Editing Platonic solid (section)

=== Topological proof ===
A purely [[topology|topological]] proof can be made using only combinatorial information about the solids. The key is [[Euler characteristic|Euler's observation]] that ''V''&nbsp;−&nbsp;''E''&nbsp;+&nbsp;''F''&nbsp;=&nbsp;2, and the fact that ''pF''&nbsp;=&nbsp;2''E''&nbsp;=&nbsp;''qV'', where ''p'' stands for the number of edges of each face and ''q'' for the number of edges meeting at each vertex. Combining these equations one obtains the equation

{{Hamiltonian_platonic_graphs.svg}}

<math display="block">\frac{2E}{q} - E + \frac{2E}{p} = 2.</math>

Simple algebraic manipulation then gives

<math display="block">{1 \over q} + {1 \over p}= {1 \over 2} + {1 \over E}.</math>

Since ''E'' is strictly positive we must have

<math display="block">\frac{1}{q} + \frac{1}{p} > \frac{1}{2}.</math>

Using the fact that ''p'' and ''q'' must both be at least 3, one can easily see that there are only five possibilities for {''p'',&nbsp;''q''}:
{{block indent|{3,&nbsp;3}, {4,&nbsp;3}, {3,&nbsp;4}, {5,&nbsp;3}, {3,&nbsp;5}.}}