Infinitely many periodic billiard paths in some irrational triangles
This is an unfolding for a stable periodic billiard path in the triangle with angles (Pi/4,3 Pi/10,9 Pi/10). That's (1/2, 3/5) in McBilliards Coordinates. The word is a palindrome of length 12. We identify opposite edges of the unfolding by a translation building an annulus.
Notice that
  1. There are two leading vertices on both the top and the bottom. (Marked A and B)
  2. The line segment AB on the top is contained in a set of triangles which differ by only a translation from those triangles containing AB on the bottom.
These facts are invariant under (small enough) perturbations of the triangle. We think of this cylinder as embedded in the minimal translation surface for the triangle. The two boundaries of the cylinder share a common interval. Any segment which starts on one interval AB and ends at the same point on the other interval AB is therefore a closed geodesic loop on the minimal translation surface cover. This loop corresponds to a periodic billiard path.
Example 1:
Example 2 differs from example 1 by a Dehn twist in the cylinder: