Friezes from surfaces and Farey triangulation

We provide a classification of positive integral friezes on marked bordered surfaces in the style of Conway and Coxeter. More precisely, we prove that positive integral friezes are in one-to-one correspondence with ideal triangulations supplied with a collection of rescaling constants assigned to punctures. For every triangulation the set of the collections of constants is finite and is completely determined by the valencies of vertices in the triangulation. In particular, it follows that the number of non-equivalent friezes on bordered surfaces is finite, and all friezes on unpunctured surfaces are unitary.