The algebra and geometry of continued fractions with integer quaternion coefficients
| dc.contributor.author | Mennen, Carminda Margaretha | |
| dc.date.accessioned | 2015-05-06T11:36:04Z | |
| dc.date.available | 2015-05-06T11:36:04Z | |
| dc.date.issued | 2015-05-06 | |
| dc.description | A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. 2015. | |
| dc.description.abstract | We consider continued fractions with coe cients that are in K, the quaternions. In particular we consider coe cients in the Hurwitz integers H in K. These continued fractions are expressed as compositions of M¨obius maps in M R4 1 that act, by Poincar´e extension, as isometries on H5. This dissertation explores groups of 2 2 matrices over K and two particular determinant type functions acting on these groups. On the one hand we find M R4 1 , the group of orientation preserving M¨obius transformations acting on R4 1 in terms of a determinant D [19],[38]. On the other hand K may be considered as a Cli ord algebra C3 based on two generators i and j, or more generally i1 and i2, where i j = k or i1i2 = k. It is shown this group of matrices over C4 defined in terms of a pseudo-determinant [1],[37] can also be used to establish M R4 1 . Through this relationship we are able to connect the determinant D to the pseudo-determinant when acting on the matrices that generate M R4 1 . We explore and build on the results of Schmidt [30] on the subdivision of a Farey simplex into 31 Farey simplices. These results are reinterpreted in H5 with boundary K1 using the group of M¨obius transformations on R4 1 [19], [38]. We investigate the unimodular group G = PS DL(2;K) with its generators and derive a fundamental domain for this group in H5. We relate this domain to the 24-cells PU and r that tessellate K. We define the concepts of Farey neighbours, Farey geodesics and Farey simplices in the Farey tessellation of H5. This tessellation of H5 by a Farey pentacross under a discrete subgroup G of M R4 1 is analogous to the Farey tessellation by Farey triangles of H2 under the modular group [31]. The result in Schmidt [30], that for each quaternion there is a chain of Farey simplices that converge to , is reinterpreted as a continued fraction, with entries from H, that converges to . We conclude with a review of Pringsheim’s theorem on convergence of continued fractions in higher dimensions [5]. | |
| dc.identifier.uri | http://hdl.handle.net/10539/17643 | |
| dc.language.iso | en | en_ZA |
| dc.subject.lcsh | Continued fractions. | |
| dc.subject.lcsh | Quaternions. | |
| dc.subject.lcsh | Algebra. | |
| dc.subject.lcsh | Geometry. | |
| dc.title | The algebra and geometry of continued fractions with integer quaternion coefficients | en_ZA |
| dc.type | Thesis | en_ZA |