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.
@article{1175789737,
author = {Schwartz, Richard Evan},
title = {Obtuse Triangular Billiards I: Near the $(2,3,6)$ Triangle},
journal = {Experiment. Math.},
volume = {15},
number = {1},
year = {2006},
pages = { 161-182},
language = {en},
url = {http://dml.mathdoc.fr/item/1175789737}
}
Schwartz, Richard Evan. Obtuse Triangular Billiards I: Near the $(2,3,6)$ Triangle. Experiment. Math., Tome 15 (2006) no. 1, pp. 161-182. http://gdmltest.u-ga.fr/item/1175789737/