Triangles with Optimal Dynamics

In my previous post, I explored the close-to-equilateral integer triangles and today I was reading another mathematical article about triangles, so-called triangles with optimal dynamics. I thought the topic worthy of a blog post. Here is a quote from the article:

When you set a ball in motion on a billiard table, it may seem as if anything is possible, but when a table has optimal dynamics, only two things truly are. The first is complete chaos, which is to say that the ball’s path will cover the entire table as time wears on. The second is periodicity — a repeating path like a ball pinging back and forth between two sides.

In tables without optimal dynamics, a wider range of possibilities exists, which makes the full analysis of all possible paths impossible: A ball could end up bouncing chaotically in one part of the table forever, never retracing its path, but also never covering the whole table.

What mathematicians do know is that there are at least eight kinds of triangles with optimal dynamics; the first was discovered in 1989 and the last in 2013. Whether there are more is anyone’s guess. 

 Let's look at each in turn:

• 1 : 1 : n means isosceles triangles with angles 10°, 10°, 160° or 20°, 20°, 140° etc.
  
• 1 : 2 : n means triangles with angles 10°, 20°, 150° or 20°, 40°, 120° etc.
  
• 3 : 4 : 5 represents a triangle with angles of 45°, 60° and 75°
  
• 2 : 3 : 4 represents a triangle with angles of 40°, 60° and 80°
  
• 3 : 5 : 7 represents a triangle with angles of 36°, 60° and 84°
  
• 1 : 4 : 7 represents a triangle with angles of 15°, 60° and 105°
  
• 2 : (n-2) : n represents a right angled triangles with angles of 2°, 88° or 90°, 4°, 86° or 6°, 84°, 90° or 10°, 80°, 90° or 12°, 78°, 90° or 18°, 72°, 90° or 20°, 70°, 90° or 30°, 60°, 90° or 36°, 54°, 90°
  
• 2 : n : n represents triangles like 2°, 89°,89° or 4°, 88°, 88° or 6°, 86°, 86° etc.

The article goes on to identify two quadrilaterals with optimal dynamics (shown below):