Spheres & Flow · 3D
Torus
Woven from Villarceau circles
A doughnut drawn only with perfect circles. Slice a torus with a tilted plane that touches it twice and you get two circles: the Villarceau circles. Rotate them and they weave the whole surface.
- Genus
- 1
- Circles per point
- 4
Turn it in your hands, strike it to hear the chord of its edges, slice it with a plane or lift it into four dimensions.
Facts
- Through every point of a torus pass four circles: meridian, parallel and two Villarceau.
- Villarceau described them in 1848.
- The same circles appear when a 4D Clifford torus is projected into 3D.
History and meaning
Many teachers describe the torus as the fundamental shape of flow in nature: smoke rings, magnetic fields, whirlpools, the heart’s field.
Across the dimensions
- Lifts into Clifford TorusIn four dimensions a torus can be perfectly flat: the Clifford torus.
