Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in ℝ²

In this paper we study the properties of the emph{Triangular tree}, a complete tree of rational pairs introduced in cite{cas}, in analogy with the main properties of the Farey tree (or Stern-Brocot tree). To our knowledge the Triangular tree is the first generalisation of the Farey tree constructed using the mediant operation. In particular we introduce a two-dimensional representation for the pairs in the tree, a coding which describes how to reach a pair by motions on the tree, and its description in terms of $SL(3,Z)$ matrices. The tree and the properties we study are then used to introduce rational approximations of non-rational pairs.