image of the rectangle corresponds to the ball bouncing off a wall, for the ball to return to its starting point traveling in the same direction, its trajectory must cross the table an even number of times in both directions.

In 2016, Samuel Lelièvre of Paris-Saclay University, Thierry Monteil of the French National Center for Scientific Research and Barak Weiss of Tel Aviv University applied a number of Mirzakhani’s results to show that any point in a rational polygon illuminates all points except finitely many. Proving results in the other direction has been a lot harder. It takes time and skill to master English largely because it’s hard to know just how much deflection is necessary to achieve the desired results. A key method for analyzing polygonal billiards is not to think of the ball as bouncing off the table’s edge, but instead to imagine that every time the ball hits a wall, it keeps on traveling into a fresh copy of the table that is flipped over its edge, producing a mirror image. Lay out a grid of identical rectangles, each viewed as a mirror image of its neighbors.

How Halloween Works: Check out this article for all things Halloween. This mathematical trick makes it possible to prove things about the trajectory that would otherwise be challenging to see. To find a periodic trajectory in an acute triangle, draw a perpendicular line from each vertex to the opposite side, as seen to the left, below. Draw a line segment from a point on the original table to the identical point on a copy n tables away in the long direction and m tables away in the short direction. In 2014, Maryam Mirzakhani, a mathematician at Stanford University, became the first woman to win the Fields medal, math’s most prestigious award, for her work on the moduli spaces of Riemann surfaces - a sort of generalization of the doughnuts that Masur used to show that all polygonal tables with rational angles have periodic orbits. In 1958, Roger Penrose, a mathematician who went on to win the 2020 Nobel Prize in Physics, found a curved table in which any point in one region couldn’t illuminate any point in another region. One simple way to show this is to reflect the triangle about one leg and then the other, as shown below.

In 2019 Amit Wolecki, then a graduate student at Tel Aviv University, applied this same technique to produce a shape with 22 sides (shown below). But in 1995, Tokarsky used a simple fact about triangles to create a blockish 26-sided polygon with two points that are mutually inaccessible, shown below. Then, in 2008, Richard Schwartz at Brown University showed that all obtuse triangles with angles of 100 degrees or less contain a periodic trajectory. Another approach has been used to show that if all the angles are rational - that is, they can be expressed as fractions - obtuse triangles with even bigger angles must have periodic trajectories. Billiard tables shaped like acute and right triangles have periodic trajectories. In a landmark 1986 article, Howard Masur used this technique to show that all polygonal tables with rational angles have periodic orbits. For example, it can be used to show why simple rectangular tables have infinitely many periodic trajectories through every point.

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