Torus

The torus is the second simplest closed surface. It can be parametrised as $$ (x,y,z) = ((2+\cos(\theta))\cos(\phi), (2+\cos(\theta))\sin(\phi),\sin(\theta)). $$ The Euler characteristic is zero, and the genus is one.

We can build a torus by pasting the sides of a square, as shown in the middle diagram above.