Billiards!!
On the following page, you will find fourteen different billiard tables, each witha different size. We are going to play “mathematical billiards”. For each table, we will shoot a ball around the table and trace its path. The ball will travel under the following rules:
1)We will always start at the lower left corner.
2)We will always launch to the upper right in a 45 degree angle.
3)Every time we bounce, we will leave the wall at a 45 degree angle (in terms of physics, we are following Snell’s law).
4)We will stop when we get to another corner.
Your first task: Draw the path of the ball for each of the billiard tables. A PDF file of these tables is on the web page, in case you mess up.
Your second task: Based on your drawings, make a chart listing the following information for each table:
a)The number of squares the ball passed through
b)The shortest distance from a bounce point to a corner of the table (if the ball never bounced, then use the length of the shortest wall)
c)The number of times you touched the walls (include the beginning and end points of the path)
d)Which corner you end up in (upper left, upper right, or lower left)
e)The number of times the ball’s path intersects itself
f)What symmetries do you have? You could have
- A vertical line of reflection (the left half looks like a mirror image of the right half)
- A horizontal line of reflection (the top half looks like a mirror image of the bottom half)
- A 180 degree rotation (if you turn the picture upside down, the path looks the same)
- None of the above
Your third task: Needless to say, all of the pieces of information you collected in your chart depend on the dimensions of the billiard table. So, given an m x n dimensional table, determine the values of a. through f. in terms of m and n.
Your fourth task: Explain (coherently) why it is impossible for the ball to end up in the lower left corner. Keep in mind; “Because it started there” is not an answer.