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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. 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. Proving results in the other direction has been a lot harder. This results in a one-point score. 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. In 2018, Jacob Garber, Boyan Marinov, Kenneth Moore and George Tokarsky at the University of Alberta extended this threshold to 112.3 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.

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. And yet analyzing billiard trajectories shows how even the most abstract mathematics can connect to the world we live in. Rather than asking about trajectories that return to their starting point, this problem asks whether trajectories can visit every point on a given table. That is, a laser beam shot from one point, regardless of its direction, cannot hit the other point. The aim of this game is to hit the balls one by one with your billiard cue. According to John Wesly Hyatt, billiard balls are made of this mixture because these materials were formed into pool balls using excessive pressure. His approach involved breaking the problem down into multiple cases and verifying each case using traditional mathematics and computer assistance. 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 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.

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. 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. Start with a trajectory that’s at a right angle to the hypotenuse (the long side of the triangle). Join the points where the right angles occur to form a triangle, what is billiards as seen on the right. In a landmark 1986 article, Howard Masur used this technique to show that all polygonal tables with rational angles have periodic orbits. 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.

Pool tables are smaller than billiards tables. Snooker tables are usually lower-set and bigger than pool tables, but their pockets are smaller than those of pool tables. Snooker is organised into frames, meaning the player wins one by one by getting the most points. Snooker is played with 15 red balls, 6 coloured balls and 1 cue ball - they are all slightly larger than pool balls. Both balls will roll. The referee will remove pocketed object balls from full or nearly full pockets, but it is the shooter’s responsibility to see that this duty is performed. Balls come in different sizes and colors depending on what type of billiards you're playing. For decades, nobody could come up with a polygon that had the same property. This story originally said that 22 was the smallest number of sides a polygon containing two interior points that don’t illuminate one another could have. The story has been updated to reflect that though the smallest such polygon known to exist has 22 sides, it remains unknown if a smaller one can be constructed. 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.