The Hopf fibration describes a relationship between the one-dimensional sphere (a circle), the two-dimensional sphere (an ordinary sphere), and the three-dimensional sphere (a hypersphere in 4D space) as a fibration with as the fiber, as the base space, and as the total space. This mapping has the property that when viewed locally, is indistinguishable from the Cartesian product . However, this is not true globally, since a fibration has a "twist" that distinguishes it from a regular product space.

This Demonstration allows you to manipulate a set of points in (shown in the bottom-left corner) and view the corresponding circles in with stereographic projection, revealing much of the interesting structure induced by the Hopf map.

While manipulating this Demonstration, here are some features of the Hopf fibration (and in general) to look for:

• In the stereographic projection, points closer to the north pole of are mapped to larger circles, with the north pole itself mapping to a circle of infinite radius, resulting in a straight line.

• Any given pair of circles are linked together in what is known as a Hopf link.

• The parallels of are mapped to nested tori (use the slider to compress points to a circle).

• A torus embedded in can be turned inside out without intersecting itself.