Obtuse Triangular Billiards I: Near the $(2,3,6)$ Triangle

Abstract

Let $S_{\epsilon}$ denote the set of Euclidean triangles whose two small angles are within $\epsilon$ radians of $\frac{\pi}{6}$ and $\frac{\pi}{3}$ respectively. In this paper we prove two complementary theorems: (1) For any $\epsilon>0$ there exists a triangle in $S_{\epsilon}$ 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.