Math Trivia: Triangular Billiards

Imagine you are playing billiards on a triangular table. The ball  travels in a straight line with no friction, and when it hits an edge, it reflects in the standard way. One might ask whether or not there exists a way to hit the ball so that it follows a closed periodic path. (i.e. the ball eventually returns to where it started moving in the same direction that it started.)

One way to achieve this is by drawing the three altitudes of the triangle. For each edge, draw a line perpendicular to that edge, passing through the third vertex not on that edge. Then, take the three points where the altitudes intersect the edges perpendicularly, and connect them up. You can check that this simple triangular path, bouncing off of each once, is a closed periodic path for the ball.

There is one problem. In order for the altitudes to intersect the edges, the triangle must be acute. What happens when the triangle is not acute? This is actually a 200 year old open problem!

It can be shown that a different method works for all right triangles. This paper gave a computer assisted, but still rigorous, proof in 2006 that there is a closed periodic path for all triangles where all of the angles are less than 100 degrees. However, if one of the angles is can be greater than 100 degrees, this question is shockingly still open!

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