The billiard in a rational polygon has the property that any trajectory that hits a side perpendicularly is either a saddle connection or periodic. In other words, this trajectory belongs to either a cylinder or the boundary of a cylinder in the translation surface unfolding.
The (7, 8, 15) and (7, 7, 16) triangles are the smallest triangle with N=a+b+c even (N=30 for both examples) for which one of the direction orthogonal to a side is not completely periodic. It would be nice to have this example in the documentation. This example actually disprove Conjecture 2.2(a) and(e) of Boshernitzan 1992 (note that (7, 8, 15) is indeed right-angle).
Conjecture 2.2(b) resists up to N=51 (we only see minimal components with directions parallel to one side).
Conjecture 2.2(c) can be proven with flatsurf !
Conjecture 2.2(d) is about complete periodicity (saddle connection direction => completely periodic) for small triangles.
The billiard in a rational polygon has the property that any trajectory that hits a side perpendicularly is either a saddle connection or periodic. In other words, this trajectory belongs to either a cylinder or the boundary of a cylinder in the translation surface unfolding.
The (7, 8, 15) and (7, 7, 16) triangles are the smallest triangle with N=a+b+c even (N=30 for both examples) for which one of the direction orthogonal to a side is not completely periodic. It would be nice to have this example in the documentation. This example actually disprove Conjecture 2.2(a) and(e) of Boshernitzan 1992 (note that (7, 8, 15) is indeed right-angle).
Conjecture 2.2(b) resists up to N=51 (we only see minimal components with directions parallel to one side).
Conjecture 2.2(c) can be proven with flatsurf !
Conjecture 2.2(d) is about complete periodicity (saddle connection direction => completely periodic) for small triangles.