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In this paper a Farey graph of Farey octahedrons that tessellate hyperbolic space $${\mathbb {H}}^{3}$$ under the action of the Picard groups is introduced. The graph is used to find a discrete subgroup of $$PSL(2, {\mathbb {C}})$$ that has the fundamental Farey octahedron as a fundamental domain. The graph is also used to find a Gaussian integer continued fraction expansion for any complex number $$\xi $$ that may be interpreted as a cutting sequence of a geodesic in $${\mathbb {H}}^{3}$$ across the Farey tessellation of $${\mathbb {H}}^{3}$$ by Farey octahedrons that has $$\xi $$ as an endpoint.

The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship | a d − b c | = 1 for a , b , c , d integers as it arises in the Farey tessellation of the hyperbolic plane H 2 and the array of Ford circles in the upper-half of the real plane R 2 . Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H 2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition | α γ − β δ | = 1 for α , β , γ , δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H 3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers.

This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph $\mathcal{F}$ , whose vertices are reduced Gaussian rationals in $\mathbb{Q}_{\infty }(i)$ and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in $\mathbb{H}^{3}$ where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of $\mathbb{H}^{3}$ by Farey octahedrons. A natural distance $\varrho $ on $\mathbb{Q}_{\infty }(i)$ is also recalled, where $\varrho (\infty ,w)=n$ is the least number of edges in $\mathcal{F}$ from $\infty $ to $w\in \mathbb{Q}(i)$ , where $n$ is called the generation of w and a relevant path in $\mathcal{F}$ is called a geodesic expansion for $w$ . The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation $n-1$ , $n$ , and $n+1$ . Subsequently, it is seen that there can be at most four Farey neighbors of generation $n-1$ in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any $w$ is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.

This paper explores M ( R ∞ 4 ) , the group of orientation preserving Mobius transformations acting in R ∞ 4 . On the one hand M ( R ∞ 4 ) is given by the group of 2 × 2 matrices over the quaternions H with determinant D derived from the corresponding 4 × 4 matrices over the complex numbers C . On the other hand we know that, in general, M ( R ∞ n ) may be given in terms of 2 × 2 matrices over the Clifford algebra C n with n − 1 generators. Thus when n = 4 , M ( R ∞ 4 ) is given in terms of 2 × 2 matrices with entries drawn from C 4 and determinant Δ defined in terms of the entries of the given matrix. We note that the skew field H may be considered as a Clifford algebra C 3 based on two generators i and j or more generally i 1 and i 2 where i j = k or i 1 i 2 = k , while the set of elements { 1 , i , j , k } form a basis of H regarded as a 4-dimensional real vector space. Thus H is embedded in C 4 . In the paper we reconcile the two representations of M ( R ∞ 4 ) by comparing the generating sets of the underlying groups of matrices. A relationship between a determinants D and Δ is also exposed.

The past decade has seen enormous changes in the quality and quantity of mathematics taught at secondary school. These changes have had a cumulative effect on the readiness of mathematics students for tertiary education. Furthermore the need for university staff to be accountable to both community and students adds to the responsibility of university mathematics departments to adapt and change to suit this environment and create new educational frameworks. This paper reviews the changes in a third year (qualifying course) in mathematics at a South African university over a period of 10 years. These changes are interrogated in the light of the international call for changes at tertiary level and the changing dynamics of the student cohort. Suggestions are made for alternate models for teaching and learning mathematics at this level.

Click on the link to view the abstract. Keywords: Continued fractions, hyperbolic geometry Quaestiones Mathematicae 36(2013), 437-448

Click on the link to view the abstract Keywords: continued fractions, hyperbolic geometry, modular group, Stern-Brocot tree Quaestiones Mathematicae 29 (2006), 427–446