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Jasdeep and I had given a seminar on Euler’s Formula and further on the basis of Euler`s Formula we had proved why there are exactly 5 Platonic solids. <br> First We will prove Euler’s Formula which is V + F  E = 2. (where E=Edges, V=Vertices and F = Faces). <br> <ul> <li>First take a cube (the proof for an arbitrary polyhedron is the same) and cut one face and spread it flat on a plane. <li>Now we have same number of V,E and one Face is reduced. <br> i.e. If we can prove VE+F=1 then it is also true for VE+F=2. <li>First "Triangulate" to a Simplicial Complex (as shown in fig3). <li>So now E = E + 1 , F = F + 1 and V = V. <li>Now remove the edges of the boundary triangles that do not belong to the inside triangle.(as shown in fig5) <li>E becomes E 1 ,F becomes F1 and V remains same. <li>Further eliminate the edges of the boundary triangle that do not belong to the inside triangle (as shown in fig6). <li>Now count the number of Edges ,Vertices and Faces and Place it in the Equation.<br> You will get V E+F=33+1=1 Hence Proved. </ul> <center><img SRC="seminar_25.gif"></center>
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Prof. Ashay Dharwadker