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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. A crazy person might get by accident into some of these states, and get so lost that he wouldn’t know how to come back to the world of ordinary normal conventions. Two weeks is believed to be the limit we could ever achieve however much better computers and software get. 24. The greatest common divisor is 3. Dividing through by 3, we get 3 and 8, what is billiards the numbers used in the example above. One fascinating aspect of mathematical billiards is that it gives us a geometrical method to determine the least common multiple and the greatest common divisor of two natural numbers. 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.

Ball and mallet games are mentioned as early as the 13th century in French texts. Some texts state the name of this game was "Pall Mall", others do not name it. Fortunately, this intricate state of synchronisation is an attractor of the system - but it is not the only one. It also allows us to accurately predict how the system will respond if it is jolted off its attractor. 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. A correctly keyed wafer is flush with the plug on the top and the bottom and allows rotation. Picks probe and lift the individual pin tumblers through the keyway, while torque tools control the degree and force of plug rotation. Rotation of the plug within the shell operates the locking mechanism. The plug can rotate freely only if the key lifts every pin stack's cut to align at the border between the plug and shell. Each disk has a notch cut in its edge. For example, Medeco locks use special wedge-shaped bottom pins that are rotated into one of several possible positions by the key cuts (which can be cut at different angles).

As with pin tumbler locks, because the levers, gates, and fence are slightly out of alignment, it is usually possible to raise and pick the levers one at time. For most locks, especially as you're starting out, a workable compromise is often the smaller Peterson hook. The three hook picks in this kit are sufficient to manipulate the vast majority of pin tumbler locks found in the US. This pick is a LAB double-ended "hook/rake" (held for use with the hook end). Spend more time on this exercise than you think you need to; most people never learn to properly apply the light touch needed to pick better quality locks. First, apply light torque (as you practiced in the previous exercises) to the two pin cylinder and gently feel each pin. 1. If one of the two given numbers is a multiple of the other, what is the shape of the arithmetic billiard path? 3. What are the symmetries of the arithmetic billiard path (as a geometrical figure)?

2. For which numbers does the arithmetic billiard path end in the corner opposite to the starting point? 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 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). 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 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.