The players will shoot at about the same time to make each ball contact the foot cushion with the goal of returning the ball closer to the head cushion than the opponent. If a player either hits the cue ball into a pocket, does not make contact with any of the balls on the table, or hits the opponent’s ball first, it is considered a scratch. However, when the cue ball is next to the spotted ball, the spotted ball should not be placed in contact with the cue ball; a small separation must be maintained. Note that these two other balls are already on the table, and remember that white and yellow are the cue balls. We ask if, given two points on a particular table, you can always shine a laser (idealized as an infinitely thin ray of light) from one point to the other. That is, a laser beam shot from one point, regardless of its direction, cannot hit the other point. 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. As you might remember from high school geometry, there are several kinds of triangles: acute triangles, where all three internal angles are less than 90 degrees; right triangles, which have a 90-degree angle; and obtuse triangles, which have one angle that is more than 90 degrees.

Then, in 2008, Richard Schwartz at Brown University showed that all obtuse triangles with angles of 100 degrees or less contain a periodic trajectory. In 2018, Jacob Garber, Boyan Marinov, Kenneth Moore and George Tokarsky at the University of Alberta extended this threshold to 112.3 degrees. 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. But in 1995, Tokarsky used a simple fact about triangles to create a blockish 26-sided polygon with two points that are mutually inaccessible, what is billiards shown below. There have been two main lines of research into the problem: finding shapes that can’t be illuminated and proving that large classes of shapes can be. 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. Not all of their benefits have fared as well. 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.

If a stalemate occurs the original breaker of the rack will break again. Upon recognition by either player or the referee that the groups have been reversed, the rack will be halted and will be replayed with the original player executing the break shot. Adjust the original point slightly if the path passes through a corner. 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. They typically assume that their billiard ball is an infinitely small, dimensionless point and that it bounces off the walls with perfect symmetry, departing at the same angle as it arrives, as seen below. 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. Suppose you want to find a periodic orbit that crosses the table n times in the long direction and m times in the short direction.

Proving results in the other direction has been a lot harder. In Wolecki’s 2019 article, he strengthened this result by proving that there are only finitely many pairs of unilluminable points. Whereas finding oddball shapes that can’t be illuminated can be done through a clever application of simple math, proving that a lot of shapes can be illuminated has only been possible through the use of heavy mathematical machinery. 17. An application to the WPA Board must use the WPA’s published application form unless otherwise agreed by the WPA Board. So m and n must be even. 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. 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.