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A Proof of Euler Theorem

  1. Aug 28, 2009 #1
    Tetrahedron is the simplest polyhedron, it is just formed from four triangles, so it has V0=E0=4*3=12, and F0=F=4, of course. Then we find that V0+E0+F0=2(V+E+F) or V0+F0=2V+2F (Because of E0 is always equal to 2E)=4F.
    Assume that other polyhedrons are the "evolution" product of a tetrahedron. I mean that they are formed from four triangles of tetrahedron and additional faces such as rectangle, hexagon, etc. For instead, a cube consists of 4 triangles and 4 rectangles. So V0+F0=2V+2F=4(4+a), where a is the number of additional faces. Now let's calculate. It's 32. So we conclude that F0 must be 4.
    Finally, we can find Euler Theorem:
    2V+2E+2F=V0+E0+4
    V+E+F=E0+2
    V-E+F=2.

    What's your comment about my proof?, I am sorry if you think that it is too poor.
     
  2. jcsd
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