Is This an Accurate Proof of Euler's Theorem for Polyhedrons?

[PlutoniumBoy]

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.

 

[jvicens]

As a fellow scientist, I appreciate your effort in trying to understand and explain the concept of polyhedrons and Euler's theorem. However, I must point out that your proof is not entirely accurate and may lead to some incorrect conclusions.

Firstly, your calculation of V0 and E0 for a tetrahedron is correct, but F0 should be 6 instead of 4. This is because a tetrahedron has 4 faces, not 3. Therefore, the correct equation should be V0+E0+F0=2(V+E+F)+2, where 2 is the number of additional edges that are not present in a tetrahedron.

Secondly, your assumption that other polyhedrons are formed from a tetrahedron and additional faces is not entirely accurate. While some polyhedrons may be formed in this way, there are also many that cannot be constructed by adding faces to a tetrahedron. For example, a dodecahedron cannot be formed from a tetrahedron and additional faces.

Furthermore, your calculation of V0+F0=2V+2F=4(4+a) is incorrect. This equation only holds true for a specific type of polyhedron called a tetrahedral-octahedral honeycomb, where a=2. For other polyhedrons, the value of a will be different and cannot be assumed to be 2.

In conclusion, while your proof may provide some insights into the relationship between V, E, and F in polyhedrons, it is not a comprehensive or accurate proof of Euler's theorem. I suggest further research and study on the topic to gain a better understanding of this important theorem in geometry.