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No matter how consistent you are with the first shot (the break), the smallest of differences in the speed and angle with which you strike the white ball will cause the pack of billiards to scatter in wildly different directions every time. What at first glance appears to be random behaviour is completely deterministic - it only seems random because imperceptible changes are making all the difference. A difference of one part in a thousand: the same sort of difference that a flap of a butterfly’s wing might make to the breeze on your face. And how can we tell the difference between pure randomness and orderly patterns that are cloaked in chaos? Commercial aircraft are aerodynamically stable, so that a small turbulent nudge (possibly butterfly-related) won’t push the plane out of a level flightpath. On the other hand, this stability is somewhat of an inconvenience to fighter pilots who prefer their aircraft to make rapid changes with minimal effort. The millions of cells that make up your heart are constantly contracting and relaxing separately as part of an intricate chaotic system with complicated attractors.

There are many ways to attack this problem, some of which are heftier than others (coordinate geometry!). Once there it clings to its attractor as it is buffeted to and fro in a literal sea of chaos, and quickly moves back to the surface if temporarily thrown above or dumped below the waves. 24. The greatest common divisor is 3. Dividing through by 3, we get 3 and 8, the numbers used in the example above. It shares a lot in common with the aforementioned games. Interestingly, what we get is an elliptical caustic curve that shares the same foci as the elliptical table, and so these are confocal ellipses. To prove our claims above, we are going to exploit this simple idea, the mirror being one side of the billiard table. Play continues until only the six colours remain on the table. Play of the game moved indoors to a wooden table with green cloth draped on it to simulate grass, and edges added to keep the balls in play. This has 2 player mode; you can invite your friends to play and create emotional bonding.

You can visualise it like this: put a loop of string around pins located at the foci, then pull the string taut at one point using a pencil. The game was played on a huge strip of land, in this case about 1000 yards long and so was more like golf than Croquet - players took great swings at the balls in an effort to hoof them as far along the pitch as possible. Phase space is not (always) like regular space - each location in phase space corresponds to a different configuration of the system. Mathematical billiards is an idealisation of what we experience on a regular pool table. Carom, or French, billiards is played with three balls on a table that has no pockets. It is worth noting that the laws of physics that determine how the billiard balls move are precise and unambiguous: they allow no room for randomness. 3. What are the symmetries of the arithmetic billiard path (as a geometrical figure)?

Clouds are not spheres, mountains are not cones … Yet other sources indicate that a variant of a ball-and-mallet game evolved into Billiards. By the 15th century, a ball-and-mallet game is noted as being played in England. A stained glass window in England's Gloucester Cathedral, dating from the mid-fourteenth century, shows a figure wielding a stick in the middle of a distinctly golf-like backswing or high-powered croquet shot. The game of carom billiards is still played primarily in France and other European countries and to a lesser degree in the United States and has many players in Japan, Indonesia, the Philippines, Taiwan, and South Korea and in Central America, South America, Africa, and the Middle East. Chalk in small cubes is applied uniformly to the cue tip permitting the players to strike the cue ball off centre on purpose in order to impart a spinning motion, called "side" in Great Britain and "English" in the United States. Each time the ball passes through one of the foci, it reflects off the elliptical table and passes through the other focus.

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