Triangulations of Surfaces: minimum number of triangles

Edited 4 Sep. search on “edit”.

Take a cube. If you cut it along a few edges, you could lay it out flat. To restore the cube, we identify certain edges with each other. Similarly for a tetrehedron, or a theoretical soccer ball (with flat faces), or any polyhedron. For studying surfaces by looking at polyhedra (i.e. picewise linear structure), it is convenient to use only triangular faces (2D simplices) rather than arbitrary polygonal faces. The analog of our cut and flattened cube is called a triangulation. As before, we want to identify certain edges with each other.

In particular, the following

is offered as a triangulation of a torus (with the top & bottom edges identified, and the left & right, as we’ve seen before).

Why are there so many triangles? I have wondered that since the first time I saw it.