It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path, one obeying the reflection law that the angle of incidence equals the angle of reflection. A rational triangle has angles that are rational multiples of $\pi$. It remains an open question of whether all triangles have a periodic billiard path.

Q. Is there some analogous theorem for hyperbolic triangles in the hyperbolic plane?

A billiard path in a hyperbolic triangle consists of geodesics which reflect from the sides of the triangle by the same reflection law. Here is a possible start of a billiard path inside a triangle drawn in the Poincare half-plane model, where the geodesics are circular arcs and vertical line segments:

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          [![HyperbolicTri][1]][1]

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Is there some analog of the Euclidean rational angle condition that yields a theorem for hyperbolic triangles?

It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path, one obeying the reflection law that the angle of incidence equals the angle of reflection. A rational triangle has angles that are rational multiples of $\pi$. It remains an open question of whether all triangles have a periodic billiard path.

Q. Is there some analogous theorem for hyperbolic triangles in the hyperbolic plane?

A billiard path in a hyperbolic triangle consists of geodesics which reflect from the sides of the triangle by the same reflection law. Here is a possible start of a billiard path inside a triangle drawn in the Poincare half-plane model, where the geodesics are circular arcs and vertical line segments:

---

          [![HyperbolicTri][1]][1]

---

Is there some analog of the Euclidean rational angle condition that yields a theorem for hyperbolic triangles?

It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path, one obeying the reflection law that the angle of incidence equals the angle of reflection. A rational triangle has angles that are rational multiples of $\pi$. It remains an open question of whether all triangles have a periodic billiard path.

Q. Is there some analogous theorem for hyperbolic triangles in the hyperbolic plane?

A billiard path in a hyperbolic triangle consists of geodesics which reflect from the sides of the triangle by the same reflection law. Here is the possible start of a billiard path inside a triangle drawn in the Poincare half-plane model, where the geodesics are circular arcs and vertical line segments:

---

          [![HyperbolicTri][1]][1]

---

Is there some analog of the Euclidean rational angle condition that yields a theorem for hyperbolic triangles?

It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path, one obeying the reflection law that the angle of incidence equals the angle of reflection. A rational triangle has angles that are rational multiples of $\pi$. It remains an open question of whether all triangles have a periodic billiard path.

Q. Is there some analogous theorem for hyperbolic triangles in the hyperbolic plane?

A billiard path in a hyperbolic triangle consists of geodesics which reflect from the sides of the triangle by the same reflection law. Here is a possible start of a billiard path inside a triangle drawn in the Poincare half-plane model, where the geodesics are circular arcs and vertical line segments:

---

          [![HyperbolicTri][1]][1]

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Is there some analog of the Euclidean rational angle condition that yields a theorem for hyperbolic triangles?