The angles in an m-gon and genrealizations

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Question from Michael:

Hello:

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.

Michael

There are indeed generalizations which fit what you want.

It is easier to work with the sum of the exterior angles, and then get the interior angles by a simple subtraction. For every (convex) polygon in the plane, the sum of the exterior angles is 360 degree (or 2\pi in radians).
I said convex, though the result generalizes to simple (not self-intersecting if there is care about signs and 'directions' for a walk around a polygon.
The best know generalization in 3-space again uses exterior angles - the defect at a vertex between the sum of the angles in all the faces at the vertex, and the whole of a plane angle - 360 degrees. If you do that at every vertex, the sum of all the defect is 720 degrees (4\pi - the surface area of a unit sphere). Consider a standard cube. At each vertex, there is 90 degrees missing to make it flat. 8x(90) = 720 degrees.

This is a version of the Gauss Bonnet theorem in 3-space also called Descarte's Theorem - see http://en.wikipedia.org/wiki/Descartes%27_theorem_on_total_angular_defect#Descartes.27_theorem

Since Gauss Bonnet works in all dimensions, there should be a discrete version in all dimensions.

Walter Whiteley
York University

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