Proof:
Consider the number of vertices (V>3), edges (E>3), and
faces (F>2) involved in the figure.
Since the figure is topologically equivalent to the sphere,
Here is a table showing the only possible integer values that satisfy this equation and therefore there can be only 5 corresponding regular convex polyhedra- the five platonic solids:
|
V |
|
|
|
|
|
4 |
|
|
|
|
|
8 |
|
|
|
|
|
6 |
|
|
|
|
|
12 |
|
|
|
|
|
20 |
|
|
|
|
Here is the proof based on Euler's Formula briefly:
If the problem can be solved
then it would form a planar graph with V=6 and E=9.
Thus there would have to be
exactly 5 regions (including the unbounded region of the plane in the graph).
Each of these regions must have
at least 4 edges. Since each edge can be counted twice once for each region
it bounds-there will be at least 10 edges in the graph. BUT we only have
9 edges!