Quote Originally Posted by gugi View Post
It the math is too abstract you can take a piece of paper and convince yourself that you cannot fold it to form a 'smiling' edge without crumpling or tearing it.
True, I cannot make a smile from a single piece of paper, but I can get a bunch of papers and make a smile, and that's my point.
Quote Originally Posted by gugi View Post
Any three points form a plane, whole lines belonging to the same plane puts a lot of restrictions. If your edge (a line) and the spine on one side of the razor (another line) belong in a single plane, the spine on the other side of the razor (third line) is not in the same plane and therefore cannot be in a plane with the edge.
Sorry, I am not a math geek (though I understand what you say). I am speaking of what I am remembering from primary school, hence I admit I might be wrong.

But, if we can imagine a half-moon 3D object and we cut it with 2D planes at any angle, we will end up with 2 edges made by the intersection of the 2 planes and the object. If we can narrow those 2 edges to a minimum (just a line), that new edge will share the same plane from the 2 sides of the spine.