The algebra and geometry of using continued fractions for approximating real and complex numbers

Abstract

Through geometric analysis we forge a new interpretation of the link between nearest integer continued fractions and the Farey tessellation of hyperbolic space. Instead of just truncating the continued fraction to generate approximations, we focus on • how to parse the product of maps derived from the nearest integer continued fraction into a product of parabolic and elliptic Mobius maps and ¨ • on the collection of points on which these maps act. It turns out that we need to set apart the elliptic maps that permute a collection of six vertices, three values in R, namely ∞, 0 and 1, and three values in C, namely i, 1 + i and 1+i 2 . The action of the remaining parabolic maps on the same six vertices results in the creation of a sequence of nested Farey quadrilaterals, containing the target, whose boundaries are based in the Schmidt arrangement formed by the Farey sets and dual Farey sets of Schmidt.