Euler's Relation For Polygonal Bases: Simplified

          Based on my original method for producing bases by creating and translating points in a plane, a much simpler proof of Euler's relation for these bases can be achieved.  A base is said to be produced when a point meets an existing edge or vertice.  By aid of a diagram (Fig. 23) I demonstrate that from a starting point, here named the elemental we can produce any polygon.  The first stage in this process is to create and translate a second point from the elemental, thus adding one edge and two vertices.  For the relation (F + V = E + C), we see that 0 + 2 = 1 + C, so C = 1.  From this we translate a third point to produce another vertice and edge as shown.  This continues on keeping the relation constant while no faces are produced.  The last stage (shown in the last two figures) is the occasion on which the n^th point reaches an existing vertice or edge.

          The first case produces one edge and two faces (one on each side of the figure), but no vertices as one already exists at that point.  Thus (as F_initial = 0):

                      V + F - E = 2 + V_initial - (E_initial + 1)

                                      = V_initial - E_initial + 1

                                      = 1 + 1               (as V_initial - E_initial = 1)

                                      = 2

          The second case is similar to the first, however an additional edge and vertice are produced, which cancel each other out.

Fig. 23 Polygonal base as translation of points in a plane

          This satisfies the relation for polygonal bases.  Unfortunately this process cannot be replicated for further production of surfaces as it becomes much more complicated when considering the lost of inner edges, faces and vertices.