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A vous de jouer! This gives Steinhaus's result that there are always two distinct periodic triangular orbits Croft and Swinnerton as a special case.
Analysis of billiards path can involve sophisticated use of ergodic theory and dynamical systems. Given a rectangular billiard table with only corner pockets and sides of integer lengths and with and relatively prime , a ball sent at a angle from a corner will be pocketed in another corner after bounces Steinhaus , p.
Steinhaus , p. Alhazen's billiard problem seeks to find the point at the edge of a circular "billiards" table at which a cue ball at a given point must be aimed in order to carom once off the edge of the table and strike another ball at a second given point.
This problem is insoluble using a compass and ruler construction Elkin , Riede , Neumann On an elliptical billiard table, the envelope of a trajectory is a smaller ellipse , a hyperbola , a line through the foci of the ellipse , or a closed polygon Steinhaus , pp.
The closed polygon case is related to Poncelet's porism. One can also consider billiard paths on polygonal billiard tables.
The only closed billiard path of a single circuit in an acute triangle is the pedal triangle. There are an infinite number of multiple-circuit paths, but all segments are parallel to the sides of the pedal triangle.
There exists a closed billiard path inside a cyclic quadrilateral if its circumcenter lies inside the quadrilateral Wells There are four identical closed billiard paths inside and touching each face of a cube such that each leg on the path has the same length Hayward ; Steinhaus , ; Gardner , pp.
This path is in the form of a chair-shaped hexagon, and each leg has length. Lewis Carroll Charles Dodgson also considered this problem Weaver There are three identical closed billiard paths inside and touching each face of a tetrahedron such that each leg of the path has the same length Gardner , pp.
These were discovered by J. Conway and independently by Hayward For a tetrahedron with unit side lengths, each leg has length. Conway has shown that period orbits exist in all tetrahedra , but it is not known if there are periodic orbits in every polyhedron Croft et al.
Altshiller Court, N. New York: Dial Press, pp. Bakst, A. Mathematical Puzzles and Pastimes. New York: Van Nostrand, pp. Skilled players may use more than one cue during a game, including a separate cue often with a hard phenolic resin tip for the opening break shot and another, shorter cue with a special tip for jump shots.
The mechanical bridge, sometimes called a "rake", "crutch", "bridge stick" or simply "bridge", and in the UK a "rest", is used to extend a player's reach on a shot where the cue ball is too far away for normal hand bridging.
It consists of a stick with a grooved metal or plastic head which the cue slides on. Many amateurs refuse to use the mechanical bridge based on the perception that to do so is unmanly or cheating.
Some players, especially current or former snooker players, use a screw-on cue butt extension instead of or in addition to the mechanical bridge.
Bridge head design is varied, and not all designs especially those with cue shaft-enclosing rings, or wheels on the bottom of the head , are broadly tournament-approved.
Chalk is applied to the tip of the cue stick , ideally before every shot, to increase the tip's friction coefficient so that when it impacts the cue ball on a non-center hit, no miscue unintentional slippage between the cue tip and the struck ball occurs.
Chalk is an important element to make good shots in pool or snooker. Cue tip chalk is not actually the substance typically referred to as " chalk " generally calcium carbonate , but any of several proprietary compounds, with a silicate base.
It was around the time of the Industrial Revolution that newer compounds formed that provided better grip for the ball. This is when the English began to experiment with side spin or applying curl to the ball.
This was shortly introduced to the American players and is how the term "putting English on the ball" came to be. Some brands of hand chalk actually are made of compressed talc.
Tip chalk is not used for this purpose because it is abrasive, hand-staining and difficult to apply. Many players prefer a slick pool glove over hand chalk or talc because of the messiness of these powders; buildup of particles on the cloth will affect ball behavior and necessitate more-frequent cloth cleaning.
Cue tip chalk invented in its modern form by straight rail billiard pro William A. Spinks and chemist William Hoskins in   is made by crushing silica and the abrasive substance corundum or aloxite  aluminium oxide ,   into a powder.
High humidity can also impair the effectiveness of chalk. Harder, drier compounds are generally considered superior by most players.
The main carom billiards games are straight rail , balkline and three cushion billiards. All are played on a pocketless table with three balls; two cue balls and one object ball.
In all, players shoot a cue ball so that it makes contact with the opponent's cue ball as well as the object ball.
Others of multinational interest are four-ball and five-pins. The most globally popular of the large variety of pocket games are Pool and snooker.
A third, English billiards , has some features of carom billiards. English billiards used to be one of the two most-competitive cue sports along with the carom game balkline, at the turn of the 20th century and is still enjoyed today in Commonwealth countries.
Another pocket game, Russian pyramid and its variants like kaisa are popular in the former Eastern bloc.
In straight rail, a player scores a point and may continue shooting each time his cue ball makes contact with both other balls.
Some of the best players of straight billiards developed the skill to gather the balls in a corner or along the same rail for the purpose of playing a series of nurse shots to score a seemingly limitless number of points.
The first straight rail professional tournament was held in where Jacob Schaefer Sr. With the balls repetitively hit and barely moving in endless "nursing", there was little for the fans to watch.
In light of these skill developments in straight rail, the game of balkline soon developed to make it impossible for a player to keep the balls gathered in one part of the table for long, greatly limiting the effectiveness of nurse shots.
A balkline is a line parallel to one end of a billiards table. In the game of balkline, the players have to drive at least one object ball past a balkline parallel to each rail after a specified number of points have been scored.
Another solution was to require a player's cue ball to make contact with the rail cushions in the process of contacting the other balls.
This in turn saw the three-cushion version emerge, where the cue ball must make three separate cushion contacts during a shot.
This is difficult enough that even the best players can only manage to average one to two points per turn.
This is sometimes described as "hardest to learn" and "require most skill" of all billiards. There are many variations of games played on a standard pool table.
Popular pool games include eight-ball , nine-ball , straight pool and one-pocket. Even within games types e. A few of the more popular examples of pool games are given below.
In eight-ball and nine-ball, the object is to sink object balls until one can legally pocket the winning eponymous " money ball ". Well-known but waning in popularity is straight pool, in which players seek to continue sinking balls, rack after rack if they can, to reach a pre-determined winning score typically Related to nine-ball, another well-known game is rotation , where the lowest-numbered object ball on the table must be struck first, although any object ball may be pocketed i.
Each pocketed ball is worth its number, and the player with the highest score at the end of the rack is the winner.
In both one-pocket and bank pool, the players must sink a set number of balls; respectively, all in a particular pocket , or all by bank shots.
In snooker, players score points by alternately potting red balls and various special " colour balls ". Speed pool is a standard billiards game where the balls must be pocketed in as little time as possible.
Rules vary greatly from tournament to tournament. The International Speed Pool Challenge has been held annually since Dating to approximately , English billiards, called simply billiards  in many former British colonies and in the UK where it originated, was originally called the winning and losing carambole game , folding in the names of three predecessor games, the winning game , the losing game and the carambole game an early form of straight rail , that combined to form it.
English billiards requires two cue balls and a red object ball. The object of the game is to score either a fixed number of points, or score the most points within a set time frame, determined at the start of the game.
Snooker is a pocket billiards game originated by British officers stationed in India during the 19th century, based on earlier pool games such as black pool and life pool.
The name of the game became generalized to also describe one of its prime strategies: to " snooker " the opposing player by causing that player to foul or leave an opening to be exploited.
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