Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles are used as they are, however, so would like some help trying to understand this
I am having a lot of trouble understanding triangulations. I know that a triangulation involves decomposing a 2-manifold into triangular regions.
A common example is the torus, which can be constructed from the square. I understand this representation:
Since the torus is homeomorphic to the space obtained by identifying those edges together.
What I do not understand is the triangulation given:
Why is this triangulation given in all the books and resources etc?
I do not understand what all the triangles mean. Why could we not just split the square into 2 triangles?
Many thanks for your help on this one
Best Answer
In a triangulation (specifically, a simplicial complex), the three vertices of a triangle are distinct. (Technically, the two 0-cells at the boundary of each 1-cell are distinct, the three 1-cells at the boundary of each 2-cell are distinct, et c. This leads to: the vertex set of a $k$-cell contains $k-1$ distinct vertices.) That is, if I tell you three vertices you can immediately tell me that either "there is no triangle with those three vertices" or "there is exactly one triangle with those three vertices, and its that one". (More generally, given a list of $k$ vertices, you can tell me whether there is no $k-1$-cell with those vertices or there is exactly one such $k-1$-cell and it's that one.) The intention is to make the vertex set of a $k$-cell into a unique label for the cell.
If you divide the square in half, there is only one vertex after the identification of the labelled edges. This fails "distinctness". Call the one vertex, $v$. Both triangles have the same vertex collection, "$v,v,v$", so giving a valid vertex set does not pick out a single triangle.