Question from Michael:
In answer to a student's question, someone named Penny from
your organization provided a proof that the sum of the interior
angles of a triangle in the plane is pi radians (or 180 degrees).
I am interested (and I'm sure many other people would be too) in
3 potential generalizations of this basic fact in plane geometry:
1) What is the corresponding result for an m-gon in the plane. I
believe the answer is that the sum of the interior angles in a 2D
m-gon (whether it is regular or not, or convex or not) is (m-2)*pi.
2) I feel intuitively that there an analogous result for m-polytopes
in 3 dimensions where the solid angles are measured in steradians,
but I cannot prove it Can you? [The proof that Penny provided for
the 2D case does not seem to generalize easily.]
3) Are there analogous results of m-polytopes in n dimensions? What
is the analogue of a steradian in n dimensions?
I feel intuitively that there are analogous results. It would be very
elegant if the results could be stated in terms of the coordinates
of the vertices of the polytope.
The reason why I'm asking these questions is that I am working on
some unsolved problems in geometry for which answers to them
would be helpful.