The rule for stereographic projection has a nice geometric description. Think of the earth as transparent sphere, sitting on the plane of the paper. (The south pole touches the paper.) Now imagine a light bulb at the north pole, which shines through the sphere. Each point on the sphere casts a shadow on the paper, and that is where it is drawn on the map.
The south pole appears at the center point of the map. Lines of lattitude appear as circles around this center. Things near the south pole are not stretched very much in the map, but the equator is twice as big on the map as on the sphere. The northern hemisphere is stretched quite a bit, and the north pole gets sent off to infinity.
In this demo, instead of showing a map of the continents on earth, we choose from five more symmetric patterns on a sphere. These correspond to the five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The selected polyhedron is drawn with straight edges in blue. Then these are bent out to lie on the surface of the sphere, as purple arcs. For instance, the twelve edges of the octahedron join to become the equator and two perpendicular meridians on the sphere.
The purple arcs are stereographically projected to the tan arcs in the plane. Note that circles map to circles in the plane, and angles are preserved. This angle property means that stereographic projection (like the Mercator map but unlike many other common map projections) is conformal: shapes of small regions are the same on the map as on the earth.
When the "Rotate Sphere" mode is selected, dragging the mouse in the image area will rotate the pattern on the sphere, so that the stereographic projection is from a different point. When "Orbit Camera" is selected, mouse motions will change the viewpoint for the whole configuration. We start out looking straight down from the north pole, through the sphere, onto the plane, but by orbiting to a different view, we can see the plane from the side, with the sphere above it. The "Reset View" button will reset both of these motions. The slider zooms in and out.
When the selected polyhedron is the tetrahedron, cube, or dodecahedron, then the purple net of arcs on the sphere has three-fold vertices, with equal 120-degree angles. This property is again true of the projected brown net in the plane (by conformality). Thus the brown arcs form what could be a soap-bubble cluster in the plane. (To see this best, zoom out completely and use the default view from the top, while rotating the sphere.)
In a completely analogous manner, we could look at a hypercube living in the sphere in four-dimensional space. This sphere can be mapped to three dimensions by a stereographic projection, and the image of the hypercube has the geometry of a soap-bubble cluster, as seen below.
The code for the applet above is online.