Classifying ?????????-tilings

Recently there has been significant progress in classifying integer friezes and S mathvariant="normal">L 2 \mathrm {SL}_2 -tilings. Typically, combinatorial methods are employed, involving triangulations of regions inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation hyperbolic plane by ideal triangles induced Farey graph. We demonstrate that geometric, numeric properties graph perfectly suited tame -tilings, positive – both finite infinite. In so doing, obtain geometric analogues certain known models for tilings triangulations, prove several new results similar type too. For instance, determine those bi-infinite sequences integers quiddity sequence some infinite frieze, give simple model which generalises classical construction Conway Coxeter friezes.