The set of nodes in the Farey graph is the set of all pairs
,
whereby is understood as.
Two nodes and are connected by an edge if and only if
applies.
Applications
Farey sequences are described by Farey diagrams , the Farey graph is the union of all Farey diagrams.
In the theory of continued fractions , the Farey graph is used to prove that every periodic continued fraction is a square irrational number .
The modular group and its quotient act by broken-linear transformations , forming adjazente node of the Farey graph back to the node from adjazente.
The embedding of the Farey graph in the compactification of the hyperbolic plane by means of the identification and realization of the edges as geodesics gives the Farey tessellation of the hyperbolic plane.