52482862675 http://1ucasvb.tumblr.com/post/52482862675/villarceau-circles-on-a-torus-next-time-you-have 618

## Villarceau circles on a torus

Next time you have a donut or bagel at hand, give this a try. Slice it diagonally in a way that the cutting plane is tangent to the donut both below and above its inner hole. If you do it just right, what you get is two perfectly symmetric pieces whose boundaries are the union of two perfect circles (for a sufficiently round donut).

Mathematically speaking, if you have a torus (what mathematicians call that donut shape) and cut it diagonally at the correct angle, you will reveal a pair of perfect circles on the surface of the torus, known as Villarceau circles.

For every point on the surface of a torus, you can trace exactly 4 distinct perfect circles, on the surface of the torus, that pass through that point: one is around the hole of the torus, and the other around its circumference. The other two are Villarceau circles, but not really the pair shown in the animation.

Villarceau circles play an important role in Hopf fibrations. Roughly speaking, you can fill the entire 3D space with an infinite number of such circles. Apparently, this sort of thing even shows up in quantum physics, but I can’t offer any information on that.

I was first introduced to the concept of Villarceau circles through this great POV-Ray render by Tor Olav Kristensen.

This was one of the first animations I did for Wikipedia, and I’m still rather proud of it.

On a side note, I’ve been very busy with college lately, that’s why I haven’t been posting much. But I’m working on a series of animations on vector calculus and electromagnetism, and I think they’ll turn out great. Stay tuned!

43003413648 http://1ucasvb.tumblr.com/post/43003413648/last-one-of-the-polygonal-torii-just-trying-a-new 204

Last one of the polygonal torii. Just trying a new point cloud renderer.

42959934018 http://1ucasvb.tumblr.com/post/42959934018/another-polygonal-torus-with-a-twist-this-is 121

Another polygonal torus - with a twist!

This is a torus made out of pentagons instead of circles, and the pentagons make 1/5th of a twist as they go around.

42956223102 http://1ucasvb.tumblr.com/post/42956223102/a-ghostly-hexagonal-pentagonal-torus-rotating 63

A ghostly hexagonal-pentagonal torus rotating simultaneously around its main axis and along its circumference.

Gotta love point clouds. They look so much better and “physical” than simple triangle meshes.

Here’s a twisted variaton.