Obtuse triangular billiards I: Near the (2,3,6) triangle

摘要

Let S, denote the set of Euclidean triangles whose two small angles are within e radians of pi/6 and pi/3 respectively. In this paper we prove two complementary theorems: (1) For any epsilon > 0 there exists a triangle in S, that has no periodic billiard path of combinatorial length less than 1/epsilon. (2) Every triangle in S-1/400 has a periodic billiard path.