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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. 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. The purpose of a defibrillator - the device that applies a large voltage of electricity across the heart - is not to "restart" the heart cells as such, but rather to give the chaotic system enough of a kick to move it off the fibrillating attractor and back to the healthy heartbeat attractor. Phase space is not (always) like regular space - each location in phase space corresponds to a different configuration of the system.

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). 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. Mathematicians use the concept of a "phase space" to describe the possible behaviours of a system geometrically. 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. Phase space may seem fairly abstract, but one important application lies in understanding your heartbeat. All you've managed to do is take up a great amount of space in the yard that could have been a play area for their children, or something that they can actually use. You may want to use one of the smaller torque tools as well, or put your torque tool in the bottom part of the keyway instead of the (curvy) top.

Billiards is a cue sport played on a rectangular table with pockets, where players use cue sticks to strike balls into the pockets. Uber-beautiful usherettes will be on hand to assist players with registration, table assignments, and more! Mathematical billiards is an idealisation of what we experience on a regular pool table. Kozoom, a French billiards company, was UMB's media partner from 2010 to 2019 before the federation switched to working with South Korean company Five&Six. In 1887, the French mathematician Henri Poincaré showed that while Newton’s theory of gravity could perfectly predict how two planetary bodies would orbit under their mutual attraction, adding a third body to the mix rendered the equations unsolvable. 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. Though we may not be able to predict exactly how a chaotic system will behave moment to moment, knowing the attractor allows us to narrow down the possibilities. It also allows us to accurately predict how the system will respond if it is jolted off its attractor.

Fortunately, this intricate state of synchronisation is an attractor of the system - but it is not the only one. 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. The mathematician Ian Stewart used the following example to illustrate an attractor. 24. The greatest common divisor is 3. Dividing through by 3, we get 3 and 8, the numbers used in the example above. If this happens, you get a point; when no ball gets hit, a point gets deducted. No matter where it starts, what is billiards the ball will immediately move in a very predictable way towards its attractor - the ocean surface. 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. The best we can do for three bodies is to predict their movements moment by moment, and feed those predictions back into our equations … This makes snooker the youngest of the three.