How to understand an triangulation of torus?

Question

In the paper, it is said that Figure 1.1 in Example 1.2.3 is a triangulation of a torus.

How to see that every face in this triangulation is an triangle? How many triangles are there in this triangulation? It is a little hard to see this directly.

Thank you very much.

Answer 1

This is the same as

under the usual identifications. But beware, I think it depends on the definitions, but usually this is not considered as a valid triangulation of a torus. For example see here (roughly it depends on the fact that the triangulation only has to be a CW-complex or a simplicial complex). To answer your question, there are either $0$ or $2$ triangles.