Remark on Continued Fractions, Moebius Transformations and Cycles

TL;DR

This paper explores the geometric interpretation of continued fractions through chains of horocycles and Moebius transformations, extending the approach to higher dimensions using Clifford algebras.

Contribution

It reviews and extends the use of horocycle chains in describing continued fractions, incorporating Moebius transformations and higher-dimensional generalizations.

Findings

Chains of tangent horocycles relate to continued fractions.

The approach is extended to multiple dimensions with Clifford algebras.

Connections between horocycle chains and Moebius transformations are clarified.

Abstract

In a recent paper A.Beardon and I.Short proposed to use chains of tangent horocycles as an extended tool describing continued fractions. We review the origin of such construction from the Moebius transformations point of view. Related descriptions with chains of orthogonal horocycles emerged as a result. The approach is extended to several dimensions in a way which is compatible to the early proposition of A.Beardon based on Clifford algebras.