A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013. / Continued fractions have been extensively studied in number theoretic ways.
These continued fractions are expressed as compositions of M¨obius
maps in the Picard group PS L(2;C) that act, by Poincar´e’s extension, as isometries
on H3. We investigate the Picard group with its generators and derive the fundamental
domain using a direct method. From the fundamental domain, we produce
an ideal octahedron, O0, that generates the Farey tessellation of H3. We explore
the properties of Farey neighbours, Farey geodesics and Farey triangles that arise
from the Farey tessellation and relate these to Ford spheres. We consider the Farey
addition of two rationals in R as a subdivision of an interval and hence are able
to generalise this notion to a subdivision of a Farey triangle with Gaussian Farey
neighbour vertices. This Farey set allows us to revisit the Farey triangle subdivision
given by Schmidt [44] and interpret it as a theorem about adjacent octahedra in
the Farey tessellation of H3. We consider continued fraction algorithms with Gaussian
integer coe cients. We introduce an analogue of Series [45] cutting sequence
across H2 in H3. We derive a continued fraction expansion based on this cutting
sequence generated by a geodesic in H3 that ends at the point in C that passes
through O0.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/15145 |
| Date | 11 August 2014 |
| Creators | Hayward, Grant Paul |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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