![]() If you get the initial conditions of the ball - the place and direction it starts out with - ever so slightly wrong, then the little error will generally snowball, rendering your prediction inaccurate. But there is another, independent feature that makes things difficult. That might not appear too bad, after all you could program a fast computer to do the work for you. By "vast majority" mathematicians mean that if you pick a direction at random, it will almost certainly behave in this ergodic way.Įrgodicity of billiards means that it's really hard to predict where a ball will be after a given amount of time: in order to find out, you have to literally trace its path on piece of paper, meticulously measuring angle after angle, because you can't rely on any regular pattern to kick in. This behaviour is a consequence of billiards being ergodic. Table whose areas are equal, then the trajectory will spend an equalĪmount of time in both. Part of the table in equal measure: if you take two regions of the What is more, a typical trajectory will visit each ![]() Majority of initial directions the trajectory willīe much wilder: not only will it not retrace its steps, but it willĮventually explore the whole of the table, getting arbitrarily close In the 1980s mathematicians proved that for the vast These are examples of periodic trajectories.īut it turns out that this regular behaviour is Right: similarly, you can make the ball travel between four points. Left: if you shoot a ball so it meets the wall at right angles, it will bounce between opposite points on the table forever. Similarly, by getting your initial direction just right, you can make sure that the ball bounces off the mid point of all four edges in turn and then returns to where it started, travelling along the same four line segments forever. It will carry on to hit the opposite edge at the exact opposite point, retrace its path, and keep bouncingīetween the two opposite points along the same What will the trajectory of a ball typically look like? "If you shoot the ball so that it hits one of sides of the table head-on (its trajectory forming a right angle with the side), then it will return along the same trajectory it arrived on, but travelling in the opposite direction. That can swallow the ball, and since it doesn't experience anyįriction the ball will keep going forever. But unlike in real billiard, there are no pockets Line until it hits the edge of the table where it bounces offįollowing the law of reflection (see the figure on the right). To the same rules as an ordinary ball." That is, it moves along a straight "The idea is similar: you have a table and a ball, but theīall has no mass so there is no friction. "Mathematical billiards is an idealisation of real billiards," explainsĬorinna Ulcigrai, a mathematician at the University of Bristol who has studied mathematical billiards. The law of reflection: the angle of incidence equals the angle of reflection.
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