# 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.

você vai me matar, pedroATL, and 15 other people added this photo to their favorites.

1. These are very cool. How did you model this?

2. In POV-ray, with a torus emptied of a center torus intersected with smaller slanted toruses (I don't know if it's very clear), and a similar construct inside with the toruses slanted the other way.

3. Cartier trinity :)

4. So, there are two toruses (tori?) here. One is red and one is green, the red one is inside the green one, and using a boolean intersect, the red one cuts out the inside of the green one. Then the the green one is intersected with 12 smaller toruses, and the red one is intersected with 12 different smaller toruses? Is that right?

Are the 12 smaller toruses all slanted and rotated at different angles?

5. Josh: the only thing missing is that the red torus has also the inside cut out. The rest is good!

The 12 smaller toruses are slanted with the same angle; the angle depends on the ratio of the radii of the torus (r and R).

6. Any chance to interweave the red and green lines?
:-)

7. Something like this? (ok this is cheating)

8. Very nice! I'm really happy Villarceau circles have made it to flickr. They deserve all the press they can get. Is your POV source code available?

9. jbuddenh: thanks! I don't think I kept the pov source code of this particlar version. It wasn't particularly hard to model.

10. Thanks anyway, Seb_Przd. It looks like it might be fun to try.

11. Greetings Fellow Human...
i'm currently administering a collective that is interested in the subtleties of Geometrical & Numerical Magick!
We would like this item of yours to be included in our library of Pictoglyphs.