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Remote Ready Biology Learning Activities

Remote Ready Biology Learning Activities has 50 remote-ready activities, which work for either your classroom or remote teaching.

Playing Billiards

From simplicity to pattern to disorder

Simple interactions of simple things can lead to a surprising degree of order, as shown by the Game of Life. The simple situation of an ideal billiard ball moving on an ideal billiard table provides an additional set of insights into the emergence of patterns from simple sets of rules, and into the appearance of disorder in such cases as well. The ease with which disorder can be generated in simple deterministic systems is one of the major intellectual insights of recent decades. And the problem of providing useful descriptions of such disorder and the circumstances in which it arises is a major intellectual challenges on which many people are currently working. All of which is to say that playing billiards can be both fun and instructive.

Starting simple

In the real world, the path of a billiard ball on a table is influenced by imperfections in ball and table, by friction which eventually brings the ball to a stop, and by spin, which skilled players can use to influence the ball's trajectory. In the ideal world shown to the right, all these complexities have been eliminated, and we're left with a situation in which the ball's path depends only on its initial direction and the simple rule that when it hits a side of the table from a particular angle it leaves at that same angle.
To start a ball moving, click at a point on the table, drag to define an initial path for the ball, and then click on the start button. You will see the path of the ball traced through ten rebounds. The start button will change to a continue button, which you can click repeatedly to trace the path of the ball through additional sets of rebounds. You can increase or decrease the size of the set of rebounds traced by altering the number in the iterations box. You can also vary the speed of the ball with the spd slider.
Try it out. You'll find that the path of the ball often traces out some surprisingly interesting and attractive pattern. That in itself is worth thinking about a bit. Neither the ball, nor you, nor we planned that pattern. It emergedsimply as a consequence of the interaction of three simple characteristics: the initial ball location and direction, the locations of the walls of the table, and the reflection rule. Given those three characteristics, the path of the ball is fully determined and, with those three characteristics constant, the same pattern will always emerge. You can verify this for yourself by more precisely setting the initial ball location and direction using the x, y, vx, and vx text entry boxes. (Notice that these vary when you click and drag to define the intial ball location and direction). Click in the x box, backspace over the numbers there, type 0, and hit return (be sure your cursor stays within the window). Do the same for the y box. This sets the initial location of the ball to the center of the square. You can set other intial locations by entering different numbers in the boxes (-150 is all the way to the left and to the bottom, 150 is all the way to the right and to the top). Now set the initial movement direction by similarly entering numbers in the vx and vy boxes (positive numbers are to the right and up, negative numbers to the left and down, try 3 in vx and 1 in vy, remember to hit return after entering numbers in each box and to keep your cursor in the window). When you click on start, a pattern will be traced. If you click on erase and then again on start, the same pattern will again be traced. Changing the numbers will cause a different pattern to be traced. So too will changing the locations of the walls, even when the initial location and direction is unchanged. You can do this by changing the number of vertices using the slider. We're not set up to change the reflection rule, but if we were you could use the same approach (varying one parameter while holding the other two constant) to show that the pattern is not determined by any one of the parameters but instead depends on the interaction of all three.

The bottom line: Without a plan, interesting and attractive patterns emerge from simple and fully deterministic interactions of simple characteristics no one of which by itself determines the pattern. The paths traced on a square, or any polygonal, billiard table are are esthetically satisfying, and orderly in at least four more concrete ways. They display symmetry, will repeat after sufficient time (and hence are predictable from their history), remain similar if the starting points are similar, and yield phase space plots consisting of continuous lines.

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