The angle defect at a vertex of a polygon was defined to be the amount by which the sum of the angles at the corners of the faces at that vertex falls short of and the total angle defect of the polyhedron was defined to be what one got when one added up the angle defects at all the vertices of the polyhedron. We call the total defect . Descartes discovered that there is a connection between the total defect, , and the Euler Number . Namely,
Here are two proofs. They both use the fact that the sum of the angles of a polygon with sides is .
Think of as putting at each vertex, on each edge, and on each face.
We will try to cancel out the terms as much as possible, by grouping within polygons.
For each edge, there is to allocate. An edge has a polygon on each side: put on one side, and on the other.
For each vertex, there is to allocate: we will do it according to the angles of polygons at that vertex. If the angle of a polygon at the vertex is , allocate of the to that polygon. This leaves something at the vertex: the angle defect.
In each polygon, we now have a total of the sum of its angles minus (where is the number of sides) plus . Since the sum of the angles of any polygon is , this is 0. Therefore,
We begin to compute:
denotes the number of edges on the face
If we sum the number of edges on each face over all of the faces, we will have counted each edge twice. Thus
Listen to both proofs given in class.