Rotating the Hopf Fibration

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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.

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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.

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Contributed by: Richard Hennigan (August 2014)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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.



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