Thread: Triangulation of a Torus

I need help doing a triangulation of a torus. I don't need to use any formulas or anything, as we have just started doing this kind of thing. What I have to do is do the triangulation and then compute V-E+F and see if it equals 2.

My professor said a couple ways we can do this is use a square with opposite sides paired with one another for gluing.

Another way he described (and the way I think I want to go) is taking a cube with a square cut out through the middle. Since the top face can't have a hole, I would create 4 more edges from the corners of the object to the corners of the square hole, thus creating 4 faces on the top. The same process would happen on the bottom.

However, I don't understand how the triangulation is supposed to work... I know I can sorta just mold this figure into a torus, and I can figure out V,E, and F before I mold it, but how is this a "triangulation"? Do I need to turn the figure into smaller triangles or something, because I don't get how I would be able to compute V,E, and F for all of it.

I pretty much just need help figuring out what I need to do. My professor said using the square with opposite sides paired with one another would be the easiest way, so if someone can explain this to me that would be a big help as well.

I don't know if this will help. It's a diagram that I drew a few years ago (when teaching a geometry course) to illustrate how to triangulate a torus as a cube with a hole through the middle from top to bottom. It has 16 vertices, 32 edges and 16 quadrilateral faces. If you want triangular faces, you can divide each quadrilateral into two triangles by joining two opposite corners. That will double the number of faces, and also add 16 new edges.