Forme des Connexes de Farey

Abstract

The aim of this work is to prove that the connected parts of Farey complex structure in plane are triangles or quadrangles. To do this work we go back to plane convex polygones with oriented edge for wich we prove that if two consecutive vectors of the edge never are in the same quadrant, then the polygone is a triangle or a quadrangle. Then we prove that the connected parts of Farey complex have this property.