May 18, 2020

Mirror Bounces

In a game of mathematical billiards, one ball moves across the table at a constant speed forever. There's no friction to slow the ball down. It simply travels in a straight line until it hits a cushion and rebounds according to the rule that the angle of reflection equals the angle of incidence.

What makes the game interesting is the table's geometry. Depending on the ball's initial position and direction, its path can vary considerably within the confines of tables having different shapes. For example, on a circular table, a ball can follow paths that never penetrate an inner circular region of a certain diameter in the middle of the table. A stadium-shaped billiard table leads to unpredictable paths reminiscent of chaos (see "Billiards in the Round").

Even billiard motion inside a rectangular table—indeed, any table bounded by straight lines—can exhibit intriguing regularities and apparent irregularities. Suppose, for example, that a rectangular billiard table is m units long and n units wide, and it has pockets at the corners, labeled A, B, C, and D. It turns out that a ball struck at 45 degrees to the table's sides from a position at A will eventually arrive in either pocket B, C, or D, depending on whether the ratio of m and n to the highest common factor, d, of m and n is even or odd.

Rectangular billiard table of dimensions m x n, with corner pockets at A, B, C. and D.

Consider a table 2 units long and 1 unit wide. In this case, the ball will bounce once—right into pocket B. Here, m = 2, n = 1, and d = 1. So, m/d = 2/1 = 2 (even) and n/d = 1/1 = 1 (odd). Indeed, you can use such a billiard shot to prove that the square root of 2 is an irrational number!

In general, a ball will eventually drop into pocket B if m/d is even and n/d is odd, pocket C if m/d is odd and n/d is odd, and pocket D if m/d is odd and n/d is even. The number of bounces is given by the value of the expression [(m + n) divided by the highest common factor of m and n] − 2. Thus, for a table 5 units long and 3 units wide, the ball bounces six times before it lands in pocket C.

The trick to solving such puzzles is to use the technique of mirror reflection. Every time the ball hits a side, it continues in its original direction across a virtual table created by reflecting the rectangle along that side. In effect, the broken course of the billiard ball is replaced by a straight line.

Example of mirror reflection applied to a rectangular table 5 units long and 3 units wide. The ball ends up in pocket C and takes 6 bounces to get there.

The same technique can be applied to solving a wide variety of ball-bouncing problems, including finding paths in which a ball hits each side just once as it cycles forever inside a polygon or even inside a polyhedron.

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