Rotating the Hopf Fibration
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.[more]
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.[less]
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.