We present a relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two. This allows one to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for the entries of the Cartan matrices.
"Weyl groupoids of rank two and continued fractions." Algebra Number Theory 3 (3) 317 - 340, 2009. https://doi.org/10.2140/ant.2009.3.317