Billiards and mathematics
Billiards is a cue sport played on a rectangular table equipped with pockets, where players aim to hit billiard balls into these pockets using a cue stick. The game involves various forms, including pool (or pocket billiards), which features specific rules and setups, such as the popular game of Eight Ball. Mathematics plays a critical role in billiards, influencing gameplay through the principles of geometry and physics. Players must understand angles, momentum, and spin to execute successful shots, with techniques like banking balls off the sides of the table being essential for strategic play. Historical figures in mathematics, such as Gaspard Coriolis and Charles Dodgson (Lewis Carroll), have contributed to the mathematical understanding of billiards, exploring concepts like the parabolic paths of balls and the geometry of different table shapes. Additionally, mathematical billiards—an area of study focusing on the dynamics of ball motion—investigates how various starting positions and angles affect gameplay. The sport continues to inspire mathematical research, making it a fascinating intersection of leisure and academic inquiry.
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Billiards and mathematics
Summary: Playing billiards depends on an understanding of spin, momentum, and angles.
Billiards is a cue sport game that involves the use of a rectangular table, billiard balls, and a stick called a “cue.” Mathematics and physics are two important components of playing the game well. There are many different games within the cue sports that Americans typically name “billiards.” Billiard tables with pockets comprise games that are termed as “pool” or pocket billiards. The rectangular table has two long sides (twice the short side) and two short sides with six pockets—one at each corner, and one midway along the longer two sides of the table. The object of the game is to hit the billiard balls into the pockets using a cue ball (the lone white ball in the set). Gaspard Coriolis, known today for the Coriolis effect, wrote a work on the mathematics and physics of billiards in 1835. He stated that the curved path followed by the cue ball after striking another ball is always parabolic because of top or bottom spin. Further, the maximum side spin on a cue ball is achieved by striking it half a radius off-center with the tip of the cue.
The game of billiards is also a source for interesting mathematical problems, which are connected to dynamical systems, ergodic theory, geometry, physics, and optics. In mathematical billiards, the angle of incidence is the same as the angle of reflection for a point mass on a frictionless domain with a boundary. The dynamics depend on the starting position, angle, and geometry of the boundary and the table. Mathematicians investigate the motion and the path of the ball on a variety of differently shaped flat and curved tables, like triangular or elliptical boundaries or hyperbolic tables. In 1890, mathematician Charles Dodgson, better known as Alice in Wonderland author Lewis Carroll, published rules for circular billiards and may have also had a table built. In 2007, mathematician Alex Eskin won the Research Prize from the Clay Mathematics Institute for his work on rational billiards and geometric group theory.
Eight Ball
Eight ball is the pool game most commonly played in the United States, and it involves 16 billiard balls. To begin the game of Eight Ball, the numbered balls are placed in a triangular rack that sets the 8-ball in the middle position of the third row of balls with a single lead ball opposite the cue ball. The cue ball is placed on the midpoint of the line parallel to the short side at one-quarter of the long side known as the head spot. The point of the triangular-shaped racked set of billiard balls is placed on the opposite short end at one-quarter of the length of the long side from the other short side and is known as the foot spot. After one player “breaks” by hitting the cue ball from the head spot into the racked set of balls, the player then hits a set of balls into the pockets. A shot that does not cause a ball of his or her set to go into the pocket results in the next shot going to the other player.
Billiards Geometry and Physics
Shooting the balls into the pockets requires an understanding of angles and momentum, as well as placement of the cue so that the correct spin is achieved to place the cue ball where it can achieve the target ball going into a pocket. Coriolis investigated 90-degree and 30-degree rules of various shots and measured the largest deflection angle the cue ball can experience. Both skill and geometric understanding contribute to successful shots. Some shots require straight shooting; some shots need to be “banked” in by using the table sides. Players can use transformational geometry to approximate where on the table to hit the ball for it to return to a pocket. By measuring the angle from the ball to the side being used to bank off and reflecting the same angle with the cue stick, one can see the most viable spot to aim for so that the path of the caromed ball ends in a pocket. Using the diamonds found on the sides of most tables is one way of measuring these angles, and some systems for pool and billiards play use the diamonds. Using the diamond system for a different billiard game, Three Cushion Billiards is demonstrated on the 1959 Donald Duck Disney cartoon Donald in Mathmagic Land. The demonstration shows that it is possible to use subtraction to know where to aim the ball in relation to a diamond to make sure that all three balls are hit.
Bibliography
Alciatore, David. “The Amazing World of Billiards Physics.” http://billiards.colostate.edu/physics/Alciatore‗SCIAM‗article‗posted‗version.pdf.
Tabachnikov, Serge. Geometry and Billiards. Providence, RI: American Mathematical Society, 2005.