Applications and Extensions

In this final section we wish to reveal connections between what has already been discussed and a significant tool that is interesting in its own right, namely, stereographic projection. Further, this application in a new context may provide a deeper understanding of the modular group's action.

Stereographic Projection

Stereographic projection maps the extended complex plane to a sphere, called the Riemann sphere, that provides a means to see the entire complex plane all at once and supplies a point to represent the mysterious "point at infinity."

We can think of the Riemann sphere as a sphere of radius one resting on the complex plane with its "south pole" at the origin. Any nonhorizontal line through the "north pole" (or (0,0,2) in three-space) will intersect the complex plane in a single point. Conveniently, any such line will also intersect the sphere in one other point besides the north pole. We have a one-to-one correspondence between points in the plane and points on the sphere, minus the north pole. The north pole is the point at infinity! In the following animations, we color arcs in the complex plane the same as in previous animations, while arcs on the sphere are colored magenta or a dark cyan. Corresponding points on the sphere and plane are marked with large orange and cyan dots. In the animation that follows, a single track in the plane is shown in yellow and its projection on the sphere is colored blue. Two views of the same scene, one from directly above the sphere's north pole, reveal the shapes in the plane and on the sphere.

Figure 15. Stereographic projection associates points in the plane with points on the sphere.

The Riemann sphere model provides a convenient method for understanding the geometry of complex space near the point at infinity. A circle passing through the point at infinity on the Riemann sphere is merely a line in the complex plane. In the following animation, this is most clearly seen by observing the action on the stereographic projections of the blue regions that are colored in cyan on the sphere.

Figure 16. The action of in the half-plane and on the sphere, connected through stereographic projection.

Hyperbolic Geometry: Past and Present

In 1882 Poincaré helped establish the reputation of Mittag-Leffler's new journal, Acta Mathematica. In its inaugural issue, Poincaré announced his discovery of a model of the so-called Lobachevskian geometry, now commonly referred to as hyperbolic geometry or, popularly, as non-Euclidean geometry. More recently Bill Thurston, in work that led to his Fields Medal award, described 3-manifolds as being Euclidean, spherical, hyperbolic, or a hybrid of these three for almost all possible 3-manifolds. Jeffrey Weeks, a student of Thurston, has fascinating information about how data is being gathered over the next two years that will provide empirical evidence for the first time regarding the global shape of our universe. News updates for this project can be found at the following web address: www.geometrygames.org/weeks/ESoS/CosmologyNews.html.

Riemann Surfaces and the Modular Group's Name

Where did the name "modular group" come from? In general, the parameters that determine any specific Riemann surface are complex numbers and are referred to as the moduli of the surface. A Riemann surface of genus , a donut-shaped object with holes instead of the usual single hole, is determined by moduli (complex numbers).

Each Riemann surface of genus one (a torus or donut with one hole) is known to be equivalent to a parallelogram in the plane with opposite sides identified. A widely known version of this torus model is the flat screen for a Pac-Man video game. In the video game, when a figure departs from the screen, it simultaneously re-enters the screen from a point on the opposite side. It is as though the opposite sides were "glued together" in some magical way.

We may also assume that similar parallelograms represent equivalent surfaces. Thus, we can normalize a parallelogram so that one of its sides has unit length and lies on the x-axis between the origin and the point . A parallelogram placed in this normalized position is completely determined by the position of its left-hand vertex lying off the x-axis. In this manner, each point in the complex plane determines a single parallelogram. Each parallelogram represents a genus-one Riemann surface (torus), once we identify opposite sides. Many different points represent parallelograms that can be cut and resewn together to be equivalent to each other, either congruent or similar. The remarkable fact here is that the points in a single fundamental region of the modular group are precisely those needed to account for every possible genus-one Riemann surface and to account for each one just once. Stated another way, each point in the fundamental region is the modulus (determining parameter) for a single genus-one Riemann surface that is distinct from all others determined by other points in the same fundamental region. This is why certain boundary points are excluded from each fundamental region in the strict definition. In other words, each boundary point determines a surface equivalent to the surface determined by a boundary point on an opposite side.