How to Cut a Bagel (Annuli II)

A torus is obtained by rotating a circle around a axis in the same plane. As such, it has two families of circles on it: the ones coming from the generating circle, and the orbits of the rotation. This allows you to slice the torus open using vertical or horizontal cuts, with the cross sections being perfectly round circles,

Of course, when you do this to your bagel, you do not really expect circles. But neither would you expect the bagel to be hollow.

The surprise, however, is that there is yet another way to slice a torus, still with perfectly circular cross sections. These are the Villarceau circles.

Here is how to do it. Looking at a vertical cross section, cut along a plane that’s perpendicular to your cross section and touches the two circles just above and below. The deeper reason for their existence lies in the Hopf fibration of the 3-dimensional sphere; these curves are stereographic images of Hopf circles.

Even more surprising is that there are certain cyclides that have six circle families on them.