Hopf fibration



In 3 dimensions, any two diametric circles drawn on a sphere will intersect at two points. However, this is not necessarily the case in 4 dimensions. The Hopf fibration refers to a particularly symmetric arrangement of great circles on a 3-sphere such that every point on the 3-sphere belongs to exactly one of these circles.

The high symmetry of the arrangement gives rise to many non-trivial symmetry groups, known as the swirl groups after Bowers. In particular, swirlchora and polytwisters can be thought of as discrete versions of the Hopf fibration.

The Hopf fibration is closely related to the complex numbers, and incidentally to the quaternions. As such, higher-dimensional generalizations of the Hopf fibration involving the quaternions and octonions exist. However, these have not been thoroughly studied from the perspective of symmetry groups or polytopes.

The Hopf fibration is most often studied at a high level in topology. However, this article aims only to give an elementary geometric introduction to it.

Construction
The usual form of the Hopf fibration can be constructed as follows. Identify all points (x, y, z, w) on a 3-sphere with the pair of complex numbers (x + yi, z + wi). For any point (a, b) on this hypersphere, we can build a circle whose points are given by
 * $$\{(ka, kb):k\in\mathbb C, |k|=1\}.$$

The set of all of these circles makes up the Hopf fibration. It can be verified that any point of the 3-sphere belongs to exactly one great circle.

There's also an alternate construction involving quaternions.

Symmetry
The Hopf fibration is preserved under the symmetry of rotating all of its constituent circles at once by the same angle θ, no matter the value of θ. This symmetry is an isoclinic double rotation of angle θ. In terms of coordinates, this symmetry just amounts to multiplying both complex coordinates of each point on the 3-sphere by exp(iθ). Alternatively, it can be seen as left multiplication by the quaternion exp(iθ).

There are also other symmetries of the Hopf fibration that don't just fix circles in place. These symmetries are given precisely by right multiplication by a unit quaternion, now identifying every point of the 3-sphere with a unit quaternion.

As a consequence, one may create finite symmetry subgroups of the Hopf fibration by starting from a finite quaternion group (these are intimately related to finite 3-dimensional rotation groups) and appending some rotation of finite order preserving the Hopf fibration circles. This is exactly the construction that gives swirl groups.