Editing 3-sphere (section)

===One-point compactification===
After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. 
An extremely useful way to see this is via [[stereographic projection]]. We first describe the lower-dimensional version.

Rest the south pole of a unit 2-sphere on the {{mvar|xy}}-plane in three-space.  We map a point {{math|P}} of the sphere (minus the north pole {{math|N}}) to the plane by sending {{math|P}} to the intersection of the line {{math|NP}} with the plane.  Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner.  (Notice that, since stereographic projection is [[conformal map projection|conformal]], round spheres are sent to round spheres or to planes.)

A somewhat different way to think of the one-point compactification is via the [[exponential map (Riemmanian geometry)|exponential map]].  Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole.  Under this map all points of the circle of radius {{pi}} are sent to the north pole.  Since the open [[unit disk]] is homeomorphic to the Euclidean plane, this is again a one-point compactification.

The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the [[Lie group]] of unit quaternions.