Update Seminar Form
Jasdeep and I had given a seminar on Eulerís Formula and further on the basis of Euler`s Formula we had proved why there are exactly 5 Platonic solids.
First We will prove Eulerís Formula which is V + F - E = 2. (where E=Edges, V=Vertices and F = Faces).
First take a cube (the proof for an arbitrary polyhedron is the same) and cut one face and spread it flat on a plane.
Now we have same number of V,E and one Face is reduced.
i.e. If we can prove V-E+F=1 then it is also true for V-E+F=2.
First "Triangulate" to a Simplicial Complex (as shown in fig-3).
So now E = E + 1 , F = F + 1 and V = V.
Now remove the edges of the boundary triangles that do not belong to the inside triangle.(as shown in fig-5)
E becomes E -1 ,F becomes F-1 and V remains same.
Further eliminate the edges of the boundary triangle that do not belong to the inside triangle (as shown in fig-6).
Now count the number of Edges ,Vertices and Faces and Place it in the Equation.
You will get V- E+F=3-3+1=1 Hence Proved.
Prof. Ashay Dharwadker