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
- There are two leading vertices on both the top and the bottom. (Marked
*A*and*B*) - 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.

Example 1:

Example 2 differs from example 1 by a Dehn twist in the cylinder: