**
The Key-Ring Model of the Hopf Fibration
**

The 3-dimensional sphere S^{3} serves as a fundamental example in several areas
of geometry and topology. The simplest way to define it, perhaps, is to say
that it is the set of unit vectors in 4-dimensional space:

Key-Ring Model of the Hopf Fibration

The model pictured has 50 steel split key rings, each with a diameter of 2.25 in (5.7 cm), and, of course, assembled so that each pair of rings constitutes a Hopf link. Another defining characteristic of the model is that each triple of key rings has the same topological orientation, as an imbedding in 3-space, as every other such triple. Of course, since the Hopf fibration is comprised of a continuum of infinitely many key rings, the model represents merely an approximation of the Hopf fibration. One may also argue that it is not a particularly faithful model because it does not respect the metrical properties of stereographic projection. Nevertheless, it is a remarkably simple and intuitive model, faithful to several of the critical properties of the Hopf fibration of the 3-sphere.

Notice finally that one may build two distinct orientations:

Both Orientations of the Key-Ring Model

One way to interpret the existence of these distinct
orientations goes as follows. The 3-sphere is a Lie group and
the circle is a closed subgroup. The Hopf fibration corresponds
to the partition of S^{3} into cosets by left- or right-orbits of the
circle subgroup. Since the group S^{3} is non-abelian, the
partitions into left cosets and right cosets are distinct, thus yielding
the two distinct orientations as pictured.

January 23, 2005.