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
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.