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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. 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. Billiard balls used to be made of strange materials -- wood, clay, and even elephant ivory. They can create it with wood, clay, ivory, or polymers. 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. 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.

There may be isolated dark spots (as in Tokarsky’s and Wolecki’s examples) but no dark regions as there are in the Penrose example, which has curved walls rather than straight ones. For example, it can be used to show why simple rectangular tables have infinitely many periodic trajectories through every point. For example, if you just pocketed a red ball, the next one should be a coloured ball. Hit the cue ball using the other cue ball and then target the red ball, which should not be pocketed to earn two points. He must not give advice on the application of the rules, or other points of play on which he is not required by the rules to speak. For a complete outline of the rules of One Pocket click here. What are the rules of Billiards? 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. 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. 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. Ten-Ball is a rotation game and a call shot game played with 10 object balls numbered one through 10. On each shot, the shooter must call his shot and make contact with the lowest numbered ball on the table. Today most balls are made of resin. If both balls interfere, what is billiards all fifteen balls are re-racked and the cue ball is in hand behind the head string. The referee will place a ball on each side of the table behind the head string and near the head string. Start with a trajectory that’s at a right angle to the hypotenuse (the long side of the triangle).

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. Somewhat remarkably, the existence of one periodic orbit in a polygon implies the existence of infinitely many; shifting the trajectory by just a little bit will yield a family of related periodic trajectories. 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. 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. Instead of just copying a polygon on a flat plane, this approach maps copies of polygons onto topological surfaces, doughnuts with one or more holes in them.