Visualizing imaginary quadratic fields

Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean algorithm (in Z) expresses any element ofSL2(Z) as a product of elementarymatrices inSL2(Z). It is remarkable that among number fields K (whose rings of integers we denote OK ), SL2(OK) fails to be generated by elementary matrices exactly whenK is a non-Euclidean imaginary quadratic field [1, 10]. A particularly useful way to visualize the group SL2(Z), or PSL2(Z), is to study its action as Möbius transformations on the upper half plane, as in Figure 1. To study the Bianchi group PSL2(OK), whenK is imaginary quadratic, consider instead the upper half space H3 lying above the complex plane. This is a model of hyperbolic space with boundary Ĉ = C∪ {∞}. The hyperbolic isometries of this model are exactly the Möbius transformations, extended from Ĉ. Each Bianchi group forms a discrete subgroup of hyperbolic isometries; in other words it is a Kleinian group. In analogy to Figure 1, each has a 3-dimensional fundamental region. For today, however, let us focus on the boundary: consider the orbit of R̂ = R∪{∞} ⊆ Ĉ. Möbius transformations take circles (including R̂, a circle through∞) to other circles. The full orbit of R̂ is dense in the plane, but if we restrict ourselves to drawing only those circles having bounded curvature (recall that curvature is the reciprocal of radius), we obtain intricate images such as in Figure 2.