The geometry of Gaussian integer continued fractions

Abstract

The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship |ad−bc|=1 for a,b,c,d integers as it arises in the Farey tessellation of the hyperbolic plane H2 and the array of Ford circles in the upper-half of the real plane R2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition |αγ−βδ|=1 for α,β,γ,δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers.