Torus from Villarceau Circles

A model of a torus

If you slice a torus (a doughnut-shaped surface) in half with a plane parallel to its axis or perpendicular to its axis, the cross section is two circles. However, there is one more pair of circles hidden inside a torus that goes through any point on the torus. If you cut the torus in half at just the right angle, your cross section will be two interlocking circles called Villarceau circles. The animated image below (from Wikipedia) shows how this is possible.

Villarceau circles

The paper model in this project uses crescent-shaped pieces defined by Villarceau circles to build a torus.

Video Instructions