Continued fractions and distorted chequer boards

Abstract We describe how the partial quotients of a nearest integer continued fraction for a complex irrational can be used to generate a sequence of nested Farey quadrilaterals in the complex plane $$\mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , each of which contains the target point. Our approach begins by establishing a connection between the Farey tessellation of hyperbolic space $$\mathbb {H}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> and the Schmidt arrangement on the boundary of $$\mathbb {H}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . We show that Farey octahedra are represented uniquely by Farey quadrilaterals among the Farey circles and dual Farey circles in the Schmidt arrangement. Next, we interpret continued fractions in terms of products of Möbius maps, making use of Beardon’s observation that Möbius maps act as isometries of hyperbolic space. Factors of the initial product of Möbius maps alternate between parabolic Möbius maps that fix infinity and those that fix zero. A closer analysis of elliptic Möbius maps that leave vertices of the fundamental octahedron invariant reveals that factors of certain elliptic Möbius maps appear naturally within the product of parabolic factors. These elliptic factors need to be isolated from the parabolic factors, adjusting the coefficients of the parabolic factors as required in the process. The structural connection between Farey quadrilaterals and the adjusted coefficients yields a new visual interpretation of how complex nearest integer continued fractions generate successive approximations for irrational complex numbers.