Bianchi and Hopf (Annuli V)

When you take a torus in Euclidean space, it will always have points of positive curvature and points of negative curvature, but the average of the curvature will be 0.

This limitation of Euclidean space disappears in the 3-sphere. Luigi Bianchi more or less completely classified all flat tori in the 3-sphere in the late 19th century, paving the way to global differential geometry.


Simple examples can be constructed using the Hopf fibration: If you take the preimage of a simple closed curve in the 2-sphere under the Hopf fibration, you obtain a torus in the 3-sphere that is intrinsically flat.

When you start with a circle in the 2-sphere, the result will be a torus of revolution (or a Dupin cyclide), which corresponds to a rectangular torus. If you start with more general curves like the spherical cycloids that I have used here, you will get tori that appear twisted.
This is because these tori will correspond to non-rectangular tori, as can be verified using an elegant formula due to Ulrich Pinkall. The side view below gives access to the warped core of these tori.

The image below shows the view from somebody standing inside such a Hopf torus, with an almost perfect mirror as a surface, and four differently colored light sources. This comes pretty close to my ideal of abstract 3-dimensional art. If you know what you are looking at, you can discern the same warped core as up above, and its reflections on the warped outer parts of the torus.


Hopf Fibration (Annuli III)

Hopf 8 right

When talking about tori, at some point the Hopf fibration will make its appearance.
It all begins with a few tori of revolution packed together. Think about circular wires
bundled into one thick cable.


Cut through all the wires, twist the cable by 360 degrees, and reconnect wires of equal color.


Now all wires are interlinked, and this has the advantage that you can extend all this wiring to all of space (except for the vertical axis) in an even way to het what mathematicians call a fibre bundle.


One can increase the complexity by showing nested wires by removing parts of then. The top view below is a simplified version of the picture at the top.


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.