Its surface area equals four pi squared times the product of the inner and outer radii.
Two of the four circles that can be drawn at any point on this shape are named for Villarceau. An integrable Hamiltonian system can only move on a surface with this shape according to the Kolmogorov-Arnold-Moser theorem. Like the Klein bottle, it has an Euler characteristic of zero. At most, seven colors are used be used to color a map of this shape, according to the Heawood conjecture. Its surface area equals four pi squared times the product of the inner and outer radii. It is topologically defined as the product of two circles. For 10 points, name this three-dimensional surface shaped like a doughnut.
Two of the four circles that can be drawn at any point on this shape are named for Villarceau.
torus [or tori; or toroidal]