variable approaches . Subgroups of the Möbius group If we require the coefficients of a Möbius transformation to be real numbers with , we obtain a subgroup of the Möbius group denoted as PSL(2,R). This is the group of those Möbius transformations that map the upper half-plane to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps . If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H, the Poincaré half-plane model, and PSL(2,R) is the group of all orientation-preserving isometries of H in this model. The subgroup of all Möbius transformations that map the open disk to itself consists of all transformations of the form with ∈ R, b ∈ C and |b| < 1. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps . By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the Poincaré disk model, and this group is the group of all orientation-preserving isometries of H in this model. Since both of the above subgroups serve as isometry groups of H, they are isomorphic. A concrete isomorphism is given by conjugation with the transformation which bijectively maps the open unit disk to the upper half plane. Alternatively, consider an open disk with radius r, centered at ri. The Poincaré disk model in this disk becomes identical to the upper-half-plane model as r approaches ∞. A maximal compact subgroup of the Möbius group is given by and corresponds under the isomorphism to the projective special unitary group PSU(2,C) which is isomorphic to the special orthogonal group SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the point groups in three dimensions. Icosahedral groups of Möbius transformations were used by Felix Klein to give an analytic solution to the quintic equation in ; a modern exposition is given in . If we require the coefficients a, b, c, d of a Möbius transformation to be integers with ad − bc = 1, we obtain the modular group PSL(2,Z), a discrete subgroup of PSL(2,R) important in the study of lattices in the complex plane, elliptic functions and elliptic curves. The discrete subgroups of