How to Cut a Bagel (Annuli II)

Annulus 1

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.


Expanding Your Mind (Spheres IV)

Circles can also intersect perpendicularly in a more complicated way than discussed in Spheres IV. Like so:


This might look complicated, but is in fact just a transformed version of the easier to grasp dart disk:


To see how these two images are related, pretend the radial lines in the second image are in fact huge circles that all intersect in the center point. Then they will also intersect in another point, which is, in the case of lines, the ominous point at infinity, but, in the case of circles, becomes just another point in the plane. This other point and the origin are the common points of one family of circles, as you can see in the first image, and the second family of circles intersects the first perpendicularly. The first image can be transformed into the second by what is called an inversion.

If we want to repeat this in three dimensions, it is maybe best to start with the second image, replacing the radial lines by vertical green planes, and the circles by concentric blue spheres. Then, something curious happens. Lines and circles are in some sense the same thing, and so are planes and spheres. But if we look for a third family of surfaces that intersect the planes and spheres orthogonally, we need to step outside the plane/sphere paradigm. It turns out that we need vertical red cones to cut both the blue spheres and the green planes perpendicularly:


Now, coming back to the 3D version of the first image, we just need to invert the above cones, planes, spheres as to become this:


The red surface is called a cyclide. It has two cusps that correspond to the tip of the cone and the (still ominous) point at infinity.

Now imagine that you are inside that cyclide, looking around…