Can you triangulate a triangle? (also, odd sided polygons to represent surfaces)

[iLoveTopology]

EDIT: My guess to the below question is that no you can't triangulate a triangle because a legitimate triangulation each edge can only be linked up to exactly two distinct faces, so if you just have one triangle, each edge would be linked up to one face (the face of the triangle)

I'm really getting confused when it comes to triangulation.I have realized now why you can't triangulate a torus with just two triangles (I had thought before you could just take the regular square and split it down the middle into two triangles - but I see now these triangles would not be coherently oriented which would mean it wouldn't be orientable but the torus is orientable, so you need a lot more triangles)but I am wondering - if you just have a single triangle, does that count as a triangulation? Like say a triangle, regular 3 sided triangle, say arrows pointing all in the same direction around the triangle giving it an orientation - does that mean it's all ready triangulated?

Everything I see talks about having to have even sides to be a surface and to be triangulted, etc. But I don't really see why and I can't find sources explaining why you can't have an odd sided polygon like a triangle as quotient map for a surface.

 

[homeomorphic]

Even sides? What's your definition of triangulation?

My definition is homeomorphism with a simplicial complex. A triangle is just a disk, and yes, the triangle itself is a triangulation.

I guess maybe your sources are talking about closed surfaces, i.e. compact without boundary. In that case, you can't have any exposed edges in a triangulation. But if you have a surface with boundary, that doesn't apply.

 

[iLoveTopology]

I'm sorry I should have been specific. Yes I was speaking of a closed, compact surface (connected 2-manifold). All the theorems I see regarding this are that you can represent a compact surface by an even sided polygon. And it was my impression that to triangulate a compact surface each edge has to be connected to exactly two distinct faces. (sorry I am very new to manifolds, etc. so my grasp on the terminology and concepts is weak)

 

[homeomorphic]

Yes, you can build the n-holed donut surface out of a polygon. Don't read what they say. Try to visualize it for yourself. How do you fold it up so that you get the surface of n-holed doughnut?

Polygons and triangulations are two different things. You would have to subdivide the polygon to get a triangulation, and you might have to subdivide more than you might think in order for it to be a simplicial complex. A triangulation, in the 2-d case has to consist of triangles, of course, or in the higher dimensional case, it's built out of simplices. A simplicial complex is something that lives in R^n (or maybe infinite-dimensional space). You can make the torus out of two triangles, but it's not a simplicial complex.

The problem with the triangle is that it is not a closed surface--it is a surface with boundary. Therefore, you don't have to glue the edges together to get a triangulation.

You need to figure out what a simplicial complex is and then you will know what a triangulation is.

 

[iLoveTopology]

thank you for the reply. Yes I have been wrecking my brain trying to understand triangulation. It sounds simple but (to me) it's not. I guess I just need to keep reading and will look to understanding simplical complexes first. What was really confusing me was why you have to have a minimum of 14 triangles in the triangulation of a torus. I can't say I understand that. But I'm just going to keep working until I do understand it!My understanding of how triangulation relates to the whole polygon issue for compact surfaces is that any triangulation can be represented as a quotient space on an even sided polygon. Is that part correct?

 

[homeomorphic]

What was really confusing me was why you have to have a minimum of 14 triangles in the triangulation of a torus. I can't say I understand that. But I'm just going to keep working until I do understand it!

I don't understand why it has to be 14, myself. But it's clear that the two won't suffice. The answer is that you need to understand what a simplicial complex is because the definition of triangulation is a homeomorphism with one of those guys. If you understood what a simplicial complex was, it would be clear that when you glue the two triangles together, it's not a simplicial complex and therefore not a triangulation. There's a theorem that tells you how to glue together two simplicial complexes to get another one. If you don't glue them carefully, the result is only a topological space, not a simplicial complex. For example, Munkres' Elements of Algebraic Topology has this material towards the beginning of the book.

My understanding of how triangulation relates to the whole polygon issue for compact surfaces is that any triangulation can be represented as a quotient space on an even sided polygon. Is that part correct?

No, that's quite right. The triangulation is not a space. The triangulation is a particular kind of space, a simplicial complex, together with a homeomorphism. To go from the polygon to the triangulation, you therefore need TWO things, the simplicial complex and the homeomorphism. Where's the simplicial complex? That is the question. The answer is that you subdivide the polygon into small enough triangles so that you can glue and actually get that quotient space you are talking about as a simplicial complex. So, yes, the simplicial complex is a quotient of the polygon, but it's more than that because you first have to subdivide it into small triangles and then take the quotient in a CAREFUL way, so that the result is a simplicial complex, rather than just a topological space, which is what you get if you just glue the sides together normally.