How many charts are needed to cover a 2-torus?

Can you please answer this question with explanation ? I just learned about the charts needed to cover 1 and 2 spheres but got confused for the case of torus. It would be great if you guys could help.

Solution 1:

If the images of the charts are not restricted to be simply connected (which they are not according to the Wikipedia definition of "chart"), then there is a two-chart-system covering the torus.

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And even with simply connected open sets, there is a way to do it with only three charts. Here is a picture showing how these charts look like on the flat torus (the right one shows that they indeed cover everything).

Solution 2:

Hint $\mathbb T^2=\mathbb S^1\times\mathbb S^1$

It is easily to see that $\mathbb S^1$ can be charted by two covers, then $\mathbb T^2$ is four.

Solution 3:

Four will do it. Wrap one around the outside of the doughnut and another on the inside. Let them overlap a little. Both of these are diffeomorphic to hollow cylinders, which require two patches to cover.