Torus with pairs of Villarceau circles

On each point on a torus there are four circles embedded on the surface of the torus: two trivial ones (vertical and horizontal) and two non-trivial, oblique circles called the Villarceau circles. This torus shows the family of pairs of Villarceau circles, one in red and one in green.

The Villarceau circles play a particular role in the homology of the 3-sphere on the 2-sphere (known as the Hopf fibration), when presented in a stereographic projection. This is a bit of Greek to me, but it seems rather cool. I'll try to explain what I understand.

The 3-sphere is the equivalent of the 2-sphere but with one dimension more. This would be more helpful if I told you that the 2-sphere is our normal 3D sphere (and similarly the 1-sphere is a circle). The 3-sphere only lives in 4 dimensional space, so visualizing it is a bit complicated. The Hopf fibration is a way to map the 3-sphere unto the 2-sphere crossed with circles. Each point in the 3-sphere will be mapped unto the 2-sphere, and the points mapped into the same point on the 2-sphere are in a circle. The incredible thing is that these circles, which cover the whole 3-sphere, are non intersecting, and when stereographically projected into 3 dimensional space (in much the same way we project the 2-sphere unto a plane for the usual stereographic projection) it looks like the torus with Villarceau circles.