More Mathematics . . .

The Hopf Fibration Experience

The Hopf fibration is a decomposition of the 3-sphere into circles, in the same spirit as the decomposition of the plane into a family of parallel lines. The decomposition of the plane into parallel lines is a trivial fibration because you can find a cross section. For example, fix a vertical line in the plane. Each point in the plane lies on a horizontal line which intersects this vertical line. Thus, by translating this point of intersection you could recover the given point in the plane. The Hopf fibration is an example of a non-trivial fibration. Although, each point in the 3-sphere lies on only one of Hopf's circles, it is not the case that every point can be realized as a translate along its circle of a point on a smooth surface in the 3-sphere which intersects those circles only once. The following short film helps you to visualize the Hopf fibration of the 3-sphere using stereographic coordinates.

The Hopf Fibration Experience (8.5M)