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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. This means that the ball will bounce infinitely many times on the sides of the billiard table and keep going forever. To prove our claims above, we are going to exploit this simple idea, the mirror being one side of the billiard table. Phase space may seem fairly abstract, but one important application lies in understanding your heartbeat. 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). The first segment of the path contains the point of self-intersection which is closest to the starting point. The behaviour of the system can be observed by placing a point at the location representing the starting configuration and watching how that point moves through the phase space. The greatest common divisor of the two given numbers, we claim, is the distance from the starting point to the closest point of self-intersection, divided by .

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. Notice that three points are aligned: the point marking your position, the point on the mirror where you see the reflection of the object and the (imaginary) point behind the mirror where you believe the object to be. Circular boards made of cork get hit for points in which sport? 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. 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. Every time someone steps forward with information on doping, we move closer to a clean and fair playing field for all. 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 labels "British" and "UK" as applied to entries in this glossary refer to terms originating in the UK and also used in countries that were fairly recently part of the British Empire and/or are part of the Commonwealth of Nations, as opposed to US (and, often, what is billiards Canadian) terminology. 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. It is usually possible to insert the torque tool at either the top or bottom part of the keyway.

Mathematicians use the concept of a "phase space" to describe the possible behaviours of a system geometrically. The key to unlocking the hidden structure of a chaotic system is in determining its preferred set of behaviours - known to mathematicians as its attractor. Use a key or secret combination to open the safe. Many of these mechanisms involve the use of a "sidebar" that must retract before the plug can rotate. Finally a use for those 50 megabytes of skins! Phase space is not (always) like regular space - each location in phase space corresponds to a different configuration of the system. What about other high-ranking federal officials, like members of the judicial branch? The branch of fractal mathematics, pioneered by the French American mathematician Benoît Mandelbröt, allows us to come to grips with the preferred behaviour of this system, even as the incredibly intricate shape of the attractor prevents us from predicting exactly how the system will evolve once it reaches it. The mathematician Ian Stewart used the following example to illustrate an attractor. It also allows us to accurately predict how the system will respond if it is jolted off its attractor.