## The algebra and geometry of continued fractions with integer quaternion coefficients

##### Date

2015-05-06

##### Authors

Mennen, Carminda Margaretha

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##### 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].

##### 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.