## The geometry of continued fractions as analysed by considering Möbius transformations acting on the hyperbolic plane

##### Date

2012-02-24

##### Authors

van Rensburg, Richard

##### Journal Title

##### Journal ISSN

##### Volume Title

##### Publisher

##### Abstract

Continued fractions have been extensively studied in number-theoretic ways.
In this text, we will illuminate some of the geometric properties of contin-
ued fractions by considering them as compositions of MÄobius transformations
which act as isometries of the hyperbolic plane H2. In particular, we examine
the geometry of simple continued fractions by considering the action of the
extended modular group on H2. Using these geometric techniques, we prove
very important and well-known results about the convergence of simple con-
tinued fractions. Further, we use the Farey tessellation F and the method of
cutting sequences to illustrate the geometry of simple continued fractions as
the action of the extended modular group on H2. We also show that F can be
interpreted as a graph, and that the simple continued fraction expansion of
any real number can be can be found by tracing a unique path on this graph.
We also illustrate the relationship between Ford circles and the action of the
extended modular group on H2. Finally, our work will culminate in the use of
these geometric techniques to prove well-known results about the relationship
between periodic simple continued fractions and quadratic irrationals.

##### Description

M.Sc., Faculty of Science, University of the Witwatersrand, 2011

##### Keywords

Transformations (mathematics), Numbers, complex, Geometry, hyperbolic, Matrix groups