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We can repeat the construction from above for these two integers, and then scale the picture up by multiplying each axis in our coordinate system by . The vertical axis runs along the left vertical side of and the horizontal axis runs along the bottom side of . We shoot a billiard ball from one corner (the bottom left in the figure above) making a 45 degree angle with the sides. Have a look at the Geogebra animation below (the play button is in the bottom left corner) and try to figure out how the construction works. If the breaker fails to make a successful break-usually defined as at least four balls hitting cushions or an object ball being pocketed-then the opponent can opt either to play from the current position or to call for a re-rack and either re-break or have the original breaker repeat the break. If you would like to play the animation again, double click the refresh button in the top right corner. In phase space, a stable system will move predictably towards a very simple attractor (which will look like a single point in the phase space if the system settles down, or a simple loop if the system cycles between different configurations repeatedly).

Phase space is not (always) like regular space - each location in phase space corresponds to a different configuration of the system. A chaotic system will also move predictably towards its attractor in phase space - but instead of points or simple loops, we see "strange attractors" appear - complex and beautiful shapes (known as fractals) that twist and turn, intricately detailed at all possible scales. The main benefit to having a chaotic heart is that tiny variations in the way those millions of cells contract serves to distribute the load more evenly, reducing wear and tear on your heart and allowing it to pump decades longer than would otherwise be possible. 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. These millions of cells must work in sync, contracting in just the right sequence at just the right time to produce a healthy heartbeat.

If the system is jolted somehow, it may find itself on an altogether different attractor called fibrillation, in which the cells constantly contract and relax in the wrong sequence. No matter where it starts, the ball will immediately move in a very predictable way towards its attractor - the ocean surface. Imagine taking a ping-pong ball far out into the ocean and letting it go. Consequently, they are equipped with on-board computers which constantly and delicately adjust the flight surfaces to cancel out the unwanted butterfly effects, leaving the pilot free to exploit his own. Two weeks is believed to be the limit we could ever achieve however much better computers and software get. The two natural numbers are 40 and 15 in this case. Suppose we are given two positive whole numbers and , neither of which is a multiple of the other (the case where one is a multiple of the other is easy and is left to the reader). Reflecting S' in its left side. It does not lose speed and, by the law of reflection, is reflected at a 45 degree angle each time it meets a side (thus the path only makes left or right 90 degree turns).

The path of the billiard ball consists of line segments. They may be either drop pockets (baskets or cups into which the balls drop) or ball returns (gutters inside the table that funnel balls into a collection area. In 9-ball, players must pocket the balls in numerical order, with the 9-ball serving as the game-winning ball. The billiard ball bounces off the rectangle’s sides. In mathematical billiards the ball bounces around according to the same rules as in ordinary billiards, but it has no mass, which means there is no friction. The second bonus shot ("continuation shot") is an ordinary shot played from where the striker ball came to rest. There also aren't any pockets that can swallow the ball. Apart from comfort considerations, it’s important to select billiard room chairs made from durable materials that can withstand regular use. In this article, we will discuss the importance of comfort in billiard room chairs and what factors you should consider when making a purchase. This means that the ball will bounce infinitely many times on the sides of the billiard table and keep going forever.

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