In my presentation of my progress in my project on Topology, I mentioned something about how Non-Euclidean surfaces have curvatures [math]\neq 0[/math]. My math teacher pulled me aside afterward and proposed to me the following:

 

Around a point on a Euclidean surface is 360º. You said that in order for this value to change, the surface must have some curvature. However, [at this point he took a piece of paper and pointed to a point lying directly on on edge] about this point there is 180º. I interjected at this point, claiming that I was talking about an edgeless plane, and he continued:

 

[Making a crease at the point and then joining the two halves of the edge on which it lay, he formed a cone] The vertex of this cone, then, must have local angles all add up to 180º. Imagine a surface covered with such cones, troughs and valleys, such that there are an infinite amount of vertices, each having 180º in their neighbourhood. Now, what happens mathematically as the density of these peaks tends to infinity? The limit is a flat plane with zero curvature, but one on which every point has local angles all add up to only 180º.

 

This struck me as really odd. What are your thoughts?

No, that is not right. How does he intend to arrange the cones? The part where the curvature changes (to go from one cone to the other) will have curvature such that the angles add up to more than 360.