Platonic solid
Topological proof

A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that V − E + F = 2, and the fact that pF = 2 E = 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

2 E q − E + 2 E p = 2. {\displaystyle {\frac {2E}{q}}-E+{\frac {2E}{p}}=2.}

Simple algebraic manipulation then gives

1 q + 1 p = 1 2 + 1 E. {\displaystyle {1 \over q}+{1 \over p}={1 \over 2}+{1 \over E}.}

Since E is strictly positive we must have

1 q + 1 p > 1 2. {\displaystyle {\frac {1}{q}}+{\frac {1}{p}}>{\frac {1}{2}}.}

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, q }:
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