More Triangulation (comb space and dunce hat)

[drainbramaged]

Triangulation seems to be getting more and more confusing. Can I get some help understand why (and how) a Dunce Hat is triangulable?

On a similar note, the Comb Space isn't triangulable. I'm completely at a loss as to any explanation. Help?

Thanks!

[Qaanol]

Seems to me you can slice a dunce cap up the side, unroll it into a circular sector, draw a couple of radii, and make some zig-zag lines, no?

[jestingrabbit]

Well, imagine trying to make something that looked like the dunce hat using just triangles, is this doable?

The comb space is a little trickier. I imagine that you can write something down that you believe *should* be a triangulation, you just need to work out why it isn't.

[drainbramaged]

Thank you for the reply. My issue with the dunce hat is if I try and cover the dunce hat with triangles, I end up identifying two of the vertices of any 2-simplex I try to use to fill a corner to the same point. This isn't allowed, right?

[jestingrabbit]

Thank you for the reply. My issue with the dunce hat is if I try and cover the dunce hat with triangles, I end up identifying two of the vertices of any 2-simplex I try to use to fill a corner to the same point. This isn't allowed, right?

That isn't allowed, true. But what if you just use more triangles? ie cut the base triangle up into smaller triangles. So, for instance, if you are identifying two vertices, can you cut your triangle up a little more and avoid that?

[Talith]

With regard to the comb space. You might like to ask the question "what properties do all triangulable spaces have?" and then see if any of these properties fail to hold for the comb space.