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:
Here
denotes the number of edges on the face
.
Thus
If we sum the number of edges on each face over all of the faces, we will have counted each edge twice. Thus
Whence,
Listen to both proofs given in class.