The Hopf Fibration

The Hopf fibration is the standard first example of a nontrivial fibre bundle, and the standard source of a visualization mistake. Its fibres are round planar circles, not the helices the usual pictures suggest.

The Hopf fibration has each fibre a great circle of , and any two fibres are linked exactly once. Under stereographic projection to every fibre becomes a Villarceau circle, a planar round circle lying on a torus, so each fibre has torsion zero. The helical appearance of the standard rendering is an ensemble effect of many nested tori at different radii, not a property of any single fibre.

The lesson generalises. The shape one sees is a choice of projection rather than the object itself, which is the same caution that applies to the Calabi–Yau slices in carving-the-calabi-yau-quintic.