The hypotenuse and its second reflection are parallel, so a perpendicular line segment joining them corresponds to a trajectory that will bounce back and forth forever: The ball departs the hypotenuse at a right angle, bounces off both legs, returns to the hypotenuse at a right angle, and then retraces its route. A ball touching a rail at the start of a shot (said to be «frozen» to the rail) is not considered driven to that rail unless it leaves the rail and returns. Whoever’s ball gets closer to the head rail goes first. By folding the imagined tables back on their neighbors, you can recover the actual trajectory of the ball. 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. To put it another way, if we placed a light bulb, which shines in all directions at once, at some point on the table, would it light up the whole room? Nobody knows. For other, more complicated shapes, it’s unknown whether it’s possible to hit the ball from any point on the table to any other point on the table.

This mathematical trick makes it possible to prove things about the trajectory that would otherwise be challenging to see. 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. 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. 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. Start with a trajectory that’s at a right angle to the hypotenuse (the long side of the triangle). 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. This process (seen below), called the unfolding of the billiard path, allows the ball to continue in a straight-line trajectory.

This inscribed triangle is a periodic billiard trajectory called the Fagnano orbit, named for Giovanni Fagnano, who in 1775 showed that this triangle has the smallest perimeter of all inscribed triangles. 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 the early 1990s, Fred Holt at the University of Washington and Gregory Galperin and his collaborators at Moscow State University independently showed that every right triangle has periodic orbits. In their 1992 paper, Galperin and his collaborators came up with a variety of methods of reflecting obtuse triangles in a way that lets you create periodic orbits, but the methods only worked for some special cases. The major and his hostess played against Captain Livingstone Tuck and an old gentleman who came from Lambeth, with the result that the gallant captain and his partner rose up poorer and sadder men, which was rather a blow to the former, who reckoned upon clearing a little on such occasions, and had not expected to find himself opposed by such a past master of the art as the major.

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, what is billiards 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. 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. 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. Adjust the original point slightly if the path passes through a corner. 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.