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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. Billiard tables shaped like acute and right triangles have periodic trajectories. 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. 26. Finally, the host club has the comfort of knowing that that the highest authority of the sport will deal with any disciplinary issues that may be necessary. Properly executing a solid break shot can be the difference between winning and losing, so knowing how to maximize your opportunities is essential. For example, it can be used to show why simple rectangular tables have infinitely many periodic trajectories through every point. In a landmark 1986 article, Howard Masur used this technique to show that all polygonal tables with rational angles have periodic orbits.

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. His approach involved breaking the problem down into multiple cases and verifying each case using traditional mathematics and computer assistance. His approach worked not only for obtuse triangles, but for far more complicated shapes: Irregular 100-sided tables, say, or polygons whose walls zig and zag creating nooks and crannies, have periodic orbits, so long as the angles are rational. However, research mathematicians still cannot answer basic questions about the possible trajectories of billiard balls on tables in the shape of other polygons (shapes with flat sides). 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. 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.

So m and n must be even. If this does not happen, then the opposing player must be given the option of either leaving the ball where it lies or replacing the ball as near as possible to its original position to the agreement of both players. 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. 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. For example, they play snooker on a pool table, where a professional table measures seven feet long and 3.5 feet wide. In many pool games, the player who scratched has to return one previously pocketed object ball to the foot spot, and the other player has ball in hand. 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. This process (seen below), called the unfolding of the billiard path, allows the ball to continue in a straight-line trajectory.

This mathematical trick makes it possible to prove things about the trajectory that would otherwise be challenging to see. 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. Snooker is played on a table similar to a pool table but uses nine to 15 balls. Pockets. Pockets are the six holes pool players use to sink, or pocket, balls. Mechanical Bridges - The player may use up to two mechanical bridges to support the cue stick during the shot. There are two schools of thought on your focus point during the break stroke. Just make sure not to hit in the 8 Ball before all of your other balls are sunk, or else it is game over for you! A player continues to take shots until they do not make a scoring shot. Here’s what mathematicians have learned about billiards since Donald Duck’s epically tangled shot. Here’s the main differences between pool and snooker. Ivory pool balls were prone to yellowing with age and tended to crack in humid climates or if struck with excessive force. On the break shot, no balls are pocketed and fewer than four balls reach the cushions, in which case the incoming player can demand a re-rack and take the break or force the original breaker to re-break, or may take ball-in-hand behind the head string and shoot the balls as they lie.

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